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Title: Modern Machine-Shop Practice, Volumes I and II
Author: Rose, Joshua
Language: English
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Copyright Status: Not copyrighted in the United States. If you live elsewhere check the laws of your country before downloading this ebook. See comments about copyright issues at end of book.

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  +------------------------------------------------------------------+
  |                    TRANSCRIBER'S NOTES                           |
  |                                                                  |
  | Transcriptions used in this e-text:                              |
  |   italics text in the original work is presented here between    |
  |   underscores, as in _text_;                                     |
  |   bold-face text in the original work is presented here between  |
  |   equal-signs, as in =text=;                                     |
  |   small-capitals in the original work are presented here as ALL  |
  |   CAPITALS;                                                      |
  |   fractions are transcribed as, for example, 2-1/2 for 2½; where |
  |   the author uses the form 1-64, this form has been retained,    |
  |   except in tables;                                              |
  |   superscript texts are transcribed as in ^{text};               |
  |   subscript texts are transcribed as in _{text};                 |
  |   single Greek letters are transcribed as [alpha], [beta], etc.; |
  |   the (single) oe-ligature used in the book has been transcribed |
  |   as oe (Phoenixville);                                          |
  |   multi-line in-line formulas and calculations from the original |
  |   work have been transcribed as single-line in-line formulas and |
  |   calculations, where necessary with the addition of brackets.   |
  | The following transcriptions are used for special characters and |
  | symbols, where x can be any character:                           |
  |   [=x]  x-macron;                                                |
  |   [)x]  x-breve;                                                 |
  |   [x.]  x-dot-below;                                             |
  |   [<--] left-pointing hand.                                      |
  | The author uses letters from a different font to describe shapes.|
  | These are transcribed between square brackets: [V] or [V]-shaped,|
  | [T] or [T]-shaped, etc. Where the original work uses regular     |
  | letters for the same purpose, this transcription has not been    |
  | used. Special cases are __|¯¯  for a stretched S-shape, [/\] for |
  | an upside-down V, and [_|_] for an upside-down T.                |
  |                                                                  |
  | More extensive Transcriber's Notes will be found at the end of   |
  | this text.                                                       |
  +------------------------------------------------------------------+



[Illustration: _VOL. I. MODERN MACHINE-SHOP PRACTICE FRONTISPIECE_

_Copyright, 1887 by Charles Scribner's Sons._

=MODERN AMERICAN FREIGHT LOCOMOTIVE.=]



MODERN

MACHINE-SHOP PRACTICE

BY

JOSHUA ROSE, M.E.


ILLUSTRATED WITH MORE THAN 3000 ENGRAVINGS


VOLUME I.


NEW YORK

CHARLES SCRIBNER'S SONS

1887


COPYRIGHT, 1887, BY

CHARLES SCRIBNER'S SONS


Press of J. J. Little & Co.

Astor Place, New York.



PREFACE.


MODERN MACHINE-SHOP PRACTICE is presented to American mechanics as a
complete guide to the operations of the best equipped and best managed
workshops, and to the care and management of engines and boilers.

The materials have been gathered in part from the author's experience of
thirty-one years as a practical mechanic; and in part from the many
skilled workmen and eminent mechanics and engineers who have generously
aided in its preparation. Grateful acknowledgment is here made to all
who have contributed information about improved machines and details of
new methods.

The object of the work is practical instruction, and it has been written
throughout from the point of view, not of theory, but of approved
practice. The language is that of the workshop. The mathematical
problems and tables are in simple arithmetical terms, and involve no
algebra or higher mathematics. The method of treatment is strictly
progressive, following the successive steps necessary to becoming an
intelligent and skilled mechanic.

The work is designed to form a complete manual of reference for all who
handle tools or operate machinery of any kind, and treats exhaustively
of the following general topics: I. The construction and use of
machinery for making machines and tools; II. The construction and use of
work-holding appliances and tools used in machines for working metal or
wood; III. The construction and use of hand tools for working metal or
wood; IV. The construction and management of steam engines and boilers.
The reader is referred to the TABLE OF CONTENTS for a view of the
multitude of special topics considered.

The work will also be found to give numerous details of practice never
before in print, and known hitherto only to their originators, and aims
to be useful as well to master-workmen as to apprentices, and to owners
and managers of manufacturing establishments equally with their
employees, whether machinists, draughtsmen, wood-workers, engineers, or
operators of special machines.

The illustrations, over three thousand in number, are taken from modern
practice; they represent the machines, tools, appliances and methods now
used in the leading manufactories of the world, and the typical steam
engines and boilers of American manufacture.

The new PRONOUNCING AND DEFINING DICTIONARY at the end of the work, aims
to include all the technical words and phrases of the machine shop, both
those of recent origin and many old terms that have never before
appeared in a vocabulary of this kind.

The wide range of subjects treated, their convenient arrangement and
thorough illustration, with the exhaustive TABLE OF CONTENTS of each
volume and the full ANALYTICAL INDEX to both, will, the author hopes,
make the work serve as a fairly complete ready reference library and
manual of self-instruction for all practical mechanics, and will
lighten, while making more profitable, the labor of his fellow-workmen.



CONTENTS.


  VOLUME I.


  CHAPTER I.

  =THE TEETH OF GEAR-WHEELS.=

                                                                    PAGE
  =Gear-Wheels.= Spur-wheels, bevel-wheels, mitre-wheels,
  crown-wheels, annular or internal wheels                             1
    Trundle-wheels, rack and pinion-wheel and tangent screw, or
    worm and worm-wheel                                                1
    The diameter of the pitch circle of                                1
  =Gear-Wheel Teeth.= The face, the flank, the depth or height         1
    The space, the pitch line, the point, the arc pitch, the chord
    pitch, the line of centres                                         2
    Rules for finding the chord pitch from the arc pitch; table of
    natural sines; diametral pitch; finding the arc from the
    diametral pitch; table of arc and diametral pitches                3
  =Gear-Wheels.= The driver and follower, a train of gears             3
    Intermediate gears                                                 3
    The velocity of compounded wheels                                  4
    Finding the diameters of the pitch circles of                      4
    Considered as revolving levers                                     5
    Calculating the revolutions of, and power transmitted by           5
    The angular velocity of                                            6
  =Gear-Wheels.= Hunting tooth in, stop motion of                      7
  =Gear-Wheel Teeth.= The requirements and nature of the teeth
  curves                                                               7
    Cycloidal curves for the faces of; epicycloidal and involute
    curves; the hypocycloidal curve; method of forming or
    generating the epicycloidal and hypocycloidal curves for the
    faces and flanks of gear teeth                                     8
    Applications of the epicycloidal and hypocycloidal curves in
    the formation of gear teeth                                        9
    The diameter of the circle for generating the epicycloidal and
    hypocycloidal curves; graphical demonstration that the flank
    curves are correctly formed to work with the face curves of the
    other wheel                                                       10
    Graphical demonstration that the curves are correct independent
    of either the respective sizes of the wheels, or of the curve
    generating circles                                                11
  =Gear-Wheels.= Hand applications of the rolling or
  generating circle to mark the tooth curves for a pair of wheels     12
  =Gear-Wheel Teeth.= The variation of curve due to different
  diameters of wheels or of rolling circles                           12
    Tracing the path of contact of tooth upon tooth in a pair of
    gear-wheels; definition of the "arc of approach;" definition of
    the "arc of recess;" demonstration that the flanks of the teeth
    on the driver or driving-wheel have contact with the faces of
    the driven wheel during the arc of approach, and with the
    flanks of the driven wheel during the arc of recess               13
    Confining the action of the teeth to one side only of the line
    of centres, when motion rather than power is to be conveyed       13
    Demonstration that the appearance or symmetry of a tooth has no
    significance with regard to its action                            14
    Finding how many teeth will be in constant action, the diameter
    of the wheels, the pitch of the teeth, and the diameter of the
    rolling circle being given                                        15
    Example of the variation of tooth form due to variation of
    wheel diameter                                                    15
  =Gear Teeth.= Variation of shape from using different
  diameters of rolling circles                                        16
    Thrust on the wheel shafts caused by different shapes of teeth    16
  =Gear-Wheels.= Willis' system of one size of rolling circle for
  trains of interchangeable gearing                                   16
    Conditions necessary to obtain a uniform velocity of              16
  =Gear Teeth.= The amount of rolling and of sliding motion of        16
    The path of the point of contact of                               16
    The arcs of approaching and of receding contact                   16
    Lengths of the arcs of approach and of recess                     16
    The influence of the sizes of the wheels upon the arcs of
    contact                                                           17
    Influence of the size of the rolling circle upon the amount of
    flank contact                                                     18
    Demonstration that incorrectly formed teeth cannot correct
    themselves by wear                                                18
    The smaller the diameter of the rolling circle, the less the
    sliding motion                                                    18
    Influence of the size of the rolling upon the number of teeth
    in contact in a given pair of wheels                              19
    Demonstration that the degrees of angle the teeth move through
    exceed those of the path of contact, unless the tooth faces
    meet in a point                                                   19
    Influence of the height of the teeth upon the number of teeth
    in contact                                                        20
    Increasing the arc of recess without increasing the arc of
    approach                                                          20
    Wheels for transmitting motion rather than power                  21
    Clock wheels                                                      21
    Forms of teeth having generating or rolling circles, as large
    or nearly as large as the diameters of the wheels                 21
  =Gear-Wheels.= Bevel                                                21
    The principles governing the formation of the teeth of bevel-
    wheels                                                            22
    Demonstration that the faces of the wheels must be in line with
    the point of intersection of the axis of the two shafts           22
  =Gear Teeth.= Method of finding the curves of, for bevel
  gear                                                                22
  =Gear-Wheels.= Internal or annular                            23 to 27
    Demonstration that the teeth of annular wheels correspond to
    the spaces of spur-wheels                                         23
  =Gear-Wheels Internal.= Increase in the length of the path
  of contact on spur-wheels of the same diameter, and having the
  same diameter of generating or rolling circle                       23
    Demonstration that the teeth of internal wheels may interfere
    when spur-wheels would not do so                                  23
    Methods of avoiding the above interference                        23
    Comparison of, with spur-wheels                                   23
    The teeth of: demonstration that it is practicable to so form
    the teeth faces that they will have contact together as well as
    with the flanks of the other wheel                                24
    Intermediate rolling circle for accomplishing the above result    24
    The application of two rolling circles for accomplishing the
    above result                                                      24
    Demonstration that the result reached by the employment of two
    rolling circles of proper diameter is theoretically and
    practically perfect                                               24
    Limits of the diameters of the two rolling circles                25
    Increase in the arc of contact obtained by using two rolling
    circles                                                           25
    Demonstration that the above increase is on the arc of recess
    or receding contact, and therefore gives a smooth action          25
    Demonstration that by using two rolling circles each tooth has
    for a certain period two points of contact                        25
    The laws governing the diameters of the two rolling circles       25
    Practical application of two rolling circles                      26
    Demonstration that by using two rolling circles the pinion may
    contain but one tooth less than the wheel                         26
    The sliding and rolling motion of the teeth of                    27


  CHAPTER II.

  =THE TEETH OF GEAR-WHEELS (Continued).=

  =Worm and Worm-Wheel=, or wheel and tangent screw             28 to 31
    General description of                                            28
    Qualifications of                                                 28
    The wear of                                                       28
  =Worm-Wheel Teeth=, the sliding motion of                           28
    When straight have contact on the centres only of the tooth
    sides                                                             28
    That envelop a part of the worm circumference                     28
    The location of the pitch line of the worm                        28
    The proper number of teeth in the worm-wheel                      29
    Locating the pitch line of the worm so as to insure durability    29
    Rule for finding the best location for the pitch line of the
    worm                                                              29
    Increasing the face of the worm to obtain a smoother action       29
  =Worms=, to work with a square thread                               29
  =Worm-Wheels=, applications of                                      30
  =Gear-Wheels= with involute teeth                             31 to 34
  =Gear Teeth.= Generating the involute curve                         31
    Templates for marking the involute curve                          32
  =Involute Teeth=, the advantages of                                 34
  =Gear Teeth=, Pratt and Whitney's machine for cutting
  templates for                                                       35


  CHAPTER III.

  =THE TEETH OF GEAR-WHEELS (Continued).=

  =Gear Teeth=, revolving cutters for                                 37
    Pantagraph engine for dressing the cutters for                    38
    Numbers of cutters used for a train of wheels                     39
  =Gear-Wheel Teeth.= Table of equidistant value of cutters           41
    Depth of, in the Brown and Sharpe system                          42
    Cutting the teeth of worm-wheels                                  42
    Finding the angle of the cutter for cutting worm-wheels           43
    The construction of templates for rolling the tooth curves        43
    Rolling the curves for gear teeth                                 43
    Forms of templates for gear teeth                                 44
    Pivoted arms for tooth templates                                  44
    Marking the curves by hand                                        45
    Former or Template of the Corliss bevel gear-wheel engine or
    cutting machine                                                   45
    The use of extra circles in marking the curves with compasses     46
    Finding the face curves by geometrical constructions              47
    The Willis odontograph for finding the radius for striking the
    curves by hand                                                    47
    The method of using the Willis odontograph                        48
    Professor Robinson's odontograph                                  49
    Method of using Professor Robinson's odontograph                  49
    Application of Professor Robinson's odontograph for trains of
    gearing                                                           51
    Tabular values and setting numbers for Professor Robinson's
    odontograph                                                       51
    Walker's patent wheel scale for marking the curves of cast
    teeth                                                             51
    The amount of side clearance in cast teeth                        53
    Filleting the roots of epicycloidal teeth with radial flanks      53
    Scale of tooth proportions given by Professor Willis              54
    The construction of a pattern for a spur-wheel that is to be
    cast with the teeth on                                            54
    Template for planing the tooth to shape                           54
    Method of marking the curves on teeth that are to be glued on     55
    Method of getting out the teeth of                                56
    Spacing the teeth on the wheel rim                                56
    Methods of accurately spacing the pattern when it has an even
    number of teeth                                                   58
    Method of spacing the wheel rim when it has an odd number of
    teeth                                                             58
  =Gear-Wheels, Bevel Pinion=, drawings for                           59
    Getting out the body for a bevel-wheel                            59
    Template for marking the division lines on the face of the
    wheel                                                             59
    Marking the lines of the division on the wheel                    60
  =Gear-Wheels, Pinion=, with dovetail teeth                          60
    Testing the angle of bevel-wheels while in the lathe              60
  =Gear-Wheels, Skew Bevel.= Finding the line of contact              61
    Marking the inclination of the teeth                              61
  =Gear-Wheels, Bevel=, drawing for built up                          61
  =Gear-Wheels, Worm=, or endless screw                               62
    Constructing a pattern from which the worm is to be cast          62
    Tools for cutting the worm in a lathe                             62
    Cutting the teeth by hand                                         62
  =Gear-Wheels, Mortise= or cogged                                    63
    Methods of fastening cogs                                         63
    Methods of getting out cogs for                                   63
  =Gear-Wheel Teeth=, calculating the strength of epicycloidal        64
    Factors of safety for                                             64
    Tredgold's rule for calculating the strength of                   65
    Cut, calculating the strength of                                  65
  =Gear-Wheel Teeth.= The strength of cogs                            66
    The thickness of cogs                                             66
    The durability of cogs                                            66
    Table for calculating the strength of different kinds of          67
    The contact of cast teeth                                         67
    Table for determining the relation between pitch diameter,
    pitch, and number of teeth in gear-wheels                         68
    Examples of the use of the above table                            68
    With stepped teeth                                                69
    Angular or helical teeth                                          69
    End thrust of angular teeth                                       69
    Herring-bone angular teeth                                        69
    For transmitting motion at a right angle by means of angular or
    helical teeth                                                     69
    Cutting helical teeth in the lathe                                69
    For wheels whose shaft axes are neither parallel nor meeting      70
    Elliptical                                                        70
    Elliptical, marking the pitch lines of                            70
    Elliptical, drawing the teeth curves of                           73
    For variable motion                                               74
    Form of worm to give a period of rest                             74
    Various applications of                                           74
  =Gear-Wheels=, arrangement of, for periodically reversing
  the direction of motion                                             75
    Watt's sun and planet motion                                      75
    Arrangements for the rapid multiplication of motion               75
    Arrangement of, for the steering gear of steam fire-engines       75
    Various forms of mangle gearing                                   79
  =Gear-Wheel and Rack=, for reciprocating motion                     77
  =Friction Wheels.=                                                  77
    The material for                                                  77
    Paper                                                             78
    For the feed motion of machines                                   78
    The unequal wear upon grooved                                     79
    Form of, for relieving the journals of strain                     79
  =Cams=, for irregular motion                                        80
    Finding the pitch line of                                         80
    Finding the working face of                                       80
    The effect the diameter roller has upon the motion produced by
    a cam                                                             80
    Demonstration of the different motion produced by different
    diameters of rollers upon the same cam                            80
    Diagram of motion produced from the same cam with different
    diameters of rollers                                              81
    Return or backing                                                 82
    Methods of finding the shape of return or backing                 82
  =Cam Motion=, for an engine slide valve without steam lap           83
    For a slide valve with steam lap                                  83
  =Groove Cams=, proper construction of                               84
    The wear of                                                       84
    Brady's improved groove cam with rolling motion and adjustment
    for wear                                                          84


  CHAPTER IV.

  =SCREW-THREADS.=

  =Screw Threads=, the various forms of                               85
    The pitch of                                                      85
    Self-locking                                                      85
    The Whitworth                                                     86
    The United States standard                                        86
    The Common V                                                      86
    The requirements of                                               86
    Tools for cutting                                                 87
    Variation of pitch from hardening                                 87
    The wear of thread-cutting tools                                  88
    Methods of producing                                              88
    Alteration of shape of, from the wear of the tools they are cut
    by                                                                89
  =Screw Thread Cutting Tools.= The wear of the tap and the die       89
    Improved form of chaser to equalize the wear                      90
    Form of, to eliminate the effects of the wear in altering the
    fit                                                               90
    Originating standard angles for                                   91
    Standard micrometer gauge for the United States standard screw
    thread                                                            91
    Standard plug and collar gauges for                               91
    Producing gauges for                                              92
    Table of United States standard for bolts and nuts                93
    Table of standard for the V-thread                                93
    United States standard for gas and steam pipes                    93
    Taper for standard pipe threads                                   95
    Tables of the pitches and diameters at root of thread, of the
    Whitworth thread                                                  95
    Table of Whitworth's screw threads for gas, water, and
    hydraulic piping                                                  96
    Whitworth's standard gauges for watch and instrument makers       96
    Screw-cutting hand tools                                          96
  =Thread-Cutting Tools.= American and English forms of stocks
  and dies                                                            97
    Adjustable or jamb dies                                           98
    The friction of jamb dies                                         98
    The sizes of hobs that should be used on jamb dies                99
    Cutting right or left-hand thread with either single, double,
    or treble threads with the same dies                              99
    Hobs for hobbing or threading dies                               100
    Various forms of stocks with dies adjustable to take up the
    wear                                                             101
    Dies for gas and steam pipes                                     101
  =Thread-Cutting Tool Taps.= The general forms of taps              102
    Reducing the friction of                                         102
    Giving clearance to                                              102
    The friction of taper                                            103
    Improved forms of                                                103
    Professor J. E. Sweet's form of tap                              104
    Adjustable standard                                              104
    The various shapes of flutes employed on taps                    105
    The number of flutes a tap should have                           105
    Demonstration that a tap should have four cutting edges rather
    than three                                                       106
    The position of the square or driving end, with relation to the
    cutting edges                                                    106
    Taper taps for blacksmiths                                       106
    Collapsing taps for use in tapping machines                      107
    Collapsing tap for use in a screw machine                        107
    The alteration of pitch that occurs in hardening                 108
    Gauging the pitch after the hardening                            108
    Correcting the errors of pitch caused by the hardening           109
    For lead                                                         109
    Elliptical in cross section                                      109
    For very straight holes                                          109
    Tap wrenches solid and adjustable                                110
  =Thread-Cutting.= Tapping                                          110
    Appliances for tapping standard work                             111


  CHAPTER V.

  =FASTENING DEVICES.=

  =Bolts=, classification of, from the shapes of their heads         112
    Classification of, from the shapes of their bodies               112
    Countersunk                                                      112
    Holes for, classification of                                     112
    For foundations, various forms of                                113
    Hook bolts                                                       113
    The United States standard for finished bolts and nuts           113
    The United States standard for rough bolts and nuts, or black
    bolts                                                            114
    The Whitworth standard for bolts and nuts                        114
  =Screws=                                                           114
  =Studs=                                                            115
  =Set Screws=                                                       115
  =Bolts= for quick removal                                          116
    That do not pass through the work                                117
    That self-lock in grooves and are readily removable              117
    Heads and their bedding                                          117
  =Nuts=, the forms of, when they are to be steam tight              118
    Various forms of                                                 118
    Jamb nuts and lock nuts                                          119
  =Differential Threads= for locking purposes                        119
    For fine adjustments                                             119
  =Nuts=, taking up the wear of                                      120
    Securing devices                                                 120
    Securing by taper pins                                           121
    Securing by cotters                                              121
    Securing by notched plates                                       121
  =Pins.= Securing for exact adjustments                             121
    And double eyes fitting                                          121
    Fixed                                                            122
    Working                                                          122
  =Bolts=, removing corroded                                         122
  =Nuts=, removing corroded                                          122
  =Washers=, standard sizes of                                       122
  =Wrench=, the proper angles of                                     123
    Box                                                              124
    Monkey                                                           125
    Adjustable, various forms of                                     125
    Sockets                                                          125
    Novel for carriage bolts                                         125
    Pin                                                              126
    Improved form of                                                 126
  =Keys=, the various kinds of                                       126
    The bearing surfaces of                                          126
  =Set Screws=, application of, to hubs or bosses                    127
  =Keys=, with set-screws                                            127
    The draught of                                                   127
  =Feathers=, and their applications                                 127
  =Keys=, for parallel rods                                          128
  =Taper Pins=, proper position of, for locking purposes             128
    Improved method of fitting                                       128


  CHAPTER VI.

  =THE LATHE.=

  =Lathe=, the importance and advantages of                          129
    Classification of lathes                                         129
    Foot                                                             130
    Methods of designating the sizes of                              130
    Bench                                                            130
    Power                                                            130
    Hand                                                             130
    Slide Rest for                                                   131
    American form of, their advantages and disadvantages             132
    English forms of                                                 132
    For spherical work                                               132
    Methods of taking up lost motion of                              133
  =Engine Lathe=, general construction of                            133
    The construction of the shears of                                134
    Construction of the headstock                                    134
    Construction of the bearings                                     134
    Construction of the back gear                                    135
    Means of giving motion to the feed spindle                       135
    Construction of the tailstock                                    135
    Method of rapidly securing and releasing the tailstock           136
  =Lathe Tailstock=, setting over for turning tapers                 136
  =Engine Lathe=, construction of carriage                           137
    Feed motion for carriage or saddle                               137
  =Lathe Apron=, Construction of the feed traverse                   138
    Construction of the cross-feed motion                            138
  =Engine Lathe=, lead screw and change wheels of                    139
    Feed spindle and lead screw bearings                             139
    Swing frame for lead screw                                       139
    Lead screw nuts                                                  140
    With compound slide rest                                         140
    Construction of compound slide rest                              141
    Advantages of compound slide rest                                141
    For taper turning                                                142
    Taper-turning attachments                                        142
    With compound duplex slide rest                                  143
    Detachable slide rest                                            143
    Three-tool slide rest for turning shafting                       143
    With flat saddle for chucking work on                            143
  =The Sellers Lathe=                                                143
    Construction of the headstock and treble gear                    144
    Construction of the tailstock and method of keeping it in line   145
    Construction of the carriage and slide rest                      145
    Methods of engaging and disengaging the feed motions             146
  =Car Axle Lathe=, with central driving motion and two slide
  rests                                                              147
    The feed motions of                                              148
  =Self-Acting Lathe=, English form of                               148
  =Pattern Maker's Lathe=                                            148
    Brake for cone pulley                                            149
    With wooden bed                                                  149
    Slide rest for                                                   149
  =Chucking Lathe=, English                                          149
    Feed motions of                                                  150
  =Pulley Lathe=                                                     150
  =Gap or Break Lathe=                                               151
  =Extension Lathe=                                                  151
  =Wheel Lathe=                                                      151
  =Chucking Lathe= for boring purposes                               152
  =Lathe= for turning crank axles                                    152
    Construction of the headstock                                    153
    Construction of the feed motions                                 154
    For turning crank, Arrangements of the slide rests               154
    Application of the slide rest to a crank                         155


  CHAPTER VII.

  =DETAILS IN LATHE CONSTRUCTION.=

  =Live Spindle= of a lathe, the fit of                              157
    With coned journals                                              157
    Methods of taking up the end motion of                           158
    Arranging the swing frame for the change gears                   158
    Taking up the wear of the back bearing                           158
    The wear of the front bearing of                                 158
  =The Taper= for the live centre                                    159
  =Methods= of removing the lathe centres                            159
  =Tapers= for the live centres                                      159
  =Methods= of removing the dead centre                              159
  =Driving Cone=, arranging the steps of                             159
    Requirements of proportioning the steps of                       159
    Rules for proportioning the diameters of the steps of, when the
    two pulleys are exactly alike and are connected by an open
    belt                                                      159 to 161
    When the two pulleys are unlike                           161 to 164
  =Back Gear=, methods of throwing in and out                        165
  =Conveying= motion to the lead screw                               165
  =Attaching= the swing frame                                        166
  =Feed Gear.= Arrangement for cutting worm threads or tangent
  screws                                                             167
  =Feed Motion= for reversing the direction of tool traverse
  in screw cutting                                                   168
    For lathe aprons                                                 168
  =Slide Rest=, weighted elevated                                    168
    Double tool holder for                                           169
    Gibbed elevating                                                 169
  =Examples= of feed motions                                         170
  =Feed Regulators= for screw cutting                                171
    The star feed                                                    172
  =Ratchet Feeds=                                                    173
  =Tool Holding= devices, the various kinds of                       173
  =Tool Rest= swiveling                                              174
  =Tool Holder= for compound slide rests                             174
    For octagon boring tools                                         175
  =Lathe Lead and Feed Screws=                                       175
    Lead screws, supporting, long                                    176
    Position of the feed nut                                         177
    Form of threads of lead screws                                   177
    The effect the form of thread has in causing the nut to lock
    properly or improperly                                           177
    Example of a lead screw with a pitch of three threads per inch   177
    Example of a lead screw with five threads per inch               178
    Example with a lead screw of five threads per inch               179
    Device for correcting the errors of pitch of                     179
  =Table= for finding the change wheels for screw cutting when
  the teeth in the change wheels advance by four                     180
    For finding the change wheels when the teeth in the wheels
    advance by six                                                   180
    Constructing a table to cut fractional threads on any lathe      181
    Finding the change wheels necessary to enable the lathe to cut
    threads of any given pitches                                     181
    Finding the change wheels necessary to cut fractional pitches    181
  =Determining= the pitches of the teeth for change wheels           182
  =Lathe Shears= or beds                                             182
    Advantages and disadvantages of, with raised V-guide-ways        182
    Examples of various forms of                                     183
  =Lathe Shears= with one V and one flat side                        183
    Methods of ribbing                                               184
    The arrangement of the legs of                                   184
  =Lathe Tailblock=                                                  185
    With rapid spindle motion                                        185
    With rapid fastenings and releasing devices                      185
    The wear of the spindles of                                      185
    Spindles, the various methods of locking                         186
    Testing, various methods of                                      187


  CHAPTER VIII.

  =SPECIAL FORMS OF THE LATHE.=

  =Watchmaker's Lathes=                                              188
    Construction of the headstock                                    188
    Construction of chucks for                                       188
    Expanding chucks for                                             188
    Contracting chucks for                                           188
    Construction of the tailblock                                    189
    Open spindle tailstocks for                                      189
    Filing fixture for                                               189
    Fixture for wheel and pinion cutting                             189
    Jewelers' rest for                                               189
  =Watch Manufacturers' Lathe=                                       190
    Special chucks for                                               190
    Pump centre rest                                                 190
  =Lathe=, hand                                                      191
    Screw slotting                                                   192
    With variable speed for facing purposes                          192
    Cutting-off machine                                              193
    Grinding Lathes                                                  193
      With elevating rest                                            194
      Universal                                                      195
      Special chucks for                                             196
    The Morton Poole calender roll grinding lathe                    196
      The construction of the bed and carriages                      197
      Principles of action of the carriages                     197, 198
      Construction of the emery-wheel arbors and the driving
      motion                                                    198, 199
      The advantages of                                              199
      The method of driving the roll                                 200
      Construction of the headstock                                  200
      The transverse motion                                          200
  =The Brown and Sharpe Screw Machine=, or screw-making lathe        200
    Threading tools for                                              203
    Examples of the use of                                           203
  =The Secor Screw Machine=, construction of the headstock           204
    The chuck                                                        205
    The feed gear                                                    205
    The turret                                                       205
    The cross slide                                                  205
    The stop motions                                                 206
  =Pratt and Whitney's Screw Machine=                                206
    Parkhurst's wire feed, construction of the headstock, chuck and
    feed motion                                                      207
    Box tools for                                                    208
    Applications of box tools                                        208
    Threading tool for                                               208
    Cutting-off tool for                                             208
  =Special Lathe= for wood working                                   208
    The construction of the carriage and reducing knife              209
    Construction of the various feed motions                         209
    Construction of the tailstock                                    209
  =Lathes for irregular forms=                                       210
    Axe-handle                                                       210
    Back knife gauge                                                 210
    Special, for pulley turning                                      211
  =Boring and Turning= mill or lathe                                 211
    Construction of the feed motions                                 213
    Construction of the framing and means of grinding the lathe      214
    Construction of the vertical feed motions                        215
  =The Morton Poole= roll turning lathe                              215
    Construction of the slide rest                                   216
    The tools for                                                    216
  =Special Lathes= for brass work                               216, 217
  =Boring Lathe= with traversing spindle                             218
    For engine cylinders                                             219
    Cylinder, with facing slide rests                                219
    With double heads and facing rests                               220
  =Lathe for turning Wheel= hubs                                     221


  CHAPTER IX.

  =DRIVING WORK IN THE LATHE.=

  =Drivers=, carriers, dogs, or clamps, and their defects            222
    Lathe clamps                                                     222
    Equalizing drivers                                               223
    The Clements driver                                              223
    Driver and face plate for screw cutting                          223
    Forms of, for bolt heads                                         224
    Adjustable, for bolt heads                                       224
    For threaded work                                                225
    For steady rest work                                             225
    For cored work                                                   225
    For wood                                                         225
  =Centres= for hollow work                                          226
    For taper work                                                   226
  =Lathe Mandrels=, or arbors                                        227
    Drivers for                                                      227
    For tubular work                                                 227
    Expanding mandrels                                               227
    With expanding cones                                             228
    With expanding pieces                                            228
    Expanding, for large work                                        228
    For threaded work                                                228
    For nuts, various forms of                                       229
    For eccentric work                                               229
  =Centring devices= for crank axles                                 230
  =The Steady Rest= or back rest                                     231
    Steady rest, improved form of                                    232
    Cone chuck                                                       232
    Steady rest for square and taper work                            233
    The cat head                                                     233
    Clamps for                                                       233
    Follower rests                                                   234
  =Chucks and Chucking=                                              234
    Simple forms of chucks                                           234
    Adjustable chucks for true work                                  235
    Two-jawed chucks                                                 236
    Box body chucks                                                  237
    Reversible jawed chucks                                          237
    Three and four-jawed chucks                                      237
    Combination chucks                                               237
    The wear of scroll chuck threads                                 237
    Universal chucks                                                 238
    The wear of chucks                                               240
    Special forms of chucks                                          241
    Expanding chucks for ring-work                                   241
    Cement chuck                                                     241
    Chucks for wood-working lathes                                   242
  =Lathe Face Plates=                                                243
    Face plates, errors in, and their effects                        243
    Work-holding straps                                              244
    Face plate, clamping work on                                     245
      Forms of clamps for                                            245
    Examples of chucking work on                                246, 247
    For wood work                                                    247
  =Special Lathe Chuck= for cranks                                   248
  =Face Plate Work=, examples of                                     249
    Errors in chucking                                               250
    Movable dogs for                                                 250
    The angle plate                                                  251
      Applications of                                                251
    Angle plate chucking, examples of                                251
    Cross-head chucking                                          251-253


  CHAPTER X.

  =CUTTING TOOLS FOR LATHES.=

  =Principles= governing the shapes of lathe tools                   254
  =Diamond-pointed=, or front tool                                   254
  =Principles= governing use of tools                                254
    Front rake and clearance of front tools                          254
    Influence of the height of a tool upon its clearance and
    keenness                                                         255
    Tools with side rake in various directions                       256
    The effect of side rake                                          256
    The angle of clearance in lathe tools                            257
    Variation of clearance from different rates of feed and
    diameters of work                                                257
  =Round-nosed= tools                                                258
  =Utmost Duty= of cutting tools                                     258
    Judging the quantity of the tool from the shape of its cutting   259
  =Square-nosed= tools                                               260
    The height of lathe tools                                        260
    Side tools for lathe work                                        261
    Cutting-off or grooving tools                                    262
    Facing tools or knife tools                                      262
    Spring tools                                                     263
  =Brass Work=, front tools for                                      264
    Side tools for                                                   264
  =Threading= tools                                                  264
    Internal threading tools                                         264
    The length of threading tools                                    265
    The level of threading tools                                     265
    Gauges for threading tools                                       266
    Setting threading tools                                          266
    Circular threading tools                                         267
    Threading tool holders                                           267
  =Chasers=                                                          268
    Chaser holders                                                   268
    Setting chasers                                                  268
  =Square Threads=, clearance of tools for                           269
    Diameter at the roots of threads                                 269
    Cutting coarse pitch square threads                              269
    Dies for finishing square threads                                269
  =Tool Holders= for outside work                                    270
    For circular cutters                                             272
    Swiveled                                                         273
    Combined tool holders and cutting-off tools                      273
  =Power Required= to drive cutting tools                            273


  CHAPTER XI.

  =DRILLING AND BORING IN THE LATHE.=

  =The Twist Drill=                                                  274
    Twist drill holders                                              274
    The diametral clearance of twist drills                          274
    The front rake of twist drills                                   275
    The variable clearance on twist drills as usually ground         275
    Demonstration of the common error in grinding twist drills       276
    The effects of improper grinding upon twist drills               276
    Table of speeds and feeds for twist drills                       277
    Grinding twist drills by hand                                    279
    Twist drills for wood work                                       279
  =Tailstock Chucks= for drilled work                                279
  =Flat Drills= for lathe work                                       280
    Holders for lathe work                                           281
  =Half-round= bit or pod auger                                      281
    With front rake for wrought iron or steel                        281
    With adjustable cutter                                           281
    For very true work                                               281
  =Chucking Reamer=                                                  281
    The number of teeth for reamers                                  282
    Spacing the teeth of reamers                                     282
    Spiral teeth for reamers                                         282
    Grinding the teeth of reamers                                    282
    Various positions of emery-wheel in grinding reamers             282
    Chucking reamers for true work                                   283
    Shell reamers                                                    283
    Arbor for shell reamers                                          283
    Rose-bit or rose reamers                                         283
    Shell rose reamers                                               284
    Adjustable reamers                                               284
    Stepped reamers for taper work                                   285
    Half-round reamers                                               285
    Reamers for rifle barrels                                        285
  =Boring Tools= for lathe work                                      285
    Countersinks                                                     285
    Shapes of lathe boring tools                                     285
    Boring tools for brass work                                      286
    The spring of boring tools                                       286
    Boring tools for small work                                      287
    Boring tool holders                                              287
  =Boring Devices for Lathes=                                        288
  =Boring Heads=                                                     288
  =Boring Bars=                                                      289
    Boring bar cutters                                               289
    Three _versus_ four cutters for boring bars                      290
    Boring bars with fixed heads                                     290
      With sliding heads                                             290
      Bar cutters, the shapes of                                     291
    Boring head with nut feed                                        291
    Boring bars for taper work, various forms of                     292
    Boring double-coned work                                         293
    Boring bar, centres for                                          293
  =Cutting Speeds= and feeds for wrought iron                        294
    Examples of speeds taken from practice                           295


  CHAPTER XII.

  =EXAMPLES IN LATHE WORK.=

  =Technical Terms= used in the work                                 296
  =Lathe Centres=                                                    296
    Devices for truing                                               297
    Tools for testing the truth of, for fine work                    298
    Shapes of, for light and heavy work                              299
  =Centre Drilling=, attachment for lathes                           300
    The error induced by straightening work after                    300
    Machine                                                          300
    Combined centre-drill and countersink                            300
    Countersink with adjustable drill                                300
    Centring square                                                  300
    Centre-punch                                                     300
    Centre-punch guide                                               301
    Centring work with the scribing block                            301
    Finding the centre of very rough work                            301
    Centre-drill chuck                                               302
    The proper form of countersink for lathe work                    302
    Countersinks for lathe work                                      302
    Various forms of square centres                                  303
    The advantage of the square centre for countersinking            303
    Novel form of countersink for hardened work                      303
    Chucks for centre-drilling and countersinking                    303
    Recentring turned work                                           304
  =Straightening Work.= Straightening machine for bar iron           304
    Hand device for straightening lathe work                         305
    Chuck for straightening wire                                     305
  =Cutting Rods= into small pieces of exact length, tools for        305
  =Roughing cuts=, the change of shape of work that occurs
  from removing the surface by                                       306
    Feeds for                                                        306
    Rates of feed for                                                307
  =Finishing Work=, the position of the tool for                     307
    Finishing cast-iron with water                                   307
    Specks in finished cast-iron work                                307
    Scrapers for finishing cast-iron work                            307
    Method of polishing lathe work                                   308
    Filing lathe work                                                308
    The use of emery paper on lathe work                             308
    The direction of tool feed in finishing long work                309
    Forms of laps for finishing gauges or other cylindrical lathe
    work                                                             310
    Forms of laps for finishing internal work                        311
    Grinding and polishing clamps for lathe work                     311
    Burnishing lathe work                                            311
  =Taper Work=, turning                                              312
    The wear of the centres of                                       312
    Setting over the tailstock to turn                               312
    Gauge for setting over                                           313
    Fitting                                                          313
    Grinding                                                         313
    The order of procedure in turning                                313
    The influence of the height of the tool in producing true        314
  =Special Forms.= Curved work                                  314, 315
    Standard gauges for taper work                                   316
    Methods of turning an eccentric                                  317
    Turning a cylinder cover                                         318
    Turning pulleys                                                  318
    Chucking device for pulleys                                      318
  =Cutting Screws= in the lathe                                      319
    The arrangement of the change gears                              319
    The intermediate wheels                                          319
    The compounded gears                                             320
    Finding the change wheels to cut a given thread                  320
    Finding the change wheels for a lathe whose gears are
    compounded                                                       321
    Finding the change gears for cutting fractional pitches          321
    To find what pitch of thread the wheels already on the lathe
    will cut                                                         322
    Cutting left-hand threads                                        322
    Cutting double threads                                           322
    Cutting screws whose pitches are given in the terms of the
        metric system                                                322
    Cutting threads on taper work                                    323
    Errors in cutting threads on taper work                          324


  CHAPTER XIII.

  =EXAMPLES IN LATHE WORK (Continued).=

  =Ball Turning= with tubular saw                                    325
    With a single tooth on the end of a revolving tube               325
    With a removable tool on an arbor                                325
    Tool holder with worm feed                                       325
    By hand                                                          325
  =Cams=, cutting in the lathe                                       326
    Improved method of originating cams in the lathe                 326
    Motions for turning cams in the lathe                       326, 327
    Application of cam motions to special work                       327
    Cam chuck for irregular work                                     328
  =Milling= or knurling tool                                         328
    Improved forms of                                                328
  =Winding Spiral Springs= in the lathe                              329
  =Hand Turning=                                                     330
    The heel tool                                                    330
    The graver and its applications                             330, 331
    Hand side tools                                                  331
    Hand round-nosed tools for iron                                  331
    Hand finishing tool                                              331
  =Hand Tools=, for roughing out brass work                          332
    Various forms and applications of scrapers                  332, 333
    Clockmakers' hand tool for special or standard work              334
    Screw cutting with hand tools                                    334
      Outside and inside chasers                                     334
      Hobs and their uses                                            335
      The application of chasers, and errors that may arise from
      the position in which they are presented to the work           336
      Errors commonly made in cutting up inside chasers              337
      V-tool for starting outside threads                            337
      Starting outside threads                                       338
      Cutting taper threads                                          338
    Wood turning hand tools                                          338
    The gauge and how to use it                                      338
    The chisel and its use                                           339
    The skew chisel and how to use it                                339
    Wood turners' boring tools for lathe work                        340


  CHAPTER XIV.

  =MEASURING MACHINES, TOOLS AND DEVICES.=

  =Standards of Measurements=, in various countries                  341
    Use of, by sight and by the sense of feeling                     341
    Variations in standard gauges                                    341
    The necessity for accurate standards                             341
    The Rogers Bond standard measuring machine                       342
      Details of construction of                                343, 344
      The principle of construction of                               344
      The methods of using                                           345
    The Whitworth measuring machine                                  345
    The Betts Machine Company's measuring machine                    346
    Professor Sweet's measuring machine                              347
    Measuring machine for sheet metal                                348
  =Circle=, division of the                                          348
    Troughton's method of dividing the circle                   348, 349
    Ramsden's dividing engine                                        349
      The construction of                                       350, 351
    Pratt and Whitney's dividing device                              352
      Practical application of                                       353
    Index wheel, method of originating, by R. Hoe & Co.              353
      Application of the index wheel (Hoe & Co.'s system)            353
  =Classification= of the measuring tools used by workmen            354
  =Micrometer Caliper= and its principle of construction        354, 355
  =Gauges.= Standard plug and collar gauges                          356
    Methods of comparing standard plug and collar gauges             356
    The effects of variations of temperature upon standard gauges    356
    Plug and collar gauges for taper work                            357
    The Baldwin standards for taper bolts                            359
    Workmen's gauges for lathe work                                  359
  =Calipers=, outside, the various forms of                          360
    Inside calipers                                                  360
    Calipers with locking devices                                    360
    Spring calipers                                                  360
    The methods of holding and using                            361, 362
    Keyway calipers                                                  363
    The advantages of calipers                                       363
  =Fitting.= The four kinds of fit in machine work                   363
    The influence of the diameter of the work in limiting the
    application of standard gauges                                   363
    The wear of tools and its influence upon the application of the
    standard gauge system                                            364
    The influence of the smoothness of the surface upon the
    allowance to be made for drilling or hydraulic fits              365
    Examples of allowance for hydraulic fits                         365
    Parallel holes and taper plugs for hydraulic fits                365
  =Fitting.= Practicable methods of testing the fit of axle
    brasses forced in by hydraulic pressure                          366
    Shrinkage or contraction fits                                    366
      Allowances for                                                 366
      Gauge for                                                      367
      The shrinkage system at the Royal Gun Factory at Woolwich      367
      Experiments by Thomas Wrightson upon the shrinkage of iron
      under repeated heatings and coolings                    368 to 374
      Shrinking work, to refit it                               374, 375


  CHAPTER XV.

  =MEASURING TOOLS.=

  =End Measurements= of large lathe work                             376
    Template gauges for                                              376
    Trammels or Trains                                               377
    Adjustable gauges for                                            377
  =Compasses=--Dividers                                              377
    Compass calipers                                                 378
  =Key Seating= rule                                                 378
  =Surface Gauge=                                                    378
    Pattern makers' pipe gauge                                       379
  =Squares.= The try square                                          379
    The T square                                                     379
    Various methods of testing squares                          379, 380
    Bevel squares                                                    380
  =Bevel Protractors=                                                380
  =Hexagon Gauge=                                                    381
  =Straight Edge= and its applications                          381, 382
    Winding strips and their application                             382
  =Surface Plate= or planimeter                                      383
  =Templates= for curves                                             384
  =Wire Gauges=, notch                                               384
    Standard gauges for wire, &c.                               384, 386
    Gauge for music wire                                             386
    Brown and Sharpe wire gauge                                      387
    Birmingham wire gauge for rolled shell silver and gold           387
    Sheet iron gauge, Russian                                        387
    Galvanized iron                                                  387
    Belgian sheet zinc                                               387
    American sheet zinc                                              387
  =Rifle Bore= gauge                                                 387
  =Strength of Wire=, Kirkaldy's experiments                    387, 388


  CHAPTER XVI.

  =SHAPING AND PLANING MACHINES.=

  =General description= of a shaping machine                         389
  =Construction= of swivel head                                      389
    Slide                                                            390
    Vice chuck                                                       390
    Feed motion                                                      390
  =Hand= shaping machine                                             392
  =Quick Return Motion=, Whitworth's                                 392
  =Vice Chucks=, the principles of construction of plain, for
  planing machine                                                    392
    The proper methods of chucking work in                           393
    Holding taper work in                                            394
    Various forms of                                                 394
    Swiveling                                                        395
    Rapid motion                                                     396
    For vice work                                                    396
  =Centres= for shaping machines                                     397
  =Traveling Head= in shaping machine                                397
  =Planer Shapers= or shaping machines, having a tappet motion
  for reversing the direction of motion                         398, 399
  =Quick Return Motion= shaping machines, link                       399
    The Whitworth                                                    400
    Comparisons of the link motion and Whitworth                     401
  =Simple Crank=, investigating the motion of                        401
  =Planing Machines=, or planer                                      402
    The various motions of                                      402, 403
    The table driving gear                                           404
    Planing machine with double heads                                404
    Rotary planing machine                                           405


  CHAPTER XVII.

  =PLANING MACHINERY.=

  =The Sellers= planing machine                                      406
    The belt shifting mechanism                                 406, 407
    The automatic feed motions                                       408
  =Sliding Head=                                                     408
  =Cross Bar=                                                        409
  =Slides of Planers=, the various forms of construction of          410
  =Wear of the Slides= of planer heads, various methods of
  taking up the                                                      410
  =Swivel Heads=                                                     411
  =Tool Aprons=                                                      411
  =Swivel Tool-holding devices= for planers                          411
  =Planer Heads=, graduations of                                     412
    Safety devices for                                               413
    Feed motions for                                                 414
    V-guideways for                                                  414
    Flat guideways for                                               415
    Oiling devices for                                               415
  =Planing Machine Tables=                                           415
    Slots and holes in planing machine tables                        416
    Forms of bolts for planer tables                                 417
    Supplementary tables for planer tables                           417
    Angle plates for planer tables                                   418
    Chucking devices for planer tables                               418
  =Planer Centres=                                                   418
  =Planer Chucks=                                                    419
    For spiral grooved work                                          419
    For curved work                                                  420
    Chucking machine beds on planer tables                           420
    For large planing machines                                       422
    Chucking the halves of large pulleys on a planer                 423
  =Gauges= for planing V-guideways in machine beds                   421
    Planing guideways in machine beds                                422
    Gauge for planer tools                                           424
  =Planer Tools=, the shapes of                                      424
    For coarse finishing feeds                                       424
    The clearance of                                                 424
    For slotted work                                                 424
  =Planer Tool Holder=, with tool post                               425
    Various applications of                                          425
    Simple and advantageous form of                                  426
    Examples of application of                                       426


  CHAPTER XVIII.

  =DRILLING MACHINES.=

  =Drilling Machines.= General description of a power drilling
  machine                                                            428
    Lever feed                                                       428
    With automatic and quick return feed motions                     428
    Improved, with simple belt and uniform motion, two series of
    rates of automatic feed, and guide for boring bar           429, 430
    Radial                                                      430, 431
    For boiler shells                                                436
    Cotter or keyway                                                 438
    Drilling Machine, three-spindle                                  434
    Four-spindle                                                     434
  =Drilling and Boring= machine                                      431
    Feed motion of                                                   432
  =Combined Drilling Machine= and lathe                              433
  =Boring Machine=, horizontal                                       433
    For car wheels                                                   438
    For pulleys                                                      438
  =Quartering Machine=                                               434
  =Drilling and Turning Machine= for boiler makers                   435
    Feed motions of                                                  436


  CHAPTER XIX.

  =DRILLS AND CUTTERS FOR DRILLING MACHINES.=

  =Jigs or Fixtures= for drilling machines                           439
    Limits of error in                                               439
    Examples of, for simple work, as for links, &c.                  440
    Considerations in designing                                      440
    For drilling engine cylinders                             440 to 441
    For cutting out steam ports                                      441
  =Drills and Cutters= for drilling machines                         442
    Table of sizes of twist drills, and their shanks                 442
    Flat drills for drilling machines                                442
    Errors in grinding flat drills                                   443
    The tit-drill                                                    443
    The lip drill                                                    443
    Cotter or keyway drills                                          446
  =Drilling holes= true to location with flat drills                 444
    Drilling hard metal                                              444
    Table of sizes of tapping holes                                  445
  =Drill Shanks= and sockets                                         445
    Improved form of drill shank                                     446
    Square shanked drills and their disadvantages                    446
  =Drill Chucks=                                                     446
  =Stocks and Cutters= for drilling machines                         447
    Tube plate cutters                                               448
  =Stocks and Cutters.= Adjustable stock and cutter                  448
    Facing tool with reamer pin                                      449
    Counterbores for drilling machines                               449
    Drill and counterbore for wood work                              449
    Facing and countersink cutters                                   449
    Device for drilling square holes                                 450
    Device for drilling taper holes in a drilling machine            451


  CHAPTER XX.

  =HAND-DRILLING AND BORING TOOLS, AND DEVICES.=

  =The Brad-awl=                                                     452
  =Bits.= The gimlet bit                                             452
    The German bit                                                   452
    The nail bit                                                     452
    The spoon bit                                                    452
    The nose bit                                                     453
    The auger bit                                                    453
    Cook's auger bit                                                 453
    Principles governing the shapes of the cutting edges of auger
    bits                                                             453
    Auger bit for boring end grain wood                              453
    The centre bit                                                   454
    The expanding bit                                                454
  =Drills.= Drill for stone                                          454
    The fiddle drill                                                 455
    The fiddle drill with feeding device                             455
    Drill with cord and spring motion                                455
    Drill stock with spiral grooves                                  455
    Drill brace                                                      455
    Drill brace with ratchet motion                                  456
    Universal joint for drill brace                                  456
    Drill brace with multiplying gear and ratchet motion             456
    Breast drill with double gear                                    456
    Drilling levers for blacksmiths                                  457
    Drill cranks                                                     457
    Ratchet brace                                                    457
    Flexible shaft for driving drills                                458
    Drilling device for lock work                                    459
    Hand drilling machine                                            459
  =Slotting Machine=                                                 459
    Sectional view of                                                460
    Tool holders                                                460, 461
    Tools                                                       461, 462


  CHAPTER XXI.

  =THREAD-CUTTING MACHINERY AND BROACHING PRESS.=

  =Pipe Threading=, die stock for, by hand                           463
    Die stock for, by power                                          463
    Pipe threading machines, general construction of                 463
  =Bolt Threading= hand machine                                      464
    With revolving head                                              465
    Power threading machine                                          465
    With automatic stop motion                                       466
    Construction of the head                                         466
    Construction of the chasers                                      466
    Bolt threading machine with back gear                            467
    Single rapid bolt threading machine                              467
    Double rapid bolt threading machine                              467
    Construction of the heads of the rapid machines                  468
    Bolt threading machinery, the Acme                               468
    Construction of the head of                               468 to 470
    Capacity of                                                      470
  =Cutting Edges= for taps, the number of                            471
    Examples when three and when four cutting edges are used, and
    the results upon bolts that are not round                   471, 472
    Demonstration that four cutting edges are correct for bar iron   472
  =Positions of Dies=, or chasers in the heads of bolt cutting
  machine                                                            473
  =Dies=, methods of hobbing, to avoid undue friction                473
    The construction of, for bolt threading machines                 473
    Method of avoiding friction in thread cutting                    474
    Hob for threading                                                474
    Cutting speeds for threading                                     474
  =Nut Tapping= machine                                              475
    Automatic socket for                                             475
    Rotary                                                           475
    Three-spindle                                                    475
  =Pipe Threading Machine=                                    475 to 477
  =Tapping Machine= for steam pipe fittings                          478
  =Broaching Press=                                                  478
    Principles of broaching                                          478
    Examples in the construction of broaches                         479



FULL-PAGE PLATES.


  Volume I.
                                                               _Facing_

  _Frontispiece._ MODERN LOCOMOTIVE ENGINE.                   TITLE PAGE

  PLATE     I. TEMPLATE-CUTTING MACHINES FOR GEAR TEETH.              34
    "      II. FORMS OF SCREW THREADS.                                85
    "     III. MEASURING AND GAUGING SCREW THREADS.                   93
    "      IV. END-ADJUSTMENT AND LOCKING DEVICES.                   120
    "       V. EXAMPLES IN LATHE CONSTRUCTION.                       148
    "      VI. CHUCKING LATHES.                                      150
    "     VII. TOOL-HOLDING AND ADJUSTING APPLIANCES.                174
    "    VIII. WATCHMAKER'S LATHE.                                   188
    "      IX. DETAILS OF WATCHMAKER'S LATHE.                        188
    "       X. EXAMPLES OF SCREW MACHINES.                           200
    "      XI. ROLL-TURNING LATHE.                                   215
    "     XII. EXAMPLES IN ANGLE-PLATE CHUCKING.                     252
    "    XIII. METHODS OF BALL-TURNING.                              325
    "     XIV. STANDARD MEASURING MACHINES.                          341
    "      XV. DIVIDING ENGINE AND MICROMETER.                       354
    "     XVI. SHAPING MACHINES AND TABLE-SWIVELING DEVICES.         398
    "    XVII. EXAMPLES OF PLANING MACHINES.                         404
    "   XVIII. EXAMPLES IN PLANING WORK.                             422
    "     XIX. LIGHT DRILLING MACHINES.                              428
    "      XX. HEAVY DRILLING MACHINES.                              430
    "     XXI. EXAMPLES IN BORING MACHINERY.                         434
    "    XXII. BOILER-DRILLING MACHINERY.                            436
    "   XXIII. NUT-TAPPING MACHINERY.                                475



MODERN

MACHINE SHOP PRACTICE.



CHAPTER I.--THE TEETH OF GEAR-WHEELS.


A wheel that is provided with teeth to mesh, engage, or gear with
similar teeth upon another wheel, so that the motion of one may be
imparted to the other, is called, in general terms, a gear-wheel.

[Illustration: Fig. 1.]

When the teeth are arranged to be parallel to the wheel-axis, as in Fig.
1, the wheel is termed a spur-wheel. In the figure, A represents the
axial line or axis of the wheel or of its shaft, to which the teeth are
parallel while spaced equidistant around the rim, or face, as it is
termed, of the wheel.

[Illustration: Fig. 2.]

[Illustration: Fig. 3.]

When the wheel has its teeth arranged at an angle to the shaft, as in
Fig. 2, it is termed a bevel-wheel, or bevel gear; but when this angle
is one of 45°, as in Fig. 3, as it must be if the pair of wheels are of
the same diameter, so as to make the revolutions of their shafts equal,
then the wheel is called a mitre-wheel. When the teeth are arranged upon
the radial or side face of the wheel, as in Fig. 4, it is termed a
crown-wheel. The smallest wheel of a pair, or of a train or set of
gear-wheels, is termed the pinion; and when the teeth are composed of
rungs, as in Fig. 5, it is termed a lantern, trundle, or wallower; and
each cylindrical piece serving as a tooth is termed a _stave_,
_spindle_, or _round_, and by some a _leaf_.

[Illustration: Fig. 4.]

An annular or internal gear-wheel is one in which the faces of the teeth
are within and the flanks without, or outside the pitch-circle, as in
Fig. 6; hence the pinion P operates within the wheel.

[Illustration: Fig. 5.]

[Illustration: Fig. 6.]

When the teeth of a wheel are inserted in mortises or slots provided in
the wheel-rim, it is termed a mortised-wheel, or a cogged-wheel, and the
teeth are termed cogs.

When the teeth are arranged along a plane surface or straight line, as
in Fig. 7, the toothed plane is termed a _rack_, and the wheel is termed
a pinion.

A wheel that is driven by a revolving screw, or worm as it is termed, is
called a worm-wheel, the arrangement of a worm and worm-wheel being
shown in Fig. 8. The screw or worm is sometimes also called an endless
screw, because its action upon the wheel does not come to an end as it
does when it is revolved in one continuous direction and actuates a nut.
So also, since the worm is tangent to the wheel, the arrangement is
sometimes called a wheel and tangent screw.

The diameter of a gear-wheel is always taken at the pitch circle, unless
otherwise specially stated as "diameter over all," "diameter of
addendum," or "diameter at root of teeth," &c., &c.

[Illustration: Fig. 7.]

When the teeth of wheels engage to the proper distance, which is when
the pitch circles meet, they are said to be in gear, or geared together.
It is obvious that if two wheels are to be geared together their teeth
must be the same distance apart, or the same _pitch_, as it is called.

The designations of the various parts or surfaces of a tooth of a
gear-wheel are represented in Fig. 9, in which the surface A is the face
of the tooth, while the dimension F is the width of face of the wheel,
when its size is referred to. B is the flank or distance from the pitch
line to the root of the tooth, and C the point. H is the _space_, or
the distance from the side of one tooth to the nearest side of the next
tooth, the width of space being measured on the pitch circle P P. E is
the depth of the tooth, and G its thickness, the latter also being
measured on the pitch circle P P. When spoken of with reference to a
tooth, P P is called the pitch line, but when the whole wheel is
referred to it becomes the pitch circle.

[Illustration: Fig. 8.]

The points C and the surface H are true to the wheel axis.

The teeth are designated for measurement by the pitch; the height or
depth above and below pitch line; and the thickness.

The pitch, however, may be measured in two ways, to wit, around the
pitch circle A, in Fig. 10, which is called the arc or circular pitch,
and across B, which is termed the chord pitch.

[Illustration: Fig. 9.]

In proportion as the diameter of a wheel (having a given pitch) is
increased, or as the pitch of the teeth is made finer (on a wheel of a
given diameter) the arc and chord pitches more nearly coincide in
length. In the practical operations of marking out the teeth, however,
the arc pitch is not necessarily referred to, for if the diameter of the
pitch circle be made correct for the required number of teeth having the
necessary arc pitch, and the wheel be accurately divided off into the
requisite number of divisions with compasses set to the chord pitch, or
by means of an index plate, then the arc pitch must necessarily be
correct, although not referred to, save in determining the diameter of
the wheel at the pitch circle.

The difference between the width of a space and the thickness of the
tooth (both being measured on the pitch circle or pitch line) is termed
the clearance or side clearance, which is necessary to prevent the teeth
of one wheel from becoming locked in the spaces of the other. The amount
of clearance is, when the teeth are cut to shape in a machine, made just
sufficient to prevent contact on one side of the teeth when they are in
proper gear (the pitch circles meeting in the line of centres). But when
the teeth are cast upon the wheel the clearance is increased to allow
for the slight inequalities of tooth shape that is incidental to casting
them. The amount of clearance given is varied to suit the method
employed to mould the wheels, as will be explained hereafter.

The line of centres is an imaginary line from the centre or axis of one
wheel to the axis of the other when the two are in gear; hence each
tooth is most deeply engaged, in the space of the other wheel, when it
is on the line of centres.

There are three methods of designating the sizes of gear-wheels. First,
by their diameters at the pitch circle or pitch diameter and the number
of teeth they contain; second, by the number of teeth in the wheel and
the pitch of the teeth; and third, by a system known as diametral pitch.

[Illustration: Fig. 10.]

The first is objectionable because it involves a calculation to find the
pitch of the teeth; furthermore, if this calculation be made by dividing
the circumference of the pitch circle by the number of teeth in the
wheel, the result gives the arc pitch, which cannot be measured
correctly by a lineal measuring rule, especially if the wheel be a small
one having but few teeth, or of coarse pitch, as, in that case, the arc
pitch very sensibly differs from the chord pitch, and a second
calculation may become necessary to find the chord pitch from the arc
pitch.

The second method (the number and pitch of the teeth) possesses the
disadvantage that it is necessary to state whether the pitch is the arc
or the chord pitch.

If the arc pitch is given it is difficult to measure as before, while if
the chord pitch is given it possesses the disadvantage that the
diameters of the wheels will not be exactly proportional to the numbers
of teeth in the respective wheels. For instance, a wheel with 20 teeth
of 2 inch chord pitch is not exactly half the diameter of one of 40
teeth and 2 inch chord pitch.

To find the chord pitch of a wheel take 180 (= half the degrees in a
circle) and divide it by the number of teeth in the wheel. In a table of
natural sines find the sine for the number so found, which multiply by
2, and then by the radius of the wheel in inches.

Example.--What is the chord pitch of a wheel having 12 teeth and a
diameter (at pitch circle) of 8 inches? Here 180 ÷ 12 = 15; (sine of 15
is .25881). Then .25881 × 2 = .51762 × 4 (= radius of wheel) = 2.07048
inches = chord pitch.

TABLE OF NATURAL SINES.

  +--------+--------++--------+--------++--------+--------+
  |Degrees.|  Sine. ||Degrees.|  Sine. ||Degrees.|  Sine. |
  +--------+--------++--------+--------++--------+--------+
  |    1   | .01745 ||   16   | .27563 ||   31   | .51503 |
  |    2   | .03489 ||   17   | .29237 ||   32   | .52991 |
  |    3   | .05233 ||   18   | .30901 ||   33   | .54463 |
  |    4   | .06975 ||   19   | .32556 ||   34   | .55919 |
  |    5   | .08715 ||   20   | .34202 ||   35   | .57357 |
  |    6   | .10452 ||   21   | .35836 ||   36   | .58778 |
  |    7   | .12186 ||   22   | .37460 ||   37   | .60181 |
  |    8   | .13917 ||   23   | .39073 ||   38   | .61566 |
  |    9   | .15643 ||   24   | .40673 ||   39   | .62932 |
  |   10   | .17364 ||   25   | .42261 ||   40   | .64278 |
  |   11   | .19080 ||   26   | .43837 ||   41   | .65605 |
  |   12   | .20791 ||   27   | .45399 ||   42   | .66913 |
  |   13   | .22495 ||   28   | .46947 ||   43   | .68199 |
  |   14   | .24192 ||   29   | .48480 ||   44   | .69465 |
  |   15   | .25881 ||   30   | .50000 ||   45   | .70710 |
  +--------+--------++--------+--------++--------+--------+

The principle upon which diametral pitch is based is as follows:--

The diameter of the wheel at the pitch circle is supposed to be divided
into as many equal parts or divisions as there are teeth in the wheel,
and the length of one of these parts is the diametral pitch. The
relationship which the diametral bears to the arc pitch is the same as
the diameter to the circumference, hence a diametral pitch which
measures 1 inch will accord with an arc pitch of 3.1416; and it becomes
evident that, for all arc pitches of less than 3.1416 inches, the
corresponding diametral pitch must be expressed in fractions of an inch,
as 1/2, 1/3, 1/4, and so on, increasing the denominator until the
fraction becomes so small that an arc with which it accords is too fine
to be of practical service. The numerators of these fractions being 1,
in each case, they are in practice discarded, the denominators only
being used, so that, instead of saying diametral pitches of 1/2, 1/3, or
1/4, we say diametral pitches of 2, 3, or 4, meaning that there are 2,
3, or 4 teeth on the wheel for every inch in the diameter of the pitch
circle.

Suppose now we are given a diametral pitch of 2. To obtain the
corresponding arc pitch we divide 3.1416 (the relation of the
circumference to the diameter) by 2 (the diametral pitch), and 3.1416 ÷
2 = 1.57 = the arc pitch in inches and decimal parts of an inch. The
reason of this is plain, because, an arc pitch of 3.1416 inches being
represented by a diametral pitch of 1, a diametral pitch of 1/2 (or 2 as
it is called) will be one half of 3.1416. The advantage of discarding
the numerator is, then, that we avoid the use of fractions and are
readily enabled to find any arc pitch from a given diametral pitch.

Examples.--Given a 5 diametral pitch; what is the arc pitch? First
(using the full fraction 1/5) we have 1/5 × 3.1416 = .628 = the arc
pitch. Second (discarding the numerator), we have 3.1416 ÷ 5 = .628 =
arc pitch. If we are given an arc pitch to find a corresponding
diametral pitch we again simply divide 3.1416 by the given arc pitch.

Example.--What is the diametral pitch of a wheel whose arc pitch is
1-1/2 inches? Here 3.1416 ÷ 1.5 = 2.09 = diametral pitch. The reason of
this is also plain, for since the arc pitch is to the diametral pitch as
the circumference is to the diameter we have: as 3.1416 is to 1, so is
1.5 to the required diametral pitch; then 3.1416 × 1 ÷ 1.5 = 2.09 = the
required diametral pitch.

To find the number of teeth contained in a wheel when the diameter and
diametral pitch is given, multiply the diameter in inches by the
diametral pitch. The product is the answer. Thus, how many teeth in a
wheel 36 inches diameter and of 3 diametral pitch? Here 36 × 3 = 108 =
the number of teeth sought. Or, per contra, a wheel of 36 inches
diameter has 108 teeth. What is the diametral pitch? 108 ÷ 36 = 3 = the
diametral pitch. Thus it will be seen that, for determining the relative
sizes of wheels, this system is excellent from its simplicity. It also
possesses the advantage that, by adding two parts of the diametral pitch
to the pitch diameter, the outside diameter of the wheel or the diameter
of the addendum is obtained. For instance, a wheel containing 30 teeth
of 10 pitch would be 3 inches diameter on the pitch circle and 3-2/10
outside or total diameter.

Again, a wheel having 40 teeth of 8 diametral pitch would have a pitch
circle diameter of 5 inches, because 40 ÷ 8 = 5, and its full diameter
would be 5-1/4 inches, because the diametral pitch is 1/8, and this
multiplied by 2 gives 1/4, which added to the pitch circle diameter of 5
inches makes 5-1/4 inches, which is therefore the diameter of the
addendum, or, in other words, the full diameter of the wheel.

Suppose now that a pair of wheels require to have pitch circles of 5 and
8 inches diameter respectively, and that the arc pitch requires to be,
say, as near as may be 4/10 inch; to find a suitable pitch and the
number of teeth by the diametral pitch system we proceed as follows:

In the following table are given various arc pitches, and the
corresponding diametral pitch.

  +----------------+----------+----------+----------------+
  |Diametral Pitch.|Arc Pitch.|Arc Pitch.|Diametral Pitch.|
  +----------------+----------+----------+----------------+
  |                |          |   Inch.  |                |
  |       2        |   1.57   |  1.75    |      1.79      |
  |       2.25     |   1.39   |  1.5     |      2.09      |
  |       2.5      |   1.25   |  1.4375  |      2.18      |
  |       2.75     |   1.14   |  1.375   |      2.28      |
  |       3        |   1.04   |  1.3125  |      2.39      |
  |       3.5      |    .890  |  1.25    |      2.51      |
  |       4        |    .785  |  1.1875  |      2.65      |
  |       5        |    .628  |  1.125   |      2.79      |
  |       6        |    .523  |  1.0625  |      2.96      |
  |       7        |    .448  |  1.0000  |      3.14      |
  |       8        |    .392  |  0.9375  |      3.35      |
  |       9        |    .350  |  0.875   |      3.59      |
  |      10        |    .314  |  0.8125  |      3.86      |
  |      11        |    .280  |  0.75    |      4.19      |
  |      12        |    .261  |  0.6875  |      4.57      |
  |      14        |    .224  |  0.625   |      5.03      |
  |      16        |    .196  |  0.5625  |      5.58      |
  |      18        |    .174  |  0.5     |      6.28      |
  |      20        |    .157  |  0.4375  |      7.18      |
  |      22        |    .143  |  0.375   |      8.38      |
  |      24        |    .130  |  0.3125  |     10.00      |
  |      26        |    .120  |  0.25    |     12.56      |
  +----------------+----------+----------+----------------+

From this table we find that the nearest diametral pitch that will
correspond to an arc pitch of 4/10 inch is a diametral pitch of 8, which
equals an arc pitch of .392, hence we multiply the pitch circles (5 and
8,) by 8, and obtain 40 and 64 as the number of teeth, the arc pitch
being .392 of an inch. To find the number of teeth and pitch by the arc
pitch and circumference of the pitch circle, we should require to find
the circumference of the pitch circle, and divide this by the nearest
arc pitch that would divide the circumference without leaving a
remainder, which would entail more calculating than by the diametral
pitch system.

The designation of pitch by the diametral pitch system is, however, not
applied in practice to coarse pitches, nor to gears in which the teeth
are cast upon the wheels, pattern makers generally preferring to make
the pitch to some measurement that accords with the divisions of the
ordinary measuring rule.

Of two gear-wheels that which impels the other is termed the driver, and
that which receives motion from the other is termed the driven wheel or
follower; hence in a single pair of wheels in gear together, one is the
driver and the other the driven wheel or follower. But if there are
three wheels in gear together, the middle one will be the follower when
spoken of with reference to the first or prime mover, and the driver,
when mentioned with reference to the third wheel, which will be a
follower. A series of more than two wheels in gear together is termed a
train of wheels or of gearing. When the wheels in a train are in gear
continuously, so that each wheel, save the first and last, both receives
and imparts motion, it is a simple train, the first wheel being the
driver, and the last the follower, the others being termed intermediate
wheels. Each of these intermediates is a follower with reference to the
wheel that drives it, and a driver to the one that it drives. But the
velocity of all the wheels in the train is the same in fact per second
(or in a given space of time), although the revolutions in that space
of time may vary; hence a simple train of wheels transmits motion
without influencing its velocity. To alter the velocity (which is always
taken at a point on the pitch circle) the gearing must be compounded, as
in Fig. 11, in which A, B, C, E are four wheels in gear, B and C being
compounded, that is, so held together on the shaft D that both make an
equal number of revolutions in a given time. Hence the velocity of C
will be less than that of B in proportion as the diameter,
circumference, radius, or number of teeth in C, varies from the
diameter, radius, circumference, or number of teeth (all the wheels
being supposed to have teeth of the same pitch) in B, although the
rotations of B and C are equal. It is most convenient, and therefore
usual, to take the number of teeth, but if the teeth on C (and therefore
those on E also) were of different pitch from those on B, the radius or
diameters of the wheels must be taken instead of the pitch, when the
velocities of the various wheels are to be computed. It is obvious that
the compounded pair of wheels will diminish the velocity when the driver
of the compounded pair (as C in the figure) is of less radius than the
follower B, and conversely that the velocity will be increased when the
driver is of greater radius than the follower of the compound pair.

[Illustration: Fig. 11.]

The diameter of the addendum or outer circle of a wheel has no influence
upon the velocity of the wheel. Suppose, for example, that we have a
pair of wheels of 3 inch arc or circular pitch, and containing 20 teeth,
the driver of the two making one revolution per minute. Suppose the
driven wheel to have fast upon its shaft a pulley whose diameter is one
foot, and that a weight is suspended from a line or cord wound around
this pulley, then (not taking the thickness of the line into account)
each rotation of the driven wheel would raise the weight 3.1416 feet
(that being the circumference of the pulley). Now suppose that the
addendum circle of either of the wheels were cut off down to the pitch
circle, and that they were again set in motion, then each rotation of
the driven wheel would still raise the weight 3.1416 feet as before.

It is obvious, however, that the addendum circle must be sufficiently
larger than the pitch circle to enable at least one pair of teeth to be
in continuous contact; that is to say, it is obvious that contact
between any two teeth must not cease before contact between the next two
has taken place, for otherwise the motion would not be conveyed
continuously. The diameter of the pitch circle cannot be obtained from
that of the addendum circle unless the pitch of the teeth and the
proportion of the pitch allowed for the addendum be known. But if these
be known the diameter of the pitch circle may be obtained by subtracting
from that of the addendum circle twice the amount allowed for the
addendum of the tooth.

Example.--A wheel has 19 teeth of 3 inch arc pitch; the addendum of the
tooth or teeth equals 3/10 of the pitch, and its addendum circle
measures 19.943 inches; what is the diameter of the pitch circle? Here
the addendum on each side of the wheel equals (3/10 of 3 inches) = .9
inches, hence the .9 must be multiplied by 2 for the two sides of the
wheel, thus, .9 × 2 = 1.8. Then, diameter of addendum circle 19.943
inches less 1.8 inches = 18.143 inches, which is the diameter of the
pitch circle.

Proof.--Number of teeth = 19, arc pitch 3, hence 19 × 3 = 57 inches,
which, divided by 3.1416 (the proportion of the circumference to the
diameter) = 18.143 inches.

If the distance between the centres of a pair of wheels that are in gear
be divided into two parts whose lengths are in the same proportion one
to the other as are the numbers of teeth in the wheels, then these two
parts will represent the radius of the pitch circles of the respective
wheels. Thus, suppose one wheel to contain 100 and the other 50 teeth,
and that the distance between their centres is 18 inches, then the pitch
radius or pitch diameter of one will be twice that of the other, because
one contains twice as many teeth as the other. In this case the radius
of pitch circle for the large wheel will be 12 inches, and that for the
small one 6 inches, because 12 added to 6 makes 18, which is the
distance between the wheel centres, and 12 is in the same proportion to
6 that 100 is to 50.

A simple rule whereby to find the radius of the pitch circles of a pair
of wheels is as follows:--

Rule.--Divide number of teeth in the large wheel by the number in the
small one, and to the sum so obtained add 1. Take this amount and divide
it into the distance between the centres of the wheels, and the result
will be the radius of the smallest wheel. To obtain the radius of the
largest wheel subtract the radius of the smallest wheel from the
distance between the wheel centres.

Example.--Of a pair of wheels, one has 100 and the other 50 teeth, the
distance between their centres is 18 inches; what is the pitch radius of
each wheel?

Here 100 ÷ 50 = 2, and 2 + 1 = 3. Then 18 ÷ 3 = 6, hence the pitch
radius of the small wheel is 6 inches. Then 18 - 6 = 12 = pitch radius
of large wheel.

Example 2.--Of a pair of wheels one has 40 and the other 90 teeth. The
distance between the wheel centres is 32-1/2 inches; what are the radii
of the respective pitch circles? 90 ÷ 40 = 2.25 and 2.25 + 1 = 3.25.
Then 32.5 ÷ 3.25 = 10 = pitch radius of small wheel, and 32.5 - 10 =
22.5, which is the pitch radius of the large wheel.

To prove this we may show that the pitch radii of the two wheels are in
the same proportion as their numbers of teeth, thus:--

  Proof.--Radius of small wheel  =   10    × 4 =    40
                                    ----           ----
          radius of large wheel  =  22.5   × 4 =   90.0

Suppose now that a pair of wheels are constructed, having respectively
50 and 100 teeth, and that the radii of their true pitch circles are 12
and 6 respectively, but that from wear in their journals or journal
bearings this 18 inches (12 + 6 = 18) between centres (or line of
centres, as it is termed) has become 18-3/8 inches. Then the acting
effective or operative radii of the pitch circles will bear the same
proportion to the 18-3/8 as the numbers of teeth in the respective
wheels, and will be 12.25 for the large, and 6.125 for the small wheel,
instead of 12 and 6, as would be the case were the wheels 18 inches
apart. Working this out under the rule given we have 100 ÷ 50 = 2, and 2
+ 1 = 3. Then 18.375 ÷ 3 = 6.125 = pitch radius of small wheel, and
18.375 - 6.125 = 12.25 = pitch radius of the large wheel.

The true pitch line of a tooth is the line or point where the face curve
joins the flank curve, and it is essential to the transmission of
uniform motion that the pitch circles of epicycloidal wheels exactly
coincide on the line of centres, but if they do not coincide (as by not
meeting or by overlapping each other), then a false pitch circle becomes
operative instead of the true one, and the motion of the driven wheel
will be unequal at different instants of time, although the revolutions
of the wheels will of course be in proportion to the respective numbers
of their teeth.

If the pitch circle is not marked on a single wheel and its arc pitch is
not known, it is practically a difficult matter to obtain either the arc
pitch or diameter of the pitch circle. If the wheel is a new one, and
its teeth are of the proper curves, the pitch circle will be shown by
the junction of the curves forming the faces with those forming the
flanks of the teeth, because that is the location of the pitch circle;
but in worn wheels, where from play or looseness between the journals
and their bearings, this point of junction becomes rounded, it cannot be
defined with certainty.

In wheels of large diameter the arc pitch so nearly coincides with the
chord pitch, that if the pitch circle is not marked on the wheel and the
arc pitch is not known, the chord pitch is in practice often assumed to
represent the arc pitch, and the diameter of the wheel is obtained by
multiplying the number of teeth by the chord pitch. This induces no
error in wheels of coarse pitches, because those pitches advance by 1/4
or 1/2 inch at a step, and a pitch measuring about, say, 1-1/4 inch
chord pitch, would be known to be 1-1/4 arc pitch, because the
difference between the arc and chord pitch would be too minute to cause
sensible error. Thus the next coarsest pitch to 1 inch would be 1-1/8,
or more often 1-1/4 inch, and the difference between the arc and chord
pitch of the smallest wheel would not amount to anything near 1/8 inch,
hence there would be no liability to mistake a pitch of 1-1/8 for 1 inch
or _vice versâ_. The diameter of wheel that will be large enough to
transmit continuous motion is diminished in proportion as the pitch is
decreased; in proportion, also, as the wheel diameter is reduced, the
difference between the arc and chord pitch increases, and further the
steps by which fine pitches advance are more minute (as 1/4, 9/32, 5/16,
&c.). From these facts there is much more liability to err in estimating
the arc from the measured chord pitch in fine pitches, hence the
employment of diametral pitch for small wheels of fine pitches is on
this account also very advantageous. In marking out a wheel the chord
pitch will be correct if the pitch circle be of correct diameter and be
divided off into as many points of equal division (with compasses) as
there are to be teeth in the wheel. We may then mark from these points
others giving the thickness of the teeth, which will make the spaces
also correct. But when the wheel teeth are to be cut in a machine out of
solid metal, the mechanism of the machine enables the marking out to be
dispensed with, and all that is necessary is to turn the wheel to the
required addendum diameter, and mark the pitch circle. The following are
rules for the purposes they indicate.

The circumference of a circle is obtained by multiplying its diameter by
3.1416, and the diameter may be obtained by dividing the circumference
by 3.1416.

The circumference of the pitch circle divided by the arc pitch gives the
number of teeth in the wheel.

The arc pitch multiplied by the number of teeth in the wheel gives the
circumference of the pitch circle.

Gear-wheels are simply rotating levers transmitting the power they
receive, less the amount of friction necessary to rotate them under the
given conditions. All that is accomplished by a simple train of gearing
is, as has been said, to vary the number of revolutions, the speed or
velocity measured in feet moved through per minute remaining the same
for every wheel in the train. But in a compound train of gears the speed
in feet per minute, as well as the revolutions, may be varied by means
of the compounded pairs of wheels. In either a simple or a compound
train of gearing the power remains the same in amount for every wheel in
the train, because what is in a compound train lost in velocity is
gained in force, or what is gained in velocity is lost in force, the
word force being used to convey the idea of strain, pressure, or pull.

In Fig. 12, let A, B, and C represent the pitch circles of three gears
of which A and B are in gear, while C is compounded with B; let E be the
shaft of A, and G that for B and C. Let A be 60 inches, B = 30 inches,
and C = 40 inches in diameter. Now suppose that shaft E suspends from
its perimeter a weight of 50 lbs., the shaft being 4 inches in diameter.
Then this weight will be at a leverage of 2 inches from the centre of E
and the 50 must be multiplied by 2, making 100 lbs. at the centre of E.
Then at the perimeter of A this 100 will become one-thirtieth of one
hundred, because from the centre to the perimeter of A is 30.
One-thirtieth of 100 is 3-33/100 lbs., which will be the force exerted
by A on the perimeter of B. Now from the perimeter of B to its centre
(or in other words its radius) is 15 inches, hence the 3-33/100 lbs. at
its perimeter will become fifteen times as much at the centre G of B,
and 3-33/100 × 15 = 49-95/100 lbs. From the centre G to the perimeter of
C being 20 inches, the 49-95/100 lbs. at the centre will be only
one-twentieth of that amount at the perimeter of C, hence 49-95/100 ÷ 20
= 2-49/100 lbs., which is the amount of force at the perimeter of C.

Here we have treated the wheels as simple levers, dividing the weight by
the length of the levers in all cases where it is transmitted from the
shaft to the perimeter, and multiplying it by the length of the lever
when it is transmitted from the perimeter of the wheel to the centre of
the shaft. The precise same result will be reached if we take the
diameter of the wheels or the number of the teeth, providing the pitch
of the teeth on all the wheels is alike.

Suppose, for example, that A has 60 teeth, B has 30 teeth, and C has 40
teeth, all being of the same pitch. Suppose the 50 lb. weight be
suspended as before, and that the circumference of the shaft be equal to
that of a pinion having 4 teeth of the same pitch as the wheels. Then
the 50 multiplied by the 4 becomes 200, which divided by 60 (the number
of teeth on A) becomes 3-33/100, which multiplied by 30 (the number of
teeth on B) becomes 99-90/100, which divided by 40 (the number of teeth
on C) becomes 2-49/100 lbs. as before.

[Illustration: Fig. 12.]

It may now be explained why the shaft was taken as equal to a pinion
having 4 teeth. Its diameter was taken as 4 inches and the wheel
diameter was taken as being 60 inches, and it was supposed to contain 60
teeth, hence there was 1 tooth to each inch of diameter, and the 4
inches diameter of shaft was therefore equal to a pinion having 4 teeth.
From this we may perceive the philosophy of the rule that to obtain the
revolutions of wheels we multiply the given revolutions by the teeth in
the driving wheels and divide by the teeth in the driven wheels.

Suppose that A (Fig. 13) makes 1 revolution per minute, how many will C
make, A having 60 teeth, B 30 teeth, and C 40 teeth? In this case we
have but one driving wheel A, and one driven wheel B, the driver having
60 teeth, the driven 30, hence 60 ÷ 30 = 2, equals revolutions of B and
also of C, the two latter being on the same shaft.

It will be observed then that the revolutions are in the same proportion
as the numbers of the teeth or the radii of the wheels, or what is the
same thing, in the same proportion as their diameters. The number of
teeth, however, is usually taken as being easier obtained than the
diameter of the pitch circles, and easier to calculate, because the
teeth will be represented by a whole number, whereas the diameter,
radius, or circumference, will generally contain fractions.

Suppose that the 4 wheels in Fig. 14 have the respective numbers of
teeth marked beside them, and that the upper one having 40 teeth makes
60 revolutions per minute, then we may obtain the revolutions of the
others as follows:--

  Revolu-      Teeth      Teeth       Teeth       Teeth
   tions.     in first   in first   in second   in second
               driver.    driven.    driver.     driven.
    60      ×    40     ÷   60    ×    20     ÷    120    =  6-66/100

and a remainder of the reciprocating decimals. We may now prove this by
reversing the question, thus. Suppose the 120 wheel to make 6-66/100
revolutions per minute, how many will the 40 wheel make?

  Revolu-      Teeth      Teeth       Teeth       Teeth
   tions.     in first   in first   in second   in second
               driver.    driven.    driver.     driven.
   6.66     ×    120    ÷   20    ×    60    ÷     40     =  59-99/100 =

revolutions of the 40 wheel, the discrepancy of 1/100 being due to the
6.66 leaving a remainder and not therefore being absolutely correct.

That the amount of power transmitted by gearing, whether compounded or
not, is equal throughout every wheel in the train, may be shown as
follows:--

[Illustration: Fig. 13.]

Referring again to Fig. 10, it has been shown that with a 50 lb. weight
suspended from a 4 inch shaft E, there would be 30-33/100 lbs. at the
perimeter of A. Now suppose a rotation be made, then the 50 lb. weight
would fall a distance equal to the circumference of the shaft, which is
(3.1416 × 4 = 12-56/100) 12-56/100 inches. Now the circumference of the
wheel is (60 dia. × 3.1416 = 188-49/100 cir.) 188-49/100 inches, which
is the distance through which the 3-33/100 lbs. would move during one
rotation of A. Now 3.33 lbs. moving through 188.49 inches represents the
same amount of power as does 50 lbs. moving through a distance of 12.56
inches, as may be found by converting the two into inch lbs. (that is to
say, into the number of inches moved by 1 lb.), bearing in mind that
there will be a slight discrepancy due to the fact that the fractions
.33 in the one case, and .56 in the other are not quite correct. Thus:

  188.49 inches × 3.33 lbs. = 627.67 inch lbs., and
   12.56   "    ×   50  "   = 628     "    "

Taking the next wheels in Fig. 12, it has been shown that the 3.33 lbs.
delivered from A to the perimeter of B, becomes 2.49 lbs. at the
perimeter of C, and it has also been shown that C makes two revolutions
to one of A, and its diameter being 40 inches, the distance this 2.49
lbs. will move through in one revolution of A will therefore be equal to
twice its circumference, which is (40 dia. × 3.1416 = 125.666 cir., and
125.666 × 2 = 251.332) 251.332 inches. Now 2.49 lbs. moving through
251.332 gives when brought to inch lbs. 627.67 inch lbs., thus 251.332 ×
2.49 = 627.67. Hence the amount of power remains constant, but is
altered in form, merely being converted from a heavy weight moving a
short distance, into a lighter one moving a distance exactly as much
greater as the weight or force is lessened or lighter.

[Illustration: Fig. 14.]

Gear-wheels therefore form a convenient method of either simply
transmitting motion or power, as when the wheels are all of equal
diameter, or of transmitting it and simultaneously varying its velocity
of motion, as when the wheels are compounded either to reduce or
increase the speed or velocity in feet per second of the prime mover or
first driver of the train or pair, as the case may be.

[Illustration: Fig. 15.]

In considering the action of gear-teeth, however, it sometimes is more
convenient to denote their motion by the number of degrees of angle they
move through during a certain portion of a revolution, and to refer to
their relative velocities in terms of the ratio or proportion existing
between their velocities. The first of these is termed the angular
velocity, or the number of degrees of angle the wheel moves through
during a given period, while the second is termed the velocity ratio of
the pair of wheels. Let it be supposed that two wheels of equal diameter
have contact at their perimeters so that one drives the other by
friction without any slip, then the velocity of a point on the perimeter
of one will equal that of a point on the other. Thus in Fig. 15 let A
and B represent the pitch circles of two wheels, and C an imaginary line
joining the axes of the two wheels and termed the line of centres. Now
the point of contact of the two wheels will be on the line of centres
as at D, and if a point or dot be marked at D and motion be imparted
from A to B, then when each wheel has made a quarter revolution the dot
on A will have arrived at E while that on B will have arrived at F. As
each wheel has moved through one quarter revolution, it has moved
through 90° of angle, because in the whole circle there is 360°, one
quarter of which is 90°, hence instead of saying that the wheels have
each moved through one quarter of a revolution we may say they have
moved through an angle of 90°, or, in other words, their angular
velocity has, during this period, been 90°. And as both wheels have
moved through an equal number of degrees of angle their velocity ratio
or proportion of velocity has been equal.

Obviously then the angular velocity of a wheel represents a portion of a
revolution irrespective of the diameter of the wheel, while the velocity
ratio represents the diameter of one in proportion to that of the other
irrespective of the actual diameter of either of them.

[Illustration: Fig. 16.]

Now suppose that in Fig. 16 A is a wheel of twice the diameter of B;
that the two are free to revolve about their fixed centres, but that
there is frictional contact between their perimeters at the line of
centres sufficient to cause the motion of one to be imparted to the
other without slip or lost motion, and that a point be marked on both
wheels at the point of contact D. Now let motion be communicated to A
until the mark that was made at D has moved one-eighth of a revolution
and it will have moved through an eighth of a circle, or 45°. But during
this motion the mark on B will have moved a quarter of a revolution, or
through an angle of 90° (which is one quarter of the 360° that there are
in the whole circle). The angular velocities of the two are, therefore,
in the same ratio as their diameters, or two to one, and the velocity
ratio is also two to one. The angular velocity of each is therefore the
number of degrees of angle that it moves through in a certain portion of
a revolution, or during the period that the other wheel of the pair
makes a certain portion of a revolution, while the velocity ratio is the
proportion existing between the velocity of one wheel and that of the
other; hence if the diameter of one only of the wheels be changed, its
angular velocity will be changed and the velocity ratio of the pair will
be changed. The velocity ratio may be obtained by dividing either the
radius, pitch, diameter, or number of teeth of one wheel into that of
the other.

Conversely, if a given velocity ratio is to be obtained, the radius,
diameter, or number of teeth of the driver must bear the same relation
to the radius, diameter, or number of teeth of the follower, as the
velocity of the follower is desired to bear to that of the driver.

If a pair of wheels have an equal number of teeth, the same pairs of
teeth will come into action at every revolution; but if of two wheels
one is twice as large as the other, each tooth on the small wheel will
come into action twice during each revolution of the large one, and will
work during each successive revolution with the same two teeth on the
large wheel; and an application of the principle of the hunting tooth is
sometimes employed in clocks to prevent the overwinding of their
springs, the device being shown in Fig. 17, which is from "Willis'
Principles of Mechanism."

For this purpose the winding arbor C has a pinion A of 19 teeth fixed to
it close to the front plate. A pinion B of 18 teeth is mounted on a stud
so as to be in gear with the former. A radial plate C D is fixed to the
face of the upper wheel A, and a similar plate F E to the lower wheel B.
These plates terminate outward in semicircular noses D, E, so
proportioned as to cause their extremities to abut against each other,
as shown in the figure, when the motion given to the upper arbor by the
winding has brought them into the position of contact. The clock being
now wound up, the winding arbor and wheel A will begin to turn in the
opposite direction. When its first complete rotation is effected the
wheel B will have gained one tooth distance from the line of centres, so
as to place the stop D in advance of E and thus avoid a contact with E,
which would stop the motion. As each turn of the upper wheel increases
the distance of the stops, it follows from the principle of the hunting
cog, that after eighteen revolutions of A and nineteen of B the stops
will come together again and the clock be prevented from running down
too far. The winding key being applied, the upper wheel A will be
rotated in the opposite direction, and the winding repeated as above.

[Illustration: Fig. 17.]

Thus the teeth on one wheel will wear to imbed one upon the other. On
the other hand the teeth of the two wheels may be of such numbers that
those on one wheel will not fall into gear with the same teeth on the
other except at intervals, and thus an inequality on any one tooth is
subjected to correction by all the teeth in the other wheel. When a
tooth is added to the number of teeth on a wheel to effect this purpose
it is termed a hunting cog, or hunting tooth, because if one wheel have
a tooth less, then any two teeth which meet in the first revolution are
distant, one tooth in the second, two teeth in the third, three in the
fourth, and so on. The odd tooth is on this account termed a hunting
tooth.

It is obvious then that the shape or form to be given to the teeth must,
to obtain correct results, be such that the motion of the driver will be
communicated to the follower with the velocity due to the relative
diameters of the wheels at the pitch circles, and since the teeth move
in the arc of a circle it is also obvious that the sides of the teeth,
which are the only parts that come into contact, must be of same curve.
The nature of this curve must be such that the teeth shall possess the
strength necessary to transmit the required amount of power, shall
possess ample wearing surface, shall be as easily produced as possible
for all the varying conditions, shall give as many teeth in constant
contact as possible, and shall, as far as possible, exert a pressure in
a direction to rotate the wheels without inducing undue wear upon the
journals of the shafts upon which the wheels rotate. In cases, however,
in which some of these requirements must be partly sacrificed to
increase the value of the others, or of some of the others, to suit the
special circumstances under which the wheels are to operate, the
selection is left to the judgment of the designer, and the
considerations which should influence his determinations will appear
hereafter.

[Illustration: Fig. 18.]

Modern practice has accepted the curve known in general terms as the
cycloid, as that best filling all the requirements of wheel teeth, and
this curve is employed to produce two distinct forms of teeth,
epicycloidal and involute. In epicycloidal teeth the curve forming the
face of the tooth is designated an epicycloid, and that forming the
flank an hypocycloid. An epicycloid may be traced or generated, as it is
termed, by a point in the circumference of a circle that rolls without
slip upon the circumference of another circle. Thus, in Fig. 18, A and B
represent two wooden wheels, A having a pencil at P, to serve as a
tracing or marking point. Now, if the wheels are laid upon a sheet of
paper and while holding B in a fixed position, roll A in contact with B
and let the tracing point touch the paper, the point P will trace the
curve C C. Suppose now the diameter of the base circle B to be
infinitely large, a portion of its circumference may be represented by a
straight line, and the curve traced by a point on the circumference of
the generating circle as it rolls along the base line B is termed a
cycloid. Thus, in Fig. 19, B is the base line, A the rolling wheel or
generating circle, and C C the cycloidal curve traced or marked by the
point D when A is rolled along B. If now we suppose the base line B to
represent the pitch line of a rack, it will be obvious that part of the
cycloid at one end is suitable for the face on one side of the tooth,
and a part at the other end is suitable for the face of the other side
of the tooth.

[Illustration: Fig. 19.]

A hypocycloid is a curve traced or generated by a point on the
circumference of a circle rolling within and in contact (without slip)
with another circle. Thus, in Fig. 20, A represents a wheel in contact
with the internal circumference of B, and a point on its circumference
will trace the two curves, C C, both curves starting from the same
point, the upper having been traced by rolling the generating circle or
wheel A in one direction and the lower curve by rolling it in the
opposite direction.

[Illustration: Fig. 20.]

To demonstrate that by the epicycloidal and hypocycloidal curves,
forming the faces and flanks of what are known as epicycloidal teeth,
motion may be communicated from one wheel to another with as much
uniformity as by frictional contact of their circumferential surfaces,
let A, B, in Fig. 21, represent two plain wheel disks at liberty to
revolve about their fixed centres, and let C C represent a margin of
stiff white paper attached to the face of B so as to revolve with it.
Now suppose that A and B are in close contact at their perimeters at the
point G, and that there is no slip, and that rotary motion commenced
when the point E (where as tracing point a pencil is attached), in
conjunction with the point F, formed the point of contact of the two
wheels, and continued until the points E and F had arrived at their
respective positions as shown in the figure; the pencil at E will have
traced upon the margin of white paper the portion of an epicycloid
denoted by the curve E F; and as the movement of the two wheels A, B,
took place by reason of the contact of their circumferences, it is
evident that the length of the arc E G must be equal to that of the arc
G F, and that the motion of A (supposing it to be the driver) would be
communicated uniformly to B.

[Illustration: Fig. 21.]

Now suppose that the wheels had been rotated in the opposite direction
and the same form of curve would be produced, but it would run in the
opposite direction, and these two curves may be utilized to form teeth,
as in Fig. 22, the points on the wheel A working against the curved
sides of the teeth on B.

To render such a pair of wheels useful in practice, all that is
necessary is to diminish the teeth on B without altering the nature of
the curves, and increase the diameter of the points on A, making them
into rungs or pins, thus forming the wheels into what is termed a wheel
and lantern, which are illustrated in Fig. 23.

[Illustration: Fig. 22.]

A represents the pinion (or lantern), and B the wheel, and C, C, the
primitive teeth reduced in thickness to receive the pins on A. This
reduction we may make by setting a pair of compasses to the radius of
the rung and describing half-circles at the bottom of the spaces in B.
We may then set a pair of compasses to the curve of C, and mark off the
faces of the teeth of B to meet the half-circles at the pitch line, and
reduce the teeth heights so as to leave the points of the proper
thickness; having in this operation maintained the same epicycloidal
curves, but brought them closer together and made them shorter. It is
obvious, however, that such a method of communicating rotary motion is
unsuited to the transmission of much power; because of the weakness of,
and small amount of wearing surface on, the points or rungs in A.

[Illustration: Fig. 23.]

[Illustration: Fig. 24.]

In place of points or rungs we may have radial lines, these lines,
representing the surfaces of ribs, set equidistant on the radial face of
the pinion, as in Fig. 24. To determine the epicycloidal curves for the
faces of teeth to work with these radial lines, we may take a generating
circle C, of half the diameter of A, and cause it to roll in contact
with the internal circumference of A, and a tracing point fixed in the
circumference of C will draw the radial lines shown upon A. The
circumstances will not be altered if we suppose the three circles, A, B,
C, to be movable about their fixed centres, and let their centres be in
a straight line; and if, under these circumstances, we suppose rotation
to be imparted to the three circles, through frictional contact of their
perimeters, a tracing point on the circumference of C would trace the
epicycloids shown upon B and the radial lines shown upon A, evidencing
the capability of one to impart uniform rotary motion to the other.

[Illustration: Fig. 25.]

To render the radial lines capable of use we must let them be the
surfaces of lugs or projections on the face of the wheel, as shown in
Fig. 25 at D, E, &c., or the faces of notches cut in the wheel as at F,
G, H, &c., the metal between F and G forming a tooth J, having flanks
only. The wheel B has the curves of each tooth brought closer together
to give room for the reception of the teeth upon A. We have here a pair
of gears that possess sufficient strength and are capable of working
correctly in either direction.

[Illustration: Fig. 26.]

But the form of tooth on one wheel is conformed simply to suit those on
the other, hence, neither two of the wheels A, nor would two of B, work
correctly together.

They may be qualified to do so, however, by simply adding to the tops
of the teeth on A, teeth of the form of those on B, and adding to those
on B, and within the pitch circle, teeth corresponding to those on A, as
in Fig. 26, where at K´ and J´ teeth are provided on B corresponding to
J and K on A, while on A there are added teeth O´, N´, corresponding to
O, N, on B, with the result that two wheels such as A or two such as B
would work correctly together, either being the driver or either the
follower, and rotation may occur in either direction. In this operation
we have simply added faces to the teeth on A, and flanks to those on B,
the curves being generated or obtained by rolling the generating, or
curve marking, circle C upon the pitch circles P and P´. Thus, for the
flanks of the teeth of A, C is rolled upon, and within the pitch circle
P of A; while for the face curves of the same teeth C is rolled upon,
but without or outside of P. Similarly for the teeth of wheel B the
generating circle C is rolled within P´ for the flanks and without for
the faces. With the curves rolled or produced with the same diameter of
generating circle the wheels will work correctly together, no matter
what their relative diameter may be, as will be shown hereafter.

[Illustration: Fig. 27.]

In this demonstration, however, the curves for the faces of the teeth
being produced by an operation distinct from that employed to produce
the flank curves, it is not clearly seen that the curves for the flanks
of one wheel are the proper curves to insure a uniform velocity to the
other. This, however, may be made clear as follows:--

In Fig. 27 let _a_ _a_ and _b_ _b_ represent the pitch circles of two
wheels of equal diameters, and therefore having the same number of
teeth. On the left, the wheels are shown with the teeth in, while on the
right-hand side of the line of centres A B, the wheels are shown blank;
_a_ _a_ is the pitch line of one wheel, and _b_ _b_ that for the other.
Now suppose that both wheels are capable of being rotated on their
shafts, whose centres will of course be on the line A B, and suppose a
third disk, Q, be also capable of rotation upon its centre, C, which is
also on the line A B. Let these three wheels have sufficient contact at
their perimeters at the point _n_, that if one be rotated it will rotate
both the others (by friction) without any slip or lost motion, and of
course all three will rotate at an equal velocity. Suppose that there is
fixed to wheel Q a pencil whose point is at _n_. If then rotation be
given to _a_ _a_ in the direction of the arrow _s_, all three wheels
will rotate in that direction as denoted by their respective arrows _s_.

Assume, then, that rotation of the three has occurred until the pencil
point at _n_ has arrived at the point _m_, and during this period of
rotation the point _n_ will recede from the line of centres A B, and
will also recede from the arcs or lines of the two pitch circles _a_
_a_, _b_ _b_. The pencil point being capable of marking its path, it
will be found on reaching _m_ to have marked inside the pitch circle _b_
_b_ the curve denoted by the full line _m_ _x_, and simultaneously with
this curve it has marked another curve outside of _a_ _a_, as denoted by
the dotted line _y_ _m_. These two curves being marked by the pencil
point at the same time and extending from _y_ to _m_, and _x_ also to
_m_. They are prolonged respectively to _p_ and to K for clearness of
illustration only.

The rotation of the three wheels being continued, when the pencil point
has arrived at O it will have continued the same curves as shown at O
_f_, and O _g_, curve O _f_ being the same as _m_ _x_ placed in a new
position, and O _g_ being the same as _m_ _y_, but placed in a new
position. Now since both these curves (O _f_ and O _g_) were marked by
the one pencil point, and at the same time, it follows that at every
point in its course that point must have touched both curves at once.
Now the pencil point having moved around the arc of the circle Q from
_n_ to _m_, it is obvious that the two curves must always be in contact,
or coincide with each other, at some point in the path of the pencil or
describing point, or, in other words, the curves will always touch each
other at some point on the curve of Q, and between _n_ and O. Thus when
the pencil has arrived at _m_, curve _m_ _y_ touches curve K _x_ at the
point _m_, while when the pencil had arrived at point O, the curves O
_f_ and O _g_ will touch at O. Now the pitch circles _a_ _a_ and _b_
_b_, and the describing circle Q, having had constant and uniform
velocity while the traced curves had constant contact at some point in
their lengths, it is evident that if instead of being mere lines, _m_
_y_ was the face of a tooth on _a_ _a_, and _m_ _x_ was the flank of a
tooth on _b_ _b_, the same uniform motion may be transmitted from _a_
_a_, to _b_ _b_, by pressing the tooth face _m_ _y_ against the tooth
flank _m_ _x_. Let it now be noted that the curve _y_ _m_ corresponds to
the face of a tooth, as say the face E of a tooth on _a_ _a_, and that
curve _x_ _m_ corresponds to the flank of a tooth on _b_ _b_, as say to
the flank F, short portions only of the curves being used for those
flanks. If the direction of rotation of the three wheels was reversed,
the same shape of curves would be produced, but they would lie in an
opposite direction, and would, therefore, be suitable for the other
sides of the teeth. In this case, the contact of tooth upon tooth will
be on the other side of the line of centres, as at some point between
_n_ and Q.

[Illustration: Fig. 28.]

In this illustration the diameter of the rolling or describing circle Q,
being less than the radius of the wheels _a_ _a_ or _b_ _b_, the flanks
of the teeth are curves, and the two wheels being of the same diameter,
the teeth on the two are of the same shape. But the principles governing
the proper formation of the curve remain the same whatever be the
conditions. Thus in Fig. 28 are segments of a pair of wheels of equal
diameter, but the describing, rolling, or curve-generating circle is
equal in diameter to the radius of the wheels. Motion is supposed to
have occurred in the direction of the arrows, and the tracing point to
have moved from _n_ to _m_. During this motion it will have marked a
curve _y_ _m_, a portion of the _y_ end serving for the face of a tooth
on one wheel, and also the line _k_ _x_, a continuation of which serves
for the flank of a tooth on the other wheel. In Fig. 29 the pitch
circles only of the wheels are marked, _a_ _a_ being twice the diameter
of _b_ _b_, and the curve-generating circle being equal in diameter to
the radius of wheel _b_ _b_. Motion is assumed to have occurred until
the pencil point, starting from _n_, had arrived at _o_, marking curves
suitable for the face of the teeth on one wheel and for the flanks of
the other as before, and the contact of tooth upon tooth still, at every
point in the path of the teeth, occurring at some point of the arc _n_
_o_. Thus when the point had proceeded as far as point _m_ it will have
marked the curve _y_ and the radial line _x_, and when the point had
arrived at _o_, it will have prolonged _m_ _y_ into _o_ _g_ and _x_ into
_o_ _f_, while in either position the point is marking both lines. The
velocities of the wheels remain the same notwithstanding their different
diameters, for the arc _n_ _g_ must obviously (if the wheels rotate
without slip by friction of their surfaces while the curves are traced)
be equal in length to the arc _n_ _f_ or the arc _n_ _o_.

[Illustration: Fig. 29.]

[Illustration: Fig. 30.]

In Fig. 30 _a_ _a_ and _b_ _b_ are the pitch circles of two wheels as
before, and _c_ _c_ the pitch circle of an annular or internal gear, and
D is the rolling or describing circle. When the describing point arrived
at _m_, it will have marked the curve _y_ for the face of a tooth on _a_
_a_, the curve _x_ for the flank of a tooth on _b_ _b_, and the curve
_e_ for the face of a tooth on the internal wheel _c_ _c_. Motion being
continued _m_ _y_ will be prolonged to _o_ _g_, while simultaneously _x_
will be extended into _o_ _f_ and _e_ into _h_ _v_, the velocity of all
the wheels being uniform and equal. Thus the arcs _n_ _v_, _n_ _f_, and
_n_ _g_, are of equal length.

[Illustration: Fig. 31.]

In Fig. 31 is shown the case of a rack and pinion; _a_ _a_ is the pitch
line of the rack, _b_ _b_ that of the pinion, A B at a right angle to
_a_ _a_, the line of centres, and D the generating circle. The wheel and
rack are shown with teeth _n_ on one side simply for clearness of
illustration. The pencil point _n_ will, on arriving at _m_, have traced
the flank curve _x_ and the curve _y_ for the face of the rack teeth.

[Illustration: Fig. 32.]

It has been supposed that the three circles rotated together by the
frictional contact of their perimeters on the line of centres, but the
circumstances will remain the same if the wheels remain at rest while
the generating or describing circle is rolled around them. Thus in Fig.
32 are two segments of wheels as before, _c_ representing the centre of
a tooth on _a_ _a_, and _d_ representing the centre of a tooth on _b_
_b_. Now suppose that a generating or rolling circle be placed with its
pencil point at _e_, and that it then be rolled around _a_ _a_ until it
had reached the position marked 1, then it will have marked the curve
from _e_ to _n_, a part of this curve serving for the face of tooth _c_.
Now let the rolling circle be placed within the pitch circle _a_ _a_ and
its pencil point _n_ be set to _e_, then, on being rolled to position 2,
it will have marked the flank of tooth _c_. For the other wheel suppose
the rolling wheel or circle to have started from _f_ and rolled to the
line of centres as in the cut, it will have traced the curve forming the
face of the tooth _d_. For the flank of _d_ the rolling circle or wheel
is placed within _b_ _b_, its tracing point set at _f_ on the pitch
circle, and on being rolled to position 3 it will have marked the flank
curve. The curves thus produced will be precisely the same as those
produced by rotating all three wheels about their axes, as in our
previous demonstrations.

The curves both for the faces and for the flanks thus obtained will vary
in their curvature with every variation in either the diameter of the
generating circle or of the base or pitch circle of the wheel. Thus it
will be observable to the eye that the face curve of tooth _c_ is more
curved than that of _d_, and also that the flank curve of _d_ is more
spread at the root than is that for _c_, which has in this case resulted
from the difference between the diameter of the wheels _a_ _a_ and _b_
_b_. But the curves obtained by a given diameter of rolling circle on a
given diameter of pitch circle will be correct for any pitch of teeth
that can be used upon wheels having that diameter of pitch circle. Thus,
suppose we have a curve obtained by rolling a wheel of 20 inches
circumference on a pitch circle of 40 inches circumference--now a wheel
of 40 inches in circumference may contain 20 teeth of 2 inch arc pitch,
or 10 teeth of 4 inch arc pitch, or 8 teeth of 5 inch arc pitch, and the
curve may be used for either of those pitches.

If we trace the path of contact of each tooth, from the moment it takes
until it leaves contact with a tooth upon the other wheel, we shall find
that contact begins at the point where the flank of the tooth on the
wheel that drives or imparts motion to the other wheel, meets the face
of the tooth on the driven wheel, which will always be where the point
of the driven tooth cuts or meets the generating or rolling circle of
the driving tooth. Thus in Fig. 33 are represented segments of two
spur-wheels marked respectively the driver and the driven, their
generating circles being marked at G and G´, and X X representing the
line of centres. Tooth A is shown in the position in which it commences
its contact with tooth B at B. Secondly, we shall find that as these
two teeth approach the line of centres X, the point of contact between
them moves or takes place along the thickened arc or curve C X, or along
the path of the generating circle G.

Thus we may suppose tooth D to be another position of tooth A, the
contact being at F, and as motion was continued the contact would pass
along the thickened curve until it arrived at the line of centres X. Now
since the teeth have during this path of contact approached the line of
centres, this part of the whole arc of action or of the path of contact
is termed the arc of approach. After the two teeth have passed the line
of centres X, the path of contact of the teeth will be along the dotted
arc from X to L, and as the teeth are during this period of motion
receding from X this part of the contact path is termed the arc of
recess.

That contact of the teeth would not occur earlier than at C nor later
than at L, is shown by the dotted teeth sides; thus A and B would not
touch when in the position denoted by the dotted teeth, nor would teeth
I and K if in the position denoted by their dotted lines.

[Illustration: Fig. 33.]

If we examine further into this path of contact we find that throughout
its whole path the face of the tooth of one wheel has contact with the
flank only of the tooth of the other wheel, and also that the flank only
of the driving-wheel tooth has contact before the tooth reaches the line
of centres, while the face of only the driving tooth has contact after
the tooth has passed the line of centres.

Thus the flanks of tooth A and of tooth D are in driving contact with
the faces of teeth B and E, while the face of tooth H is in contact with
the flank of tooth I.

These conditions will always exist, whatever be the diameters of the
wheels, their number of teeth or the diameter of the generating circle.
That is to say, in fully developed epicycloidal teeth, no matter which
of two wheels is the driver or which the driven wheel, contact on the
teeth of the driver will always be on the tooth flank during the arc of
approach and on the tooth face during the arc of recess; while on the
driven wheel contact during the arc of approach will be on the tooth
face only, and during the arc of recess on the tooth flank only, it
being borne in mind that the arcs of approach and recess are reversed in
location if the direction of revolution be reversed. Thus if the
direction of wheel motion was opposite to that denoted by the arrows in
Fig. 33 then the arc of approach would be from M to X, and the arc of
recess from X to N.

It is laid down by Professor Willis that the motion of a pair of
gear-wheels is smoother in cases where the path of contact begins at the
line of centres, or, in other words, when there is no arc of approach;
and this action may be secured by giving to the driven wheel flanks
only, as in Fig. 34, in which the driver has fully developed teeth,
while the teeth on the driven have no faces.

In this case, supposing the wheels to revolve in the direction of arrow
P, the contact will begin at the line of centres X, move or pass along
the thickened arc and end at B, and there will be contact during the arc
of recess only. Similarly, if the direction of motion be reversed as
denoted by arrow Q, the driver will begin contact at X, and cease
contact at H, having, as before, contact during the arc of recess only.

But if the wheel W were the driver and V the driven, then these
conditions would be exactly reversed. Thus, suppose this to be the case
and the direction of motion be as denoted by arrow P, the contact would
occur during the arc of approach, from H to X, ceasing at X.

Or if W were the driver, and the direction of motion was as denoted by
Q, then, again, the path of contact would be during the arc of approach
only, beginning at B and ceasing at X, as denoted by the thickened arc B
X.

The action of the teeth will in either case serve to give a
theoretically perfect motion so far as uniformity of velocity is
concerned, or, in other words, the motion of the driver will be
transmitted with perfect uniformity to the driven wheel. It will be
observed, however, that by the removal of the faces of the teeth, there
are a less number of teeth in contact at each instant of time; thus, in
Fig. 33 there is driving contact at three points, C, F, and J, while in
Fig. 34 there is driving contact at two points only. From the fact that
the faces of the teeth work with the flanks only, and that one side only
of the teeth comes into action, it becomes apparent that each tooth may
have curves formed by four different diameters of rolling or generating
circles and yet work correctly, no matter which wheel be the driver, or
which the driven wheel or follower, or in which direction motion occurs.
Thus in Fig. 35, suppose wheel V to be the driver, having motion in the
direction of arrow P, then faces a on the teeth of V will work with
flanks B of the teeth on W, and so long as the curves for these faces
and flanks are obtained with the same diameter of rolling circle, the
action of the teeth will be correct, no matter what the shapes of the
other parts of the teeth. Now suppose that V still being the driver,
motion occurs in the other direction as denoted by Q, then the faces C
of the teeth on V will drive the flanks C of the teeth on W, and the
motion will again be correct, providing that the same diameter (whatever
it may be) of rolling circle be used for these faces and flanks,
irrespective, of course, of what diameter of rolling circle is used for
any other of the teeth curves. Now suppose that W is the driver, motion
occurring in the direction of P, then faces E will drive flanks F, and
the motion will be correct as before if the curves E and F are produced
with the same diameter of rolling circle. Finally, let W be the driving
wheel and motion occur in the direction of Q, and faces G will drive
flanks H, and yet another diameter of rolling circle may be used for
these faces and flanks. Here then it is shown that four different
diameters of rolling circles may be used upon a pair of wheels, giving
teeth-forms that will fill all the requirements so far as correctly
transmitting motion is concerned. In the case of a pair of wheels having
an equal number of teeth, so that each tooth on one wheel will always
fall into gear with the same tooth on the other wheel, every tooth may
have its individual curves differing from all the others, providing that
the corresponding teeth on the other wheel are formed to match them by
using the same size of rolling circle for each flank and face that work
together.

[Illustration: Fig. 34.]

[Illustration: Fig. 35.]

It is obvious, however, that such teeth would involve a great deal of
labor in their formation and would possess no advantage, hence they are
not employed. It is not unusual, however, in a pair of wheels that are
to gear together and that are not intended to interchange with other
wheels, to use such sizes as will give to for the face of the teeth on
the largest wheel of the pair and for the flanks of the teeth of the
smallest wheel, a generating circle equal in diameter to the radius of
the smallest wheel, and for the faces of the teeth of the small wheel
and the flanks of the teeth of the large one, a generating circle whose
diameter equals the radius of the large wheel.

[Illustration: Fig. 36.]

It will now be evident that if we have planned a pair or a train of
wheels we may find how many teeth will be in contact for any given
pitch, as follows. In Fig. 36 let A, B, and C, represent three blanks
for gear-wheels whose addendum circles are M, N and O; P representing
the pitch circles, and Q representing the circles for the roots of the
teeth. Let X and Y represent the lines of centres, and A, H, I and K the
generating or rolling circle, whose centres are on the respective lines
of centres--the diameter of the generating circle being equal to the
radius of the pinion, as in the Willis system, then, the pinion M being
the driver, and the wheels revolving in the direction denoted by the
respective arrows, the arc or path of contact for the first pair will be
from point D, where the generating circle G crosses circle N to E, where
generating circle H crosses the circle M, this path being composed of
two arcs of a circle. All that is necessary, therefore, is to set the
compasses to the pitch the teeth are to have and step them along these
arcs, and the number of steps will be the number of teeth that will be
in contact. Similarly, for the second pair contact will begin at R and
end at S, and the compasses applied as before (from R to S) along the
arc of generating circle I to the line of centres, and thence along the
arc of generating circle K to S, will give in the number of steps, the
number of teeth that will be in contact. If for any given purpose the
number of teeth thus found to be in contact is insufficient; the pitch
may be made finer.

[Illustration: Fig. 37.]

When a wheel is intended to be formed to work correctly with any other
wheel having the same pitch, or when there are more than two wheels in
the train, it is necessary that the same size of generating circle be
used for all the faces and all the flanks in the set, and if this be
done the wheels will work correctly together, no matter what the number
of the teeth in each wheel may be, nor in what way they are
interchanged. Thus in Fig. 37, let A represent the pitch line of a rack,
and B and C the pitch circles of two wheels, then the generating circle
would be rolled within B, as at 1, for the flank curves, and without it,
as at 2, for the face curves of B. It would be rolled without the pitch
line, as at 3, for the rack faces, and within it, as at 4, for the rack
flanks, and without C, as at 5, for the faces, and within it, as at 6,
for flanks of the teeth on C, and all the teeth will work correctly
together however they be placed; thus C might receive motion from the
rack, and B receive motion from C. Or if any number of different
diameters of wheels are used they will all work correctly together and
interchange perfectly, with the single condition that the same size of
generating circle be used throughout. But the curves of the teeth so
formed will not be alike. Thus in Fig. 38 are shown three teeth, all
struck with the same size of generating circle, D being for a wheel of
12 teeth, E for a wheel of 50 teeth, and F a tooth of a rack; teeth E,
F, being made wider so as to let the curves show clearly on each side,
it being obvious that since the curves are due to the relative sizes of
the pitch and generating circles they are equally applicable to any
pitch or thickness of teeth on wheels having the same diameters of pitch
circle.

[Illustration: Fig. 38.]

In determining the diameter of a generating circle for a set or train
of wheels, we have the consideration that the smaller the diameter of
the generating circle in proportion to that of the pitch circle the more
the teeth are spread at the roots, and this creates a pressure tending
to thrust the wheels apart, thus causing the axle journals to wear. In
Fig. 39, for example, A A is the line of centres, and the contact of the
curves at B C would cause a thrust in the direction of the arrows D, E.
This thrust would exist throughout the whole path of contact save at the
point F, on the line of centres. This thrust is reduced in proportion as
the diameter of the generating circle is increased; thus in Fig. 40, is
represented a pair of pinions of 12 teeth and 3 inch pitch, and C being
the driver, there is contact at E, and at G, and E being a radial line,
there is obviously a minimum of thrust.

[Illustration: Fig. 39.]

[Illustration: Fig. 40.]

What is known as the Willis system for interchangeable gearing, consists
of using for every pitch of the teeth a generating circle whose diameter
is equal to the radius of a pinion having 12 teeth, hence the pinion
will in each pitch have radial flanks, and the roots of the teeth will
be more spread as the number of teeth in the wheel is increased. Twelve
teeth is the least number that it is considered practicable to use;
hence it is obvious that under this system all wheels of the same pitch
will work correctly together.

[Illustration: Fig. 41.]

Unless the faces of the teeth and the flanks with which they work are
curves produced from the same size of generating circle, the velocity of
the teeth will not be uniform. Obviously the revolutions of the wheels
will be proportionate to their numbers of teeth; hence in a pair of
wheels having an equal number of teeth, the revolutions will per force
be equal, but the driver will not impart uniform motion to the driven
wheel, but each tooth will during the path of contact move irregularly.

The velocity of a pair of wheels will be uniform at each instant of
time, if a line normal to the surfaces of the curves at their point of
contact passes through the point of contact of the pitch circles on the
line of centres of the wheels. Thus in Fig. 41, the line A A is tangent
to the teeth curves where they touch, and D at a right angle to A A, and
meets it at the point of the tooth curves, hence it is normal to the
point of contact, and as it meets the pitch circles on the line of
centres the velocity of the wheels will be uniform.

The amount of rolling motion of the teeth one upon the other while
passing through the path of contact, will be a minimum when the tooth
curves are correctly formed according to the rules given. But
furthermore the sliding motion will be increased in proportion as the
diameter of the generating circle is increased, and the number of teeth
in contact will be increased because the arc, or path, of contact is
longer as the generating circle is made larger.

[Illustration: Fig. 42.]

Thus in Fig. 42 is a pair of wheels whose tooth curves are from a
generating circle equal to the radius of the wheels, hence the flanks
are radial. The teeth are made of unusual depth to keep the lines in the
engraving clear. Suppose V to be the driver, W the driven wheel or
follower, and the direction of motion as at P, contact upon tooth A will
begin at C, and while A is passing to the line of centres the path of
contact will pass along the thickened line to X. During this time the
whole length of face from C to R will have had contact with the length
of flank from C to N, and it follows that the length of face on A that
rolled on C N can only equal the length of C N, and that the amount of
sliding motion must be represented by the length of R N on A, and the
amount of rolling motion by the length N C. Again, during the arc of
recess (marked by dots) the length of flank that will have had contact
is the depth from S to Ls, and over this depth the full length of tooth
face on wheel V will have swept, and as L S equals C N, the amount of
rolling and of sliding motion during the arc of recess is equal to that
during the arc of approach, and the action is in both cases partly a
rolling and partly a sliding one. The two wheels are here shown of the
same diameter, and therefore contain an equal number of teeth, hence the
arcs of approach and of recess are equal in length, which will not be
the case when one wheel contains more teeth than the other. Thus in Fig.
43, let A represent a segment of a pinion, and B a segment of a
spur-wheel, both segments being blank with their pitch circles, the
tooth height and depth being marked by arcs of circles. Let C and D
represent the generating circles shown in the two respective positions
on the line of centres. Let pinion A be the driver moving in the
direction of P, and the arc of approach will be from E to X along the
thickened arc, while the arc of recess will be as denoted by the dotted
arc from X to F. The distance E X being greater than distance X F,
therefore the arc of approach is longer than that of recess.

But suppose B to be the driver and the reverse will be the case, the arc
of approach will begin at G and end at X, while the arc of recess will
begin at X and end at H, the latter being farther from the line of
centres than G is. It will be found also that, one wheel being larger
than the other, the amount of sliding and rolling contact is different
for the two wheels, and that the flanks of the teeth on the larger wheel
B, have contact along a greater portion of their depths than do the
flanks of those on the smaller, as is shown by the dotted arc I being
farther from the pitch circle than the dotted arc J is, these two dotted
arcs representing the paths of the lowest points of flank contact,
points F and G, marking the initial lowest contact for the two
directions of revolution.

[Illustration: Fig. 43.]

Thus it appears that there is more sliding action upon the teeth of the
smaller than upon those of the larger wheel, and this is a condition
that will always exist.

In Fig. 44 is represented portion of a pair of wheels corresponding to
those shown in Fig. 42, except that in this case the diameter of the
generating circle is reduced to one quarter that of the pitch diameter
of the wheels. V is the driver in the direction the teeth of V that
will have contact is C N, which, the wheels, being of equal diameter,
will remain the same whichever wheel be the driver, and in whatever
direction motion occurs. The amount of rolling motion is, therefore, C
N, and that of sliding is the difference between the distance C N and
the length of the tooth face.

[Illustration: Fig. 44.]

If now we examine the distance C N in Fig. 42, we find that reducing the
diameter of generating circle in Fig. 44 has increased the depth of
flank that has contact, and therefore increased the rolling motion of
the tooth face along the flank, and correspondingly diminished the
sliding action of the tooth contact. But at the same time we have
diminished the number of teeth in contact. Thus in Fig. 42 there are
three teeth in driving contact, while in Fig. 44 there are but two,
viz., D and E.

[Illustration: Fig. 45.]

In an article by Professor Robinson, attention is called to the fact
that if the teeth of wheels are not formed to have correct curves when
new, they cannot be improved by wear; and this will be clearly perceived
from the preceding remarks upon the amount of rolling and sliding
contact. It will also readily appear that the nearer the diameter of the
generating to that of the base circle the more the teeth wear out of
correct shape; hence, in a train of gearing in which the generating
circle equals the radius of the pinion, the pinion will wear out of
shape the quickest, and the largest wheel the least; because not only
does each tooth on the pinion more frequently come into action on
account of its increased revolutions, but furthermore the length of
flank that has contact is less, while the amount of sliding action is
greater. In Fig. 45, for example, are a wheel and pinion, the latter
having radial flanks and the pinion being the driver, the arc of
approach is the thickened arc from C to the line of centres, while the
arc of recess is denoted by the dotted arc. As contact on the pinion
flank begins at point C and ends at the line of centres, the total depth
of flank that suffers wear from the contact is that from C to N; and as
the whole length of the wheel tooth face sweeps over this depth C N, the
pinion flanks must wear faster than the wheel faces, and the pinion
flanks will wear underneath, as denoted by the dotted curve on the
flanks of tooth W. In the case of the wheel, contact on its tooth flanks
begins at the line of centres and ends at L, hence that flank can only
wear between point L and the pitch line L; and as the whole length of
pinion face sweeps on this short length L S, the pinion flank will wear
most, the wear being in the direction of the dotted arc on the left-hand
side V of the tooth. Now the pinion flank depth C N, being less than the
wheel flank depth S L, and the same length of tooth face sweeping
(during the path of contact) over both, obviously the pinion tooth will
wear the most, while both will, as the wear proceeds, lose their proper
flank curve. In Fig. 46 the generating arcs, G and G´, and the wheel are
the same, but the pinion is larger. As a result the acting length C N,
of pinion flank is increased, as is also the acting length S L, of wheel
flank; hence, the flanks of both wheels would wear better, and also
better preserve their correct and original shapes.

It has been shown, when referring to Figs. 42 and 44, when treating of
the amount of sliding and of rolling motion, that the smaller the
diameter of rolling circle in proportion to that of pitch circle, the
longer the acting length of flank and the more the amount of rolling
motion; and it follows that the teeth would also preserve their original
and true shape better. But the wear of the teeth, and the alteration of
tooth form by reason of that wear, will, in any event, be greater upon
the pinion than upon the wheel, and can only be equal when the two
wheels are of equal diameter, in which case the tooth curves will be
alike on both wheels, and the acting depths of flank will be equal, as
shown in Fig. 47, the flanks being radial, and the acting depths of
flank being shown at J K. In Fig. 48 is shown a pair of wheels with a
generating circle, G and G´, of one quarter the diameter of the base
circle or pitch diameter, and the acting length of flank is shown at L
M. The wear of the teeth would, therefore, in this latter case, cause it
in time to assume the form shown in Fig. 49. But it is to be noted that
while the acting depth of flank has been increased the arcs of contact
have been diminished, and that in Fig. 47 there are two teeth in
contact, while in Fig. 48 there is but one, hence the pressure upon each
tooth is less in proportion as the diameter of the generating circle is
increased. If a train of wheels are to be constructed, or if the wheels
are to be capable of interchanging with other combinations of wheels of
the same pitch, the diameter of the generating circle must be equal to
the smallest wheel or pinion, which is, under the Willis system, a
pinion of 12 teeth; under the Pratt and Whitney, and Brown and Sharpe
systems, a pinion of 15 teeth.

[Illustration: Fig. 46.]

[Illustration: Fig. 47.]

But if a pair or a particular train of gears are to be constructed, then
a diameter of generating circle may be selected that is considered most
suitable to the particular conditions; as, for example, it may be equal
to the radius of the smallest wheel giving it radial flanks, or less
than that radius giving parallel or spread flanks. But in any event, in
order to transmit continuous motion, the diameter of generating circle
must be such as to give arcs of action that are equal to the pitch, so
that each pair of teeth will come into action before the preceding pair
have gone out of action.

[Illustration: Fig. 48.]

It may now be pointed out that the degrees of angle that the teeth move
through always exceeds the number of degrees of angle contained in the
paths of contact, or, in other words, exceeds the degrees contained in
the arcs of approach and recess combined.

[Illustration: Fig. 49.]

In Fig. 50, for example, are a wheel A and pinion B, the teeth on the
wheel being extended to a point. Suppose that the wheel A is the driver,
and contact will begin between the two teeth D and F on the dotted arc.
Now suppose tooth D to have moved to position C, and F will have been
moved to position H. The degrees of angle the pinion has been moved
through are therefore denoted by I, whereas the degrees of angle the
arcs of contact contain are therefore denoted by J.

The degrees of angle that the wheel A has moved through are obviously
denoted by E, because the point of tooth D has during the arcs of
contact moved from position D to position C. The degrees of angle
contained in its path of contact are denoted by K, and are less than E,
hence, in the case of teeth terminating in a point as tooth D, the
excess of angle of action over path of contact is as many degrees as are
contained in one-half the thickness of the tooth, while when the points
of the teeth are cut off, the excess is the number of degrees contained
in the distance between the corner and the side of the tooth as marked
on a tooth at P.

[Illustration: Fig. 50.]

With a given diameter of pitch circle and pitch diameter of wheel, the
length of the arc of contact will be influenced by the height of the
addendum from the pitch circle, because, as has been shown, the arcs of
approach and of recess, respectively, begin and end on the addendum
circle.

If the height of the addendum on the follower be reduced, the arc of
approach will be reduced, while the arc of recess will not be altered;
and if the follower have no addendum, contact between the teeth will
occur on the arc of recess only, which gives a smoother motion, because
the action of the driver is that of dragging rather than that of pushing
the follower. In this case, however, the arc of recess must, to produce
continuous motion, be at least equal to the pitch.

It is obvious, however, that the follower having no addendum would, if
acting as a driver to a third wheel, as in a train of wheels, act on its
follower, or the fourth wheel of the train, on the arc of approach only;
hence it follows that the addendum might be reduced to diminish, or
dispensed with to eliminate action, on the arc of approach in the
follower of a pair of wheels only, and not in the case of a train of
wheels.

To make this clear to the reader it may be necessary to refer again to
Fig. 33 or 34, from which it will be seen that the action of the teeth
of the driver on the follower during the arc of approach is produced by
the flanks of the driver on the faces of the follower. But if there are
no such faces there can be no such contact.

On the arc of recess, however, the faces of the driver act on the flanks
of the follower, hence the absence of faces on the follower is of no
import.

From these considerations it also appears that by giving to the driver
an increase of addendum the arc of recess may be increased without
affecting the arc of approach. But the height of addendum in machinists'
practice is made a constant proportion of the pitch, so that the wheel
may be used indiscriminately, as circumstances may require, as either a
driver or a follower, the arcs of approach and of recess being equal.
The height of addendum, however, is an element in determining the number
of teeth in contact, and upon small pinions this is of importance.

[Illustration: Fig. 51.]

In Fig. 51, for example, is shown a section of two pinions of equal
diameters, and it will be observed that if the full line A determined
the height of the addendum there would be contact either at C or B only
(according to the direction in which the motion took place).

With the addendum extended to the dotted circle, contact would be just
avoided, while with the addendum extended to D there would be contact
either at E or at F, according to which direction the wheel had motion.

This, by dividing the strain over two teeth instead of placing it all
upon one tooth, not only doubles the strength for driving capacity, but
decreases the wear by giving more area of bearing surface at each
instant of time, although not increasing that area in proportion to the
number of teeth contained in the wheel.

In wheels of larger diameter, short teeth are more permissible, because
there are more teeth in contact, the number increasing with the
diameters of the wheels. It is to be observed, however, that from having
radial flanks, the smallest wheel is always the weakest, and that from
making the most revolutions in a given time, it suffers the most from
wear, and hence requires the greatest attainable number of teeth in
constant contact at each period of time, as well as the largest possible
area of bearing or wearing surface on the teeth.

It is true that increasing the "depth of tooth to pitch line" increases
the whole length of tooth, and, therefore, weakens it; but this is far
more than compensated for by distributing the strain over a greater
number of teeth. This is in practice accomplished, _when circumstances
will permit_, by making the pitch finer, giving to a wheel, of a given
diameter, a greater number of teeth.

[Illustration: Fig. 52.]

When the wheels are required to transmit motion rather than power (as in
the case of clock wheels), to move as frictionless as possible, and to
place a minimum of thrust on the journals of the shafts of the wheels,
the generating circle may be made nearly as large as the diameter of the
pitch circle, producing teeth of the form shown in Fig. 52. But the
minimum of friction is attained when the two flanks for the tooth are
drawn into one common hypocycloid, as in Fig. 53. The difference between
the form of tooth shown in Fig. 52 and that shown in Fig. 53, is merely
due to an increase in the diameter of the generating circle for the
latter. It will be observed that in these forms the acting length of
flank diminishes in proportion as the diameter of the generating circle
is increased, the ultimate diameter of generating circle being as large
as the pitch circles.

[Illustration: Fig. 53.]

[1]This form is undesirable in that there is contact on one side only
(on the arc of approach) of the line of centres, but the flanks of the
teeth may be so modified as to give contact on the arc of recess also,
by forming the flanks as shown in Fig. 54, the flanks, or rather the
parts within the pitch circles, being nearly half circles, and the parts
without with peculiarly formed faces, as shown in the figure. The pitch
circles must still be regarded as the rolling circles rolling upon each
other. Suppose _b_ a tracing point on B, then as B rolls on A it will
describe the epicycloid _a_ _b_. A parallel line _c_ _d_ will work at a
constant distance as at _c_ _d_ from _a_ _b_, and this distance may be
the radius of that part of D that is within the pitch line, the same
process being applied to the teeth on both wheels. Each tooth is thus
composed of a spur based upon a half cylinder.

  [1] From an article by Professor Robinson.

[Illustration: Fig. 54.]

Comparing Figs. 53 and 54, we see that the bases in 53 are flattest, and
that the contact of faces upon them must range nearer the pitch line
than in 54. Hence, 53 presents a more favorable obliquity of the line of
direction of the pressures of tooth upon tooth. In seeking a still more
favorable direction by going outside for the point of contact, we see by
simply recalling the method of generating the tooth curves, that tooth
contacts outside the pitch lines have no possible existence; and hence,
Fig. 53 may be regarded as representing that form of toothed gear which
will operate with less friction than any other known form.

[Illustration: Fig. 55.]

This statement is intended to cover fixed teeth only, and not that
complicated form of the trundle wheel in which the cylinder teeth are
friction rollers. No doubt such would run still easier, even with their
necessary one-sided contacts. Also, the statement is supposed to be
confined to such forms of teeth as have good practical contacts at and
near the line of centres.

Bevel-gear wheels are employed to transmit motion from one shaft to
another when the axis of one is at an angle to that of the other. Thus
in Fig. 55 is shown a pair of bevel-wheels to transmit motion from
shafts at a right angle. In bevel-wheels all the lines of the teeth,
both at the tops or points of the teeth, at the bottoms of the spaces,
and on the sides of the teeth, radiate from the centre E, where the axes
of the two shafts would meet if produced. Hence the depth, thickness,
and height of the tooth decreases as the point E is approached from the
diameter of the wheel, which is always measured on the pitch circle at
the largest end of the cone, or in other words, at the largest pitch
diameter.

The principles governing the practical construction of the curves for
the teeth of the bevel-wheels may be explained as follows:--

In Fig. 56 let F and G represent two shafts, rotating about their
respective axes; and having cones whose greatest diameters are at A and
B, and whose points are at E. The diameter A being equal to that of B
their circumferences will be equal, and the angular and velocity ratios
will therefore be equal.

[Illustration: Fig. 56.]

Let C and D represent two circles about the respective cones, being
equidistant from E, and therefore of equal diameters and circumferences,
and it is obvious that at every point in the length of each cone the
velocity will be equal to a point upon the other so long as both points
are equidistant from the points of intersection of the axes of the two
shafts; hence if one cone drive the other by frictional contact of
surfaces, both shafts will be rotated at an equal speed of rotation, or
if one cone be fixed and the other moved around it, the contact of the
surfaces will be a rolling contact throughout. The line of contact
between the two cones will be a straight line, radiating at all times
from the point E. If such, however, is not the case, then the contact
will no longer be a rolling one. Thus, in Fig. 57 the diameters or
circumferences at A and B being equal, the surfaces would roll upon each
other, but on account of the line of contact not radiating from E (which
is the common centre of motion for the two shafts) the circumference C
is less than that of D, rendering a rolling contact impossible.

[Illustration: Fig. 57.]

We have supposed that the diameters of the cones be equal, but the
conditions will remain the same when their diameters are unequal; thus,
in Fig. 58 the circumference of A is twice that of B, hence the latter
will make two rotations to one of the former, and the contact will still
be a rolling one. Similarly the circumference of D is one half that of
C, hence D will also make two rotations to one of C, and the contact
will also be a rolling one; a condition which will always exist
independent of the diameters of the wheels so long as the angles of the
faces, or wheels, or (what is the same thing, the line of contact
between the two,) radiates from the point E, which is located where the
axes of the shafts would meet.

[Illustration: Fig. 58.]

[Illustration: Fig. 59.]

The principles governing the forms of the cones on which the teeth are
to be located thus being explained, we may now consider the curves of
the teeth. Suppose that in Fig. 59 the cone A is fixed, and that the
cone whose axis is F be rotated upon it in the direction of the arrow.
Then let a point be fixed in any part of the circumference of B (say at
_d_), and it is evident that the path of this point will be as B rolls
around the axis F, and at the same time around A from the centre of
motion, E. The curve so generated or described by the point _d_ will be
a spherical epicycloid. In this case the exterior of one cone has rolled
upon the coned surface of the other; but suppose it rolls upon the
interior, as around the walls of a conical recess in a solid body; then
a point in its circumference would describe a curve known as the
spherical hypocycloid; both curves agreeing (except in their spherical
property) to the epicycloid and hypocycloid of the spur-wheel. But this
spherical property renders it very difficult indeed to practically
delineate or mark the curves by rolling contact, and on account of this
difficulty Tredgold devised a method of construction whereby the curves
may be produced sufficiently accurate for all practical purposes, as
follows:--

[Illustration: Fig. 60.]

In Fig. 60 let A A represent the axis of one shaft, and B the axis of
the other, the axes of the two meeting at W. Mark E, representing the
diameter of one wheel, and F that of the other (both lines representing
the pitch circles of the respective wheels). Draw the line G G passing
through the point W, and the point T, where the pitch circles E, F meet,
and G G will be the line of contact between the cones. From W as a
centre, draw on each side of G G dotted lines as _p_, representing the
height of the teeth above and below the pitch line G G. At a right angle
to G G mark the line J K, and from the junction of this line with axis B
(as at Q) as a centre, mark the arc _a_, which will represent the pitch
circle for the large diameter of pinion D; mark also the arc _b_ for the
addendum and _c_ for the roots of the teeth, so that from _b_ to _c_
will represent the height of the tooth at that end.

[Illustration: Fig. 61.]

Similarly from P, as a centre, mark (for the large diameter of wheel C,)
the pitch circle _g_, root circle _h_, and addendum _i_. On these arcs
mark the curves in the same manner as for spur-wheels. To obtain these
arcs for the small diameters of the wheels, draw M M parallel to J K.
Set the compasses to the radius R L, and from P, as a centre, draw the
pitch circle _k_. To obtain the depth for the tooth, draw the dotted
line _p_, meeting the circle _h_, and the point W. A similar line from
circle _i_ to W will show the height of the addendum, or extreme
diameter; and mark the tooth curves on _k_, _l_, _m_, in the same manner
as for a spur-wheel.

Similarly for the pitch circle of the small end of the pinion teeth, set
the compasses to the radius S L, and from Q as a centre, mark the pitch
circle _d_, outside of _d_ mark _e_ for the height of the addendum and
inside of _d_ mark _f_ for the roots of the teeth at that end. The
distance between the dotted lines (as _p_) represents the full height of
the teeth, hence _h_ meets line _p_, being the root of tooth for the
large wheel, and to give clearance, the point of the pinion teeth is
marked below, thus arc _b_ does not meet _h_ or _p_. Having obtained
these arcs the curves are rolled as for a spur-wheel.

A tooth thus marked out is shown at _x_, and from its curves between _b_
_c_, a template for the large diameter of the pinion tooth may be made,
while from the tooth curves between the arcs _e_ _f_, a template for the
smallest tooth diameter of the pinion can be made.

Similarly for the wheel C the outer end curves are marked on the lines
_g_, _h_, _i_, and those for the inner end on the lines _k_, _l_, _m_.

[Illustration: Fig. 62.]

Internal or annular gear-wheels have their tooth curves formed by
rolling the generating circle upon the pitch circle or base circle, upon
the same general principle as external or spur-wheels. But the tooth of
the annular wheel corresponds with the space in the spur-wheel, as is
shown in Fig. 61, in which curve A forms the flank of a tooth on a
spur-wheel P, and the face of a tooth on the annular wheel W. It is
obvious then that the generating circle is rolled within the pitch
circle for the face of the wheel and without for its flank, or the
reverse of the process for spur-wheels. But in the case of internal or
annular wheels the path of contact of tooth upon tooth with a pinion
having a given number of teeth increases in proportion as the number of
teeth in the wheel is diminished, which is also the reverse of what
occurs in spur-wheels; as will readily be perceived when it is
considered that if in an internal wheel the pinion have as many teeth as
the wheel the contact would exist around the whole pitch circles of the
wheel and pinion and the two would rotate together without any motion of
tooth upon tooth. Obviously then we have, in the case of internal
wheels, a consideration as to what is the greatest number (as well as
what is the least number) of teeth a pinion may contain to work with a
given wheel, whereas in spur-wheels the reverse is again the case, the
consideration being how few teeth the wheel may contain to work with a
given pinion. Now it is found that although the curves of the teeth in
internal wheels and pinions may be rolled according to the principles
already laid down for spur-wheels, yet cases may arise in which internal
gears will not work under conditions in which spur-wheels would work,
because the internal wheels will not engage together. Thus, in Fig. 62,
is a pinion of 12 teeth and a wheel of 22 teeth, a generating circle
having a diameter equal to the radius of the pinion having been used for
all the tooth curves of both wheel and pinion. It will be observed that
teeth A, B, and C clearly overlap teeth D, E, and F, and would therefore
prevent the wheels from engaging to the requisite depth. This may of
course be remedied by taking the faces off the pinion, as in Fig. 63,
and thus confining the arc of contact to an arc of recess if the pinion
drives, or an arc of approach if the wheel drives; or the number of
teeth in the pinion may be reduced, or that in the wheel increased;
either of which may be carried out to a degree sufficient to enable the
teeth to engage and not interfere one with the other. In Fig. 64 the
number of teeth in the pinion P is reduced from 12 to 6, the wheel W
having 22 as before, and it will be observed that the teeth engage and
properly clear each other.

[Illustration: Fig. 63.]

By the introduction into the figure of a segment of a spur-wheel also
having 22 teeth and placed on the other side of the pinion, it is shown
that the path of contact is greater, and therefore the angle of action
is greater, in internal than in spur gearing. Thus suppose the pinion to
drive in the direction of the arrows and the thickened arcs A B will be
the arcs of approach, A measuring longer than B. The dotted arcs C D
represent the arcs of receding contact and C is found longer than D, the
angles of action being 66° for the spur-wheels and 72° for the annular
wheel.

On referring again to Fig. 62 it will be observed that it is the faces
of the teeth on the two wheels that interfere and will prevent them from
engaging, hence it will readily occur to the mind that it is possible to
form the curves of the pinion faces correct to work with the faces of
the wheel teeth as well as with the flanks; or it is possible to form
the wheel faces with curves that will work correctly with the faces, as
well as with the flanks of the pinion teeth, which will therefore
increase the angle of action, and Professor McCord has shown in an
article in the London _Engineering_ how to accomplish this in a simple
and yet exceedingly ingenious manner which may be described as
follows:--

[Illustration: Fig. 64.]

It is required to find a describing circle that will roll the curves for
the flanks of the pinion and the faces of the wheels, and also a
describing circle for the flanks of the wheel and the faces of the
pinion; the curve for the wheel faces to work correctly with the faces
as well as with the flanks of the pinion, and the curve for the pinion
faces to work correctly with both the flanks and faces of the internal
wheel.

[Illustration: Fig. 65.]

[Illustration: Fig. 66.]

In Fig. 65 let P represent the pitch circle of an annular or internal
wheel whose centre is at A, and Q the pitch circle of a pinion whose
centre is at B, and let R be a describing circle whose centre is at C,
and which is to be used to roll all the curves for the teeth. For the
flanks of the annular wheel we may roll R within P, while for the faces
of the wheel we may roll R outside of P, but in the case of the pinion
we cannot roll R within Q, because R is larger than Q, hence we must
find some other rolling circle of less diameter than R, and that can be
used in its stead (the radius of R always being greater than the radius
of the axis of the wheel and pinion for reasons that will appear
presently). Suppose then that in Fig. 66 we have a ring whose bore R
corresponds in diameter to the intermediate describing circle R, Fig. 65
and that Q represents the pinion. Then we may roll R around and in
contact with the pinion Q, and a tracing point in R will trace the curve
M N O, giving a curve a portion of which may be used for the faces of
the pinion. But suppose that instead of rolling the intermediate
describing circle R around P, we roll the circle T around P, and it will
trace precisely the same curve M N O; hence for the faces of the pinion
we have found a rolling circle T which is a perfect substitute for the
intermediate circle Q, and which it will always be, no matter what the
diameters of the pinion and of the intermediate describing circle may
be, providing that the diameter of T is equal to the difference between
the diameters of the pinion and that of the intermediate describing
circle as in the figure. If now we use this describing circle to roll
the flanks of the annular wheel as well as the faces of the pinion,
these faces and flanks will obviously work correctly together. Since
this describing circle is rolled on the outside of the pinion and on the
outside of the annular wheel we may distinguish it as the exterior
describing circle.

[Illustration: Fig. 67.]

Now instead of rolling the intermediate describing circle R within the
annular wheel P for the face curves of the teeth upon P, we may find
some other circle that will give the same curve and be small enough to
be rolled within the pinion Q for its teeth flanks. Thus in Fig. 67 P
represents the pitch circle of the annular wheel and R the intermediate
circle, and if R be rolled within P, a point on the circumference of R
will trace the curve V W. But if we take the circle S, having a diameter
equal to the difference between the diameter of R and that of P, and
roll it within P, a point in its circumference will trace the same curve
V W; hence S is a perfect substitute for R, and a portion of the curve V
W may be used for the faces of the teeth on the annular wheel. The
circle S being used for the pinion flanks, the wheel faces and pinion
flanks will work correctly together, and as the circle S is rolled
within the pinion for its flanks and within the wheel for its faces, it
may be distinguished as the interior describing circle.

To prove the correctness of the construction it may be noted that with
the particular diameter of intermediate describing circle used in Fig.
65, the interior and exterior describing circles are of equal diameters;
hence, as the same diameter of describing circle is used for all the
faces and flanks of the pair of wheels they will obviously work
correctly together, in accordance with the rules laid down for spur
gearing. The radius of S in Fig. 69 is equal to the radius of the
annular wheel, less the radius of the intermediate circle, or the radius
from A to C. The radius of the exterior describing circle T is the
radius of the intermediate circle less the radius of the pinion, or
radius C B in the figure.

[Illustration: Fig. 68.]

[Illustration: Fig. 69.]

Now the diameter of the intermediate circle may be determined at will,
but cannot exceed that of the annular wheel or be less than the pinion.
But having been selected between these two limits the interior and
exterior describing circles derived from it give teeth that not only
engage properly and avoid the interference shown in Fig. 62, but that
will also have an additional arc of action during the recess, as is
shown in Fig. 68, which represents the wheel and pinion shown in Fig.
62, but produced by means of the interior and exterior describing
circles. Supposing the pinion to be the driver the arc of approach will
be along the thickened arc of the interior describing circle, while
during the arc of recess there will be an arc of contact along the
dotted portion of the exterior describing circle as in ordinary gearing.
But in addition there will be an arc of recess along the dotted portion
of the intermediate circle R, which arc is due to the faces of the
pinion acting upon the faces as well as upon the flanks of the wheel
teeth. It is obvious from this that as soon as a tooth passes the line
of centres it will, during a certain period, have two points of contact,
one on the arc of the exterior describing circle, and another along the
arc of R, this period continuing until the addendum circle of the pinion
crosses the dotted arc of the exterior describing circle at Z.

The diameters of the interior and exterior describing circles obviously
depend upon the diameter of the intermediate circle, and as this may, as
already stated, be selected, within certain limits, at will, it is
evident that the relative diameters of the interior and exterior
describing circles will vary in proportion, the interior becoming
smaller and the exterior larger, while from the very mode of
construction the radius of the two will equal that of the axes of the
wheel and pinion. Thus in Fig. 69 the radii of S, T, equal A B, or the
line of centres, and their diameters, therefore, equal the radius of the
annular wheel, as is shown by dotting them in at the upper half of the
figure. But after their diameters have been determined by this
construction either of them may be decreased in diameter and the teeth
of the wheels will clear (and not interfere as in Fig. 62), but the
action will be the same as in ordinary gear, or in other words there
will be no arc of action on the circle R. But S cannot be increased
without correspondingly decreasing T, nor can T be increased without
correspondingly decreasing S.

[Illustration: Fig. 70.]

Fig. 70 shows the same pair of gears as in Fig. 68 (the wheel having 22
and the pinion 12 teeth), the diameter of the intermediate circle having
been enlarged to decrease the diameter of S and increase that of T, and
as these are left of the diameter derived from the construction there is
receding action along R from the line of centres to T.

[Illustration: Fig. 71.]

In Fig. 71 are represented a wheel and pinion, the pinion having but
four teeth less than the wheel, and a tooth, J, being shown in position
in which it has contact at two places. Thus at _k_ it is in contact with
the flank of a tooth on the annular wheel, while at L it is in contact
with the face of the same tooth.

As the faces of the teeth on the wheel do not have contact higher than
point _t_, it is obvious that instead of having them 3/10 of the pitch
as at the bottom of the figure, we may cut off the portion X without
diminishing the arc of contact, leaving them formed as at the top of the
figure. These faces being thus reduced in height we may correspondingly
reduce the depth of flank on the pinion by filling in the portion G,
leaving the teeth formed as at the top of the pinion. The teeth faces of
the wheel being thus reduced we may, by using a sufficiently large
intermediate circle, obtain interior and exterior describing circles
that will form teeth that will permit of the pinion having but one tooth
less than the wheel, or that will form a wheel having but one tooth more
than the pinion.

The limits to the diameter of the intermediate describing circle are as
follows: in Fig. 72 it is made equal in diameter to the pitch diameter
of the pinion, hence B will represent the centre of the intermediate
circle as well as of the pinion, and the pitch circle of the pinion will
also represent the intermediate circle R. To obtain the radius for the
interior describing circle we subtract the radius of the intermediate
circle from the radius of the annular wheel, which gives A P, hence the
pitch circle of the pinion also represents the interior circle R. But
when we come to obtain the radius for the exterior describing circle
(T), by subtracting the radius of the pinion from that of the
intermediate circle, we find that the two being equal give O for the
radius of (T), hence there could be no flanks on the pinion.

[Illustration: Fig. 72.]

Now suppose that the intermediate circle be made equal in diameter to
the pitch circle of the annular wheel, and we may obtain the radius for
the exterior describing circle T; by subtracting the radius of the
pinion from that of the intermediate circle, we shall obtain the radius
A B; hence the radius of (T) will equal that of the pinion. But when we
come to obtain the radius for the interior describing circle by
subtracting the radius of the intermediate circle from that of the
annular wheel, we find these two to be equal, hence there would be no
interior describing circle, and, therefore, no faces to the pinion.

[Illustration: Fig. 73.]

The action of the teeth in internal wheels is less a sliding and more a
rolling one than that in any other form of toothed gearing. This may be
shown as follows: In Fig. 73 let A A represent the pitch circle of an
external pinion, and B B that of an internal one, and P P the pitch
circle of an external wheel for A A or an internal one for B B, the
point of contact at the line of centres being at C, and the direction of
rotation P P being as denoted by the arrow; the two pinions being
driven, we suppose a point at C, on the pitch circle P P, to be
coincident with a point on each of the two pinions at the line of
centres. If P P be rotated so as to bring this point to the position
denoted by D, the point on the external pinion having moved to E, while
that on the internal pinion has moved to F, both having moved through an
arc equal to C D, then the distance from E to D being greater than from
D to F, more sliding motion must have accompanied the contact of the
teeth at the point E than at the point F; and the difference in the
length of the arc E D and that of F D, may be taken to represent the
excess of sliding action for the teeth on E; for whatever, under any
given condition, the amount of sliding contact may be, it will be in the
proportion of the length of E D to that of F D. Presuming, then, that
the amount of power transmitted be equal for the two pinions, and the
friction of all other things being equal--being in proportion to the
space passed (or in this case slid) over--it is obvious that the
internal pinion has the least friction.



CHAPTER II.--THE TEETH OF GEAR-WHEELS.--CAMS.


WHEEL AND TANGENT SCREW OR WORM AND WORM GEAR.

In Fig. 74 are shown a worm and worm gear partly in section on the line
of centres. The worm or tangent screw W is simply one long tooth wound
around a cylinder, and its form may be determined by the rules laid down
for a rack and pinion, the tangent screw or worm being considered as a
rack and the wheel as an ordinary spur-wheel.

[Illustration: Fig. 74.]

Worm gearing is employed for transmitting motion at a right angle, while
greatly reducing the motion. Thus one rotation of the screw will rotate
the wheel to the amount of the pitch of its teeth only. Worm gearing
possesses the qualification that, unless of very coarse pitch, the worm
locks the wheel in any position in which the two may come to a state of
rest, while at the same time the excess of movement of the worm over
that of the wheel enables the movement of the latter, through a very
minute portion of a revolution. And it is evident that, when the plane
of rotation of the worm is at a right angle to that of the wheel, the
contact of the teeth is wholly a sliding one. The wear of the worm is
greater than that of the wheel, because its teeth are in continuous
contact, whereas the wheel teeth are in contact only when passing
through the angle of action. It may be noted, however, that each tooth
upon the worm is longer than the teeth on the wheel in proportion as the
circumference of the worm is to the length of wheel tooth.

[Illustration: Fig. 75.]

If the teeth of the wheel are straight and are set at an angle equal to
the angle of the worm thread to its axis, as in Fig. 75, P P
representing the pitch line of the worm, C D the line of centres, and
_d_ the worm axis, the contact of tooth upon tooth will be at the centre
only of the sides of the wheel teeth. It is generally preferred,
however, to have the wheel teeth curved to envelop a part of the
circumference of the worm, and thus increase the line of contact of
tooth upon tooth, and thereby provide more ample wearing surface.

[Illustration: Fig. 76.]

In this case the form of the teeth upon the worm wheel varies at every
point in its length as the line of centres is departed from. Thus in
Fig. 76 is shown an end view of a worm and a worm gear in section, _c_
_d_ being the line of centres, and it will be readily perceived that the
shape of the teeth if taken on the line _e_ _f_, will differ from that
on the line of centres _c_ _d_; hence the form of the wheel teeth must,
if contact is to occur along the full length of the tooth, be conformed
to fit to the worm, which may be done by taking a series of section of
the worm thread at varying distances from, and parallel to, the line of
centres and joining the wheel teeth to the shape so obtained. But if the
teeth of the wheel are to be cut to shape, then obviously a worm may be
provided with teeth (by serrating it along its length) and mounted in
position upon the wheel so as to cut the teeth of the wheel to shape as
the worm rotates. The pitch line of the wheel teeth, whether they be
straight and are disposed at an angle as in Fig. 75, or curved as in
Fig. 76, is at a right angle to the line of centres _c_ _d_, or in other
words in the plane of _g_ _h_, in Fig. 76. This is evident because the
pitch line must be parallel to the wheel axis, being at an equal radius
from that axis, and therefore having an equal velocity of rotation at
every point in the length of the pitch line of the wheel tooth.

If we multiply the number of teeth by their pitch to obtain the
circumference of the pitch circle we shall obtain the circumference due
to the radius of _g_ _h_, from the wheel axis, and so long as _g_ _h_ is
parallel to the wheel axis we shall by this means obtain the same
diameter of pitch circle, so long as we measure it on a line parallel to
the line of centres _c_ _d_. The pitch of the worm is the same at
whatever point in the tooth depth it may be measured, because the teeth
curves are parallel one to the other, thus in Fig. 77 the pitch measures
are equal at _m_, _n_, or _o_.

But the action of the worm and wheel will nevertheless not be correct
unless the pitch line from which the curves were rolled coincides with
the pitch line of the wheel on the line of centres, for although, if the
pitch lines do not so coincide, the worm will at each revolution move
the pitch line of the wheel through a distance equal to the pitch of the
worm, yet the motion of the wheel will not be uniform because,
supposing the two pitch lines not to meet, the faces of the pinion teeth
will act against those of the wheel, as shown in Fig. 78, instead of
against their flanks, and as the faces are not formed to work correctly
together the motion will be irregular.

[Illustration: Fig. 77.]

The diameter of the worm is usually made equal to four times the pitch
of the teeth, and if the teeth are curved as in figure 76 they are made
to envelop not more than 30° of the worm.

The number of teeth in the wheel should not be less than thirty, a
double worm being employed when a quicker ratio of wheel to worm motion
is required.

[Illustration: Fig. 78.]

[Illustration: Fig. 79.]

When the teeth of the wheel are curved to partly envelop the worm
circumference it has been found, from experiments made by Robert Briggs,
that the worm and the wheel will be more durable, and will work with
greatly diminished friction, if the pitch line of the worm be located to
increase the length of face and diminish that of the flank, which will
decrease the length of face and increase the length of flank on the
wheel, as is shown in Fig. 79; the location for the pitch line of the
worm being determined as follows:--

[Illustration: Fig. 80.]

The full radius of the worm is made equal to twice the pitch of its
teeth, and the total depth of its teeth is made equal to .65 of its
pitch. The pitch line is then drawn at a radius of 1.606 of the pitch
from the worm axis. The pitch line is thus determined in Fig. 76, with
the result that the area of tooth face and of worm surface is equalized
on the two sides of the pitch line in the figure. In addition to this,
however, it may be observed that by thus locating the pitch line the
arcs both of approach and of recess are altered. Thus in Fig. 80 is
represented the same worm and wheel as in Fig. 79, but the pitch lines
are here laid down as in ordinary gearing. In the two figures the arcs
of approach are marked by the thickened part of the generating circle,
while the arcs of recess are denoted by the dotted arc on the generating
circle, and it is shown that increasing the worm face, as in Fig. 79,
increases the arc of recess, while diminishing the worm flank diminishes
the arc of approach, and the action of the worm is smoother because the
worm exerts more pulling than pushing action, it being noted that the
action of the worm on the wheel is a pushing one before reaching, and a
pulling one after passing, the line of centres.

[Illustration: Fig. 81.]

It may here be shown that a worm-wheel may be made to work correctly
with a square thread. Suppose, for example, that the diameter of the
generating circle be supposed to be infinite, and the sides of the
thread may be accepted as rolled by the circle. On the wheel we roll a
straight line, which gives a cycloidal curve suitable to work with the
square thread. But the action will be confined to the points of the
teeth, as is shown in Fig. 81, and also to the arc of approach. This is
the same thing as taking the faces off the worm and filling in the
flanks of the wheel. Obviously, then, we may reverse the process and
give the worm faces only, and the wheel, flanks only, using such size of
generating circle as will make the spaces of the wheel parallel in their
depths and rolling the same generating circle upon the pitch line of the
worm to obtain its face curve. This would enable the teeth on the wheel
to be cut by a square-threaded tap, and would confine the contact of
tooth upon tooth to the recess.

The diameter of generating circle used to roll the curves for a worm and
worm-wheel should in all cases be larger than the radius of the
worm-wheel, so that the flanks of the wheel teeth may be at least as
thick at the root as they are at the pitch circle.

To find the diameter of a wheel, driven by a tangent-screw, which is
required to make one revolution for a given number of turns of the
screw, it is obvious, in the first place, that when the screw is
single-threaded, the number of teeth in the wheel must be equal to the
number of turns of the screw. Consequently, the pitch being also given,
the radius of the wheel will be found by multiplying the pitch by the
number of turns of the screw during one turn of the wheel, and dividing
the product by 6.28.

[Illustration: Fig. 82.]

When a wheel pattern is to be made, the first consideration is the
determination of the diameter to suit the required speed; the next is
the pitch which the teeth ought to have, so that the wheel may be in
accordance with the power which it is intended to transmit; the next,
the number of the teeth in relation to the pitch and diameter; and,
lastly, the proportions of the teeth, the clearance, length, and
breadth.

[Illustration: Fig. 83.]

When the amount of power to be transmitted is sufficient to cause
excessive wear, or when the velocity is so great as to cause rapid wear,
the worm instead of being made parallel in diameter from end to end, is
sometimes given a curvature equal to that of the worm-wheel, as is shown
in Fig. 82.

[Illustration: Fig. 84.]

[Illustration: Fig. 85.]

The object of this design is to increase the bearing area, and thus, by
causing the power transmitted to be spread over a larger area of
contact, to diminish the wear. A mechanical means of cutting a worm to
the required form for this arrangement is shown in Fig. 83, which is
extracted from "Willis' Principles of Mechanism." "A is a wheel driven
by an endless screw or worm-wheel, B, C is a toothed wheel fixed to the
axis of the endless screw B and in gear with another and equal toothed
gear D, upon whose axis is mounted the smooth surfaced solid E, which it
is desired to cut into Hindley's[2] endless screw. For this purpose a
cutting tooth F is clamped to the face of the wheel A. When the handle
attached to the axis of B C is turned round, the wheel A and solid wheel
E will revolve with the same relative velocity as A and B, and the tool
F will trace upon the surface of the solid E a thread which will
correspond to the conditions. For from the very mode of its formation
the section of every thread through the axis will point to the centre of
the wheel A. The axis of E lies considerably higher than that of B to
enable the solid E to clear the wheel A.

  [2] The inventor of this form of endless screw.

"The edges of the section of the solid E along its horizontal centre
line exactly fit the segment of the toothed wheel, but if a section be
made by a plane parallel to this the teeth will no longer be equally
divided as they are in the common screw, and therefore this kind of
screw can only be in contact with each tooth along a line corresponding
to its middle section. So that the advantage of this form over the
common one is not so great as appears at first sight.

"If the inclination of the thread of a screw be very great, one or more
intermediate threads may be added, as in Fig. 84, in which case the
screw is said to be double or triple according to the number of separate
spiral threads that are so placed upon its surface. As every one of
these will pass its own wheel-tooth across the line of centres in each
revolution of the screw, it follows that as many teeth of the wheel will
pass that line during one revolution of the screw as there are threads
to the screw. If we suppose the number of these threads to be
considerable, for example, equal to those of the wheel teeth, then the
screw and wheel may be made exactly alike, as in Fig. 85; which may
serve as an example of the disguised forms which some common
arrangements may assume."

[Illustration: Fig. 86.]

In Fig. 86 is shown Hawkins's worm gearing. The object of this ingenious
mechanical device is to transmit motion by means of screw or worm
gearing, either by a screw in which the threads are of equal diameter
throughout its length, or by a spiral worm, in which the threads are not
of equal diameter throughout, but increase in diameter each way from the
centre of its length, or about the centre of its length outwardly.
Parallel screws are most applicable to this device when rectilinear
motions are produced from circular motions of the driver, and spiral
worms are applied when a circular motion is given by the driver, and
imparted to the driven wheel. The threads of a spiral worm instead of
gearing into teeth like those of an ordinary worm-wheel, actuate a
series of rollers turning upon studs, which studs are attached to a
wheel whose axis is not parallel to that of the worm, but placed at a
suitable inclination thereto. When motion is given to the worm then
rotation is produced in the roller wheel at a rate proportionable to the
pitch of worm and diameter of wheel respectively.

In the arrangement for transmitting rectilinear motion from a screw,
rollers may be employed whose axes are inclined to the axis of the
driving screw, or else at right angles to or parallel to the same. When
separate rollers are employed with inclined axes, or axes at right
angles with that of the main driving screw, each thread in gear touches
a roller at one part only; but when the rollers are employed with axes
parallel to that of the driving screw a succession of grooves are turned
in these rollers, into which the threads of the driving screw will be in
gear throughout the entire length of the roller. These grooves may be
separate and apart from each other, or else form a screw whose pitch is
equal to that of the driving screw or some multiple thereof.

In Fig. 86 the spiral worm is made of such a length that the edge of one
roller does not cease contact until the edge of the next comes into
contact; a wheel carries four rollers which turn on studs, the latter
being secured by cottars; the axis of the worm is at right angles with
that of the wheel. The edges of the rollers come near together, leaving
sufficient space for the thread of the worm to fit between any two
contiguous rollers. The pitch line of the screw thread forms an arc of a
circle, whose centre coincides with that of the wheel, therefore the
thread will always bear fairly against the rollers and maintain rolling
contact therewith during the whole of the time each roller is in gear,
and by turning the screw in either direction the wheel will rotate.

[Illustration: Fig. 87.]

To prevent end thrust on a worm shaft it may have a right-hand worm A,
and a left-hand one C (Fig. 87), driving two wheels B and D which are in
gear, and either of which may transmit the power. The thrust of the two
worms A and C, being in opposite directions, one neutralizes the other,
and it is obvious that as each revolution of the worm shaft moves both
wheels to an amount equal to the pitch of the worms, the two wheels B D
may, if desirable, be of different diameters.

[Illustration: Fig. 88.]

[Illustration: Fig. 89.]

Involute teeth.--These are teeth having their whole operative surfaces
formed of one continuous involute curve. The diameter of the generating
circle being supposed as infinite, then a portion of its circumference
may be represented by a straight line, such as A in Fig. 88, and if this
straight line be made to roll upon the circumference of a circle, as
shown, then the curve traced will be involute P. In practice, a piece of
flat spring steel, such as a piece of clock spring, is used for tracing
involutes. It may be of any length, but at one end it should be filed so
as to leave a scribing point that will come close to the base circle or
line, and have a short handle, as shown in Fig. 89, in which S
represents the piece of spring, having the point P´, and the handle H.
The operation is, to make a template for the base circle, rest this
template on drawing paper and mark a circle round its edge to represent
on the paper the pitch circle, and to then bend the spring around the
circle B, holding the point P´ in contact with the drawing paper,
securing the other end of the piece of steel, so that it cannot slip
upon B, and allowing the steel to unwind from the cylinder or circle B.
The point P´ will mark the involute curve P. Another way to mark an
involute is to use a piece of twine in place of the spring and a pencil
instead of the tracing point; but this is not so accurate, unless,
indeed, a piece of wood be laid on the drawing-board and the pencil held
firmly against it, so as to steady the pencil point and prevent the
variation in the curve that would arise from variation in the vertical
position of the pencil.

The flanks being composed of the same curve as the faces of the teeth,
it is obvious that the circle from which the tracing point starts, or
around which the straight line rolls, must be of less diameter than the
pitch circle, or the teeth would have no flanks.

A circle of less diameter than the pitch circle of the wheel is,
therefore, introduced, wherefrom to produce the involute curves forming
the full side of the tooth.

[Illustration: Fig. 90.]

The depth below pitch line or the length of flank is, therefore, the
distance between the pitch circle and the base circle. Now even
supposing a straight line to be a portion of the circumference of a
circle of infinite diameter or radius, the conditions would here appear
to be imperfect, because the generating circle is not rolled upon the
pitch circle but upon a circle of lesser diameter. But it can be shown
that the requirements of a proper velocity ratio will be met,
notwithstanding the employment of the base instead of the pitch circle.
Thus, in Fig. 90, let A and B represent the respective centres of the
two pitch circles, marked in dotted lines. Draw the base circle for B as
E Q, which may be of any radius less than that of the pitch circle of
B. Draw the straight line Q D R touching this base circle at its
perimeter and passing through the point of contact on the pitch circles
as at D. Draw the circle whose radius is A R forming the base circle for
wheel A. Thus the line R P Q will meet the perimeters of the two circles
while passing through the point of contact D at the line of centres (a
condition which the relative diameters of the base circles must always
be so proportioned as to attain).

If now we take any point on R Q, as P in the figure, as a tracing point,
and suppose the radius or distance P Q to represent the steel spring
shown in Fig. 89, and move the tracing point back to the base circle of
B, it will trace the involute E P. Again we may take the tracing point P
(supposing the line P R to represent the steel spring), and trace the
involute P F, and these two involutes represent each one side of the
teeth on the respective wheels.

[Illustration: Fig. 91.]

The line R P Q is at a right angle to the curves P E and P F, at their
point of contact, and, therefore, fills the conditions referred to in
Fig. 41. Now the line R P Q denotes the path of contact of tooth upon
tooth as the wheels revolve; or, in other words, the point of contact
between the side of a tooth on one wheel, and the side of a tooth on the
other wheel, will always move along the line Q R, or upon a similar line
passing through D, but meeting the base circles upon the opposite sides
of the line of centres, and since line Q R always cuts the line of
centres at the point of contact of the pitch circles, the conditions
necessary to obtain a correct angular velocity are completely fulfilled.
The velocity ratio is, therefore, as the length of B Q is to that of A
R, or, what is the same thing, as the radius of the base circle of one
wheel is to that of the other. It is to be observed that the line Q R
will vary in its angle to the line of centres A B, according to the
diameter of the base circle from which it is struck, and it becomes a
consideration as to what is its most desirable angle to produce the
least possible amount of thrust tending to separate the wheels, because
this thrust (described in Fig. 39) tends to wear the journals and
bearings carrying the wheel shafts, and thus to permit the pitch circles
to separate. To avoid, as far as possible, this thrust the proportions
between the diameters of the base circles D and E, Fig. 91, must be such
that the line D E passes through the point of contact on the line of
centres, as at C, while the angles of the straight line D E should be as
nearly 90° to a radial line, meeting it from the centres of the wheels
(as shown in the figure, by the lines B E and D E), as is consistent
with the length of D E, which in order to impart continuous motion must
at least equal the pitch of the teeth. It is obvious, also, that, to
give continuous motion, the length of D E must be more than the pitch in
proportion, as the points of the teeth come short of passing through the
base circles at D and E, as denoted by the dotted arcs, which should
therefore represent the addendum circles. The least possible obliquity,
or angle of D E, will be when the construction under any given
conditions be made such by trial, that the base circles D and E coincide
with the addendum circles on the line of centres, and thus, with a given
depth of both beyond, the pitch circle, or addenda as it is termed, will
cause the tooth contacts to extend over the greatest attainable length
of line between the limits of the addendum circles, thus giving a
maximum number of teeth in contact at any instant of time. These
conditions are fulfilled in Fig. 92,[3] the addendum on the small wheel
being longer than the depth below pitch line, while the faces of the
teeth are the narrowest.

  [3] From an article by Prof. Robinson.

[Illustration: Fig. 92.]

In seeking the minimum obliquity or angle of D E in the figure, it is to
be observed that the less it is, the nearer the base circle approaches
the pitch circle; hence, the shorter the operative length of tooth flank
and the greater its wear.

In comparing the merits of involute with those of epicycloidal teeth,
the direction of the line of pressure at each point of contact must
always be the common perpendicular to the surfaces at the point of
contact, and these perpendiculars or normals must pass through the pitch
circles on the line of centres, as was shown in Fig. 41, and it follows
that a line drawn from C (Fig. 91) to any point of contact, is in the
direction of the pressure on the surfaces at that point of contact. In
involute teeth, the contact will always be on the line D E (Fig. 92),
but in epicycloidal, on the line of the generating circle, when that
circle is tangent at the line of centres; hence, the direction of
pressure will be a chord of the circle drawn from the pitch circle at
the line of centres to the position of contact considered. Comparing
involute with radial flanked epicycloidal teeth, let C D A (Fig. 91)
represent the rolling circle for the latter, and D C will be the
direction of pressure for the contact at D; but for point of contact
nearer C, the direction will be much nearer 90°, reaching that angle as
the point of contact approaches C. Now, D is the most remote legitimate
contact for involute teeth (and considering it so far as epicycloidal
struck with a generating circle of infinite diameter), we find that the
aggregate directions of the pressures of the teeth upon each other is
much nearer perpendicular in epicycloidal, than in involute gearing;
hence, the latter exert a greater pressure, tending to force the wheels
apart. Hence, the former are, in this respect, preferable.

It is to be observed, however, that in some experiments made by Mr.
Hawkins, he states that he found "no tendency to press the wheels apart,
which tendency would exist if the angle of the line D E (Fig. 92)
deviated more than 20° from the line of centres A B of the two wheels."

A method commonly employed in practice to strike the curves of involute
teeth, is as follows:--

In Fig. 93 let C represent the centre of a wheel, D D the full diameter,
P P the pitch circle, and E the circle of the roots of the teeth, while
R is a radial line. Divide on R, the distance between the pitch circle
and the wheel centre, into four equal parts, by 1, 2, 3, &c. From point
or division 2, as a centre, describe the semicircle S, cutting the wheel
centre and the pitch circle at its junction with R (as at A). From A,
with compasses set to the length of one of the parts, as A 3, describe
the arc B, cutting S at F, and F will be the centre from which one side
of the tooth may be struck; hence from F as a centre, with the compasses
set to the radius A B, mark the curve G. From the centre C strike,
through F, a circle T T, and the centres wherefrom to strike all the
teeth curves will fall on T T. Thus, to strike the other curve of the
tooth, mark off from A the thickness of the tooth on the pitch circle P
P, producing the point H. From H as a centre (with the same radius as
before,) mark on T T the point I, and from I, as a centre, mark the
curve J, forming the other side of the tooth.

[Illustration: Fig. 93.]

[Illustration: Fig. 94.]

In Fig. 94 the process is shown carried out for several teeth. On the
pitch circle P P, divisions 1, 2, 3, 4, &c., for the thickness of teeth
and the width of the spaces are marked. The compasses are set to the
radius by the construction shown in Fig. 93, then from _a_, the point
_b_ on T is marked, and from _b_ the curve _c_ is struck.

In like manner, from _d_, _g_, _j_, the centres _e_, _h_, _k_, wherefrom
to strike the respective curves, _f_, _i_, _l_, are obtained.

Then from _m_ the point _n_, on T T, is marked, giving the centre
wherefrom to strike the curve at _h_ _m_, and from _o_ is obtained the
point _p_, on T T, serving as a centre for the curve _e_ _o_.

A more simple method of finding point F is to make a sheet metal
template, C, as in Fig. 95, its edges being at an angle one to the other
of 75° and 30'. One of its edges is marked off in quarters of an inch,
as 1, 2, 3, 4, &c. Place one of its edges coincident with the line R,
its point touching the pitch circle at the side of a tooth, as at A, and
the centre for marking the curve on that side of the tooth will be found
on the graduated edge at a distance from A equal to one-fourth the
length of R.

[Illustration: Fig. 95.]

The result obtained in this process is precisely the same as that by the
construction in Fig. 93, as will be plainly seen, because there are
marked on Fig. 93 all the circles by which point F was arrived at in
Fig. 95; and line 3, which in Fig. 95 gives the centre wherefrom to
strike curve _o_, is coincident with point F, as is shown in Fig. 95. By
marking the graduated edge of C in quarter-inch divisions, as 1, 2, 3,
&c., then every division will represent the distance from A for the
centre for every inch of wheel radius. Suppose, for example, that a
wheel has 3 inches radius, then with the scale C set to the radial line
R, the centre therefrom to strike the curve _o_ will be at 3; were the
radius of the wheel 4 inches, then the scale being set the same as
before (one edge coincident with R), the centre for the curve _o_ would
be at 4, and arc T would require to meet the edge of C at 4. Having
found the radius from the centre of the wheel of point F for one tooth,
we may mark circle T, cutting point F, and mark off all the teeth by
setting one point of the compasses (set to radius A F) on one side of
the tooth and marking on circle T the centre wherefrom to mark the curve
(as _o_), continuing the process all around the wheel and on both sides
of the tooth.

This operation of finding the location for the centre wherefrom to
strike the tooth curves, must be performed separately for each wheel,
because the distance or radius of the tooth curves varies with the
radius of each wheel.

In Fig. 96 this template is shown with all the lines necessary to set
it, those shown in Fig. 95 to show the identity of its results with
those given in Fig. 93 being omitted.

The principles involved in the construction of a rack to work correctly
with a wheel or pinion, having involute teeth, are as in Fig. 97, in
which the pitch circle is shown by a dotted circle and the base circle
by a full line circle. Now the diameter of the base circle has been
shown to be arbitrary, but being assumed the radius B Q will be
determined (since it extends from the centre B to the point of contact
of D Q, with the base circle); B D is a straight line from the centre B
of the pinion to the pitch line of the rack, and (whatever the angle of
Q D to B D) the sides of the rack teeth must be straight lines inclined
to the pitch line of the rack at an angle equal to that of B D Q.

Involute teeth possess four great advantages--1st, they are thickest at
the roots, where they should be to have a maximum of strength, which is
of great importance in pinions transmitting much power; 2nd, the action
of the teeth will remain practically perfect, even though the wheels are
spread apart so that the pitch circles do not meet on the line of
centres; 3rd, they are much easier to mark, and truth in the marking is
easier attained; and 4th, they are much easier to cut, because the full
depth of the teeth can, on spur-wheels, in all cases be cut with one
revolving cutter, and at one passage of the cutter, if there is
sufficient power to drive it, which is not the case with epicycloidal
teeth whenever the flank space is wider below than it is at the pitch
circle. On account of the first-named advantage, they are largely
employed upon small gears, having their teeth cut true in a gear-cutting
machine; while on account of the second advantage, interchangeable
wheels, which are merely required to transmit motion, may be put in gear
without a fine adjustment of the pitch circle, in which case the wear of
the teeth will not prove destructive to the curves of the teeth. Another
advantage is, that a greater number of teeth of equal strength may be
given to a wheel than in the epicycloidal form, for with the latter the
space must at least equal the thickness of the tooth, while in involute
the space may be considerably less in width than the tooth, both
measured, of course, at the pitch circle. There are also more teeth in
contact at the same time; hence, the strain is distributed over more
teeth.

[Illustration: Fig. 96.]

These advantages assume increased value from the following
considerations.

In a train of epicycloidal gearing in which the pinion or smallest wheel
has radial flanks, the flanks of the teeth will become spread as the
diameters of the wheels in the train increase. Coincident with spread at
the roots is the thrust shown with reference to Fig. 39, hence under the
most favorable conditions the wear on the journals of the wheel axles
and the bearings containing them will take place, and the pitch circles
will separate. Now so soon as this separation takes place, the motion of
the wheels will not be as uniformly equal as when the pitch circles were
in contact on the line of centres, because the conditions under which
the tooth curves, necessary to produce a uniform velocity of motion,
were formed, will have become altered, and the value of those curves to
produce constant regularity of motion will have become impaired in
proportion as the pitch circles have separated.

[Illustration: Fig. 97.]

In a single pair of epicycloidal wheels in which the flanks of the teeth
are radial, the conditions are more favorable, but in this case the
pinion teeth will be weaker than if of involute form, while the wear of
the journals and bearings (which will take place to some extent) will
have the injurious effect already stated, whereas in involute teeth, as
has been noted, the separation of the pitch circles does not affect the
uniformity of the motion or the correct working of the teeth.

If the teeth of wheels are to be cut to shape in a gear-cutting machine,
either the cutters employed determine from their shapes the shapes or
curves of the teeth, or else the cutting tool is so guided to the work
that the curves are determined by the operations of the machine. In
either case nothing is left to the machine operator but to select the
proper tools and set them, and the work in proper position in the
machine. But when the teeth are to be cast upon the wheel the pattern
wherefrom the wheel is to be moulded must have the teeth proportioned
and shaped to proper curve and form.

Wheels that require to run without noise or jar, and to have uniformity
of motion, must be finished in gear-cutting machines, because it is
impracticable to cast true wheels.

When the teeth are to be cast upon the wheels the pattern-maker makes
templates of the tooth curves (by some one of the methods to be
hereafter described), and carefully cuts the teeth to shape. But the
production of these templates is a tedious and costly operation, and one
which is very liable to error unless much experience has been had. The
Pratt and Whitney Company have, however, produced a machine that will
produce templates of far greater accuracy than can be made by hand work.
These templates are in metal, and for epicycloidal teeth from 15 to a
rack, and having a diametral pitch ranging from 1-1/2 to 32.

The principles of action of the machine are that a segment of a ring
(representing a portion of the pitch circle of the wheel for whose teeth
a template is to be produced) is fixed to the frame of the machine. Upon
this ring rolls a disk representing the rolling, generating, or
describing circle, this disk being carried by a frame mounted upon an
arm representing the radius of the wheel, and therefore pivoted at a
point central to the ring. The describing disk is rolled upon the ring
describing the epicycloidal curve, and by suitable mechanical devices
this curve is cut upon a piece of steel, thus producing a template by
actually rolling the generating upon the base circle, and the rolling
motion being produced by positive mechanical motion, there cannot
possibly be any slip, hence the curves so produced are true epicycloids.

The general construction of the machine is shown in the side view, Fig.
98 (Plate I.), and top view, Fig. 99 (Plate I.), details of construction
being shown in Figs. 100, 101 (Plate I.), 102, 103, 104, 105, and 106. A
A is the segment of a ring whose outer edge represents a part of the
pitch circle. B is a disk representing the rolling or generating circle
carried by the frame C, which is attached to a rod pivoted at D. The
axis of pivot D represents the axis of the base circle or pitch circle
of the wheel, and D is adjustable along the rod to suit the radius of A
A, or what is the same thing, to equal the radius of the wheel for whose
teeth a template is to be produced.

When the frame C is moved its centre or axis of motion is therefore at D
and its path of motion is around the circumference of A A, upon the edge
of which it rolls. To prevent B from slipping instead of rolling upon A
A, a flexible steel ribbon is fastened at one end upon A A, passes
around the edge of A A and thence around the circumference of B, where
its other end is fastened; due allowance for the thickness of this
ribbon being made in adjusting the radii of A A and of B.

E´ is a tubular pivot or stud fixed on the centre line of pivots E and
D, and distant from the edge of A A to the same amount that E is. These
two studs E and E´ carry two worm-wheels F and F´ in Fig. 102, which
stand above A and B, so that the axis of the worm G is vertically over
the common tangent of the pitch and describing circles.

[Illustration: _VOL. I._ =TEMPLATE-CUTTING MACHINES FOR GEAR TEETH.=
_PLATE I._

Fig. 98.

Fig. 99.

Fig. 100.

Fig. 101.]

The relative positions of these and other parts will be most clearly
seen by a study of the vertical section, Fig. 102.[4] The worm G is
supported in bearings secured to the carrier C and is driven by another
small worm turned by the pulley I, as seen in Fig. 101 (Plate I.); the
driving cord, passing through suitable guiding pulleys, is kept at
uniform tension by a weight, however C moves; this is shown in Figs. 98
and 99 (Plate I.).

  [4] From "The Teeth of Spur Wheels," by Professor McCord.

[Illustration: Fig. 102.]

Upon the same studs, in a plane still higher than the worm-wheels turn
the two disks H, H´, Figs. 103, 104, 105. The diameters of these are
equal, and precisely the same as those of the describing circles which
they represent, with due allowance, again, for the thickness of a steel
ribbon, by which these also are connected. It will be understood that
each of these disks is secured to the worm-wheel below it, and the outer
one of these, to the disk B, so that as the worm G turns, H and H´ are
rotated in opposite directions, the motion of H being identical with
that of B; this last is a rolling one upon the edge of A, the carrier C
with all its attached mechanism moving around D at the same time.
Ultimately, then, the motions of H, H´, are those of two equal
describing circles rolling in external and internal contact with a fixed
pitch circle.

[Illustration: Fig. 103.]

[Illustration: Fig. 104.]

In the edge of each disk a semicircular recess is formed, into which is
accurately fitted a cylinder J, provided with flanges, between which the
disks fit so as to prevent end play. This cylinder is perforated for the
passage of the steel ribbon, the sides of the opening, as shown in Fig.
103, having the same curvature as the rims of the disks. Thus when these
recesses are opposite each other, as in Fig. 104, the cylinder J fills
them both, and the tendency of the steel ribbon is to carry it along
with H when C moves to one side of this position, as in Fig. 105, and
along with H´ when C moves to the other side, as in Fig. 103.

This action is made positively certain by means of the hooks K, K´,
which catch into recesses formed in the upper flange of J, as seen in
Fig. 104. The spindles, with which these hooks turn, extend through the
hollow studs, and the coiled springs attached to their lower ends, as
seen in Fig. 102, urge the hooks in the directions of their points;
their motions being limited by stops _o_, _o´_, fixed, not in the disks
H, H´, but in projecting collars on the upper ends of the tubular studs.
The action will be readily traced by comparing Fig. 104 with Fig. 105;
as C goes to the left, the hook K´ is left behind, but the other one, K,
cannot escape from its engagement with the flange of J; which,
accordingly, is carried along with H by the combined action of the hook
and the steel ribbon.

On the top of the upper flange of J, is secured a bracket, carrying the
bearing of a vertical spindle L, whose centre line is a prolongation of
that of J itself. This spindle is driven by the spur-wheel N, keyed on
its upper end, through a flexible train of gearing seen in Fig. 99; at
its lower end it carries a small milling cutter M, which shapes the edge
of the template T, Fig. 105, firmly clamped to the framing.

[Illustration: Fig. 105.]

When the machine is in operation, a heavy weight, seen in Fig. 98 (Plate
I.), acts to move C about the pivot D, being attached to the carrier by
a cord guided by suitably arranged pulleys; this keeps the cutter M up
to its work, while the spindle L is independently driven, and the duty
left for the worm G to perform is merely that of controlling the motions
of the cutter by the means above described, and regulating their speed.

The centre line of the cutter is thus automatically compelled to travel
in the path R S, Fig. 105, composed of an epicycloid and a hypocycloid
if A A be the segment of a circle as here shown; or of two cycloids, if
A A be a straight bar. The radius of the cutter being constant, the edge
of the template T is cut to an outline also composed of two curves;
since the radius M is small, this outline closely resembles R S, but
particular attention is called to the fact that it is _not identical
with it, nor yet composed of truly epicycloidal curves of any generation
whatever:_ the result of which will be subsequently explained.


NUMBER AND SIZES OF TEMPLATES.

With a given pitch every additional tooth increases the diameter of the
wheel, and changes the form of the epicycloid; so that it would appear
necessary to have as many different cutters, as there are wheels to be
made, of any one pitch.

But the proportional increment, and the actual change of form, due to
the addition of one tooth, becomes less as the wheel becomes larger; and
the alteration in the outline soon becomes imperceptible. Going still
farther, we can presently add more teeth without producing a sensible
variation in the contour. That is to say, several wheels can be cut with
the same cutter, without introducing a perceptible error. It is obvious
that this variation in the form is least near the pitch circle, which is
the only part of the epicycloid made use of; and Prof. Willis many years
ago deduced theoretically, what has since been abundantly proved by
practice, that instead of an infinite number of cutters, 24 are
sufficient of one pitch, for making all wheels, from one with 12 teeth
up to a rack.

[Illustration: Fig. 106.]

Accordingly, in using the epicycloidal milling engine, for forming the
template, segments of pitch circles are provided of the following
diameters (in inches):

  12,   16,   20,   27,   43,   100,
  13,   17,   21,   30,   50,   150,
  14,   18,   23,   34,   60,   300.
  15,   19,   25,   38,   75,

In Fig. 106, the edge T T is shaped by the cutter T T, whose centre
travels in the path R S, therefore these two lines are at a constant
normal distance from each other. Let a roller P, of any reasonable
diameter, be run along T T, its centre will trace the line U V, which is
at a constant normal distance from T T, and therefore from R S. Let the
normal distance between U V and R S be the radius of another milling
cutter N, having the same axis as the roller P, and carried by it, but
in a different plane as shown in the side view; then whatever N cuts
will have R S for its contour, if it lie upon the same side of the
cutter as the template.

The diameter of the disks which act as describing circles is 7-1/2
inches, and that of the milling cutter which shapes the edge of the
template is 1/8 of an inch.

Now if we make a set of 1-pitch wheels with the diameters above given,
the smallest will have twelve teeth, and the one with fifteen teeth will
have radial flanks. The curves will be the same whatever the pitch; but
as shown in Fig. 106, the blank should be adjusted in the epicycloidal
engine, so that its lower edge shall be 1/16th of an inch (the radius of
the cutter M) above the bottom of the space; also its relation to the
side of the proposed tooth should be as here shown. As previously
explained, the depth of the space depends upon the pitch. In the system
adopted by the Pratt & Whitney Company, the whole height of the tooth is
2-1/8 times the diametral pitch, the projection outside the pitch circle
being just equal to the pitch, so that diameter of blank = diameter of
pitch circle + 2 × diametral pitch.

We have now to show how, from a single set of what may be called 1-pitch
templates, complete sets of cutters of the true epicycloidal contour may
be made of the same or any less pitch.

Now if T T be a 1-pitch template as above mentioned, it is clear that N
will correctly shape a cutting edge of a gear cutter for a 1-pitch
wheel. The same figure, reduced to half size, would correctly represent
the formation of a cutter for a 2-pitch wheel of the same number of
teeth; if to quarter size, that of a cutter for a 4-pitch wheel, and so
on.

But since the actual size and curvature of the contour thus determined
depend upon the dimensions and motion of the cutter N, it will be seen
that the same result will practically be accomplished, if these only be
reduced; the size of the template, the diameter and the path of the
roller remaining unchanged.

The nature of the mechanism by which this is effected in the Pratt &
Whitney system of producing epicycloidal cutters will be hereafter
explained in connection with cutters.



CHAPTER III.--THE TEETH OF GEAR-WHEELS (continued).


The revolving cutters employed in gear-cutting machines, gear-cutters,
or cutting engines (as the machines for cutting the teeth of gear-wheels
to shape are promiscuously termed), are of the form shown in Fig. 107,
which represents what is known as a Brown and Sharpe patent cutter,
whose peculiarities will be explained presently. This class of cutters
is made as follows:--

[Illustration: Fig. 107.]

A cast steel disk is turned in the lathe to the required form and
outline. After turning, its circumference is serrated as shown, so as to
provide protuberances, or teeth, on the face of which the cutting edges
may be formed. To produce a cutting edge it is necessary that the metal
behind that edge should slope or slant away leaving the cutting edge to
project. Two methods of accomplishing this are employed: in the first,
which is that embodied in the Brown and Sharpe system, each tooth has
the curved outline, forming what may be termed its circumferential
outline, of the same curvature and shape from end to end, and from front
to back, as it may more properly be termed, the clearance being given by
the back of the tooth approaching the centre of the cutter, so that if a
line be traced along the circumference of a tooth, from the cutting edge
to the back, it will approach the centre of the cutter as the back is
approached, but the form of the tooth will be the same at every point in
the line. It follows then that the radial faces of the teeth may be
ground away to sharpen the teeth without affecting the shape of the
tooth, which being made correct will remain correct.

This not only saves a great deal of labor in sharpening the teeth, but
also saves the softening and rehardening process, otherwise necessary at
each resharpening.

The ordinary method of producing the cutting edges after turning the
cutter and serrating it, is to cut away the metal with a file or rotary
cutter of some kind forming the cutting edge to correct shape, but
paying no regard to the shape of the back of the tooth more than to give
it the necessary amount of clearance. In this case the cutter must be
softened and reset to sharpen it. To bring the cutting edge up to a
sharp edge all around its profile, while still preserving the shape to
which it was turned, the pantagraphic engine, shown in Fig. 108, has
been made by the Pratt and Whitney Company. Figs. 109 and 110 show some
details of its construction.[5] "The milling cutter N is driven by a
flexible train acting upon the wheel O, whose spindle is carried by the
bracket B, which can slide from right to left upon the piece B, and this
again is free to slide in the frame F. These two motions are in
horizontal planes, and perpendicular to each other.

  [5] From "The Teeth of Spur Wheels," by Professor McCord.

[Illustration: Fig. 108.]

"The upper end of the long lever P C is formed into a ball, working in a
socket which is fixed to P C. Over the cylindrical upper part of this
lever slides an accurately fitted sleeve D, partly spherical externally,
and working in a socket which can be clamped at any height on the frame
F. The lower end P of this lever being accurately turned, corresponds to
the roller P in Fig. 109, and is moved along the edge of the template T,
which is fastened in the frame in an invariable position.

"By clamping D at various heights, the ratio of the lever arms P D, P D,
may be varied at will, and the axis of N made to travel in a path
similar to that of the axis of P, but as many times smaller as we
choose; and the diameter of N must be made less than that of P in the
same proportion.

"The template being on the left of the roller, the cutter to be shaped
is placed on the right of N, as shown in the plan view at Z, because the
lever reverses the movement.

"This arrangement is not mathematically perfect, by reason of the
angular vibration of the lever. This is, however, very small, owing to
the length of the lever; it might have been compensated for by the
introduction of another universal joint, which would practically have
introduced an error greater than the one to be obviated, and it has,
with good judgment, been omitted.

"The gear-cutter is turned nearly to the required form, the notches are
cut in it, and the duty of the pantagraphic engine is merely to give the
finishing touch to each cutting edge, and give it the correct outline.
It is obvious that this machine is in no way connected with, or
dependent upon, the epicycloidal engine; but by the use of proper
templates it will make cutters for any desired form of tooth; and by its
aid exact duplicates may be made in any numbers with the greatest
facility.

[Illustration: Fig. 109.]

"It forms no part of our plan to represent as perfect that which is not
so, and there are one or two facts, which at first thought might seem
serious objections to the adoption of the epicycloidal system. These
are:

"1. It is physically impossible to mill out a _concave_ cycloid, by any
means whatever, because at the pitch line its radius of curvature is
zero, and a milling cutter must have a sensible diameter.

"2. It is impossible to mill out even a _convex_ cycloid or epicycloid,
by the means and in the manner above described.

[Illustration: Fig. 110.]

"This is on account of a hitherto unnoticed peculiarity of the curve at
a constant normal distance from the cycloid. In order to show this
clearly, we have, in Fig. 110, enormously exaggerated the radius C D, of
the milling cutter (M of Figs. 105 and 106). The outer curve H L,
evidently, could be milled out by the cutter, whose centre travels in
the cycloid C A; it resembles the cycloid somewhat in form, and presents
no remarkable features. But the inner one is quite different; it starts
at D, and at first goes down, _inside the circle whose radius is_ C D,
forms a cusp at E, then begins to rise, crossing this circle at G, and
the base line at F. It will be seen, then, that if the centre of the
cutter travel in the cycloid A C, its edge will cut away the part G E D,
leaving the template of the form O G I. Now if a roller of the same
radius C D, be rolled along this edge, its centre will travel in the
cycloid from A, to the point P, where a normal from G, cuts it; then the
roller will turn upon G as a fulcrum, and its centre will travel from P
to C, in a circular arc whose radius G P = C D.

"That is to say even a roller of the same size as the original milling
cutter, will not retrace completely the cycloidal path in which the
cutter travelled.

"Now in making a rack template, the cutter, after reaching C, travels in
the reversed cycloid C R, its left-hand edge, therefore, milling out a
curve D K, similar to H L. This curve lies wholly _outside_ the circle D
I, and therefore cuts O G at a point between F and G, but very near to
G. This point of intersection is marked S in Fig. 110, where the actual
form of the template O S K is shown. The roller which is run along this
template is _larger_, as has been explained, than the milling cutter.
When the point of contact reaches S (which so nearly corresponds to G
that they practically coincide), this roller cannot now swing about S
through an angle so great as P G C of Fig. 110; because at the root D,
the radius of curvature of D K is only equal to that of the cutter, and
G and S are so near the root that the curvature of S K, near the latter
point, is greater than that of the roller. Consequently there must be
some point U in the path of the centre of the roller, such, that when
the centre reaches it, the circumference will pass through S, and be
also tangent to S K. Let T be the point of tangency; draw S U and T U,
cutting the cycloidal path A R in X and Y. Then, U Y being the radius of
the new milling cutter (corresponding to N of Fig. 109), it is clear
that in the outline of the gear cutter shaped by it, the circular arc X
Y will be substituted for the true cycloid.

[Illustration: Fig. 111.]


THE SYSTEM PRACTICALLY PERFECT.

"The above defects undeniably exist; now, what do they amount to? The
diagram is drawn purposely with these sources of error greatly
exaggerated, in order to make their nature apparent and their existence
sensible. The diameters used in practice, as previously stated, are:
describing circle, 7-1/2 inches; cutter for shaping template, 1/8 of an
inch; roller used against edge of template, 1-1/8 inches; cutter for
shaping a 1-pitch gear cutter, 1 inch.

"With these data the writer has found that the _total length_ of the arc
X Y of Fig. 110, which appears instead of the cycloid in the outline of
a cutter for a 1-pitch rack, is less than 0.0175 inch; the real
_deviation_ from the true form, obviously, must be much less than that.
It need hardly be stated that the effect upon the velocity ratio of an
error so minute, and in that part of the contour, is so extremely small
as to defy detection. And the best proof of the practical perfection of
this system of making epicycloidal teeth is found in the smoothness and
precision with which the wheels run; a set of them is shown in gear in
Fig. 111, the rack gearing as accurately with the largest as with the
smallest. To which is to be added, finally, that objection taken, on
whatever grounds, to the epicycloidal form of tooth, has no bearing upon
the method above described of producing duplicate cutters for teeth of
any form, which the pantagraphic engine will make with the same facility
and exactness, if furnished with the proper templates.

"The front faces of the teeth of rotary cutters for gear-cutting are
usually radial lines, and are ground square across so as to stand
parallel with the axis of the cutter driving spindle, so that to
whatever depth the cutter may have entered the wheel, the whole of the
cutting edge within the wheel will meet the cut simultaneously. If this
is not the case the pressure of the cut will spring the cutter, and also
the arbor driving it, to one side. Suppose, for example, that the tooth
faces not being square across, one side of the teeth meets the work
first, then there will be as each tooth meets its cut an endeavour to
crowd away from the cut until such time as the other side of the tooth
also takes its cut."

It is obvious that rotating cutters of this class cannot be used to cut
teeth having the width of the space wider below than it is at the pitch
line. Hence, if such cutters are required to be used upon epicycloidal
teeth, the curves to be theoretically correct must be such as are due to
a generating circle that will give at least parallel flanks. From this
it becomes apparent that involute teeth being always thicker at the root
than at the pitch line, and the spaces being, therefore, narrower at the
root, may be cut with these cutters, no matter what the diameter of the
base circle of the involute.

To produce with revolving cutters teeth of absolutely correct
theoretical curvature of face and flank, it is essential that the cutter
teeth be made of the exact curvature due to the diameter of pitch circle
and generating circle of the wheel to be cut; while to produce a tooth
thickness and space width, also theoretically correct, the thickness of
the cutter must also be made to exactly answer the requirements of the
particular wheel to be cut; hence, for every different number of teeth
in wheels of an equal pitch a separate cutter is necessary if
theoretical correctness is to be attained.

This requirement of curvature is necessary because it has been shown
that the curvatures of the epicycloid and hypocycloid, as also of the
involute, vary with every different diameter of base circle, even
though, in the case of epicycloidal teeth, the diameter of the
generating circle remain the same. The requirement of thickness is
necessary because the difference between the arc and the chord pitch is
greater in proportion as the diameter of the base or pitch circle is
decreased.

But the difference in the curvature on the short portions of the curves
used for the teeth of fine pitches (and therefore of but little height)
due to a slight variation in the diameter of the base circle is so
minute, that it is found in practice that no sensible error is produced
if a cutter be used within certain limits upon wheels having a different
number of teeth than that for which the cutter is theoretically correct.

The range of these limits, however, must (to avoid sensible error) be
more confined as the diameter of the base circle (or what is the same
thing, the number of the teeth in the wheel) is decreased, because the
error of curvature referred to increases as the diameters of either the
base or the generating circles decrease. Thus the difference in the
curve struck on a base circle of 20 inches diameter, and one of 40
inches diameter, using the same diameter of generating circle, would be
very much less than that between the curves produced by the same
diameter of generating circle on base circles respectively 10 and 5
inches diameter.

For these reasons the cutters are limited to fewer wheels according as
the number of teeth decreases, or, per contra, are allowed to be used
over a greater range of wheels as the number of teeth in the wheels is
increased.

Thus in the Brown and Sharpe system for involute teeth there are 8
cutters numbered numerically (for convenience in ordering) from 1 to 8,
and in the following table the range of the respective cutters is shown,
and the number of teeth for which the cutter is theoretically correct is
also given.

BROWN AND SHARPE SYSTEM.

  No. of cutter.                Involute teeth.                   Teeth.
  1 Used upon all wheels having from 135 teeth to a rack correct for 200
  2   "      "      "      "      "   55   "   to 134 teeth,          68
  3   "      "      "      "      "   35   "   to 54     "            40
  4   "      "      "      "      "   26   "   to 34     "            29
  5   "      "      "      "      "   21   "   to 25     "            22
  6   "      "      "      "      "   17   "   to 20     "            18
  7   "      "      "      "      "   14   "   to 16     "            16
  8   "      "      "      "      "   12   "   to 14     "            13

Suppose that it was required that of a pair of wheels one make twice the
revolutions of the other; then, knowing the particular number of teeth
for which the cutters are made correct, we may obtain the nearest
theoretically true results as follows: If we select cutters Nos. 8 and 4
and cut wheels having respectively 13 and 26 teeth, the 13 wheel will be
theoretically correct, and the 26 will contain the minute error due to
the fact that the cutter is used upon a wheel having three less teeth
than the number it is theoretically correct for. But we may select the
cutters that are correct for 16 and 29 teeth respectively, the 16th
tooth being theoretically correct, and the 29th cutter (or cutter No. 4
in the table) being used to cut 32 teeth, this wheel will contain the
error due to cutting 3 more teeth than the cutter was made correct for.
This will be nearer correct, because the error is in a larger wheel,
and, therefore, less in actual amount. The pitch of teeth may be
selected so that with the given number of teeth the diameters of the
wheels will be that required.

We may now examine the effect of the variation of curvature in
combination with that of the thickness, upon a wheel having less and
upon one having more teeth than the number in the wheel for which the
cutter is correct.

First, then, suppose a cutter to be used upon a wheel having less teeth
and it will cut the spaces too wide, because of the variation of
thickness, and the curves too straight or insufficiently curved because
of the error of curvature. Upon a wheel having more teeth it will cut
the spaces too narrow, and the curvature of the teeth too great; but, as
before stated, the number of wheels assigned to each cutter may be so
apportioned that the error will be confined to practically unappreciable
limits.

If, however, the teeth are epicycloidal, it is apparent that the spaces
of one wheel must be wide enough to admit the teeth of the other to a
depth sufficient to permit the pitch lines to coincide on the line of
centres; hence it is necessary in small diameters, in which there is a
sensible difference between the arc and the chord pitches, to confine
the use of a cutter to the special wheel for which it is designed, that
is, having the same number of teeth as the cutter is designed for.

Thus the Pratt and Whitney arrangement of cutters for epicycloidal teeth
is as follows:--

PRATT AND WHITNEY SYSTEM.

EPICYCLOIDAL TEETH.

[All wheels having from 12 to 21 teeth have a special cutter for each
number of teeth.][6]

  Cutter correct for
  No. of teeth.
  23   Used on wheels having from  22 to 24  teeth.
  25      "      "      "      "   25 to 26    "
  27      "      "      "      "   26 to 29    "
  30      "      "      "      "   29 to 32    "
  34      "      "      "      "   32 to 36    "
  38      "      "      "      "   36 to 40    "
  43      "      "      "      "   40 to 46    "
  50      "      "      "      "   46 to 55    "
  60      "      "      "      "   55 to 67    "
  76      "      "      "      "   67 to 87    "
  100     "      "      "      "   87 to 123   "
  150     "      "      "      "  123 to 200   "
  300     "      "      "      "  200 to 600   "
  Rack    "      "      "      "  600 to rack.

  [6] For wheels having less than 12 teeth the Pratt and Whitney Co. use
  involute cutters.

Here it will be observed that by a judicious selection of pitch and
cutters, almost theoretically perfect results may be obtained for
almost any conditions, while at the same time the cutters are so
numerous that there is no necessity for making any selection with a view
to taking into consideration for what particular number of teeth the
cutter is made correct.

For epicycloidal cutters made on the Brown and Sharpe system so as to
enable the grinding of the face of the tooth to sharpen it, the Brown
and Sharpe company make a separate cutter for wheels from 12 to 20
teeth, as is shown in the accompanying table, in which the cutters are
for convenience of designation denoted by an alphabetical letter.

24 CUTTERS IN EACH SET.

  Letter A  cuts                12 teeth.
         B   "                  13   "
         C   "                  14   "
         D   "                  15   "
         E   "                  16   "
         F   "                  17   "
         G   "                  18   "
         H   "                  19   "
         I   "                  20   "
         J   "       21  to     22   "
         K   "       23   "     24   "
         L   "       25   "     26   "
         M   "       27   "     29   "
         N   "       30   "     33   "
         O   "       34   "     37   "
         P   "       38   "     42   "
         Q   "       43   "     49   "
         R   "       50   "     59   "
         S   "       60   "     74   "
         T   "       75   "     99   "
         U   "      100   "    149   "
         V   "      150   "    249   "
         W   "      250   "  Rack.
         X   "    Rack.

In these cutters a shoulder having no clearance is placed on each side
of the cutter, so that when the cutter has entered the wheel until the
shoulder meets the circumference of the wheel, the tooth is of the
correct depth to make the pitch circles coincide.

In both the Brown and Sharpe and Pratt and Whitney systems, no side
clearance is given other than that quite sufficient to prevent the teeth
of one wheel from jambing into the spaces of the other. Pratt and
Whitney allow 1/8 of the pitch for top and bottom clearance, while Brown
and Sharpe allow 1/10 of the thickness of the tooth for top and bottom
clearance.

It may be explained now, why the thickness of the cutter if employed
upon a wheel having more teeth than the cutter is correct for,
interferes with theoretical exactitude.

[Illustration: Fig. 112.]

[Illustration: Fig. 113.]

First, then, with regard to the thickness of tooth and width of space.
Suppose, then, Fig. 112 to represent a section of a wheel having 12
teeth, then the pitch circle of the cutter will be represented by line
A, and there will be the same difference between the arc and chord pitch
on the cutter as there is on the wheel; but suppose that this same
cutter be used on a wheel having 24 teeth, as in Fig. 113, then the
pitch circle on the cutter will be more curved than that on the wheel as
denoted at C, and there will be more difference between the arc and
chord pitches on the cutter than there is on the wheel, and as a result
the cutter will cut a groove too narrow.

The amount of error thus induced diminishes as the diameter of the pitch
circle of the cutter is increased.

But to illustrate the amount. Suppose that a cutter is made to be
theoretically correct in thickness at the pitch line for a wheel to
contain 12 teeth, and having a pitch circle diameter of 8 inches, then
we have

                          3.1416 = ratio of circumference to diameter.
                               8 = diameter.
                         -------
  Number of teeth = 12 ) 25.1328 = circumference.
                         -------
                          2.0944 = arc pitch of wheel.

If now we subtract the chord pitch from the arc pitch, we shall obtain
the difference between the arc and the chord pitches of the wheel; here

  2.0944 = arc pitch.
  2.0706 = chord pitch.
  ------
   .0238 = difference between the arc and the chord pitch.

Now suppose this cutter to be used upon a wheel having the same pitch,
but containing 18 teeth; then we have

  2.0944 = arc pitch.
  2.0836 = chord pitch.
  ------
   .0108 = difference between the arc and the chord pitch.

Then

  .0238 = difference on wheel with 12 teeth.
  .0108 =       "      "       "   18   "
  -----
  .0130 = variation between the differences.

And the thickness of the tooth equalling the width of the space, it
becomes obvious that the thickness of the cutter at the pitch line being
correct for the 12 teeth, is one half of .013 of an inch too thin for
the 18 teeth, making the spaces too narrow and the teeth too thick by
that amount.

Now let us suppose that a cutter is made correct for a wheel having 96
teeth of 2.0944 arc pitch, and that it be used upon a wheel having 144
teeth. The proportion of the wheels one to the other remains as before
(for 96 bears the proportion to 144 as 12 does to 18).

Then we have for the 96 teeth

  2.0944 = arc pitch.
  2.0934 = chord pitch.
  ------
   .0010 = difference.

For the 144 teeth we have

  2.0944 = arc pitch.
  2.0937 = chord pitch.
  ------
   .0007 = difference.

We find, then, that the variation decreases as the size of the wheels
increases, and is so small as to be of no practical consequence.

If our examples were to be put into practice, and it were actually
required to make one cutter serve for wheels having, say, from 12 to 18
teeth, a greater degree of correctness would be obtained if the cutter
were made to some other wheel than the smallest. But it should be made
for a wheel having less than the mean diameter (within the range of 12
and 18), that is, having less than 15 teeth; because the difference
between the arc and chord pitch increases as the diameter of the pitch
circle increases, as already shown.

A rule for calculating the number of wheels to be cut by each cutter
when the number of cutters in the set and the number of teeth in the
smallest and largest wheel in the train are given is as follows:--

Rule.--Multiply the number of teeth in the smallest wheel of the train
by the number of cutters it is proposed to have in the set, and divide
the amount so obtained by a sum obtained as follows:--

From the number of cutters in the set subtract the number of the cutter,
and to the remainder add the sum obtained by multiplying the number of
the teeth in the smallest wheel of the set or train by the number of the
cutter and dividing the product by the number of teeth in the largest
wheel of the set or train.

Example.--I require to find how many wheels each cutter should cut,
there being 8 cutters and the smallest wheel having 12 teeth, while the
largest has 300.

  Number of teeth in      Number of cutters
    smallest wheel.          in the set.
          12           ×         8            = 96

Then

  Number of cutters         Number of
       in set.               cutter.
          8           -         7             = 1

Then

  Number of teeth in    The number of the    The number of the teeth
   smallest wheel.           cutter.             in largest wheel.
       12             ×        8           ÷           300

                              12
                               8
                              ---
                        300 ) 960 ( 0.32
                              900
                              ---
                               600
                               600

Now add the 1 to the .32 and we have 1.32, which we must divide into the
96 first obtained.

Thus

  1.32 ) 96.00 ( 72
         924
         ----
          360
          264
          ---
           96

Hence No. 8 cutter may be used for all wheels that have between 72 teeth
and 300 teeth.

To find the range of wheels to be cut by the next cutter, which we will
call No. 7, proceed again as before, but using 7 instead of 8 as the
number of the cutter.

Thus

  Number of teeth in    Number of cutters in
   smallest wheel.           the set.
         12           ×         8             =          96

Then

  Number of cutters        Number of
    in the set.             cutters.
         8            -        6           =           2

And

  Number of teeth in    The number of the     The number of teeth
   smallest wheel.           cutter.         in the largest wheel.
         12           ×         8          ÷         300

Here

         12
          8
        ---
  300 ) 960 ( 0.32
        900
        ---
         600
         600

Add the 2 to the .32 and we have 2.32 to divide into the 96.

Thus

  2.32 ) 96.00 ( 41
         928
         ---
          320
          232
          ---
           88

Hence this cutter will cut all wheels having not less than the 41 teeth,
and up to the 72 teeth where the other cutter begins. For the range of
the next cutter proceed the same, using 6 as the number of the cutter,
and so on.

By this rule we obtain the lowest number of teeth in a wheel for which
the cutter should be used, and it follows that its range will continue
upwards to the smallest wheel cut by the cutter above it.

Having by this means found the range of wheels for each cutter, it
remains to find for what particular number of teeth within that range
the cutter teeth should be made correct, in order to have whatever error
there may be equal in amount on the largest and smallest wheel of its
range. This is done by using precisely the same rule, but supposing
there to be twice as many cutters as there actually are, and then taking
the intermediate numbers as those to be used.

Applying this plan to the first of the two previous examples we have--

  Number of teeth in the    Number of cutters in
    smallest wheel.              the set.
           12            ×         16            =         192

Then

  Number of cutters      Number of the
     in the set.            cutter.
         16          -        15           =         1

And

  Number of teeth in    The number of the    The number of the teeth in
   smallest wheel.           cutter.           the largest wheel.
         12          ×         15           ÷          300

                               12
                               15
                              ---
                               60
                              12
                              -----
                        300 ) 180.0 ( 0.6
                               1800

Then add the 1 to the .6 = 1.6, and this divided into 192 = 120.

By continuing this process for each of the 16 cutters we obtain the
following table:--

  Number of     Number of
  Cutter.        Teeth.
     1             12
    *2             13
     3             14
    *4             15
     5             17
    *6             18
     7             20.61
    *8             23
     9             26
   *10             30
    11             35
   *12             42
    13             54
   *14             75
    15            120
   *16            300

Suppose now we take for our 8 cutters those marked by an asterisk, and
use cutter 2 for all wheels having either 12, 13, or 14 teeth, then the
next cutter would be that numbered 4, cutting 14, 15, or 16 toothed
wheels, and so on.

A similar table in which 8 cutters are required, but 16 are used in the
calculation, the largest wheel having 200 teeth in the set, is given
below.

  Number of     Number of
  Cutter.        Teeth.
     1            12.7
     2            13.5
     3            14.5
     4            15.6
     5            16.9
     6            18
     7            21
     8            23.5
     9            26.5
    10            29
    11            35
    12            40.6
    13            52.9
    14            67.6
    15           101
    16           200

To assist in the selections as to what wheels in a given set the
determined number of cutters should be made correct for, so as to obtain
the least limit of error, Professor Willis has calculated the following
table, by means of which cutters may be selected that will give the same
difference of form between any two consecutive numbers, and this table
he terms the table of equidistant value of cutters.

TABLE OF EQUIDISTANT VALUE OF CUTTERS.

Number of Teeth.

  Rack--300, 150, 100, 76, 60, 50, 43, 38, 34, 30, 27, 25, 23, 21, 20, 19,
        17, 16, 15, 14, 13, 12.

The method of using the table is as follows:--Suppose it is required to
make a set of wheels, the smallest of which is to contain 50 teeth and
the largest 150, and it is determined to use but one cutter, then that
cutter should be made correct for a wheel containing 76; because in the
table 76 is midway between 50 and 150.

But suppose it were determined to employ two cutters, then one of them
should be made correct for a wheel having 60 teeth, and used on all the
wheels having between 50 and 76 teeth, while the other should be made
correct for a wheel containing 100 teeth, and used on all wheels
containing between 76 and 150 teeth.

In the following table, also arranged by Professor Willis, the most
desirable selection of cutters for different circumstances is given, it
being supposed that the set of wheels contains from 12 teeth to a rack.

  +-----------+------------------------------------------------------+
  |Number of  |                                                      |
  |cutters in | Number of Teeth in Wheel for which the Cutter is to  |
  |the set.   | be made correct.                                     |
  +-----------+----+----+--------------------------------------------+
  |     2     | 50 | 16 |                                            |
  |         --+----+----+----+                                       |
  |     3     | 75 | 25 | 15 |                                       |
  |         --+----+----+----+----+                                  |
  |     4     | 100| 34 | 20 | 14 |                                  |
  |         --+----+----+----+----+----+----+                        |
  |     6     | 150| 50 | 30 | 21 | 16 | 13 |                        |
  |         --+----+----+----+----+----+----+----+----+              |
  |     8     | 200| 67 | 40 | 29 | 22 | 18 | 15 | 13 |              |
  |         --+----+----+----+----+----+----+----+----+----+----+    |
  |    10     | 200| 77 | 50 | 35 | 27 | 22 | 19 | 16 | 14 | 13 |    |
  |         --+----+----+----+----+----+----+----+----+----+----+----+
  |           | 300| 100| 60 | 43 | 34 |27  | 23 | 20 | 17 | 15 | 14 |
  |    12     +----+----+----+----+----+----+----+----+----+----+----+
  |           | 13 |                                                 |
  |         --+----+----+----+----+----+----+----+----+----+----+----+
  |           | 300| 150| 100| 70 | 50 | 40 | 30 | 26 | 24 | 22 | 20 |
  |    18     +----+----+----+----+----+----+----+----+----+----+----+
  |           | 18 | 16 | 15 | 14 | 13 | 12 |                        |
  |         --+----+----+----+----+----+----+----+----+----+----+----+
  |           |Rack| 300| 150| 100| 76 | 60 | 50 | 43 | 38 | 34 | 30 |
  |           +----+----+----+----+----+----+----+----+----+----+----+
  |    24     | 27 | 25 | 23 | 21 | 20 | 19 | 18 | 17 | 16 | 15 | 14 |
  |           +----+----+----+----+----+----+----+----+----+----+----+
  |           | 13 | 12 |                                            |
  +-----------+----+----+--------------------------------------------+

Suppose now we take the cutters, of a given pitch, necessary to cut all
the wheels from 12 teeth to a rack, then the thickness of the teeth at
the pitch line will for the purposes of designation be the thickness of
the teeth of all the wheels, which thickness may be a certain proportion
of the pitch.

But in involute teeth while the depth of tooth on the cutter may be
taken as the standard for all the wheels in the range, and the actual
depth for the wheel for which the cutter is correct, yet the depth of
the teeth in the other wheels in the range may be varied sufficiently on
each wheel to make the thickness of the teeth equal the width of the
spaces (notwithstanding the variation between the arc and chord
pitches), so that by a variation in the tooth depth the error induced by
that variation may be corrected. The following table gives the
proportions in the Brown and Sharpe system.

  +------------+-----------------+-----------------+
  | Arc Pitch. | Depth of Tooth. | Depth in terms  |
  |            |                 |of the arc pitch.|
  +------------+-----------------+-----------------+
  |  inches.   |      inches.    |     inches.     |
  |   1.570    |       1.078     |      .686       |
  |   1.394    |        .958     |      .687       |
  |   1.256    |        .863     |      .686       |
  |   1.140    |        .784     |      .697       |
  |   1.046    |        .719     |      .687       |
  |    .896    |        .616     |      .686       |
  |    .786    |        .539     |      .685       |
  |    .628    |        .431     |      .686       |
  |    .524    |        .359     |      .685       |
  |    .448    |        .307     |      .685       |
  |    .392    |        .270     |      .686       |
  |    .350    |        .240     |      .686       |
  |    .314    |        .216     |      .687       |
  +------------+-----------------+-----------------+

To avoid the trouble of measuring, and to assist in obtaining accuracy
of depth, a gauge is employed to mark on the wheel face a line denoting
the depth to which the cutter should be entered.

Suppose now that it be required to make a set of cutters for a certain
range of wheels, and it be determined that the cutters be so constructed
that the greatest permissible amount of error in any wheel of the set be
1/100 inch. Then the curves for the smallest wheel, and those for the
largest in the set, and the amount of difference between them
ascertained, and assuming this difference to amount to 1/16 inch, which
is about 6/100, then it is evident that 6 cutters must be employed for
the set.

It has been shown that on bevel-wheels the tooth curves vary at every
point in the tooth breadth; hence it is obvious that the cutter being of
a fixed curve will make the tooth to that curve. Again, the thickness of
the teeth and breadth of the spaces vary at every point in the breadth,
while with a cutter of fixed thickness the space cut will be parallel
from end to end. To overcome these difficulties it is usual to give to
the cutter a curve corresponding to the curve required at the middle of
the wheel face and a thickness equal to the required width of space at
its smallest end, which is at the smallest face diameter.

The cutter thus formed produces, when passed through the wheel once, and
to the required depth, a tooth of one curve from end to end, having its
thickness and width of space correct at the smaller face diameter only,
the teeth being too thick and the spaces too narrow as the outer
diameter of the wheel is approached. But the position and line of
traverse of the cutter may be altered so as to take a second cut,
widening the space and reducing the tooth thickness at the outer
diameter.

By moving the cutter's position two or three times the points of contact
between the teeth may be made to occur at two or three points across the
breadth of the teeth and their points of contact; the wear will soon
spread out so that the teeth bear all the way across.

Another plan is to employ two or three cutters, one having the correct
curve for the inner diameter, and of the correct thickness for that
diameter, another having the correct curve for the pitch circle, and
another having the correct curve at the largest diameter of the teeth.

The thickness of the first and second cutters must not exceed the
required width of space at the small end, while that for the third may
be the same as the others, or equal to the thickness of the smallest
space breadth that it will encounter in its traverse along the teeth.

The second cutter must be so set that it will leave the inner end of the
teeth intact, but cut the space to the required width in the middle of
the wheel face. The third cutter must be so set as to leave the middle
of the tooth breadth intact, and cut the teeth to the required thickness
at the outer or largest diameter.


CUTTING WORM-WHEELS.

The most correct method of cutting the teeth of a worm-wheel is by means
of a worm-cutter, which is a worm of the pitch and form of tooth that
the working worm is intended to be, but of hardened steel, and having
grooves cut lengthways of the worm so as to provide cutting edges
similar to those on the cutter shown in Fig. 107.

The wheel is mounted on an arbor or mandril free to rotate on its axis
and at a right angle to the cutter worm, which is rotated and brought to
bear upon the perimeter of the worm-wheel in the same manner as the
working worm-wheel when in action. The worm-cutter will thus cut out the
spaces in the wheel, and must therefore be of a thickness equal to those
spaces. The cutter worm acting as a screw causes the worm-wheel to
rotate upon its axis, and therefore to feed to the cutter.

In wheels of fine pitch and small diameter this mode of procedure is a
simple matter, especially if the form of tooth be such that it is
thicker, as the root of the tooth is approached from the pitch line,
because in that case the cutter worm may be entered a part of the depth
in the worm-wheel and a cut be taken around the wheel. The cutter may
then be moved farther into the wheel and a second cut taken around the
wheel, so that by continuing the process until the pitch line of the
cutter worm coincides with that of the worm-cutter, the worm-wheel may
be cut with a number of light cuts, instead of at one heavy cut.

But in the case of large wheels the strain due to such a long line of
cutting edge as is possessed by the cutter worm-teeth springs or bends
the worm-wheel, and on account of the circular form of the breadth of
the teeth this bending or spring causes that part of the tooth arc above
the centre of the wheel thickness to lock against the cutter.

To prevent this, several means may be employed. Thus the grooves forming
the cutting edges of the worm-cutter may wind spirally along instead of
being parallel to the axis of the cutter.

The distance apart of these grooves may be greater than the breadth of
tooth a width of worm-wheel face, in which case the cutting edge of one
tooth only will meet the work at one time. In addition to this two
stationary supports may be placed beneath the worm-wheel (one on each
side of the cutter). But on coarse pitches with their corresponding
depth of tooth, the difficulty presents itself, that the arbor driving
the worm-cutter will spring, causing the cutter to lift and lock as
before; hence it is necessary to operate on part of the space at a time,
and shape it out to so nearly the correct form that the finishing cut
may be a very light one indeed, in which case the worm-cutter will
answer for the final cut.

The removal of the surplus metal preparatory to the introduction of the
worm-cutter to finish, may be made with a cutter-worm that will cut out
a narrow groove being of the thickness equal to the bottom of the tooth
space and cutting on its circumference only. This cutter may be fed into
the wheel to the permissible depth of cut, and after the cut is taken
all around the wheel, it may be entered deeper and a second cut taken,
and so on until it has entered the wheel to the necessary depth of
tooth. A second cutter-worm may then be used, it being so shaped as to
cut the face curve only of the teeth. A third may cut the flank curve
only, and finally a worm-cutter of correct form may take a finishing cut
over both the faces and the flanks. In this manner teeth of any pitch
and depth may be cut. Another method is to use a revolving cutter such
as shown in Fig. 107, and to set it at the required angle to the wheel,
and then take a succession of cuts around the wheel, the first cut
forming a certain part of the tooth depth, the second increasing this
depth, and so on until the final cut forms the tooth to the requisite
depth. In this case the cutter operates on each space separately, or on
one space only at a time, and the angle at which to set the cutter may
be obtained as follows in Fig. 114. Let the length of the line A A equal
the diameter of the worm at the pitch circle, and B B (a line at a right
angle to A A) represent the axial line of the worm. Let the distance C
equal the pitch of the teeth, and the angle of the line D with A A or B
B according to circumstances, will be that to which the cutter must be
set with reference to the tooth.

[Illustration: Fig. 114.]

If then a piece of sheet metal be cut to the lines A, D, and the cutter
so set that with the edge D of the piece held against the side face of
the cutter (which must be flat or straight across), the edge A will
stand truly vertical, and the cutter will be at the correct angle
supposing the wheel to be horizontal.

[Illustration: Fig. 115.]

[Illustration: Fig. 116.]

In making patterns wherefrom gear-wheels may be cast in a mould, the
true curves are frequently represented by arcs of circles struck from
the requisite centres and of the most desirable radius with compasses,
and this will be treated after explaining the pattern maker's method of
obtaining true curves by rolling segments by hand. If, then, the wheels
are of small diameter, as say, less than 12 inches in diameter, and
precision is required, it is best to turn in the lathe wooden disks
representing in their diameters the base and generating circles. But
otherwise, wooden segments to answer the same purpose may be made as
from a piece of soft wood, such as pine or cedar, about three-eighths
inch thick, make two pieces A and B, in Fig. 115, and trim the edges C
and D to the circle of the pitch line of the required wheel. If the
diameter of the pitch circle is marked on a drawing, the pieces may be
laid on the drawing and sighted for curvature by the eye. In the absence
of a drawing, strike a portion of the pitch circle with a pair of
sharp-pointed compasses on a piece of zinc, which will show a very fine
line quite clear. After the pieces are filed to the circle, try them
together by laying them flat on a piece of board, bringing the curves in
contact and sweeping A against B, and the places of contact will plainly
show, and may be filed until continuous contact along the curves is
obtained. Take another similar piece of wood and form it as shown in
Fig. 116, the edge E representing a portion of the rolling circle. In
preparing these segments it is an excellent plan to file the convex
edges, as shown in Fig. 117, in which P is a piece of iron or wood
having its surface S trued; F is a file held firmly to S, while its
surface stands vertical, and T is the template laid flat on S, while
swept against the file. This insures that the edge shall be square
across or at least at the same angle all around, which is all that is
absolutely necessary. It is better, however, that the edges be square.
So likewise in fitting A and B (Fig. 115) together, they should be laid
flat on a piece of board. This will insure that they will have contact
clear across the edge, which will give more grip and make slip less
likely when using the segments. Now take a piece of stiff drawing paper
or of sheet zinc, lay segment A upon it, and mark a line coincident with
the curved edge. Place the segment representing the generating circle
flat on the paper or zinc, hold its edge against segment A, and roll it
around a sufficient distance to give as much of the curve as may be
required; the operation being illustrated in Fig. 118, in which A is the
segment representing the pitch or base circle, E is the segment
representing the generating circle, P is the paper, C the curve struck
by the tracing point or pencil O.

[Illustration: Fig. 117.]

[Illustration: Fig. 118.]

This tracing point should be, if paper be used to trace on, a piece of
the _hardest_ pencil obtainable, and should be filed so that its edge,
if flat, shall stand as near as may be in the line of motion when
rolled, thus marking a fine line. If sheet zinc be used instead of paper
a needle makes an excellent tracing point. Several of the curves, C,
should be struck, moving the position of the generating segment a little
each time.

[Illustration: Fig. 119.]

On removing the segments from the paper, there will appear the lines
shown in Fig. 119; A representing the pitch circle, and O O O the curves
struck by the tracing point.

Cut out a piece of sheet zinc so that its edge will coincide with the
curve A and the epicycloid O, trying it with all four of the epicycloids
to see that no slip has occurred when marking them; shape a template as
shown in Fig. 120. Cutting the notches at _a_ _b_, acts to let the file
clear well when filing the template, and to allow the scriber to go
clear into the corner. Now take the segment A in Fig. 118, and use it as
a guide to carry the pitch circle across the template as at P, in Fig.
120. A zinc template for the flank curve is made after the same manner,
using the rolling segment in conjunction with the segment B in Fig. 115.

[Illustration: Fig. 120.]

But the form of template for the flank should be such as shown in Fig.
121, the curve P representing, and being of the same radius as the pitch
circle, and the curve F being that of the hypocycloid. Both these
templates are set to the pitch circles and to coincide with the marks
made on the wheel teeth to denote the thickness, and with a hardened
steel point a line is traced on the tooth showing the correct curve for
the same.

[Illustration: Fig. 121.]

An experienced hand will find no difficulty in producing true templates
by this method, but to avoid all possibility of the segments slipping on
coarse pitches, and with large segments, the segments may be connected,
as shown in Fig. 122, in which O represents a strip of steel fastened at
one end into one segment and at the other end to the other segment.
Sometimes, indeed, where great accuracy is requisite, two pieces of
steel are thus employed, the second one being shown at P P, in the
figure. The surfaces of these pieces should exactly coincide with the
edge of the segments.

[Illustration: Fig. 122.]

[Illustration: Fig. 123.]

[Illustration: Fig. 124.]

[Illustration: Fig. 125.]

[Illustration: Fig. 126.]

The curve templates thus produced being shaped to apply to the pitch
circle may be correctly applied to that circle independently of its
concentricity to the wheel axis or of the points of the teeth, but if
the points of the teeth are turned in the lathe so as to be true (that
is, concentric to the wheel axis) the form of the template may be such
as shown in Fig. 123, the radius of the arc A A equalling that of the
addendum circle or circumference at the points of the teeth, and the
width at B (the pitch circle) equaling the width of a space instead of
the thickness of a tooth. The curves on each side of the template may in
this case be filed for the full side of a tooth on each side of the
template so that it will completely fill the finished space, or the
sides of two contiguous teeth may be marked at one operation. This
template may be set to the marks made on the teeth at the pitch circle
to denote their requisite thickness, or for greater accuracy, a similar
template made double so as to fill two finished tooth spaces may be
employed, the advantage being that in this case the template also serves
to mark or test the thickness of the teeth. Since, however, a double
template is difficult to make, a more simple method is to provide for
the thickness of a tooth, the template shown in Fig. 124, the width from
A to B being either the thickness of tooth required or twice the
thickness of a tooth plus the width of a space, so that it may be
applied to the outsides of two contiguous teeth. The arc C may be made
both in its radius and distance from the pitch circle D D to equal that
of the addendum circle, so as to serve as a gauge for the tooth points,
if the latter are not turned true in the lathe, or to rest on the
addendum circle (if the teeth points are turned true), and adjust the
pitch circle D D to the pitch circle on the wheel.

The curves for the template must be very carefully filed to the lines
produced by the rolling segments, because any error in the template is
copied on every tooth marked from it. Furthermore, instead of drawing
the pitch circle only, the addendum circle and circle for the roots of
the teeth or spaces should also be drawn, so that the template may be
first filed to them, and then adjusted to them while filing the edges to
the curves.

Another form of template much used is shown in Fig. 125. The curves A
and B are filed to the curve produced by rolling segments as before, and
the holes C, D, E, are for fastening the template to an arm, such as
shown in Fig. 126, which represents a section of a wheel W, with a plug
P, fitting tightly into the hub H of the wheel. This plug carries at its
centre a cylindrical pin on which pivots the arm A. The template T is
fastened to the arm by screws, and set so that its pitch circle
coincides with the pitch circle P on the wheel, when the curves for one
side of all the teeth may be marked. The template must then be turned
over to mark the other side of the teeth.

The objection to this form of template is that the length of arc
representing the pitch circle is too short, for it is absolutely
essential that the pitch line on the template (or line representing the
arc of the addendum if that be used) be greater than the width of a
single tooth, because an error of the thickness of a line (in the
thickness of a tooth), in the coincidence of the pitch line of the
template with that of the tooth, would throw the tooth curves out to an
extent altogether inadmissible where true work is essential.

[Illustration: Fig. 127.]

To overcome this objection the template may be made to equal half the
thickness of a tooth and its edge filed to represent a radial line on
the wheel. But there are other objections, as, for example, that the
template can only be applied to the wheel when adjusted on the arm shown
in Fig. 126, unless, indeed, a radial line be struck on every tooth of
the wheel. Again, to produce the template a radial line representing the
radius of the wheel must be produced, which is difficult where segments
only are used to produce the curves. It is better, therefore, to form
the template as shown in Fig. 127, the projections at A B having their
edges filed to coincide with the pitch circle P, so that they may be
applied to a length of one arc of pitch circle at least equal to the
pitch of the teeth.

The templates for the tooth curves being obtained, the wheel must be
divided off on the pitch circle for the thickness of the teeth and the
width of the spaces, and the templates applied to the marks or points of
division to serve as guides to mark the tooth curves. Since, however, as
already stated, the tooth curves are as often struck by arcs of circles
as by templates, the application of such arcs and their suitability may
be discussed.


MARKING THE CURVES BY HAND.

In the employment of arcs of circles several methods of finding the
necessary radius are found in practice.

[Illustration: Fig. 128.]

In the best practice the true curve is marked by the rolling segments
already described, and the compass points are set by trial to that
radius which gives an arc nearest approaching to the true face and flank
curves respectively. The degree of curve error thus induced is
sufficient that the form of tooth produced cannot with propriety be
termed epicycloidal teeth, except in the case of fine pitches in which
the arc of a circle may be employed to so nearly approach the true curve
as to be permissible as a substitute. But in coarse pitches the error is
of much importance. Thus in Fig. 128 is shown the curve of the _former_
or _template_ attachment used on the celebrated Corliss Bevel Gear
Cutting Machine, to cut the teeth on the bevel-wheels employed upon the
line shafting at the Centennial Exhibition. These gears, it may be
remarked, were marvels of smooth and noiseless running, and attracted
wide attention both at home and abroad. The engraving is made from a
drawing marked direct from the _former_ itself, and kindly furnished me
by Mr. George H. Corliss. A A is the face and B B the flank of the
tooth, C C is the arc of a circle nearest approaching to the face curve,
and D D the arc of a circle nearest approaching the flank curve. In the
face curve, there are but two points where the circle coincides with the
true curve, while in the flank there are three such points; a circle of
smaller radius than C C would increase the error at _b_, but decrease it
at _a_; one of a greater radius would decrease it at _b_, and increase
it at _a_. Again, a circle larger in radius than D D would decrease the
error at _e_ and increase it at _f_; while one smaller would increase it
at _e_ and decrease it at _f_. Only the working part of the tooth is
given in the illustration, and it will be noted that the error is
greatest in the flank, although the circle has three points of
coincidence.

[Illustration: Fig. 129.]

In this case the depth of the _former_ tooth is about three and
three-quarter times greater than the depth of tooth cut on the
bevel-wheels; hence, in the figure the actual error is magnified three
and three-quarter times. It demonstrates, however, the impropriety of
calling coarsely pitched teeth that are found by arcs of circles
"epicycloidal" teeth.

When, however, the pitches of the teeth are fine as, say an inch or
less, the coincidence of an arc of a circle with the true curve is
sufficiently near for nearly all practical purposes, and in the case of
cast gear the amount of variation in a pitch of 2 inches would be
practically inappreciable.

To obtain the necessary set of the compasses to mark the curves, the
following methods may be employed.

First by rolling the true curves with segments as already described, and
the setting the compass points (by trial) to that radius which gives an
arc nearest approaching the true curves. In this operation it is not
found that the location for the centre from which the curve must be
struck always falls on the pitch circle, and since that location will
for every tooth curve lie at the same radius from the wheel centre it is
obvious that after the proper location for one of the curves, as for the
first tooth face or tooth flank as the case may be, is found, a circle
may be struck denoting the radius of the location for all the teeth. In
Fig. 129, for example, P P represents the pitch circle, A B the radius
that will produce an arc nearest approaching the true curve produced by
rolling segments, and A the location of the centre from which the face
arc B should be struck. The point A being found by trial with the
compasses applied to the curve B, the circle A C may be struck, and the
location for the centres from which the face arcs of each tooth must be
struck will also fall on this circle, and all that is necessary is to
rest one point of the compasses on the side of the tooth as, say at E,
and mark on the second circle A C the point C, which is the location
wherefrom to mark the face arc D.

If the teeth flanks are not radial, the locations of the centre
wherefrom to strike the flank curves are found in like manner by trial
of the compasses with the true curves, and a third circle, as I in Fig.
130, is struck to intersect the first point found, as at G in the
figure. Thus there will be upon the wheel face three circles, P P the
pitch circle, J J wherefrom to mark the face curves, and I wherefrom to
mark the flank curves.

When this method is pursued a little time may be saved, when dividing
off the wheel, by dividing it into as many divisions as there are teeth
in the wheel, and then find the locations for the curves as in Fig. 131,
in which 1, 2, 3 are points of divisions on the pitch circle P P, while
A, B, struck from point 2, are centres wherefrom to strike the arcs E,
F; C, D, struck also from point 2 are centres wherefrom to strike the
flank curves G, H.

[Illustration: Fig. 130.]

It will be noted that all the points serving as centres for the face
curves, in Fig. 130, fall within a space; hence if the teeth were rudely
cast in the wheel, and were to be subsequently cut or trimmed to the
lines, some provision would have to be made to receive the compass
points.

To obviate the necessity of finding the necessary radius from rolling
segments various forms of construction are sometimes employed.

[Illustration: Fig. 131.]

Thus Rankine gives that shown in Fig. 132, which is obtained as follows.
Draw the generating circle D, and A D the line of centres. From the
point of contact at C, mark on circle D, a point distance from C
one-half the amount of the pitch, as at P, and draw the line P C of
indefinite length beyond C. Draw a line from P, passing through the line
of centres at E, which is equidistant between C and A. Then multiply
the length from P to C by the distance from A to D, and divide by the
distance between D and E. Take the length and radius so found, and mark
it upon P C, as at F, and the latter will be the location of centre for
compasses to strike the face curve.

[Illustration: Fig. 132.]

Another method of finding the face curve, with compasses, is as follows:
In Fig. 133, let P P represent the pitch circle of the wheel to be
marked, and B C the path of the centre of the generating or describing
circle as it rolls outside of P P. Let the point B represent the centre
of the generating circle when that circle is in contact with the pitch
circle at A. Then from B, mark off on B C any number of equidistant
points, as D, E, F, G, H, and from A, mark on the pitch circle, points
of division, as 1, 2, 3, 4, 5, at the intersection of radial lines from
D, E, F, G, and H. With the radius of the generating circle, that is, A
B, from B, as a centre, mark the arc I, from D the arc J, from E the arc
K, &c., to M, marking as many arcs as there are points of division on B
C. With the compasses set to the radius of divisions 1, 2, step off on
arc M the five divisions, N, O, S, T, V, and V will be a point in the
epicycloidal curves. From point of division 4, step off on L four points
of division, as _a_, _b_, _c_, _d_, and _d_ will be another point in the
epicycloidal curve. From point 3 set off three divisions on K, from
point 2 two dimensions on L, and so on, and through the points so
obtained, draw by hand or with a scroll the curve represented in the cut
by curve A V.

[Illustration: Fig. 133.]

Hypocycloids for the flanks of the teeth may be traced in a similar
manner. Thus in Fig. 134 P P is the pitch circle, and B C the line of
motion of the centre of the generating circle to be rolled within P P,
and R a radial line. From 1 to 6 are points of equal division on the
pitch circle, and D to I are arc locations for the centre of the
generating circle. Starting from A, which represents the supposed
location for the centre of the generating circle, the point of contact
between the generating and base circles will be at B. Then from 1 to 6
are points of equal division on the pitch circle, and from D to I are
the corresponding locations for the centres of the generating circle.
From these centres the arcs J, K, L, M, N, O, are struck. From 6 mark
the six points of division from _a_ to _f_, and _f_ is a point in the
curve. Five divisions on N, four on M, and so on, give respectively
points in the curve which is marked in the figure from A to _f_.

There is this, however, to be noted concerning the constructions of the
last two figures. Since the circle described by the centre of the
generating circle is of different arc or curve to that of the pitch
circle, the chord of an arc having an equal length on each will be
different. The amount is so small as to be practically correct. The
direction of the error is to give to the curves a less curvature, as
though they had been produced by a generating circle of larger diameter.
Suppose, for example, that the difference between the arc N 5 (Fig. 133)
and its chord is .1, and that the difference between the arc 4 5, and
its chord is .01, then the error in one step is .09, and, as the point V
is formed in 5 steps, it will contain this error multiplied five times.
Point _d_ would contain it multiplied four times, because it has 4
steps, and so on.

The error will increase in proportion as the diameter of the generating
is less than that of the pitch circle, and though in large wheels,
working with large wheels (so that the difference between the radius of
the generating circle and that of the smallest wheel is not excessive),
it is so small as to be practically inappreciable, yet in small wheels,
working with large ones, it may form a sensible error.

[Illustration: Fig. 134.]

An instrument much employed in the best practice to find the radius
which will strike an arc of a circle approximating the true epicycloidal
curve, _and for finding at the same time_ the location of the centre
wherefrom that curve should be struck, is found in the Willis'
odontograph. This is, in reality, a scale of centres or radii for
different and various diameters of wheels and generating circles. It
consists of a scale, shown in Fig. 135, and is formed of a piece of
sheet metal, one edge of which is marked or graduated in divisions of
one-twentieth of an inch. The edge meeting the graduated edge at O is at
angle of 75° to the graduated edge.

On one side of the odontograph is a table (as shown in the cut), for the
flanks of the teeth, while on the other is the following table for the
faces of the teeth:

TABLE SHOWING THE PLACE OF THE CENTRES UPON THE SCALE.

CENTRES FOR THE FACES OF THE TEETH.

Pitch in Inches and Parts.

  +------+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
  |No. of|1/4|3/8|1/2|5/8|3/4|  1|1- |1- |1- |  2|2- |2- |  3|3- |
  |Teeth |   |   |   |   |   |   |1/4|1/2|3/4|   |1/4|1/2|   |1/2|
  |------+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
  |   12 |  1|  2|  2|  3|  4|  5|  6|  7|  9| 10| 11| 12| 15| 17|
  |   15 | ..| ..|  3| ..| ..| ..|  7|  8| 10| 11| 12| 14| 17| 19|
  |   20 |  2| ..| ..|  4|  5|  6|  8|  9| 11| 12| 14| 15| 18| 21|
  |   30 | ..|  3|  4| ..| ..|  7|  9| 10| 12| 14| 16| 18| 21| 25|
  |   40 | ..| ..| ..| ..|  6|  8| ..| 11| 13| 15| 17| 19| 23| 26|
  |      |   |   |   |   |   |   |   |   |   |   |   |   |   |   |
  |   60 | ..| ..| ..|  5| ..| ..| 10| 12| 14| 16| 18| 20| 25| 29|
  |   80 | ..| ..| ..| ..| ..|  9| 11| 13| 15| 17| 19| 21| 26| 30|
  |  100 | ..| ..| ..| ..|  7| ..| ..| ..| ..| 18| 20| 22| ..| 31|
  |  150 | ..| ..|  5|  6| ..| ..| ..| 14| 16| 19| 21| 23| 27| 32|
  |Rack. | ..|  4| ..| ..| ..| 10| 12| 15| 17| 20| 22| 25| 30| 34|
  +------+---+---+---+---+---+---+---+---+---+---+---+---+---+---+

The method of using the instrument is as follows: In Fig. 136, let C
represent the centre, and P the pitch circle of a wheel to contain 30
teeth of 3 inch arc pitch. Draw the radial line L, meeting the pitch
circle at A. From A mark on the pitch circle, as at B, a radius equal to
the pitch of the teeth, and the thickness of the tooth as A _k_. Draw
from B to C the radial line E. Then for the flanks place the slant edge
of the odontograph coincident and parallel with E, and let its corners
coincide with the pitch circle as shown. In the table headed _centres
for the flanks of the teeth_, look down the column of 3 inch pitch, and
opposite to the 30 in the column of numbers of teeth, will be found the
number 49, which indicates that the centre from which to draw an arc for
the flank is at 49 on the graduated edge of the odontograph, as denoted
in the cut by _r_. Thus from _r_ to the side _k_ of the tooth is the
radius for the compasses, and at _r_, or 49, is the location for the
centre to strike the flank curve _f_. For the face curve set the slant
edge of the odontograph coincident with the radial line L, and in the
table of centres for the faces of teeth, look down the column of 3-inch
pitch, and opposite to 30 in the number of teeth column will be found
the number 21, indicating that at 21 on the graduated edge of the
odontograph, is the location of the centre wherefrom to strike the curve
_d_ for the face of the tooth, this location being denoted in the cut at
R.

[Illustration: Fig. 135.

TABLE SHOWING THE PLACE OF THE CENTRES UPON THE SCALE.

  +--------------------------------------------------------------+
  |            CENTRES FOR THE FLANKS OF THE TEETH.              |
  +--------------------------------------------------------------+
  |                 PITCH IN INCHES AND PARTS.                   |
  +------+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
  |Number|   |   |   |   |   |   |   |   |   |   |   |   |   |   |
  |  of  |   |   |   |   |   | 1 | 1-| 1-| 1-| 2 | 2-| 2-| 3 | 3-|
  |teeth.|1/4|3/8|1/2|5/8|3/4|   |1/4|1/2|3/4|   |1/4|1/2|   |1/2|
  +------+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
  |    13| 32| 48| 64| 80| 96|129|160|193|225|257|289|321|386|450|
  |    14| 17| 26| 35| 43| 52| 69| 87|104|121|139|156|173|208|242|
  |    15| 12| 18| 25| 31| 37| 49| 62| 74| 86| 99|111|123|148|173|
  |    16| 10| 15| 20| 25| 30| 40| 50| 59| 69| 79| 89| 99|119|138|
  |    17|  8| 13| 17| 21| 25| 34| 43| 50| 59| 67| 75| 84|101|117|
  |    18|  7| 11| 15| 19| 22| 30| 37| 45| 52| 59| 67| 74| 89|104|
  |    19|...| 10| 13| 17| 20| 27| 35| 40| 47| 54| 60| 67| 80| 94|
  |    20|  6|  9| 12| 16| 19| 25| 31| 37| 43| 49| 56| 62| 74| 86|
  |    22|  5|  8| 11| 14| 16| 22| 27| 33| 39| 43| 49| 54| 65| 76|
  |    24|...|  7| 10| 12| 15| 20| 25| 30| 35| 40| 45| 49| 59| 69|
  |    26|...|...|  9| 11| 14| 18| 23| 27| 32| 37| 41| 46| 55| 64|
  |    28|  4|  6|...|...| 13|...| 22| 26| 30| 35| 40| 43| 52| 60|
  |    30|...|...|  8| 10| 12| 17| 21| 25| 29| 33| 37| 41| 49| 58|
  |    35|...|...|...|  9| 11| 16| 19| 23| 26| 30| 34| 38| 45| 53|
  |    40|...|  5|  7|...|...| 15| 18| 21| 25| 28| 32| 35| 42| 49|
  |    60|  3|...|  6|  8|  9| 13| 15| 19| 22| 25| 28| 31| 37| 43|
  |    80|...|  4|...|  7|...| 12|...| 17| 20| 23| 26| 29| 35| 41|
  |   100|...|...|...|...|  8| 11| 14|...|...| 22| 25| 28| 34| 39|
  |   150|...|...|  5|...|...|...| 13| 16| 19| 21| 24| 27| 32| 38|
  | Rack.|  2|...|...|  6|  7| 10| 12| 15| 17| 20| 22| 25| 30| 34|
  +------+---+---+---+---+---+---+---+---+---+---+---+---+---+---+]

The requisite number on the graduated edge for pitches beyond 3-1/2 (the
greatest given in the tables), may be obtained by direct proportion from
those given in the tables. Thus for 4 inch pitch, by doubling the
numbers given for a 2 inch pitch, containing the same number of teeth,
for 4-1/2 inch pitch by doubling the numbers given for a 2-1/4 inch
pitch. If the pitch be a fraction that cannot be so obtained, no serious
error will be induced if the nearest number marked be taken.

[Illustration: Fig. 136.]

An improved form of template odontograph, designed by Professor Robinson
of the Illinois School of Industry, is shown in Fig. 137.

In this instrument the curved edge, having graduated lines, approaches
more nearly to the curves produced by rolling circles than can be
obtained from any system in which an arc of a circle is taken to
represent the curve; hence, that edge is applied direct to the teeth and
used as a template wherefrom to mark the curve. The curve is a
logarithmic spiral, and the use of the instrument involves no other
labor than that of setting it in position. The applicability of this
curve, for the purpose, arises from two of its properties: first, that
the involute of the logarithmic spiral is another like spiral with poles
in common; and, second, that the obliquity or angle between a normal and
radius sector is constant, the latter property being possessed by this
curve only. By the first property it is known that a line, lying tangent
to the curve C E H, will be normal or perpendicular to the curve C D B;
so that when the line D E F is tangent to the pitch line, the curve A D
B will coincide very closely with the true epicycloidal curve, or,
rather, with that portion of it which is applied to the tooth curve of
the wheel. By the second quality, all sectors of the spiral, with given
angle at the poles, are similar figures which admit of the same degree
of coincidence for all similar epicycloids, whether great or small, and
nearly the same for epicycloids in general; thus enabling the
application of the instrument to epicycloids in general.

To set the instrument in position for drawing a tooth face a table which
accompanies the instrument is used. From this table a numerical value is
taken, which value depends upon the diameters of the wheels, and the
number of teeth in the wheel for which the curve is sought. This tabular
value, when multiplied by the pitch of the teeth, is to be found on the
graduated edge on the instrument A D B in Fig. 137. This done, draw the
line D E F tangent to the pitch line at the middle of the tooth, and
mark off the half thickness of the tooth, as E, D, either on the tangent
line or the pitch line. Then place the graduated edge of the odontograph
at D, and in such a position that the number and division found as
already stated shall come precisely on the tangent line at D, and at the
same time so set the curved edge H F C so that it shall be tangent to
the tangent line, that is to say, the curved edge C H must just meet the
tangent line at some one point, as at F in the figure. A line drawn
coincident with the graduated edge will then mark the face curve
required, and the odontograph may be turned over, and the face on the
other side of the tooth marked from a similar setting and process.

For the flanks of the teeth setting numbers are obtained from a separate
table, and the instrument is turned upside down, and the tangent line D
F, Fig. 137, is drawn from the side of the tooth (instead of from the
centre), as shown in Fig. 138.

It is obvious that this odontograph may be set upon a radial arm and
used as a template, as shown in Fig. 126, in which case the instrument
would require but four settings for the whole wheel, while rolling
segments and the making of templates are entirely dispensed with, and
the degree of accuracy is greater than is obtainable by means of the
employment of arcs of circles.

The tables wherefrom to find the number or mark on the graduated edge,
which is to be placed coincident with the tangent line in each case, are
as follows:--

TABLE OF TABULAR VALUES WHICH, MULTIPLIED BY THE ARC PITCH OF THE TEETH,
GIVES THE SETTING NUMBER ON THE GRADUATED EDGE OF THE INSTRUMENT.

  +--------------------+-----------------------------------------------------+
  |                    |    Number of Teeth in Wheel Sought; or, Wheel for   |
  |                    |               Which Teeth are Sought.               |
  |                    +-----+-----+-----+-----+-----+-----+-----+-----+-----+
  |                    |  8  | 12  | 16  | 20  | 30  | 40  | 50  | 60  | 70  |
  |                    +-----+-----+-----+-----+-----+-----+-----+-----+-----+
  |                    |        _For Faces: Flanks Radial or Curved._        |
  |     RATIOS.[7]     |      Draw Setting Tangent at Middle of Tooth.--     |
  |                    |         Epicycloidal Spur or Bevel Gearing.         |
  +--------------------+-----+-----+-----+-----+-----+-----+-----+-----+-----+
  |     1/12 = .083    | .32 | .39 | .46 | .51 |     |     |     |     |     |
  |      1/4 = .250    | .31 | .37 | .44 | .49 | .61 | .70 | .78 | .85 | .92 |
  |      1/2 = .500    | .28 | .34 | .41 | .46 | .57 | .66 | .73 | .80 | .87 |
  |      2/3 = .667    | .27 | .32 | .38 | .43 | .54 | .62 | .70 | .77 | .83 |
  |        1           | .23 | .28 | .34 | .39 | .49 | .58 | .65 | .72 | .78 |
  |      3/2 = 1.50    | .19 | .25 | .29 | .34 | .44 | .51 | .58 | .64 | .69 |
  |        2           | .17 | .22 | .26 | .30 | .38 | .46 | .53 | .59 | .63 |
  |        3           |     | .16 | .19 | .23 | .31 | .38 | .44 | .49 | .53 |
  |        4           |     | .14 | .17 | .20 | .26 | .33 | .38 | .42 | .46 |
  |        6           |     |     |     |     | .22 | .26 | .30 | .34 | .37 |
  |       12           |     |     |     |     |     | .20 | .23 | .25 | .28 |
  |       24           |     |     |     |     |     |     |     |     |     |
  +--------------------+-----+-----+-----+-----+-----+-----+-----+-----+-----+
  |                    | Number of Teeth in Wheel Sought; or, Wheel for|
  |                    |            Which Teeth are Sought.            |
  |                    +-----+-----+-----+-----+-----+-----+-----+-----+
  |                    | 80  | 90  | 100 | 120 | 150 | 200 | 300 | 500 |
  |                    +-----+-----+-----+-----+-----+-----+-----+-----+
  |                    |     _For Faces: Flanks Radial or Curved._     |
  |     RATIOS.[7]     |   Draw Setting Tangent at Middle of Tooth.--  |
  |                    |      Epicycloidal Spur or Bevel Gearing.      |
  +--------------------+-----+-----+-----+-----+-----+-----+-----+-----+
  |     1/12 = .083    |     |     |     |     |     |     |     |     |
  |      1/4 = .250    | .99 | 1.05| 1.11| 1.22| 1.36| 1.55| 1.94| 2.54|
  |      1/2 = .500    | .93 | 1.00| 1.06| 1.15| 1.29| 1.50| 1.86| 2.41|
  |      2/3 = .667    | .89 |  .95| 1.01| 1.11| 1.24| 1.45| 1.79| 2.32|
  |        1           | .83 |  .89|  .94| 1.03| 1.15| 1.36| 1.65| 2.10|
  |      3/2 = 1.50    | .74 |  .79|  .84|  .93| 1.05| 1.25| 1.53| 1.94|
  |        2           | .68 |  .72|  .76|  .84|  .95| 1.13| 1.40| 1.81|
  |        3           | .57 |  .60|  .63|  .71|  .82|  .97| 1.23| 1.60|
  |        4           | .49 |  .53|  .56|  .63|  .73|  .87| 1.08| 1.42|
  |        6           | .41 |  .44|  .47|  .53|  .61|  .71|  .90| 1.20|
  |       12           | .30 |  .32|  .34|  .37|  .42|  .49|  .60|  .82|
  |       24           |     |  .19|  .21|  .23|  .26|  .31|  .40|  .57|
  +--------------------+-----+-----+-----+-----+-----+-----+-----+-----+
  +--------------------+-----------------------------------------------------+
  |                    |    Number of Teeth in Wheel Sought; or, Wheel for   |
  |                    |               Which Teeth are Sought.               |
  |                    +-----+-----+-----+-----+-----+-----+-----+-----+-----+
  |                    |  8  | 12  | 16  | 20  | 30  | 40  | 50  | 60  | 70  |
  |                    +-----+-----+-----+-----+-----+-----+-----+-----+-----+
  |                    |             _For Flanks, when Curved._              |
  |                    |       Draw Setting Tangent at Side of Tooth.--      |
  |                    |         Epicycloidal Spur and Bevel Gearing.        |
  |D   C               |    Faces of Internal, and Flanks of Pinion Teeth.   |
  |e   u               +-----+-----+-----+-----+-----+-----+-----+-----+-----+
  |g F r  { 1.5 slight.| .77 | .98 | 1.18| 1.36| 1.75| 2.05| 2.31| 2.56| 2.75|
  |r l v  { 2 good.    | .44 | .54 |  .63| .72 |  .92| 1.09| 1.24| 1.38| 1.49|
  |e a a  { 3 more.    | .20 | .28 |  .35| .40 |  .54|  .65|  .76|  .86|  .95|
  |e n t  { 4 much.    |     | .20 |  .23| .25 |  .34|  .42|  .51|  .59|  .66|
  |  k u  { 6          |     |     |  .16| .17 |  .26|  .32|  .38|  .43|  .48|
  |o   r  {12          |     |     |     |     |  .19|  .24|  .28|  .31|  .34|
  |f   e  {24          |     |     |     |     |     |     |     |     |     |
  +--------------------+-----+-----+-----+-----+-----+-----+-----+-----+-----+
  |                    | Number of Teeth in Wheel Sought; or, Wheel for|
  |                    |            Which Teeth are Sought.            |
  |                    +-----+-----+-----+-----+-----+-----+-----+-----+
  |                    | 80  | 90  | 100 | 120 | 150 | 200 | 300 | 500 |
  |                    +-----+-----+-----+-----+-----+-----+-----+-----+
  |                    |          _For Flanks, when Curved._           |
  |                    |    Draw Setting Tangent at Side of Tooth.--   |
  |                    |      Epicycloidal Spur and Bevel Gearing.     |
  |D   C               | Faces of Internal, and Flanks of Pinion Teeth.|
  |e   u               +-----+-----+-----+-----+-----+-----+-----+-----+
  |g F r  { 1.5 slight.| 2.92| 3.08| 3.24| 3.52| 3.87| 4.51| 5.50| 7.20|
  |r l v  { 2 good.    | 1.59| 1.79| 1.79| 1.98| 2.23| 2.67| 3.22| 4.50|
  |e a a  { 3 more.    | 1.02| 1.10| 1.18| 1.31| 1.46| 1.67| 2.08| 2.76|
  |e n t  { 4 much.    |  .71|  .77|  .82|  .92| 1.06| 1.25| 1.64| 2.15|
  |  k u  { 6          |  .52|  .56|  .60|  .66|  .76|  .93| 1.20| 1.54|
  |o   r  {12          |  .36|  .38|  .40|  .45|  .52|  .63|  .80|  .98|
  |f   e  {24          |     |     |  .22|  .25|  .28|  .33|  .47|  .60|
  +--------------------+-----+-----+-----+-----+-----+-----+-----+-----+
  +--------------------+-----------------------------------------------------+
  |                    |    Number of Teeth in Wheel Sought; or, Wheel for   |
  |                    |               Which Teeth are Sought.               |
  |                    +-----+-----+-----+-----+-----+-----+-----+-----+-----+
  |                    |  8  | 12  | 16  | 20  | 30  | 40  | 50  | 60  | 70  |
  |                    +-----+-----+-----+-----+-----+-----+-----+-----+-----+
  |  _For Faces of Racks; and of Pinions for Racks and Internal Gears; for   |
  |            Flanks of Internal and Sides of Involute Teeth._              |
  |   Draw Setting Tangent at Middle of Tooth, regarding Space as Tooth in   |
  |          Internal Teeth. For Rack use Number of Teeth in Pinion.         |
  +--------------------+-----+-----+-----+-----+-----+-----+-----+-----+-----+
  |        Pinion.     | .31 | .39 | .48 | .57 | .73 | .88 | 1.00| 1.10| 1.20|
  |        Rack.       | .32 | .38 | .44 | .50 | .62 | .72 |  .80|  .87|  .93|
  +--------------------+-----+-----+-----+-----+-----+-----+-----+-----+-----+
  |                    | Number of Teeth in Wheel Sought; or, Wheel for|
  |                    |            Which Teeth are Sought.            |
  |                    +-----+-----+-----+-----+-----+-----+-----+-----+
  |                    | 80  | 90  | 100 | 120 | 150 | 200 | 300 | 500 |
  |                    +-----+-----+-----+-----+-----+-----+-----+-----+
  | _For Faces of Racks; and of Pinions for Racks and Internal Gears;  |
  |         for Flanks of Internal and Sides of Involute Teeth._       |
  |Draw Setting Tangent at Middle of Tooth, regarding Space as Tooth in|
  |       Internal Teeth. For Rack use Number of Teeth in Pinion.      |
  +--------------------+-----+-----+-----+-----+-----+-----+-----+-----+
  |       Pinion.      | 1.30| 1.40| 1.48| 1.65| 1.85| 2.15| 2.65| 3.50|
  |       Rack.        |  .99| 1.03| 1.08| 1.16| 1.27| 1.49| 1.86| 2.44|
  +--------------------+-----+-----+-----+-----+-----+-----+-----+-----+

  [7] These ratios are obtained by dividing the radius of the wheel
  sought by the diameter of the generating circle.

From these tables may be found a tabular value which, multiplied by the
pitch of the wheel to be marked (as stated at the head of the table),
will give the setting number on the graduated edge of the instrument,
the procedure being as follows:--

For the teeth of a pair of wheels intended to gear together only (and
not with other wheels having a different number of teeth).

For the face of such teeth where the flanks are to be radial lines.

Rule.--Divide the pitch circle radius of the wheel to have its teeth
marked by the pitch circle radius of the wheel with which it is to gear:
or, what is the same thing, divide the number of teeth in the wheel to
have its teeth marked by the number of teeth in the wheel with which it
is to gear, and the quotient is the "ratio." In the ratio column find
this number, and look along that line, and in the column at the head of
which is the number of teeth contained in the wheel to be marked, is a
number termed the tabular value, which, multiplied by the arc pitch of
the teeth, will give the number on the graduated edge by which to set
the instrument to the tangent line.

Example.--What is the setting number for the face curves of a wheel to
contain 12 teeth, of 3-inch arc pitch, and to gear with a wheel having
24 teeth?

Here number of teeth in wheel to be marked = 12, divided by the number
of teeth (24) with which it gears; 12 ÷ 24 = .5. Now in column of ratios
may be found 1/2 = .500 (which is the same thing as .5), and along the
same horizontal line in the table, and in the column headed 12 (the
number of teeth in the wheel) is found .34. This is the tabular value,
which, multiplied by 3 (the arc pitch of the teeth), gives 1.02, which
is the setting number on the graduated edge. It will be noted, however,
that the graduated edge is marked 1, 2, 3, &c., and that between each
consecutive division are ten subdivisions; hence, for the decimal .02 an
allowance may be made by setting the line 1 a proportionate amount below
the tangent line marked on the wheel to set the instrument by.

[Illustration: Fig. 137. NEW ODONTOGRAPH Full Size]

Required now the setting number for the wheel to have the 24 teeth.

Here number of teeth on the wheel = 24, divided by the number of teeth
(12) on the wheel with which it gears; 24 ÷ 12 = 2. Now, there is no
column in the "number of teeth sought" for 24 teeth; but we may find the
necessary tabular value from the columns given for 20 teeth and 30
teeth, thus:--opposite ratio 2, and under 20 teeth is given .30, and
under 30 teeth is given .38--the difference between the two being .08.
Now the difference between 20 teeth and 24 teeth is 4/10; hence, we take
4/10 of the .08 and add it to the tabular value given for 20 teeth,
thus: .08 × 4 ÷ 10 = .032, and this added to .30 (the tabular value
given for 20 teeth = .33, which is the tabular value for 24 teeth). The
.33 multiplied by arc pitch (3) gives .99. This, therefore, is the
setting number for the instrument, being sufficiently near to the 1 on
the graduated edge to allow that 1 to be used instead of .99.

[Illustration: Fig. 138.]

It is to be noted here that the pinion, having radial lines, the other
wheel must have curved flanks; the rule for which is as follows:--


CURVED FLANKS FOR A PAIR OF WHEELS.

Note.--When the flanks are desired to be curved instead of radial, it is
necessary to the use of the instrument to select and assume a value for
the degree of curve, as is done in the table in the column marked
"Degree for flank curving;" in which

  1.5 slight--a slight curvature of flank.
  2 good--an increased curvature of flank.
  3 more--a degree of pronounced spread at root.
  4 much--spread at root is a distinguishing feature of tooth form.
  6--still increased spread in cases where the strength at root of
  pinion is of much importance to give strength.
  12--as above, under aggravated conditions.
  24--undesirable (unless requirement of strength compels this degree),
  because of excessive strain on pinion.

Rule.--For faces of teeth to have curved flanks.

Divide the number of teeth in the wheel to be marked by the number of
teeth in the wheel with which it gears, and multiply by the degree of
flank curve selected for the wheel with which that to be marked is to
gear, and this will give the ratio. Find this number in ratio column,
and the tabular number under the column of number of teeth of wheel to
be marked; multiply tabular number so found by arc pitch of wheel to be
marked, and the product will be the setting number for the instrument.

Example.--What is the setting number on the graduated edge of the
odontograph for the faces of a wheel (of a pair) to contain 12 teeth of
2-inch arc pitch, and to gear with a wheel having 24 teeth and a flank
curvature represented by 3 in "Degree of flank curving" column?

Here teeth in wheel to be marked (12) divided by number of teeth in the
wheel it is to gear with (24), 12 ÷ 24 = .5, which multiplied by 3
(degree of curvature selected for flanks of 24-teeth wheel), .5 × 3 =
1.5. In column of ratio numbers find 1.5, and in 12-teeth column is .25,
which multiplied by pitch (2) gives .5 as the setting number for the
instrument; this being the fifth line on the instrument, and half way
between the end and mark 1.


FOR CURVED FLANKS.

Rule.--Assume the degree of curve desired for the flanks to be marked,
select the corresponding value in the column of "Degrees of flank
curving," and find the tabular value under the number of teeth column.

Multiply tabular value so found by the arc pitch of the teeth, and the
product is the setting number on the instrument.

Example.--What is the setting number on the odontograph for the flanks
of a wheel to contain 12 teeth and gear with one having 24 teeth, the
degree of curvature for the flanks being represented by 4 in the column
of "Degree of flank curvature?"

Here in column of degrees of flank curvature on the 3 line and under 12
teeth is .20, which multiplied by pitch of teeth (2) is .20 × 2 = 40, or
4/10; hence, the fourth line of division on the curved corner is the
setting line, it representing 4/10 of 1.


FOR INTERCHANGEABLE GEARING (THAT IS, A TRAIN OF GEARS ANY ONE OF WHICH
WILL WORK CORRECTLY WITH ANY OTHER OF THE SAME SET).

Rule--both for the faces and for the flanks. For each respective wheel
divide the number of teeth in that wheel by some one number not greater
than the number of teeth in the smallest wheel in the set, which gives
the ratio number for the wheel to be marked. On that line of ratio
numbers, and in the column of numbers of teeth, find the tabular value
number; multiply this by the arc pitch of the wheel to be marked, and
the product is the setting number of the instrument.

Example.--A set of wheels is to contain 10 wheels; the smallest is to
contain 12 teeth; the arc pitch of the wheels is four inches. What is
the setting number for the smallest wheel?

Here number of teeth in smallest wheel of set is 10; divide this by any
number smaller than itself (as say 5), 10 ÷ 5 = 2 = the ratio number on
ratio line for 2; and under column for 12 is .17, which is the tabular
value, which multiplied by pitch (4) is .17 × 4 = 68, or 6/10 and 8/100;
hence, the instrument must be set with its seventh line of division just
above the tangent line marked on the wheel. It will be noted that, if
the seventh line were used as the setting, the adjustment would be only
the 2/100 of a division out, an amount scarcely practically appreciable.

Both for the faces and flanks, the second number is obtained in
_precisely_ the same manner for every wheel in the set, except that
instead of 10 the number of teeth in each wheel must be substituted.

RACK AND PINION.--_For radial flanks_ use for faces the two lower lines
of table. _For curved flanks_ find tabular value for pinion faces in
lowest line. For flanks of pinion choose degree of curving, and find
tabular value under "flanks," as for other wheels. For faces of rack
divide number of teeth in pinion by degree of curving, which take for
number of teeth in looking opposite "rack." Flanks of rack are still
parallel, but may be arbitrarily curved beyond half way below pitch
line.

INTERNAL GEARS.--For tooth curves within the pitch lines, divide radius
of each wheel by any number not greater than radius of pinion, and look
in the table under "flanks." For curves outside pitch line use lower
line of table; or, divide radii by any number and look under "faces." In
applying instrument draw tangents at middle and side of _space_, for
internal teeth.

INVOLUTE TEETH.--For tabular values look opposite "Pinion," under proper
number of teeth, for each wheel. Draw setting tangent from "base circle"
of involute, at middle of tooth. For this the instrument gives the whole
side of tooth at once.

In all cases multiply the tabular value by the pitch in inches.

BEVEL-WHEELS.--Apply above rules, using the developed normal cone bases
as pitch lines. For right-angled axes this is done by using in place of
the actual ratio of radii, or of teeth numbers, the square of that
ratio; and for number of teeth, the actual number multiplied by the
square root of one plus square of ratio or radii; the numerator of
ratio, and number of teeth, belonging to wheel sought.

When the first column ratio and teeth numbers fall between those given
in the table, the tabular values are found by interpolating as seen in
the following examples:

EXAMPLES OF TABULAR VALUES AND SETTING NUMBERS.

_Take a pair of 16 and 56 teeth; radii 5.09 and 17.82 inches
respectively; and 2 inches pitch._

  +----------------+------+----------------+------+---------------+-------+
  |                |Number}                |      |  First Column | Tab.  |
  |Kind of Gearing.|  of  } Kind of Flank. |Ratio |     Ratio.    | Val.  |
  |                |Teeth.}                |Radii.+--------+------+---+---+
  |                |      |                |      | Flank. |Face. |[A]|[B]|
  +----------------+------+----------------+------+--------+------+---+---+
  |Epicycloidal,  }|Small |Radial          |  .29 |Radial  |  .29 |.. |.44|
  |Radial Flanks  }|Large |Radial          | 3.5  |Radial  | 3.5  |.. |.44|
  |Epicycloidal, } |Small |Curved 2 deg.   |  .29 |  2     |  .87 |.63|.36|
  |Curved Flanks.} |Large |Curved 3 deg. } | 3.5  |  3     | 7.   |.82|.30|
  |Epicycloidal,  }|Small |"Sets," Divide} | 2.   |  2     | 2.   |.63|.26|
  |Interchange'bl.}|Large |Radii by 2.55 } | 7.   |  7     | 7.   |.40|.30|
  |Epicycloidal, } |Pinion|Curved 2 deg.   |      |  2     |Pinion|.63|.44|
  |Internal.     } |Wheel |Int. face 7 deg.| 3.5  |Pinion  | 7[8] |.84|.39|
  |Epicycloidal,  }|Pinion|Curved 2 deg.   |      |  2     |Pinion|.63|.44|
  |Rack & Pinion. }|Rack  |Parallel        |      |Parallel|Rack  |.. |.31|
  |Involute }      |Small |Face and Flank  |      |     Pinion.   |  .44  |
  |Gearing. }      |Large |One Curve       |      |     Pinion.   |  .84  |
  +----------------+------+----------------+------+--------+------+-------+

  Legend: A = Flank.
          B = Face.

  [8] The face being here internal, the tabular value is to be found
  under "flanks." If bevels, use ratio radii .082 and 12.25; and teeth
  numbers 16.6 and 203.8 respectively.

WALKER'S PATENT WHEEL SCALE.--This scale is used in many manufactories
in the United States to mark off the teeth for patterns, wherefrom to
mould cast gears, and consists of a diagram from which the compasses may
be set to the required radius to strike the curves of the teeth.

[Illustration: Fig. 139.]

The general form of this diagram is shown in Fig. 139. From the portion
A the length of the teeth, according to the pitch, is obtained. From the
portion B half the thickness of the tooth at the pitch line is obtained.
From the part C half the thickness at the root is obtained, and from the
part D half the thickness at the point is obtained.

[Illustration: Fig. 140.]

Each of these parts is marked with the number of teeth the wheel is to
contain, and with the pitch of the teeth as shown in Fig. 140, which
represents part C full size. Now suppose it is required to find the
thickness at the root, for a tooth of a wheel having 60 teeth of one
inch pitch, the circles from the point A, pitch line B and root C being
drawn, and a radial line representing the middle of the tooth being
marked, as is shown in Fig. 142, the compass points are set to the
distance F B, Fig. 140--F being at the junction of line 1 with line 60;
the compasses are then rested at G, and the points H I are marked. Then,
from the portion B, Fig. 139 of the diagram, which is shown full-size in
Fig. 141, the compasses may be set to half the thickness at the pitch
circle, as in this case (for ordinary teeth) from E to E, and the points
J K, Fig. 142, are marked. By a reference to the portion D of the
diagram, half the thickness of the tooth at the point is obtained, and
marked as at L M in Fig. 142. It now remains to set compasses to the
radius for the face and that for the flank curves, both of which may be
obtained from the part A of the diagram. The locations of the centres,
wherefrom to strike these curves, are obtained as in Fig. 142. The
compasses set for the face curve are rested at H, and the arc N is
struck; they are then rested at J and the arc O struck; and from the
intersection of N O, as a centre, the face curve H J is marked. By a
similar process, reference to the portion D of the diagram, half the
thickness of the tooth at the point is obtained, and marked as at L M in
Fig. 142. It now remains to set the compasses to the radius to strike
the respective face and flank curves, and for this purpose the operator
turns to the portion A, Fig. 139, of the diagram or scale, and sets the
compasses from the marks on that portion to the required radii.

[Illustration: Fig. 141.]

It now remains to find the proper location from which to strike the
curves.

[Illustration: Fig. 142.]

The face curve on the other side of the tooth is struck. The compasses
set to the flank radius is then rested at M, and the arc P is marked and
rested at K to mark the arc Q; and from the intersection of P Q, as a
centre, the flank curve K M is marked: that on the other side of the
tooth being marked in a similar manner.

Additional scales or diagrams, not shown in Fig. 139, give similar
distances to set the compasses for the teeth of internal wheels and
racks.

It now remains to explain the method whereby the author of the scale has
obtained the various radii, which is as follows: A wheel of 200 teeth
was given the form of tooth curve that would be obtained by rolling it
upon another wheel, containing 200 teeth of the same pitch. It was next
given the form of tooth that would be obtained by rolling upon it a
wheel having 10 teeth of the same pitch, and a line intermediate between
the two curves was taken as representing the proper curve for the large
wheel. The wheel having 10 teeth was then given the form of tooth that
would be obtained by rolling upon it another wheel of the same diameter
of pitch circle and pitch of teeth. It was next given the form of tooth
that would be given by rolling upon it a wheel having 200 teeth, and a
curve intermediate between the two curves thus obtained was taken as
representing the proper curve for the pinion of 10 teeth. By this means
the inventor does not claim to produce wheels having an exactly equal
velocity ratio, but he claims that he obtains a curve that is the
nearest approximation to the proper epicycloidal curve. The radii for
the curves for all other numbers of teeth (between 10 and 200) are
obtained in precisely the same manner, the pinion for each pitch being
supposed to contain 10 teeth. Thus the scale is intended for
interchangeable cast gears.

The nature of the scale renders it necessary to assume a constant height
of tooth for all wheels of the same pitch, and this Mr. Walker has
assumed as .40 of the pitch, from the pitch line to the base, and .35
from the pitch line to the point.

The curves for the faces obtained by this method have rather more
curvature than would be due to the true epicycloid, which causes the
points to begin and leave contact more easily than would otherwise be
the case.

For a pair of wheels Mr. Walker strikes the face curve by a point on the
pitch rolling circle, and the flanks by a point on the addendum circle,
fastening a piece of wood to the pitch circle to carry the tracing
point. The flank of each wheel is struck with a tracing point, thus
attached to the pitch circle of the other wheel.

The proportions of teeth and of the spaces between them are usually
given in turns of the pitch, so that all teeth of a given pitch shall
have an equal thickness, height, and breadth, with an equal addendum and
flank, and the same amount of clearance.

The term "clearance" as applied to gear-wheel teeth means the amount of
space left between the teeth of one wheel, and the spaces in the other,
or, in other words, the difference between the width of the teeth and
that of the spaces between the teeth.

This clearance exists at the sides of the teeth, as in Fig. 143, at A,
and between the tops of the teeth and the bottoms or roots of the spaces
as at B. When, however, the simple term clearance is employed it implies
the side clearance as at A, the clearance at B being usually designated
as _top and bottom clearance_. Clearance is necessary for two purposes;
first, in teeth cut in a machine to accurate form and dimensions, to
prevent the teeth of one wheel from binding in the spaces of the other,
and second, in cast teeth, to allow for the imperfections in the teeth
which are incidental to casting in a founder's mould. In machine-cut
teeth the amount of clearance is a minimum.

In wheels which are cast with their teeth complete and on the pattern,
the amount of clearance must be a maximum, because, in the first place,
the teeth on the pattern must be made taper to enable the extraction of
the pattern from the mould without damage to the teeth in the mould, and
the amount of this taper must be greater than in machine-moulded teeth,
because the pattern cannot be lifted so truly vertical by hand as to
avoid, in all cases, damage to the mould; in which case the moulder
repairs the mould either with his moulding tools and by the aid of the
eye, or else with a tooth and a space made on a piece of wood for the
purpose. But even in this case the concentricity of the teeth is
scarcely likely to be preserved.

It is obvious that by reason of this taper each wheel is larger in
diameter on one side than on the other, hence to preserve the true
curves to the teeth the pitch circle is made correspondingly smaller.
But if in keying the wheels to their shafts the two large diameters of a
pair of wheels be placed to work together, the teeth of the pair would
have contact on that side of the wheel only, and to avoid this and give
the teeth contact across their full breadth the wheels are so placed on
their shafts that the large diameter of one shall work with the small
one of the other, the amount of taper being the same in each wheel
irrespective of their relative diameters. This also serves to keep the
clearance equal in amount both top, and bottom, and sideways.

A second imperfection is that in order to loosen the pattern in the sand
or mould, and enable its extraction by hand from the mould, the pattern
requires to be _rapped_ in the mould, the blows forcing back the sand of
the mould and thus loosening the pattern. In ordinary practice the
amount of this rapping is left entirely to the judgment of the moulder,
who has nothing to guide him in securing an equal amount of pattern
movement in each direction in the mould; hence, the finished mould may
be of increased radius at the circumference in the direction in which
the wheel moved most during the rapping. Again, the wood pattern is apt
in time to shrink and become _out of round_, while even iron patterns
are not entirely free from warping. Again, the cast metal is liable to
contract in cooling more in one direction than in another. The amount of
clearance usually allowed for pattern-moulded cast gearing is given by
Professor Willis as follows:--Whole depth of tooth 7/10, of the pitch
working depth 6/10; hence 1/10 of the pitch is allowed for top and
bottom clearance, and this is the amount shown at B in Fig. 143. The
amount of side clearance given by Willis as that ordinarily found in
practice is as follows:--"Thickness of tooth 5/11 of the pitch; breadth
of space 6/11; hence, the side clearance equals 1/11 of the pitch, which
in a 3-inch pitch equals .27 of an inch in each wheel." Calling this in
round figures, which is near enough for our purpose, 1/4 inch, we have
thickness of tooth 1-1/4, width of space 1-3/4, or 1/2 inch of clearance
in a 3-inch pitch, an amount which on wheels of coarse pitch is
evidently more than that necessary in view of the accuracy of modern
moulding, however suitable it may have been for the less perfect
practice of Professor Willis's time. It is to be observed that the
rapping of the pattern in the founder's mould reduces the thickness of
the teeth and increases the width of the spaces somewhat, and to that
extent augments the amount of side clearance allowed on the pattern, and
the amount of clearance thus obtained would be nearly sufficient for a
small wheel, as say of 2 inches diameter. It is further to be observed
that the amount of rapping is not proportionate to the diameter of the
wheel; thus, in a wheel of 2 inches diameter, the rapping would increase
the size of the mould about 1/32 inch. But in the proportion of 1/32
inch to every 2 inches of diameter, the rapping on a 6-foot wheel would
amount to 1-1/16 inches, whereas, in actual practice, a 6-foot wheel
would not enlarge the mould more than at most 1/8 inch from the rapping.

[Illustration: Fig. 143.]

It is obvious, then, that it would be more in accordance with the
requirements to proportion the amount of clearance to the diameter of
the wheel, so as to keep the clearance as small as possible. This will
possess the advantage that the teeth will be stronger, it being obvious
that the teeth are weakened both from the loss of thickness and the
increase of height due to the clearance.

It is usual in epicycloidal teeth to fill in the corner at the root of
the tooth with a fillet, as at C, D, in Fig. 143, to strengthen it.
This is not requisite when the diameter of the generating circle is so
small in proportion to the base circle as to produce teeth that are
spread at the roots; but it is especially advantageous when the teeth
have radial flanks, in which case the fillets may extend farther up the
flanks than when they are spread; because, as shown in Fig. 47, the
length of operative flank is a minimum in teeth having radial flanks,
and as the smallest pinion in the set is that with radial flanks, and
further as it has the least number of teeth in contact, it is the
weakest, and requires all the strengthening that the fillets in the
corners will give, and sometimes the addition of the flanges on the
sides of the pinion, such gears being termed "shrouded."

The proportion of the teeth to the pitch as found in ordinary practice
is given by Professor Willis as follows:--

  Depth to pitch line    3/10 of the pitch.
  Working depth          6/10   "   "
  Whole depth            7/10   "   "
  Thickness of tooth     5/11   "   "
  Breadth of space       6/11   "   "

The depth to pitch line is, of course, the same thing as the height of
the addendum, and is measured through the centre of the tooth from the
point to the pitch line in the direction of a radial line and not
following the curve of tooth face.

Referring to the working depth, it was shown in Figs. 42 and 44 that the
height of the addendum remaining constant, it varies with the diameter
of the generating circle.

[Illustration: Fig. 144. Scale of Proportions given by Willis]

From these proportions or such others as may be selected, in which the
proportions bear a fixed relation to the pitch, a scale may be made and
used as a gauge, to set the compasses by, and in marking off the teeth
for any pitch within the capacity of the scale. A vertical line A B in
Fig. 144, is drawn and marked off in inches and parts of an inch, to
represent the pitches of the teeth; at a right angle to A B, the line B
C is drawn, its length equalling the whole depth of tooth, which since
the coarsest pitch in the scale is 4 inches will be 7/10 of 4 inches.
From the end of line C we draw a diagonal line to A, and this gives us
the whole depth of tooth for any pitch up to 4 inches: thus the whole
depth for a 4-inch pitch is the full length of the horizontal line B C;
the whole depth for a 3-inch pitch will be the length of the horizontal
line running from the 3 on line A B, to line A C on the right hand of
the figure; similarly for the full depth of tooth for a 2-inch pitch is
the length of the horizontal line running from 2 to A C. The working
depth of tooth being 6/10 of the pitch a diagonal is drawn from A
meeting line C at a distance from B of 6/10 of 4 inches and we get the
working depth for any other pitch by measuring (along the horizontal
line corresponding to that pitch), from the line of pitches to the
diagonal line for working depth of tooth. The thickness of tooth is 5/11
of the pitch and its diagonal is distant 5/11 of 4 (from B) on line B C,
the thickness for other pitches being obtained on the horizontal line
corresponding to those pitches as before.

[Illustration: Fig. 145.]

The construction of a pattern wherefrom to make a foundry mould, in
which to cast a spur gear-wheel, is as shown in section, and in plan of
Fig. 145. The method of constructing these patterns depends somewhat on
their size. Large patterns are constructed with the teeth separate, and
the body of the wheel is built of separate pieces, forming the arms, the
hub, the rim, and the teeth respectively. Pinion patterns, of six inches
and less in diameter, are usually made out of a solid piece, in which
case the grain of the wood must lie in the direction of the teeth
height. The chuck or face plate of the lathe, for turning the piece,
must be of smaller diameter than the pinion, so that it will permit
access to a tool applied on both sides, so as to strike the pitch circle
on both sides. A second circle is also struck for the roots or depths of
the teeth, and also, if required, an extra circle for striking the
curves of the teeth with compasses, as was described in Fig. 130. All
these circles are to be struck on both sides of the pattern, and as the
pattern is to be left slightly taper, to permit of its leaving the
mould easily, they must be made of smaller diameter on one side than on
the other of the pattern; the reduction in diameter all being made on
the same side of the pattern. The pinion body must then be divided off
on the pitch line into as many equal divisions as there are to be teeth
in it; the curves of the teeth are then marked by some one of the
methods described in the remarks on curves of gear-teeth. The top of the
face curves are then marked along the points of the teeth by means of a
square and scribe, and from these lines the curves are marked in on the
other side of the pinion, and the spaces cut out, leaving the teeth
projecting. For a larger pinion, without arms, the hub or body is built
up of courses of quadrants, the joints of the second course _breaking
joint_ with those of the first.

[Illustration: Fig. 146.]

The quadrants are glued together, and when the whole is formed and the
glue dry, it is turned in the lathe to the diameter of the wheel at the
roots of the teeth. Blocks of wood, to form the teeth, are then planed
up, one face being a hollow curve to fit the circle of the wheel. The
circumference of the wheel is divided, or pitched off, as it is termed,
into as many points of equal division as there are to be teeth, and at
these points lines are drawn, using a square, having its back held
firmly against the radial face of the pinion, while the blade is brought
coincidal with the point of division, so as to act as a guide in
converting that point into a line running exactly true with the pinion.
All the points of division being thus carried into lines, the blocks for
the teeth are glued to the body of the pinion, as denoted by A, in Fig.
145. Another method is to dovetail the teeth into the pinion, as in Fig.
145 at B. After the teeth blocks are set, the process is, as already
described, for a solid pinion.

[Illustration: Fig. 147.]

The construction of a wheel, such as shown in Fig. 145, is as follows:
The rim R must be built up in segments, but when the courses of segments
are high enough to reach the flat sides of the arms they should be
turned in the lathe to the diameter on the inside, and the arms should
be let in, as shown in the figure at O. The rest of the courses of
segments should then be added. The arms are then put in, and the inside
of the segments last added may then be turned up, and the outside of the
rim turned. The hub should then be added, one-half on each side of the
arms, as in the figure. The ribs C of the arms are then added, and the
body is completed (ready to receive the teeth), by filleting in the
corners. An excellent method of getting out the teeth is as follows:
Shape A piece of hard wood, as in Fig. 146, making it some five or six
inches longer than the teeth, and about three inches deeper, the
thickness being not less than the thickness of the required teeth at the
pitch line. Parallel to the edge B C, mark the line A D, distant from B
C to an amount equal to the required depth of tooth. Mark off, about
midway of the piece, the lines A B and C D, distant from each other to
an amount equal to the breadth of the wheel rim, and make two saw cuts
to those lines. Take a piece of board an inch or two longer than the
radius of the gear-wheel and insert a piece of wood (which is termed a
box) tightly into the board, as shown in Fig. 147, E representing the
box. Let the point F on the board represent the centre of the wheel, and
draw a radial line R from F through the centre of the box. From the
centre F, with a trammel, mark the addendum line G G, pitch line H I,
and line J K for the depth of the teeth (and also a line wherefrom to
strike the teeth curves, as shown in Fig. 129 if necessary). From the
radial line R, as a centre, mark off on the pitch circle, points of
division for several teeth, so as to be able to test the accuracy of the
spacing across the several points, as well as from one point to the
next, and mark the curves for the teeth on the end of the box, as shown.
Turn the box end for end in the board, and mark out a tooth by the same
method on the other end of the box. The box being removed from the board
must now have its sides planed to the lines, when it will be ready to
shape the teeth in. The teeth are got out for length, breadth, and
thickness at the pitch line as follows: The lumber from which they are
cut should be very straight grained, and should be first cut into strips
of a width and thickness slightly greater than that of the teeth at the
pitch line. These strips (which should be about two feet long) should
then be planed down on the sides to very nearly the thickness of the
tooth at the pitch line, and hollow on one edge to fit the curvature of
the wheel rim. From these strips, pieces a trifle longer than the
breadth of the wheel rim are cut, these forming the teeth. The pieces
are then planed on the ends to the exact width of the wheel rim. To
facilitate this planing a number of the pieces or blank teeth may be set
in a frame, as in Figs. 148 and 149, in which A is a piece having the
blocks B B affixed to it. C is a clamp secured by the screws at S S, and
1, 2, 3, 4, 5, 6 are the ends of the blank teeth. The clamp need not be
as wide as the teeth, as in Fig. 148, but it is well to let the pieces
A and B B equal the breadth of the wheel rim, so that they will act as a
template to plane the blank teeth ends to. The ends of B B may be
blackleaded, so as to show plainly if the plane blade happens to shave
them, and hence to prevent planing B B with the teeth. The blank teeth
may now be separately placed in the box (Fig. 146) and secured by a
screw, as shown in that figure, in which S is the screw, and T the blank
tooth. The sides of the tooth must be carefully planed down equal and
level with the surface of the box. The rim of the wheel, having been
divided off into as many divisions as there are to be teeth in the
wheel, as shown in Fig. 150, at _a_, _a_, _a_, &c., the finished teeth
are glued so that the same respective side of each tooth exactly meets
one of the lines _a_. Only a few spots of glue should be applied, and
these at the middle of the root thickness, so that the glue shall not
exude and hide the line _a_, which would make it difficult to set the
teeth true to the line. When the teeth are all dry they must be
additionally secured to the rim by nails. Wheels sufficiently large to
incur difficulty of transportation are composed of a number of sections,
each usually consisting of an arm, with an equal length of the rim arc
on each side of it, so that the joint where the rim segments are bolted
together will be midway between the two arms.

[Illustration: Fig. 148.]

This, however, is not absolutely necessary so long as the joints are so
arranged as to occur in the middle of tooth spaces, and not in the
thickness of the tooth. This sometimes necessitates that the rim
sections have an unequal length of arc, in which event the pattern is
made for the longest segment, and when these are cast the teeth
superfluous for the shorter segments are stopped off by the foundry
moulder. This saves cutting or altering the pattern, which, therefore,
remains good for other wheels when required.

[Illustration: Fig. 149.]

When the teeth of wheels are to be cut in a gear-cutting machine the
accurate spacing of the teeth is determined by the index plate and
gearing of the machine itself; but when the teeth are to be cast upon
the wheel and a pattern is to be made, wherefrom to cast the wheel the
points of division denoting the thickness of the teeth and the width of
the spaces are usually marked by hand. This is often rendered necessary
from the wheels being of too large a diameter to go into dividing
machines of the sizes usually constructed.

To accurately divide off the pitch circle of a gear-wheel by hand,
requires both patience and skilful manipulation, but it is time and
trouble that well repays its cost, for in the accuracy of spaces lies
the first requisite of a good gear-wheel.

It is a very difficult matter to set the compasses so that by commencing
at any one point and stepping the compasses around the circle
continuously in one direction, the compass point shall fall into the
precise point from which it started, for if the compass point be set the
1-200th inch out, the last space will come an inch out in a circle
having 200 points of divisions. It is, therefore, almost impossible and
quite impracticable to accurately mark or divide off a circle having
many points of division in this manner, not only on account of the
fineness of the adjustment of the compass points, but because the
frequent trials will leave so many marks upon the circle that the true
ones will not be distinguishable from the false. Furthermore, the
compass points are apt to spring and fall into the false marks when
those marks come close to the true ones.

[Illustration: Fig. 150.]

In Fig. 151 is shown a construction by means of which the compass points
may be set more nearly than by dividing the circumference of the circle
by the number of divisions it is required to be marked into and setting
the compasses to the quotient, because such a calculation gives the
length of the division measured around the arc of the circle, instead of
the distance measured straight from point of division to point of
division.

[Illustration: Fig. 151.]

The construction of Fig. 151 is as follows: P P is a portion of the
circle to be divided, and A B is a line at a tangent to the point C of
the circle P P. The point D is set off distant from C, to an amount
obtained by dividing the circumference of P P by the number of divisions
it is to have. Take one-quarter of this distance C D, and mark it from
C, giving the point E, set one point of the compass at E and the other
at D, and draw the arc D F, and the distance from F to C, as denoted by
G, is the distance to which to set the compasses to divide the circle
properly. The compasses being set to this distance G, we may rest one
compass point at C, and mark the arc F H, and the distance between arc H
and arc D, measured on the line A B, is the difference between the
points C, F when measured around the circle P P, and straight across, as
at G.

[Illustration: Fig. 152.]

A pair of compasses set even by this construction will not, however, be
entirely accurate, because there will be some degree of error, even
though it be in placing the compass points on the lines and on the
points marked, hence it is necessary to step the compasses around the
circle, and the best method of doing this is as follows: Commencing at
A, Fig. 152, we mark off continuously one from the other, and taking
care to be very exact to place the compass point exactly coincident with
the line of the circle, the points B, C, D, &c., continuing until we
have marked half as many divisions as the circle is to contain, and
arriving at E, starting again at A, we mark off similar divisions (one
half of the total number), F, G, H, arriving at I, and the centre K,
between the two lines E, I, will be the true position of the point
diametrally opposite to point A, whence we started. These points are all
marked inside the circle to keep them distinct from those subsequently
marked.

[Illustration: Fig. 153.]

It will be, perhaps, observed by the reader that it would be more
expeditious, and perhaps cause less variation, were we to set the
compasses to the radius of the circle and mark off the point K, as shown
in Fig. 153, commencing at the point A, and marking off on the one side
the lines B, C, and D, and on the other side E, F, and G, the junction
or centre, between G and D, at the circle being the true position of the
point K. For circles struck upon flat surfaces, this plan may be
advantageous; and in cases where there are not at hand compasses large
enough, a pair of trammels may be used for the purpose; but our
instructions are intended to apply also to marking off equidistant
points on such circumferences as the faces of pulleys or on the outsides
of small rings or cylinders, in which cases the use of compasses is
impracticable. The experienced hand may, it is true, adjust the
compasses as instructed, and mark off three or four of the marks B, C,
&c., in Fig. 152, and then open out the compasses to the distance
between the two extreme marks, and proceed as before to find the centre
K, but as a rule, the time saved will scarcely repay the trouble; and
all that can be done to save time in such cases is, if the holes come
reasonably close together, to mark off, after the compasses are
adjusted, three or four spaces, as shown in Fig. 154. Commencing at the
point A, and marking off the points B, C, and D, we then set another
pair of compasses to the distance between A and D, and then mark, from D
on one side and from A on the other, the marks from F to L and from M to
T, thus obtaining the point K. This method, however expeditious and
correct for certain work, is not applicable to circumferential work of
small diameter and in which the distance between two of the adjacent
points is, at the most, 1/20 of the circumference of the circle; because
the angle of the surface of the metal to the compass point causes the
latter to spring wider open in consequence of the pressure necessary to
cause the compass point to mark the metal. This will be readily
perceived on reference to Fig. 155 in which A represents the stationary,
and B the scribing or marking point of the compasses.

[Illustration: Fig. 154.]

The error in the set of the compasses as shown by the distance apart of
the two marks E and I on the circle in Fig. 152 is too fine to render it
practicable to remedy it by moving the compass legs, hence we effect the
adjustment by oilstoning the points on the outside, throwing them closer
together as the figure shows is necessary.

[Illustration: Fig. 155.]

Having found the point K, we mark (on the outside of the circle, so as
to keep the marks distinct from those first marked) the division B, C,
D, Fig. 156, &c., up to G, the number of divisions between B and G being
one quarter of those in the whole circle. Then, beginning at K, we mark
off also one quarter of the number of divisions arriving at M in the
figure and producing the point 3. By a similar operation on the other
side of the circle, we get the true position of point No. 4. If, in
obtaining points 3 and 4, the compasses are not found to be set dead
true, the necessary adjustment must be made; and it will be seen that,
so far, we have obtained four true positions, and the process of
obtaining each of them has served as a justification of the distance of
the compass points. From these four points we may proceed in like
manner to mark off the holes or points between them; and the whole will
be as true as it is practicable to mark them off upon that size of
circle. In cases, however, where mathematical precision is required upon
flat and not circumferential surfaces, the marking off may be performed
upon a circle of larger diameter, as shown in Fig. 157. If it is
required to mark off the circle A, Fig. 157, into any even number of
equidistant points, and if, in consequence of the closeness together of
the points, it becomes difficult to mark them (as described) with the
compasses, we mark a circle B B of larger diameter, and perform our
marking upon it, carrying the marks across the smaller circle with a
straightedge placed to intersect the centres of the circles and the
points marked on each side of the diameter. Thus, in Fig. 157, the lines
1 and 2 on the smaller circle would be obtained from a line struck
through 1 and 4 on the outer circle; and supposing the larger circle to
be three times the size of the smaller, the deviation from truth in the
latter will be only 1/3 of whatever it is in the former.

[Illustration: Fig. 156.]

[Illustration: Fig. 157.]

In this example we have supposed the number of divisions to be an even
one, hence the point K, Fig. 152, falls diametrically opposite to A,
whereas in an odd number of points of division this would not be the
case, and we must proceed by either of the two following methods:--

[Illustration: Fig. 158.]

In Fig. 158 is shown a circle requiring to be divided by 17 equidistant
points. Starting from point 1 we mark on the outside of the
circumference points 2, 3, 4, &c., up to point 9. Starting again from
point 1 we mark points 10, 11, &c., up to 17. If, then, we try the
compasses to 17 and 9 we shall find they come too close together, hence
we take another pair of compasses (so as not to disturb the set of our
first pair) and find the centre between 9 and 17 as shown by the point
A. We then correct the set of our first pair of compasses, as near as
the judgment dictates, and from point A, we mark with the second
compasses (set to one half the new space of the first compasses) the
points B, C. With the first pair of compasses, starting from B, we mark
D, E, &c., to G; and from I, we mark divisions H, I, &c., to K, and if
the compasses were set true, K and G would meet at the circle. We may,
however, mark a point midway between K and G, as at 5. Starting again
from points C and I, we mark the other side of the circle in a similar
manner, producing the lines P and Q, midway between which (the compasses
not being set quite correct as yet) is the true point for another
division. After again correcting the compasses, we start from B and 5
respectively, and mark point 7, again correcting the compasses. Then
from C and the point between P and Q, we may mark an intermediate point,
and so on until all the points of division are made. This method is
correct enough for most practical purposes, but the method shown in Fig.
159 is more correct for an odd number of points of division. Suppose
that we have commenced at the point marked I, we mark off half the
required number of holes on one side and arrive at the point 2; and
then, commencing at the point I again, we mark off the other half of the
required number of holes, arriving at the point 3. We then apply our
compasses to the distance between the points 2 and 3; and if that
distance is not exactly the same to which the compasses are set, we make
the necessary adjustment, and try again and again until correct
adjustment is secured.

[Illustration: Fig. 159.]

It is highly necessary, in this case, to make the lines drawn at each
trial all on the same side of the circle and of equal length, but of a
different length to those marked on previous trials. For example, left
the lines A, B, C, D, in Fig. 159 represent those made on the first
trial, and E, F, G, H, those made on the second trial; and when the
adjustment is complete, let the last trial be made upon the outside or
other side of the circle, as shown by the lines I, J, K, L. Having
obtained the three true points, marked 1, 2, 3, we proceed to mark the
intermediate divisions, as described for an even number of divisions,
save that there will be a space, 2 and 3, opposite point 1, instead of a
point, as in case of a circle having an even number of divisions.

[Illustration: Fig. 160.]

The equal points of division thus obtained may be taken for the centres
of the tooth at the pitch circle or for one side of the teeth, as the
method to be pursued to mark the tooth curves may render most desirable.
If, for example, a template be used to mark off the tooth curves, the
marks may be used to best advantage as representing the side of a tooth,
and from them the thickness of the tooth may be marked or not as the
kind of template used may require. Thus, if the template shown in Fig.
21 be used, no other marks will be used, because the sides of a tooth on
each side of a space may be marked at one setting of the template to the
lines or marks of division. If, however, a template, such as shown in
Fig. 81 be used, a second set of lines marked distant from the first to
a radius equal to the thickness of a tooth becomes necessary so that the
template may be set to each line marked. If the Willis odontograph or
the Robinson template odontograph be used the second set of lines will
also be necessary. In using the Walker scale a radial line, as G in Fig.
142, will require to be marked through the points of equal division, and
the thickness of the tooth at the points on the pitch circle and at the
root must be marked as was shown in Fig. 142.

But if the arcs for the tooth curves are to be marked by compasses, the
location for the centres wherefrom to strike these arcs may be marked
from the points of division as was shown in Fig. 130.

To construct a pattern wherefrom to cast a bevel gear-wheel.--When a
pair of bevel-wheels are in gear and upon their respective shafts all
the teeth on each wheel incline, as has been shown, to a single point,
hence the pattern maker draws upon a piece of board a sketch
representing the conditions under which the wheels are to operate. A
sketch of this kind is shown in Fig. 160, in which A, B, C, D, represent
in section the body of a bevel pinion. F G is the point of a tooth on
one side, and E the point of a tooth on the other side of the pinion,
while H I are pitch lines for the two teeth. Thus, the cone surface, the
points, the pitch lines and the bottom of the spaces, projected as
denoted by the dotted lines, would all meet at X, which represents the
point where the axes of the shafts would meet.

[Illustration: Fig. 161.]

In making wooden patterns wherefrom to cast the wheels, it is usual,
therefore, to mark these lines on a drawing-board, so that they may be
referred to by the workman in obtaining the degree of cone necessary for
the body A B C D, to which the teeth are to be affixed. Suppose, then,
that the diameter of the pinion is sufficiently small to permit the body
A B C D to be formed of one piece instead of being put together in
segments, the operation is as follows: The face D C is turned off on the
lathe, and the piece is reversed on the lathe chuck, and the face A B is
turned, leaving a slight recess at the centre to receive and hold the
cone point true with the wheel. A bevel gauge is then set to the angle A
B C, and the cone of the body is turned to coincide in angle with the
gauge and to the required diameter, its surface being made true and
straight so that the teeth may bed well. While turning the face D C in
the lathe a fine line circle should be struck around the circumference
of the cone and near D C, on which line the spacing for the teeth may be
stepped off with the compasses. After this circle or line is divided off
into as many equidistant points as there are to be teeth on the wheel,
the points of division require to be drawn into lines, running across
the cone surface of the wheel, and as the ordinary square is
inapplicable for the purpose, a suitable square is improvised as
follows: In Fig. 161 let the outline in full lines denote the body of a
pinion ready to receive the teeth, and A B the circle referred to as
necessary for the spacing or dividing with the compasses. On A B take
any point, as C, as a centre, and with a pair of compasses mark
equidistant on each side of it two lines, as D, D. From D, D as
respective centres mark two lines, crossing each other as at F, and draw
a line, joining the intersection of the lines at F with C, and the last
line, so produced, will be in the place in which the teeth are to lie;
hence the wheel will require as many of these lines as it is to contain
teeth, and the sides of the teeth, being set to these lines all around
the pinion, will be in their proper positions, with the pitch lines
pointing to X, in Fig. 160.

[Illustration: Fig. 162.]

To avoid, however, the labor involved in producing these lines for each
tooth, two other plans may be adopted. The first is to make a square,
such as shown in Fig. 162, the face _f_ _f_ being fitted to the surface
C, in Fig. 161, while the edges of its blade coincide with the line
referred to; hence the edge of the blade may be placed coincident
successively with each point of division, as D D, and the lines for the
place of the length of each tooth be drawn. The second plan is to divide
off the line A B before removing the body of the pinion from the lathe,
and produce, as described, a line for one tooth. A piece of wood may
then be placed so that when it lies on the surface of the hand-rest its
upper surface will coincide with the line as shown in Fig. 163, in which
W is the piece of wood, and A, B, C, &c., the lines referred to. If the
teeth are to be glued and bradded to the body, they are first cut out in
blocks, left a little larger every way than they are to be when
finished, and the surfaces which are to bed on the cone are hollowed to
fit it. Then blocks are glued to the body, one and the same relative
side of each tooth being set fair to the lines. When the glue is dry,
the pinion is again turned on the lathe, the gauge for the cone of the
teeth being set in this case to the lines E, F, G in Fig. 160. The pitch
circles must then be struck at the ends of the teeth. The turned wheel
is then ready to have the curves of the teeth marked. The wheel must now
again be divided off on the pitch circle at the large end of the cone
into as many equidistant points as there are to be teeth on the wheel,
and from these points, and on the same relative side of them, mark off a
second series of points, distant from the points of division to an
amount equal to the thickness the teeth are required to be. From these
points draw in the outline of the teeth (upon the ends of the blocks to
form the teeth) at the large end of the cone. Then, by use of the
square, shown in Fig. 162, transfer the points of the teeth to the small
end of the cone, and trace the outline of the teeth at the small end,
taking centres and distances proportionate to the reduced diameter of
the pitch circle at the small end, as shown in Fig. 160, where at J are
three teeth so marked for the large end, and at K three for the small
end, P P representing the pitch circle, and R R a circle for the compass
points. The teeth for bevel pinions are sometimes put on by dovetails,
as shown in Fig. 164, a plan which possesses points of advantage and
disadvantage. Wood shrinks more across the grain than lengthwise with
it, hence when the grain of the teeth crosses that of the body with
every expansion or contraction of the wood (which always accompanies
changes in the humidity of the atmosphere) there will be a movement
between the two, because of the unequal expansion and contraction,
causing the teeth to loosen or to move. In the employment of dovetails,
however, a freedom of movement lengthways of the tooth is provided to
accommodate the movement, while the teeth are detained in their proper
positions. Again, if in making the founders' mould, one of the mould
teeth should break or fall down when the pattern is withdrawn, a tooth
may be removed from the pattern and used by the moulder to build up the
damaged part of the mould again. And if the teeth of a bevel pinion are
too much undercut on the flank curves to permit the whole pattern from
being extracted from the mould without damaging it, dovetailed teeth may
be drawn, leaving the body of the pattern to be extracted from the mould
last. On the other hand, the dovetail is a costly construction if
applied to large wheels. If the teeth are to be affixed by dovetails,
the construction varies as follows: Cut out a wooden template of the
dovetail, leaving it a little narrower than the thickness of the tooth
at the root, and set the template on the cone at a distance from one of
the lines A, B, C, Fig. 163, equal to the margin allowed between the
edge of the dovetail and the side of the root of the tooth, and set it
true by the employment of the square, shown in Fig. 162, and draw along
the cone surface of the body lines representing the location of the
dovetail grooves. The lines so drawn will give a taper toward X (Fig.
160), providing that, the template sides being parallel, each side is
set to the square. While the body is in the lathe, a circle on each end
may be struck for the depth of the dovetails, which should be cut out to
gauge and to template, so that the teeth will interchange to any
dovetail. The bottom of the dovetails need not be circular, but flat,
which is easier to make. Dovetail pieces or strips are fitted to the
grooves, being left to project slightly above the face of the cone or
body. They are drawn in tight enough to enable them to keep their
position while being turned in the lathe when the projecting points are
turned down level with the cone of the body. The teeth may then be got
out as described for glued teeth, and the dovetails added, each being
marked to its place, and finally the teeth are cut to shape.

[Illustration: Fig. 163.]

[Illustration: Fig. 164.]

[Illustration: Fig. 165.]

In wheels too large to have their cones tested by a bevel gauge, a
wooden gauge may be made by nailing two pieces of wood to stand at the
required angle as shown in Fig. 165, which is extracted from _The
American Machinist_, or the dead centre C and a straightedge may be used
as follows. In the figure the other wheel of the pair is shown dotted in
at B, and the dead centre is set at the point where the axes of A and B
would meet; hence if the largest diameter of the cone of A is turned to
correct size, the cone will be correct when a straightedge applied as
shown lies flat on the cone and meets the point of the dead centre E.
The pinion B, however, is merely introduced to explain the principle,
and obviously could not be so applied practically, the distance to set
_e_, however, is the radius _a_.

Skew Bevel.[9]--When the axles of the shaft are inclined to each other
instead of being in a straight line, and it is proposed to connect and
communicate motion to the shafts by means of a single pair of
bevel-gears, the teeth must be inclined to the base of the frustra to
allow them to come into contact.

  [9] From the "Engineer and Machinists' Assistant."

[Illustration: Fig. 166.]

To find the line of contact upon a given frustrum of the tangent-cone;
let the Fig. 166 be the plane of the frustrum; _a_ the centre. Set off
_a_ _e_ equal to the shortest distance between the axes (called the
_eccentricity_), and divide it in _c_, so that _a_ _c_ is to _e_ _c_ as
the mean radius of the frustrum to the mean radius of that with which it
is to work; draw _c_ _p_ perpendicular to _a_ _e_, and meeting the
circumference of the conical surface at _m_; perform a similar operation
on the base of the frustrum by drawing a line parallel to _c_ _m_ and at
the same distance _a_ _c_ from the centre, meeting the circumference in
_p_.

The line _p_ _c_ is then plainly the line of direction of the teeth. We
are also at liberty to employ the equally inclined line _c_ _q_ in the
opposite direction, observing only that, in laying out the two wheels,
the pair of directions be taken, of which the inclinations correspond.

[Illustration: Fig. 167.]

Fig. 167 renders this mode of laying off the outlines of the wheels at
once obvious. In this figure the line _a_ _e_ corresponds to the line
marked by the same letters in Fig. 166; and the division of it at _c_ is
determined in the manner directed. The line _c_ _m_ being thus found in
direction, it is drawn indefinitely to _d_. Parallel to this line and
from the point _c_ draw _e_ to _e_, and in this line take the centre of
the second wheel. The line _c_ _m_ _d_ gives the direction of the teeth;
and if from the centre _a_ with radius at _c_ a circle be described, the
direction of any tooth of the wheel will be a tangent to it, as at _c_,
and similarly if a centre _e_ be taken in the line _e_ _d_, and with
radius _e_ _d_, _c_ _e_ a circle be drawn, the direction of the teeth of
the second wheel will be tangents to this last, as at _d_.

Having thus found the direction of the teeth, these outlines may be
formed as in the case of ordinary bevel-wheels and with equal exactness
and facility, all that is necessary being to find the curves for the
teeth as described for bevel-wheels, and follow precisely the same
construction, except that the square, Fig. 162, marking the lines across
the cones, requires to be set to the angle for the tooth instead of at a
right angle, and this angle may be found by the construction shown in
Fig. 167, it being there represented by line _d_ _c_. It is obvious,
however, that the bottoms of the blocks to form the teeth must be curved
to bed on the cone along the line _d_ _c_, Fig. 167, and this may best
be done by bedding two teeth, testing them by trial of the actual
surfaces.

[Illustration: Fig. 168.]

Then two teeth may be set in as No. 1 and No. 6 in the box shown in Fig.
148, the intermediate ones being dressed down to them.

Where a bevel-wheel pattern is too large to be constructed in one piece
and requires to be built up in pieces, the construction is as in Fig.
168, in which on the left is shown the courses of segments 1, 2, 3, 4,
5, &c., of which the rim is built up (as described for spur wheels), and
on the right is shown the finished rim with a tooth, _c_, in position.

The tooth proper is of the length of face of the wheel as denoted by _b
b´_; now all the lines bounding the teeth must converge to the point X.
Suppose, then, that the teeth are to be shaped for curve of face and
flank in a box as described for spur-wheel teeth in Fig. 146, then in
Fig. 168 let _a_, _a_ represent the bottom and _b b´_ the top of the
box, and _c_ a tooth in the box, its ends filling the opening in the box
at _b b´_ then the curve on the sides of the box at _b´_ must be of the
form shown at F, and the curve on the sides of the box (at the point _b_
of its length) must be as shown at G, the teeth shown in profile at G
and U representing the forms of the teeth at their ends, on the outside
of the wheel rim at _b´_, and on the inside at _b_; having thus made a
box of the correct form on its sides, the teeth may be placed in it and
planed down to it, thus giving all the teeth the same curve.

The spacing for the teeth and their fixing may be done as described for
the bevel pinion.

[Illustration: Fig. 169.]

To construct a pattern wherefrom to cast an endless screw, worm, or
tangent screw, which is to have the worm or thread cut in a lathe.--Take
two pieces, each to form one longitudinal half of the pattern; peg and
screw them together at the ends, an excess of stuff being allowed at
each end for the accommodation of two screws to hold the two halves
together while turning them in the lathe, or dogs, if the latter are
more convenient, as they might be in a large pattern. Turn the piece
down to the size over the top of the thread, after which the core prints
are turned. The body thus formed will be ready to have the worm or
thread cut, and for this purpose the tools shown in Figs. 169 and 140
are necessary.

That shown in Fig. 169 should be flat on the face similar to a parting
tool for cast iron, but should have a great deal more bottom rake, as
strength is not so much an object, and the tool is more easily
sharpened. It has also in addition two little projections A B like the
point of a penknife, formed by filing away the steel in the centre;
these points are to cut the fibres of the wood, the severed portion
being scraped away by the flat part of the tool.

[Illustration: Fig. 170.]

The degree of side rake given to the tool must be sufficient to let the
tool sides well clear the thread or worm, and will therefore vary with
the pitch of the worm.

The width of the tool must be a shade narrower than the narrowest part
of the space in the worm. Having suitably adjusted the change wheels of
the lathe to cut the pitch required the parting tool is fed in until the
extreme points reach the bottom of the spaces, and a square nosed
parting tool without any points or spurs will finish the worm to the
required depth. This will have left a square thread, and this we have
now to cut to the required curves on the thread or worm sides, and as
the cutting will be performed on the end grain of the wood, the top face
of the tool must be made keen by piercing through the tool a slot A,
Fig. 170, and filing up the bevel faces B, C and D, and then carefully
oilstoning them. This tool should be made slightly narrower than the
width of the worm space, so that it may not cut on both sides at once,
as it would have too great a length of cutting edge.

[Illustration: Fig. 171.]

Furthermore, if the pattern is very large, it will be necessary to have
two tools for finishing, one to cut from the pitch line inwards and the
other to complete the form from the pitch line outwards. It is advisable
to use hard wood for the pattern.

If it is decided to cut the thread by hand instead of with these lathe
tools, then, the pattern being turned as before, separate the two halves
by taking out the screws at the ends; select the half that has not the
pegs, as being a little more convenient for tracing lines across. Set
out the sections of the thread, A, B, C, and D, Fig. 171, similar to a
rack; through the centres of A, B, C, and D, square lines across the
piece; these lines, where they intersect the pitch line, will give the
centres of teeth on that side: or if we draw lines, as E, F, through the
centres of the spaces, they will pass through the centres of the teeth
(so to speak) on the other side; in this position complete the outline
on that side. It will be found, in drawing these outlines, that the
centres of some of the arcs will lie outside the pattern. To obtain
support for the compasses, we must fit over the pattern a piece of board
such as shown by dotted lines at G H.

[Illustration: Fig. 172.]

It now remains to draw in the top of the thread upon the curved surface
of the half pattern; for this purpose take a piece of stiff card or
other flexible material, wrap it around the pattern and fix it
temporarily by tacks, we then trim off the edges true to the pattern,
and mark upon the edges of the card the position of the tops of the
thread upon each side; we remove the card and spread it out on a flat
surface, join the points marked on the edges by lines as in Fig. 172,
replace the card exactly as before upon the pattern, and with a fine
scriber we prick through the lines. The cutting out is commenced by
sawing, keeping, of course, well within the lines; and it is facilitated
by attaching a stop to the saw so as to insure cutting at all parts
nearly to the exact depth. This stop is a simple strip of wood and may
be clamped to the saw, though it is much more convenient to have a
couple of holes in the saw blade for the passage of screws. For
finishing, a pair of templates, P and Q, Fig. 173, right and left, will
be found useful; and finally the work should be verified and slight
imperfections corrected by the use of a form or template taking in three
spaces, as shown at R in Fig. 173. In drawing the lines on the card, we
must consider whether it is a right or left-handed worm that we desire.
In the engraving the lines are those suitable for a right-handed thread.
Having completed one half of the pattern, place the two halves together,
and trace off the half that is uncut, using again the card template for
drawing the lines on the curved surface. The cutting out will be the
same as before.

[Illustration: Fig. 173.]

As the teeth of cast wheels are, from their deviation from accuracy in
the tooth curves and the concentricity of the teeth to the wheel centre,
apt to create noise in running, it is not unusual to cast one or both
wheels with mortises in the rim to receive wooden teeth. In this case
the wheel is termed a mortise wheel, and the teeth are termed _cogs_. If
only one of a pair of wheels is to be cogged, the largest of the pair is
usually selected, because there are in that case more teeth to withstand
the wear, it being obvious that the wear is greatest upon the wheel
having the fewest teeth, and that the iron wheel or pinion can better
withstand the wear than the mortise wheel. The woods most used for cogs
are hickory, maple, hornbeam and locust. The blocks wherefrom the teeth
are to be formed are usually cut out to nearly the required dimensions,
and kept in stock, so as to be thoroughly well-seasoned when required
for use, and, therefore less liable to come loose from shrinkage after
being fitted to the mortise in the wheel. The length of the shanks is
made sufficient to project through the wheel rim and receive a pin, as
shown in Fig. 174, in which B is a blank tooth, and C a finished tooth
inserted in the wheel, the pin referred to being at P. But, if a mortise
should fall in an arm of the wheel, this pin-hole must pass through the
rim, as shown in the mortise A. The wheel, however, should be designed
so that the mortises will not terminate in the arms of the wheel.

[Illustration: Fig. 174.]

[Illustration: Fig. 175.]

Another method of securing the teeth in the mortises is to dovetail them
at the small end and drive wedges between them, as shown in Fig. 175, in
which C C are two contiguous teeth, R the wheel rim and W W two of the
wedges. On account of the dovetailing the wedges exert A pressure
pressing the teeth into the mortises. This plan is preferable to that
shown in the Fig. 174 inasmuch as from the small bearing area of the
pins they become loose quicker, and furthermore there is more elasticity
to take up the wear in the case of the wedges.

[Illustration: Fig. 176.]

The mortises are first dressed out to a uniform size and taper, using
two templates to test them with, one of which is for the breadth and the
other for the width of the mortise. The height above the wheel requires
to be considerably more than that due to the depth of the teeth, so that
the surface bruised by driving the cogs or when fitting them into the
mortises may be cut off. To avoid this damage as much as possible, a
broad-face hammer should be employed--a copper, lead, lignum vitæ, or a
raw hide hammer being preferable, and the last the best. The teeth are
got out in a box and two guides, such as shown in Figs. 176, 177, and
178, similar letters of reference denoting the same parts in all three
illustrations.

In Fig. 176, X is a frame or box containing and holding the operative
part of the tooth, and resting on two guides C D. The height of D from
the saw table is sufficiently greater than that of C to give the shank G
the correct taper, E F representing the circular saw. T is a plain piece
of the full size of the box or frame, and serving simply to close up on
that side the mortise in the frame. The grain of T should run at a right
angle to the other piece of the frame so as to strengthen it. S is a
binding screw to hold the cog on the frame, and H is a guide for the
edge of the frame to slide against. It is obvious, now, that if the
piece D be adjusted at a proper distance from the circular saw E F, and
the edge of the frame be moved in contact with the guide H, one side of
the tooth shank will be sawn. Then, by reversing the frame end for end,
the other side of the shank may be sawn. Turning the frame to a right
angle the edges of the cog shank can be sawn from the same box or frame,
and pieces C, D, as shown in Fig. 177.

[Illustration: Fig. 177.]

The frame is now stood on edge, as in Fig. 178, and the underneath
surfaces sawed off to the depth the saw entered when the shank taper was
sawn. This operation requires to be performed on all four sides of the
tooth.

After this operation is performed on one cog, it should be tried in the
wheel mortises, to test its correctness before cutting out the shanks on
all the teeth.

[Illustration: Fig. 178.]

The shanks, being correctly sawn, may then be fitted to the mortises,
and let in within 1/8 of butting down on the face of the wheel, this
amount being left for the final driving. The cogs should be numbered to
their places, and two of the mortises must be numbered to show the
direction in which the numbers proceed. To mark the shoulders (which are
now square) to the curvature of the rim, a fork scriber should be used,
and the shanks of the cogs should have marked on them a line coincident
with the inner edge of the wheel rim. This line serves as a guide in
marking the pin-holes and for cutting the shanks to length; but it is to
be remembered that the shanks will pass farther through to the amount of
the distance marked by the fork scriber. The holes for the pins which
pass through the shanks should be made slightly less in their distances
(measured from the nearest edge of the pin-hole) from the shoulders of
the cogs than is the thickness of the rim of the wheel, so that when the
cogs are driven fully home the pin-holes will appear not quite full
circles on the inside of the wheel rim; hence, the pins will bind
tightly against the inside of the wheel rim, and act somewhat as keys,
locking and drawing the shanks to their seats in the mortises.

In cases where quietness of running is of more consequence than the
durability of the teeth, or where the wear is not great, both wheels may
be cogged, but as a rule the larger wheel is cogged, the smaller being
of metal. This is done because the teeth of the smaller wheel are the
most subject to wear. The teeth of the cogged wheel are usually made the
thickest, so as to somewhat equalise the strength of the teeth on the
two wheels.

Since the power transmitted by a wheel in a given time is composed of
the pressure or weight upon the wheel, and the space a point on the
pitch circle moves through in the given time, it is obvious that in a
train of wheels single geared, the velocities of all the wheels in the
train being equal at the pitch circle, the teeth require to be of equal
pitch and thickness throughout the train. But when the gearing is
compounded the variation of velocity at the pitch circle, which is due
to the compounding, has an important bearing upon the necessary strength
of the teeth.

Suppose, for example, that a wheel receives a tooth pressure of 100 lbs.
at the pitch circle, which travels at the velocity of 100 feet per
minute, and is keyed to the same shaft with another wheel whose velocity
is 50 feet per minute. Now, in the power transmitted by the two wheels
the element of time is 50 for one wheel and 100 for the other, hence the
latter (supposing both wheels to have an equal number of teeth in
contact with their driver or follower as the case may be) will be twice
as strong in proportion to the duty, and it appears that in compounded
gearing the strength in proportion to the duty may be varied in
proportion as the velocity is modified by compounding of the wheels.
Thus, when the velocity at the pitch circle is increased its strength is
increased, and per contra when its velocity is decreased its strength is
decreased, when considered in proportion to the duty. When, however, the
wheels are upon long shafts, or when they overhang the bearing of the
shaft, the corner contact will from tension of the shaft, continue much
longer than when the shaft is maintained rigid.

It is obvious that if a wheel transmits a certain amount of power, the
pressure of tooth upon tooth will depend upon the number of teeth in
contact, but since, in the case of very small wheels, that is to say,
pinions of the smallest diameter of the given pitch that will transmit
continuous motion, it occurs that only one tooth is in continuous
contact, it is obvious that each single tooth must have sufficient
strength to withstand the whole of the pressure when worn to the limits
to which the teeth are supposed to wear. But when the pinion is so small
that it has but one tooth in continuous contact, that contact takes
place nearer the line of centres and to the root of the tooth, and
therefore at a less leverage to the line of fracture, hence the ultimate
strength of the tooth is proportionately increased. On the other hand,
however, the whole stress of the wheel being concentrated on the arc of
contact of one tooth only (instead of upon two or more teeth as in
larger wheels), the wear is proportionately greater; hence, in a short
time the teeth of the pinion are found to be thinner than those on the
other wheel or wheels. The multiplicity of conditions under which small
wheels may work with relation to the number of teeth in contact, the
average leverage of the point of contact from the root of the tooth, the
shape of the tooth, &c., renders it desirable in a general rule to
suppose that the whole strain falls upon one tooth, so that the
calculation shall give results to meet the requirements when a single
tooth only is in continuous contact.

It follows, then, that the thickness of tooth arrived at by calculation
should be that which will give to a tooth, when worn to the extreme
thinness allowed, sufficient strength (with a proper margin of safety)
to transmit the whole of the power transmitted by the wheel.

The margin (or factor) of safety, or in other words, the number of times
the strength of the tooth should exceed the amount of power transmitted,
varies (according to the conditions under which the wheels work) between
5 and 10.

The lesser factor may be used for slow speeds when the power is
continuously and uniformly transmitted. The greater factor is necessary
when the wheels are subjected to violent shocks and the direction of
revolution requires to be reversed.

[Illustration: Fig. 179.]

In pattern-cast teeth, contact between the teeth of one wheel and those
of the other frequently occurs at one corner only, as shown in Fig. 179,
and the line of fracture is in the direction denoted by the diagonal
dotted lines. The causes of this corner contact have been already
explained, but it may be added that as the wheels wear, the contact
extends across the full breadths of the teeth, and the strength in
proportion to the duty, therefore, steadily increases from the time the
new wheels have action until the wear has caused contact fully across
the breadth. Tredgold's rule for finding the proper thickness of tooth
for a given stress upon cast-iron teeth loaded at the corner as in Fig.
179 and supposed to have a velocity of three feet per second of time, is
as follows:--

Rule.--Divide the stress in pounds at the pitch circle by 1500, and the
square root of the quotient is the required thickness of tooth in inches
or parts of an inch.

In the results obtained by the employment of this rule, an allowance of
one-third the thickness for wear, and the margin for safety is included,
so that the thickness of tooth arrived at is that to be given to the
actual tooth. Further, the rule supposes the breadth of the tooth to be
not less than twice the height of the same, any extra breadth not
affecting the result (as already explained), when the pressure falls on
a corner of the tooth.

In practical application, however, the diameter of the wheel at the
pitch circle is generally, or at least often a fixed quantity, as well
as the amount of stress, and it will happen as a rule that taking the
stress as a fixed element and arriving at the thickness of the tooth by
calculation, the required diameter of wheel, or what is the same thing,
its circumference, will not be such as to contain the exact number of
teeth of the thickness found by the calculation, and still give the
desired amount of side clearance. It is desirable, therefore, to deal
with the stress upon the tooth at the pitch circle, and the diameter,
radius, or circumference of the pitch circle, and its velocity, and
deduce therefrom the required thickness for the teeth, and conform the
pitch to the requirements as to clearance from the tooth thickness thus
obtained.

To deduce the thickness of the teeth from these elements we have
Robertson Buchanan's rule, which is as follows:--

Find the amount of horse-power employed to move the wheel, and divide
such horse-power by the velocity in feet per second of the pitch line of
the wheel. Extract the square root of the quotient, and three-fourths of
this root will be the least thickness of the tooth. To the result thus
obtained, there must be added the allowance for wear of the teeth and
the width of the space including the clearance which will determine the
number of teeth in the wheel.

In conforming strictly to this rule the difficulty is met with that it
would give fractional pitches not usually employed and difficult to
measure on an existing wheel. Cast wheels kept on hand or in stock by
machinists have usually the following standard:--

Beginning with an inch pitch, the pitches increase by 1/8 inch up to
3-inch pitch, from 3 to 4-inch pitches the increase is by 1/4 inch, and
from 4-inch pitch and upwards the increase is by 1/2 inch. Now, under
the rule the pitches would, with the clearance made to bear a certain
proportion to the pitch, be in odd fractions of an inch.

It appears then, that, if in a calculation to obtain the necessary
thickness of tooth, the diameter of the pitch circle is not an element,
the rule cannot be strictly adhered to unless the diameter of the pitch
circle be varied to suit the calculated thickness of tooth; or unless
either the clearance, factor of safety, or amount of tooth thickness
allowed for wear be varied to admit of the thickness of tooth arrived at
by the calculation. But if the diameter of the pitch circle is one of
the elements considered in arriving at the thickness of tooth requisite
under given conditions, the pitch must, as a rule, either be in odd
fractions, or else the allowance for wear, factor of safety, or amount
of side clearance cannot bear a definite proportion to the pitch. But
the allowance for clearance is in practice always a constant proportion
of the pitch, and under these circumstances, all that can be done when
the circumstances require a definite circumference of pitch circle, is
to select such a pitch as will nearest meet the requirements of tooth
thickness as found by calculation, while following the rule of making
the clearance a constant proportion of the pitch. When following this
plan gives a thinner tooth than the calculation calls for, the factor of
safety and the allowance for wear are reduced. But this is of little
consequence whenever more than one tooth on each wheel is in contact,
because the rules provide for all the stress falling on one tooth. When,
however, the number of teeth in the pinion is so small that one tooth
only is in contact, it is better to select a pitch that will give a
thicker rather than a thinner tooth than called for by the calculation,
providing, of course, that the pitch be less than the arc of contact, so
that the motion shall be continuous.

But when the pinions are shrouded, that is, have flanges at each end,
the teeth are strengthened; and since the wear will continue greater
than in wheels having more teeth in contact, the shrouding may be
regarded as a provision against breakage in consequence of the reduction
of tooth thickness resulting from wear.

In the following table is given the thickness of the tooth for a given
stress at the pitch circle, calculated from Tredgold's rule for teeth
supposed to have contact when new at one corner only.

  +-------------------+--------------------+---------------------+
  | Stress in lbs. at | Thickness of tooth |  Actual pitches to  |
  |   pitch circle.   |     in inches.     | which wheels may be |
  |                   |                    |        made.        |
  +-------------------+--------------------+---------------------+
  |        400        |        .52         |   1-1/8 to 1-1/4    |
  |        800        |        .75         |   1-1/2  " 1-5/8    |
  |      1,200        |        .90         |   1-7/8  " 2        |
  |      1,600        |       1.03         |   2      " 2-1/8    |
  |      2,000        |       1.15         |   2-1/4  " 2-3/8    |
  |      2,400        |       1.26         |   2-1/2  " 2-5/8    |
  |      2,800        |       1.36         |   2-5/8  " 2-3/4    |
  |      3,200        |       1.43         |   2-7/8  " 3        |
  |      3,600        |       1.56         |   3-1/8  " 3-1/4    |
  |      4,000        |       1.63         |   3-1/4  " 3-3/8    |
  |      4,400        |       1.70         |   3-3/8  " 3-1/2    |
  |      4,800        |       1.78         |   3-1/2  " 3-5/8    |
  |      5,200        |       1.86         |   3-5/8  " 3-3/4    |
  |      5,600        |       1.93         |   3-3/4  " 4        |
  |      6,000        |       2.00         |   4      " 4-1/4    |
  +-------------------+--------------------+---------------------+

In wheels that have their teeth cut to form in a gear-cutting machine
the thickness of tooth at any point in the depth is equal at any point
across the breadth; hence, supposing the wheels to be properly keyed to
their shafts so that the pitch line across the breadth of the wheel
stands parallel to the axis of the shaft, the contact of tooth upon
tooth occurs across the full breadth of the tooth.

As the practical result of these conditions we have three important
advantages: first, that the stress being exerted along the full breadth
of the tooth instead of on one corner only, the tooth is stronger (with
a given breadth and thickness) in proportion to the duty; second, that
with a given pitch, the thickness and therefore the margin for safety
and allowance for wear are increased, because the tooth may be increased
in thickness at the expense of the clearance, which need be merely
sufficient to prevent contact on both sides of the spaces so as to
prevent the teeth from locking in the spaces; and thirdly, because the
teeth will not be subject to sudden impacts or shocks of tooth upon
tooth by reason of back lash.

[Illustration: Fig. 180.]

In determining the strength of cut gear-teeth we may suppose the weight
to be disposed along the face at the extreme height of the tooth, in
which case the theoretical shape of the tooth to possess equal strength
at every point from the addendum circle to the root would be a parabola,
as shown by the dotted lines in Fig. 180, which represents a tooth
having radial flanks. In this case it is evident that the ultimate
strength of the tooth is that due to the thickness at the root, because
it is less than that at the pitch circle, and the strength, as a whole,
is not greater than that at the weakest part. But since teeth with
radial flanks are produced, as has been shown, with a generating circle
equal in diameter to the radius of the pinion, and since with a
generating circle bearing that ratio of diameter to diameter of pitch
circle the acting part of the flank is limited, it is usual to fill in
the corners with fillets or rounded corners, as shown in Fig. 129;
hence, the weakest part of the tooth will be where the radial line of
the flank joins the fillet and, therefore, nearer the pitch circle than
is the root. But as only the smallest wheel of the set has radial flanks
and the flanks thicken as the diameter of the wheels increase, it is
usual to take the thickness of the tooth at the pitch circle as
representing the weakest part of the tooth, and, therefore, that from
which the strength of the tooth is to be computed. This, however, is not
actually the case even in teeth which have considerable spread at the
roots, as is shown in Fig. 181, in which the shape of the tooth to
possess equal strength throughout its depth is denoted by the parabolic
dotted lines.

[Illustration: Fig. 181.]

Considering a tooth as simply a beam supporting the strain as a weight
we may calculate its strength as follows:--

Multiply the breadth of the tooth by the square of its thickness, and
the product by the strength of the material, per square inch of section,
of which the teeth are composed, and divide this last product by the
distance of the pitch line from the root, and the quotient will give a
tooth thickness having a strength equal to the weight of the load, but
having no margin for safety, and no allowance for wear; hence, the
result thus obtained must be multiplied by the factor of safety (which
for this class of tooth may be taken as 6), and must have an additional
thickness added to allow for wear, so that the factor of safety will be
constant notwithstanding the wear.

Another, and in some respects more convenient method, for obtaining the
strength of a tooth, is to take the strength of a tooth having 1-inch
pitch, and 1 inch of breadth, and multiply this quantity of strength by
the pitch and the face of the tooth it is required to find the strength
of, both teeth being of the same material.

Example.--The safe working pressure for a cast-iron tooth of an inch
pitch, and an inch broad will transmit, being taken as 400 lbs., what
pressure will a tooth of 3/4-inch pitch and 3 inches broad transmit with
safety?

Here 400 lbs. × 3/4 pitch × 3 breadth = 900 = safe working pressure of
tooth 3/4-inch pitch and 3 inches broad.

Again, the safe working pressure of a cast-iron tooth, 1 inch in breadth
and of 1-inch pitch, being considered as 400 lbs., what is the safe
working pressure of a tooth of 1-inch pitch and 4-inch breadth?

Here 400 × 1 × 4 = 1600.

The philosophy of this is apparent when we consider that four wheels of
1-inch pitch and an inch face, placed together side by side, would
constitute, if welded together, one wheel of an inch pitch and 4 inches
face. (The term _face_ is applied to the wheel, and the term breadth to
the tooth, because such is the custom of the workshop, both terms,
however, mean, in the case of spur-wheels, the dimension of the tooth in
a direction parallel to the axis of the wheel shaft or wheel bore.)

The following table gives the safe working pressures for wheels having
an inch pitch and an inch face when working at the given velocities,
S.W.P. standing for "safe working pressure:"--

  +------------+------------+------------+------------+--------------+
  | Velocity of|            |            |            |              |
  |pitch circle| S.W.P. for | S.W.P. for |S.W.P. for  |   S.W.P.     |
  |  in feet   | cast-iron  |spur mortise| cast-iron  |  for bevel   |
  |per second. |spur gears. |   gears.   |bevel gears.|mortise gears.|
  +------------+------------+------------+------------+--------------+
  |     2      |     368    |    178     |    258     |      178     |
  |     3      |     322    |    178     |    225     |      157     |
  |     6      |     255    |    178     |    178     |      125     |
  |    12      |     203    |    142     |    142     |       99     |
  |    18      |     177    |    124     |    124     |       87     |
  |    24      |     161    |    113     |    113     |       79     |
  |    30      |     150    |    105     |    105     |       74     |
  |    36      |     140    |     98     |     98     |       69     |
  |    42      |     133    |     93     |     93     |       65     |
  |    48      |     127    |     88     |     88     |       62     |
  +------------+------------+------------+------------+--------------+

For velocities less than 2 feet per second, use the same value as for 2
feet per second.

The proportions, in terms of the pitch, upon which this table is based,
are as follows:--

  Thickness of iron  teeth     .395 of the pitch.
      "       wooden   "       .595     "
  Height of addendum           .28      "
  Depth below pitch line       .32      "

The table is based upon 400 lbs. per inch of face for an inch pitch, as
the safe working pressure of mortise wheel teeth or cogs; it may be
noted that there is considerable difference of opinion. They are claimed
by some to be in many cases practically stronger than teeth of cast
iron. This may be, and probably is, the case when the conditions are
such that the teeth being rigid and rigidly held (as in the case of
cast-iron teeth), there is but one tooth on each wheel in contact. But
when there is so nearly contact between two teeth on each wheel that but
little elasticity in the teeth would cause a second pair of teeth to
have contact, then the elasticity of the wood would cause this second
contact. Added to this, however, we have the fact that under conditions
where violent shock occurs the cog would have sufficient elasticity to
give, or spring, and thus break the shock which cast iron would resist
to the point of rupture. It is under these conditions, which mainly
occur in high velocities with one of the wheels having cast teeth, that
mortise wheels, or cogging, is employed, possessing the advantage that a
broken or worn-out tooth, or teeth, may be readily replaced. It is
usual, however, to assign to wooden teeth a value of strength more
nearly equal to that of its strength in proportion to that of cast iron;
hence, Thomas Box allows a wood tooth a value of about 3/10ths the
strength of cast iron; a value as high as 7/10ths is, however, assigned
by other authorities. But the strength of the tooth cannot exceed that
at the top of the shank, where it fits into the mortise of the wheel,
and on account of the leverage of the pressure the width of the mortise
should exceed the thickness of the tooth.

In some practice, the mortise teeth, or cogs, are made thicker in
proportion to the pitch than the teeth on the iron wheel; thus Professor
Unwin, in his "Elements of Machine Design," gives the following as "good
proportions":--

  Thickness of iron teeth  0.395 of the pitch.
      "        wood cogs   0.595   "      "

which makes the cogs 2/10ths inch thicker than the teeth.

The mortises in the wheel rim are made taper in both the breadth and the
width, which enables the tooth shank to be more accurately fitted, and
also of being driven more tightly home, than if parallel. The amount of
this taper is a matter of judgment, but it may be observed that the
greater the taper the more labor there is involved in fitting, and the
more strain there is thrown upon the pins when locking the teeth with a
given amount of strain. While the less the taper, the more care required
to obtain an accurate fit. Taking these two elements into consideration,
1/8th inch of taper in a length of 4 inches may be given as a desirable
proportion.

[Illustration: Fig. 182.]

As an evidence of the durability of wooden teeth, there appeared in
_Engineering_ of January 7th, 1879, the illustration shown in Fig. 182,
which represents a cog from a wheel of 14 ft. 1/2 in. diameter, and
having a 10-inch face, its pinion being 4 ft. in diameter. This cog had
been running for 26-1/2 years, day and night; not a cog in the wheel
having been touched during that time. Its average revolutions were 38
per minute, the power developed by the engine being from 90 to 100
indicated horse-power. The teeth were composed of beech, and had been
greased twice a week, with tallow and plumbago ore.

Since the width of the face of a wheel influences its wear (by providing
a larger area of contact over which the pressure may be distributed, as
well as increasing the strength), two methods of proportioning the
breadth may be adopted. First, it may be made a certain proportion of
the pitch; and secondly, it may be proportioned to the pressure
transmitted and the number of revolutions. The desirability of the
second is manifest when we consider that each tooth will pass through
the arcs of contact (and thus be subjected to wear) once during each
revolution; hence, by making the number of revolutions an element in the
calculation to find the breadth, the latter is more in proportion to the
wear than it would be if proportioned to the pitch.

It is obvious that the breadth should be sufficient to afford the
required degree of strength with a suitable factor of safety, and
allowance for wear of the smallest wheel in the pair or set, as the case
may be.

According to Reuleaux, the face of a wheel should never be less than
that obtained by multiplying the gross pressure, transmitted in lbs., by
the revolutions per minute, and dividing the product by 28,000.

In the case of bevel-wheels the pitch increases, as the perimeter of
the wheel is approached, and the maximum pitch is usually taken as the
designated pitch of the wheel. But the mean pitch is that which should
be taken for the purposes of calculating the strength, it being in the
middle of the tooth breadth. The mean pitch is also the diameter of the
pitch circle, used for ascertaining the velocity of the wheel as an
element in calculating the safe pressure, or the amount of power the
wheel is capable of transmitting, and it is upon this basis that the
values for bevel-wheels in the above table are computed.

In many cases it is required to find the amount of horse-power a wheel
will transmit, or the proportions requisite for a wheel to transmit a
given horse-power; and as an aid to the necessary calculations, the
following table is given of the amount of horse-power that may be
transmitted with safety, by the various wheels at the given velocities,
with a wheel of an inch pitch and an inch face, from which that for
other pitches and faces may be obtained by proportion.

TABLE SHOWING THE HORSE-POWER WHICH DIFFERENT KINDS OF GEAR-WHEELS OF
ONE INCH PITCH AND ONE INCH FACE WILL SAFELY TRANSMIT AT VARIOUS
VELOCITIES OF PITCH CIRCLE.

  +------------+------------+------------+-------------+-------------+
  |Velocity of |            |            |             |             |
  |Pitch Circle|Spur-Wheels.|Spur Mortise|Bevel-Wheels.|Bevel Mortise|
  |in Feet per |    H.P.    |  Wheels.   |    H.P.     |   Wheels.   |
  |Second.     |            |    H.P.    |             |     H.P.    |
  +------------+------------+------------+-------------+-------------+
  |      2     |    1.338   |     .647   |     .938    |     .647    |
  |      3     |    1.756   |     .971   |    1.227    |     .856    |
  |      6     |    2.782   |    1.76    |    1.76     |    1.363    |
  |     12     |    4.43    |    3.1     |    3.1      |    2.16     |
  |     18     |    5.793   |    4.058   |    4.058    |    2.847    |
  |     24     |    7.025   |    4.931   |    4.931    |    3.447    |
  |     30     |    8.182   |    5.727   |    5.727    |    4.036    |
  |     36     |    9.163   |    6.414   |    6.414    |    4.516    |
  |     42     |   10.156   |    7.102   |    7.102    |    4.963    |
  |     48     |   11.083   |    7.680   |    7.680    |    5.411    |
  +------------+------------+------------+-------------+-------------+

In this table, as in the preceding one, the safe working pressure for
1-inch pitch and 1-inch breadth of face is supposed to be 400 lbs.

In cast gearing, the mould for which is made by a gear moulding machine,
the element of draft to permit the extraction of the pattern is reduced:
hence, the pressure of tooth upon tooth may be supposed to be along the
full breadth of the tooth instead of at one corner only, as in the case
of pattern-moulded teeth. But from the inaccuracies which may occur from
unequal contraction in the cooling of the casting, and from possible
warping of the casting while cooling, which is sure to occur to some
extent, however small the amount may be, it is not to be presumed that
the contact of the teeth of one wheel will be in all the teeth as
perfect across the full breadth as in the case of machine-cut teeth.
Furthermore, the clearance allowed for machine-moulded teeth, while
considerably less than that allowed for pattern-moulded teeth, is
greater than that allowed for machine-cut teeth; hence, the strength of
machine-moulded teeth in proportion to the pitch lies somewhere between
that of pattern-moulded and machine-cut teeth--but exactly where, it
would be difficult to determine in the absence of experiments made for
the purpose of ascertaining.

It is not improbable, however, that the contact of tooth upon tooth
extends in cast gears across at least two-thirds of the breadth of the
tooth, in which case the rules for ascertaining the strength of cut
teeth of equal thickness may be employed, substituting 2/3rds of the
actual tooth breadth as the breadth for the purposes of the calculation.

If instead of supposing all the strain to fall upon one tooth and
calculating the necessary strength of the teeth upon that basis (as is
necessary in interchangeable gearing, because these conditions may exist
in the case of the smallest pinion that can be used in pitch), the
actual working condition of each separate application of gears be
considered, it will appear that with a given diameter of pitch circle,
all other things being equal, the arc of contact will remain constant
whatever the pitch of the teeth, or in other words is independent of the
pitch, and it follows that when the thickness of iron necessary to
withstand (with the allowances for wear and factor of safety) the given
stress under the given velocity has been determined, it may be disposed
in a coarse pitch that will give one tooth always in contact, or a finer
pitch that will give two or more teeth always in contact, the strength
in proportion to the duty remaining the same in both cases.

In this case the expense of producing the wheel patterns or in trimming
the teeth is to be considered, because if there are a train of wheels
the finer pitch would obviously involve the construction and dressing to
shape of a much greater number of teeth on each wheel in the train, thus
increasing the labor. When, however, it is required to reduce the pinion
to a minimum diameter, it is obvious that this may be accomplished by
selecting the finer pitch, because the finer the pitch, the less the
diameter of the wheel may be. Thus with a given diameter of pitch circle
it is possible to select a pitch so fine that motion from one wheel may
be communicated to another, whatever the diameter of the pitch circle
may be, the limit being bounded by the practicability of casting or
producing teeth of the necessary fineness of pitch. The durability of a
wheel having a fine pitch is greater for two reasons: first, because the
metal nearest the cast surface of cast iron is stronger than the
internal metal, and the finer pitch would have more of this surface to
withstand the wear; and second, because in a wheel of a given width
there would be two points, or twice the area of metal, to withstand the
abrasion, it being remembered that the point of contact is a line which
partly rolls and partly slides along the depth of the tooth as the wheel
rotates, and that with two teeth in contact on each wheel there are two
of such lines. There is also less sliding or rubbing action of the
teeth, but this is offset by the fact that there are more teeth in
contact, and that there are therefore a greater number of teeth
simultaneously rubbing or sliding one upon the other.

But when we deal with the number of teeth the circumstances are altered;
thus with teeth of epicycloidal form it is manifestly impossible to
communicate constant motion with a driving wheel having but one tooth,
or to receive motion on a follower having but one tooth. The number of
teeth must always be such that there is at all times a tooth of each
wheel within the arc of action, or in contact, so that one pair of teeth
may come into contact before the contact of the preceding teeth has
ceased.

In the construction of wheels designed to transmit power as well as
simple motion, as is the case with the wheels employed in machine work,
however, it is not considered desirable to employ wheels containing a
less number of teeth than 12. The diameter of the wheel bearing such a
relation to the pitch that both wheels containing the same number of
teeth (12), the motion will be communicated from one to the other
continuously.

It is obvious that as the number of teeth in one of the wheels (of a
pair in gear) is increased the number of teeth in the other may be
(within certain limits) diminished, and still be capable of transmitting
continuous motion. Thus a pinion containing, say 8 teeth, may be capable
of receiving continuous motion from a rack in continuous motion, while
it would not be capable of receiving continuous motion from a pinion
having 4 teeth; and as the requirements of machine construction often
call for the transmission of motion from one pinion to another of equal
diameters, and as small as possible, 12 teeth are the smallest number it
is considered desirable for a pinion to contain, except it be in the
case of an internal wheel, in which the arc of contact is greater in
proportion to the diameters than in spur-wheels, and continuous motion
can therefore be transmitted either with coarser pitches or smaller
diameters of pinion.

For convenience in calculating the pitch diameter at pitch circle, or
pitch diameter as it is termed, and the number of teeth of wheels, the
following rules and table extracted from the _Cincinnati Artisan_ and
arranged from a table by D. A. Clarke, are given. The first column gives
the pitch, the following nine columns give the pitch diameters of wheels
for each pitch from 1 tooth to 9. By multiplying these numbers by 10 we
have the pitch diameters from 10 to 90 teeth, increasing by _tens_; by
multiplying by 100 we likewise have the pitch diameters from 100 to 900,
increasing by _hundreds_.

TABLE FOR DETERMINING THE RELATION BETWEEN PITCH DIAMETER, PITCH, AND
NUMBER OF TEETH IN GEAR-WHEELS.

  +-----+------------------------------------------------------------------+
  |     |                       NUMBER OF TEETH.                           |
  |Pitch.------+------+------+------+------+-------+-------+-------+-------+
  |     |   1. |   2. |  3.  |   4. |   5. |   6.  |  7.   |  8.   |  9.   |
  +-----+------+------+------+------+------+-------+-------+-------+-------+
  |1    | .3183| .6366| .9549|1.2732|1.5915| 1.9099| 2.2282| 2.5465| 2.8648|
  |1-1/8| .3581| .7162|1.0743|1.4324|1.7905| 2.1486| 2.5067| 2.8648| 3.2229|
  |1-1/4| .3979| .7958|1.1937|1.5915|1.9894| 2.3873| 2.7852| 3.1831| 3.5810|
  |1-3/8| .4377| .8753|1.3130|1.7507|2.1884| 2.6260| 3.0637| 3.5014| 3.9391|
  |     |      |      |      |      |      |       |       |       |       |
  |1-1/2| .4775| .9549|1.4324|1.9099|2.3873| 2.8648| 3.3422| 3.8197| 4.2971|
  |1-5/8| .5173|1.0345|1.5517|2.0690|2.5862| 3.1035| 3.6207| 4.1380| 4.6552|
  |1-3/4| .5570|1.1141|1.6711|2.2282|2.7852| 3.3422| 3.8993| 4.4563| 5.0134|
  |1-7/8| .5968|1.1937|1.7905|2.3873|2.9841| 3.5810| 4.1778| 4.7746| 5.3714|
  |     |      |      |      |      |      |       |       |       |       |
  |2    | .6366|1.2732|1.9099|2.5465|3.1831| 3.8197| 4.4563| 5.0929| 5.7296|
  |2-1/8| .6764|1.3528|2.0292|2.7056|3.3820| 4.0584| 4.7348| 5.4112| 6.0877|
  |2-1/4| .7162|1.4324|2.1486|2.8648|3.5810| 4.2972| 5.0134| 5.7296| 6.4457|
  |2-3/8| .7560|1.5120|2.2679|3.0239|3.7799| 4.5359| 5.2919| 6.0479| 6.8038|
  |     |      |      |      |      |      |       |       |       |       |
  |2-1/2| .7958|1.5915|2.3873|3.1831|3.9789| 4.7746| 5.5704| 6.3662| 7.1619|
  |2-5/8| .8355|1.6711|2.5067|3.3422|4.1778| 5.0133| 5.8499| 6.6845| 7.5200|
  |2-3/4| .8753|1.7507|2.6260|3.5014|4.3767| 5.2521| 6.1274| 7.0028| 7.8781|
  |2-7/8| .9151|1.8303|2.7454|3.6605|4.5757| 5.4908| 6.4059| 7.3211| 8.2362|
  |     |      |      |      |      |      |       |       |       |       |
  |3    | .9549|1.9099|2.8648|3.8197|4.7746| 5.7296| 6.6845| 7.6394| 8.5943|
  |3-1/4|1.0345|2.0690|3.1035|4.1380|5.1725| 6.2070| 7.2415| 8.2760| 9.3105|
  |3-1/2|1.1141|2.2282|3.3422|4.4563|5.5704| 6.6845| 7.7986| 8.9126|10.0268|
  |3-3/4|1.1937|2.3873|3.5810|4.7746|5.9683| 7.1619| 8.3556| 9.5493|10.7429|
  |     |      |      |      |      |      |       |       |       |       |
  |4    |1.2732|2.5465|3.8197|5.0929|6.3662| 7.6394| 8.9127|10.1839|11.4591|
  |4-1/2|1.4324|2.8648|4.2972|5.7296|7.1619| 8.5943|10.0267|11.4591|12.8915|
  |5    |1.5915|3.1831|4.7746|6.3662|7.9577| 9.5493|11.1408|12.7324|14.3240|
  |5-1/2|1.7507|3.5014|5.2521|7.0028|8.7535|10.5042|12.2549|14.0056|15.7563|
  |6    |1.9099|3.8196|5.7295|7.6394|9.5493|11.4591|13.3690|15.2788|17.1887|
  +-----+------+------+------+------+------+-------+-------+-------+-------+

The following rules and examples show how the table is used:

Rule 1.--Given ---- number of teeth and pitch; to find ---- pitch
diameter.

Select from table in columns opposite the given pitch--

First, the value corresponding to the number of units in the number of
teeth.

Second, the value corresponding to the number of tens, and multiply this
by 10.

Third, the value corresponding to the number of hundreds, and multiply
this by 100. Add these together, and their sum is the pitch diameter
required.

Example.--What is the pitch diameter of a wheel with 128 teeth, 1-1/2
inches pitch?

We find in line corresponding to 1-1/2 inch pitch--

  Pitch diameter for   8 teeth                  3.8197
     "      "         20   "                    9.549
     "      "        100   "                   47.75
                     ---                       -------
     "      "        128   "                   61.1187

                    Or about 61-1/8". Answer.

Rule 2.--Given ---- pitch diameter and number of teeth; to find
---- pitch.

First, ascertain by Rule 1 the pitch diameter for a wheel of 1-_inch
pitch_, and the given _number of teeth_.

Second, divide _given pitch diameter_ by the _pitch diameter_ for
1-_inch pitch_.

The quotient is the pitch desired.

Example.--What is the pitch of a wheel with 148 teeth, the pitch
diameter being 72"?

First, pitch diameter for 148 teeth, 1-inch pitch, is--

    8 teeth        2.5465
   40   "         12.732
  100   "         31.83
  ---             -------
  148   "         47.1085

  Second, 72/47.1 = 1.53 inch equal to the pitch.

This is nearly 1-1/2-inch pitch, and if possible the diameter would be
reduced or the number of teeth increased so as to make the wheel exactly
1-1/2-inch pitch.

Rule 3.--Given ---- pitch and pitch diameter; to find ---- number of
teeth.

First, ascertain from table the _pitch diameter_ for 1 _tooth_ of the
given _pitch_.

Second, divide the _given pitch diameter_ by the _value_ found in table.

The quotient is the number required.

Example.--What is the number of teeth in a wheel whose pitch diameter is
42 inches, and pitch is 2-1/2 inches?

First, the pitch diameter, 1 tooth, 2-1/2-inch pitch, is 0.7958 inches.

            42
  Second. ------ = 52.8. Answer.
          0.7958

This gives a fractional number of teeth, which is impossible; so the
pitch diameter will have to be increased to correspond to 53 teeth, or
the pitch changed so as to have the number of teeth come an even number.

Whenever two parallel shafts are connected together by gearing, the
distance between centres being a fixed quantity, and the speeds of the
shafts being of a fixed ratio, then the pitch is generally the best
proportion to be changed, and necessarily may not be of standard size.
Suppose there are two shafts situated in this manner, so that the
distance between their centres is 84 inches, and the speed of one is
2-1/2 times that of the other, what size wheels shall be used? In this
case the pitch diameter and number of teeth of the wheel on the
slow-running shaft have to be 2-1/2 times those of the wheel on the
fast-running shaft; so that 84 inches must be divided into two parts,
one of which is 2-1/2 times the other, and these quantities will be the
pitch radii of the wheels; that is, 84 inches are to be divided into
3-1/2 equal parts, 1 of which is the radius of one wheel, and 2-1/2 of
which the radius of the other, thus 84"/3-1/2 = 24 inches. So that 24
inches is the pitch radius of pinion, pitch diameter = 48 inches; and
2-1/2 × 24 inches = 60 inches is the pitch radius of the wheel, pitch
diameter = 120 inches. The pitch used depends upon the power to be
transmitted; suppose that 2-5/8 inches had been decided as about the
pitch to be used, it is found by Rule 3 that the number of teeth are
respectively 143.6, and 57.4 for wheel and pinion. As this is
impossible, some whole number of teeth, nearest these in value, have to
be taken, one of which is 2-1/2 times the other; thus 145 and 58 are the
nearest, and the pitch for these values is found by Rule 2 to be 2.6
inches, being the best that can be done under the circumstances.

[Illustration: Fig. 183.]

[Illustration: Fig. 184.]

The forms of spur-gearing having their teeth at an angle to the axis, or
formed in advancing steps shown in Figs. 183 and 184, were designed by
Dr. Hooke, and "were intended," says the inventor, "first to make a
piece of wheel work so that both the wheel and pinion, though of never
so small a size, shall have as great a number of teeth as shall be
desired, and yet neither weaken the wheels nor make the teeth so small
as not to be practicable by any ordinary workman. Next that the motion
shall be so equally communicated from the wheel to the pinion that the
work being well made there can be no inequality of force or motion
communicated.

"Thirdly, that the point of touching and bearing shall be always in the
line that joins the two centres together.

"Fourthly, that it shall have _no manner of rubbing_, nor be more
difficult to make than common wheel work."

The objections to this form of wheel lies in the difficulty of making
the pattern and of moulding it in the foundry, and as a result it is
rarely employed at the present day. For racks, however, two or more
separate racks are cast and bolted together to form the full width of
rack as shown in Fig. 185. This arrangement permits of the adjustment of
the width of step so as to take up the lost motion due to the wear of
the tooth curves.

Another objection to the sloping of the teeth, as in Fig. 183, is that
it induces an end pressure tending to force the wheels apart
_laterally_, and this causes _end_ wear on the journals and bearings.

[Illustration: Fig. 185.]

[Illustration: Fig. 186.]

To obviate this difficulty the form of gear shown in Fig. 186 is
employed, the angles of the teeth from each side of the wheel to its
centre being made equal so as to equalize the lateral pressure. It is
obvious that the stepped gear, Fig. 184, is simply equivalent to a
number of thin wheels bolted together to form a thick one, but
possessing the advantage that with a sufficient number of steps, as in
the figure, there is always contact on the line of centres, and that the
condition of constant contact at the line of centres will be approached
in proportion to the number of steps in the wheel, providing that the
steps progress in one continuous direction across the wheel as in Fig.
184. The action of the wheels will, in this event, be smoother, because
there will be less pressure tending to force the wheels apart.

But in the form of gearing shown in Fig. 183, the contact of the teeth
will bear every instant at a single point, which, as the wheels revolve,
will pass from one end to the other of the tooth, a fresh contact always
beginning on the first side immediately before the preceding contact has
ceased on the opposite side. The contact, moreover, being always in the
plane of the centres of the pair, the action is reduced to that of
rolling, and as there is no sliding motion there is consequently no
rubbing friction between the teeth.

[Illustration: Fig. 187.]

[Illustration: Fig. 188.]

A further modification of Dr. Hooke's gearing has been somewhat
extensively adopted, especially in cotton-spinning machines. This
consists, when the direction of the motion is simply to be changed to an
angle of 90°, in forming the teeth upon the periphery of the pair at an
angle of 45° to the respective axes of the wheels, as in Figs. 187 and
188; it will then be perceived that if the sloped teeth be presented to
each other in such a way as to have exactly the same horizontal angle,
the wheels will gear together, and motion being communicated to one axis
the same will be transmitted to the other at a right angle to it, as in
a common bevel pair. Thus if the wheel A upon a horizontal shaft have
the teeth formed upon its circumference at an angle of 45° to the plane
of its axis it can gear with a similar wheel B upon a vertical axis. Let
it be upon the driving shaft and the motion will be changed in direction
as if A and B were a pair of bevel-wheels of the ordinary kind, and, as
with bevels generally, the direction of motion will be changed through
an equal angle to the sum of the angles which the teeth of the wheels of
the pair form with their respective axes. The objection in respect of
lateral or end pressure, however, applies to this form equally with that
shown in Fig. 183, but in the case of a vertical shaft the end pressure
may be (by sloping the teeth in the necessary direction) made to tend to
lift the shaft and not force it down into the step bearing. This would
act to keep the wheels in close contact by reason of the weight of the
vertical shaft and at the same time reduce the friction between the end
of that shaft and its step bearing. This renders this form of gearing
preferable to skew bevels when employed upon vertical shafts.

It is obvious that gears, such as shown in Figs. 187 and 188 may be
turned up in the lathe, because the teeth are simply portions of spirals
wound about the circumference of the wheel. For a pair of wheels of
equal diameter a cylindrical piece equal in length to the required
breadth of the two wheels is turned up in the lathe, and the teeth may
be cut in the same manner as cutting a thread in the lathe, that is to
say, by traversing the tool the requisite distance per lathe revolution.
In pitches above about 1/4 inch, it will be necessary to shape one side
of the tooth at a time on account of the broadness of the cutting edges.
After the spiral (for the teeth are really spirals) is finished the
piece may be cut in two in the lathe and each half will form a wheel.

To find the full diameter to which to turn a cylinder for a pair of
these wheels we proceed as in the following example: Required to cut a
spiral wheel 5 inches in diameter and to have 30 teeth. First find the
diametral pitch, thus 30 (number of teeth) ÷ 5 (diameter of wheel at
pitch circle) = 6; thus there are 6 teeth or 6 parts to every inch of
the wheel's diameter at the pitch circle; adding 2 of these parts to the
diameter of the wheel, at the pitch circle we have 5 and 2/6 of another
inch, or 5-2/6 inches, which is the full diameter of the wheel, or the
diameter of the addendum, as it is termed.

[Illustration: Fig. 189.]

[Illustration: Fig. 190.]

It is now necessary to find what change wheels to put on the lathe to
cut the teeth out the proper angle. Suppose then the axes of the shafts
are at a right angle one to the other, and that the teeth therefore
require to be at an angle of 45° to the axes of the respective wheels,
then we have the following considerations. In Fig. 189 let the line A
represent the circumference of the wheel, and B a line of equal length
but at a right angle to it, then the line C, joining A, B, is at an
angle of 45°. It is obvious then that if the traverse of the lathe tool
be equal at each lathe revolution to the circumference of the wheel at
the pitch circle, the angle of the teeth will be 45° to the axis of the
wheel.

Hence, the change wheels on the lathe must be such as will traverse the
tool a distance equal to the circumference at pitch circle of the wheel,
and the wheels may be found as for ordinary screw cutting.

If, however, the axes of the shafts are at any other angle we may find
the distance the lathe tool must travel per lathe revolution to give
teeth of the required angle (or in other words the pitch of the spiral)
by direct proportion, thus: Let it be required to find the angle or
pitch for wheels to connect shafts at an angle of 25°, the wheels to
have 20 teeth, and to be of 10 diametral pitch.

Here, 20 ÷ 10 = 2 = diameter of wheel at the pitch circle. The
circumference of 2 inches being 6.28 inches we have, as the degrees of
angle of the axes of the shafts are to 45°, so is 6.28 inches (the
circumference of the wheels, to the pitch sought).

Here, 6.28 inches × 45° ÷ 25° = 11.3 inches, which is the required pitch
for the spiral.

When the axes of the shafts are neither parallel nor meeting, motion
from one shaft to another may be transmitted by means of a double gear.
Thus (taking rolling cones of the diameters of the respective pitch
circles as representing the wheels) in Fig. 190, let A be the shaft of
gear _h_, and B _b_ that of wheel _e_. Then a double gear-wheel having
teeth on _f_, _g_ may be placed as shown, and the face _f_ will gear
with _e_, while face _g_ will gear with _h_, the cone surfaces meeting
in a point as at C and D respectively, hence the velocity will be equal.

When the axial line of the shafts for two gear-wheels are nearly in line
one with the other, motion may be transmitted by gearing the wheels as
in Fig. 191. This is a very strong method of gearing, because there are
a large number of teeth in contact, hence the strain is distributed by a
larger number of teeth and the wear is diminished.

[Illustration: Fig. 191.]

[Illustration: Fig. 192.]

Fig. 192 (from Willis's "Principles of Mechanism") is another method of
constructing the same combination, which admits of a steady support for
the shafts at their point of intersection, A being a spherical bearing,
and B, C being cupped to fit to A.

Rotary motion variable at different parts of a rotation may be obtained
by means of gear-wheels varied in form from the true circle.

[Illustration: Fig. 193.]

The commonest form of gearing for this purpose is elliptical gearing,
the principles governing the construction of which are thus given by
Professor McCord. "It is as well to begin at the foundation by defining
the ellipse as a closed plane-curve, generated by the motion of a point
subject to the condition that the sum of its distances from two fixed
points within shall be constant: Thus, in Fig. 193, A and B are the two
fixed points, called the _foci_; L, E, F, G, P are points in the curve;
and A F + F B = A E + E B. Also, A L + L B = A P + P B = A G + G B. From
this it follows that A G = L O, O being the centre of the curve, and G
the extremity of the minor axis, whence the foci may be found if the
axes be assumed, or, if the foci and one axis be given, the other axis
may be determined. It is also apparent that if about either focus, as B,
we describe an arc with a radius greater than B P and less than B L, for
instance B E, and about A another arc with radius A E = L P-B E, the
intersection, E, of these arcs will be on the ellipse; and in this
manner any desired number of points may be found, and the curve drawn by
the aid of sweeps.

"Having completed this ellipse, prolong its major axis, and draw a
similar and equal one, with its foci, C, D, upon that prolongation, and
tangent to the first one at P; then B D = L P. About B describe an arc
with any radius, cutting the first ellipse at Y and the line L at Z;
about D describe an arc with radius D Z, cutting the second ellipse in
X; draw A Y, B Y, C X, and D X. Then A Y = D X, and B Y = C X, and
because the ellipses are alike, the arcs P Y and P X are equal. If then
B and D are taken as fixed centres, and the ellipses turn about them as
shown by the arrows, X and Y will come together at Z on the line of
centres; and the same is true of any points equally distant from P on
the two curves. But this is the condition of rolling contact. We see,
then, that in order that two ellipses may roll together, and serve as
the pitch-lines of wheels, they must be equal and similar, the fixed
centres must be at corresponding foci, and the distance between these
centres must be equal to the major axis. Were they to be toothless
wheels, if would evidently be essential that the outlines should be
truly elliptical; but the changes of curvature in the ellipse are
gradual, and circular arcs may be drawn so nearly coinciding with it,
that when teeth are employed, the errors resulting from the substitution
are quite inappreciable. Nevertheless, the rapidity of these changes
varies so much in ellipses of different proportions, that we believe it
to be practically better to draw the curve accurately first, and to find
the radii of the approximating arcs by trial and error, than to trust to
any definite rule for determining them; and for this reason we give a
second and more convenient method of finding points, in connection with
the ellipse whose centre is R, Fig. 193. About the centre describe two
circles, as shown, whose diameters are the major and minor axes; draw
any radius, as R T, cutting the first circle in T, and the second in S;
through T draw a parallel to one axis, through S a parallel to the
other, and the intersection, V, will lie on the curve. In the left hand
ellipse, the line bisecting the angle A F B is normal to the curve at F,
and the perpendicular to it is tangent at the same point, and bisects
the angles adjacent to A F B, formed by prolonging A F, B F.

[Illustration: Fig. 194.]

"To mark the pitch line we proceed as follows:--

"In Fig. 194, A A and B B are centre lines passing through the major and
minor axes of the ellipse, of which _a_ is the axis or centre, _b_ _c_
is the major and _a_ _e_ half of the minor axis. Draw the rectangle _b_
_f_ _g_ _c_, and then the diagonal line _b_ _e_; at a right angle to _b_
_e_ draw line _f_ _h_ cutting B B at _i_. With radius _a_ _e_ and from
_a_ as a centre draw the dotted arc _e_ _j_, giving the point _j_ on the
line B B. From centre _k_, which is on line B B, and central between _b_
and _j_, draw the semicircle _b_ _m_ _j_, cutting A A at _l_. Draw the
radius of the semicircle _b_ _m_ _j_ cutting _f_ _g_ at _n_. With radius
_m_ _n_ mark on A A, at and from _a_ as a centre, the point _o_. With
radius _h_ _o_ and from centre _h_ draw the arc _p_ _o_ _q_. With radius
_a_ _l_ and from _b_ and _c_ as centres draw arcs cutting _p_ _o_ _q_ at
the points _p_ _q_. Draw the lines _h_ _p_ _r_ and _h_ _q_ _s_, and also
the lines _p_ _i_ _t_ and _q_ _v_ _w_. From _h_ as centre draw that part
of the ellipse lying between _r_ and _s_. With radius _p_ _r_ and from
_p_ as a centre draw that part of the ellipse lying between _r_ and _t_.
With radius _q_ _s_ and from _q_ draw the ellipse from _s_ to _w_. With
radius _i_ _t_ and from _i_ as a centre draw the ellipse from _t_ to
_b_. With radius _v_ _w_ and from _v_ as a centre draw the ellipse from
_w_ to _c_, and one half the ellipse will be drawn. It will be seen that
the whole construction has been performed to find the centres _h_ _p_
_q_ _i_ and _v_, and that while _v_ and _i_ may be used to carry the
curve around the other side or half of the ellipse, new centres must be
provided for _h_ _p_ and _q_; these new centres correspond in position
to _h_ _p_ _q_.

"If it were possible to subdivide the ellipse into equal parts it would
be unnecessary to resort to these processes of approximately
representing the two curves by arcs of circles; but unless this be done,
the spacing of the teeth can only be effected by the laborious process
of stepping off the perimeter into such small subdivisions that the
chords may be regarded as equal to the arcs, which after all is but an
approximation; unless, indeed, we adopt the mechanical expedient of
cutting out the ellipse in metal or other substance, measuring and
subdividing it with a strip of paper or a steel tape, and wrapping back
the divided measure in order to find the points of division on the
curve.

"But these circular arcs may be rectified and subdivided with great
facility and accuracy by a very simple process, which we take from Prof.
Rankine's "Machinery and Mill Work," and is illustrated in Fig. 195. Let
O B be tangent at O to the arc O D, of which C is the centre. Draw the
chord D O, bisect it in E, and produce it to A, making O A = O E; with
centre A and radius A D describe an arc cutting the tangent in B; then O
B will be very nearly equal in length to the arc O D, which, however,
should not exceed about 60°; if it be 60°, the error is theoretically
about 1/900 of the length of the arc, O B being so much too short; but
this error varies with the fourth power of the angle subtended by the
arc, so that for 30° it is reduced to 1/16 of that amount, that is, to
1/14400. Conversely, let O B be a tangent of given length; make O F =
1/4 O B; then with centre F and radius F B describe an arc cutting the
circle O D G (tangent to O B at O) in the point D; then O D will be
approximately equal to O B, the error being the same as in the other
construction and following the same law.

[Illustration: Fig. 195.]

"The extreme simplicity of these two constructions and the facility with
which they may be made with ordinary drawing instruments make them
exceedingly convenient, and they should be more widely known than they
are. Their application to the present problem is shown in Fig. 196,
which represents a quadrant of an ellipse, the approximate arcs C D, D
E, E F, F A having been determined by trial and error. In order to space
this off, for the positions of the teeth, a tangent is drawn at D, upon
which is constructed the rectification of D C, which is D G, and also
that of D E in the opposite direction, that is, D H, by the process just
explained. Then, drawing the tangent at F, we set off in the same manner
F I = F E, and F K = F A, and then measuring H L = I K, we have finally
G L, equal to the whole quadrant of the ellipse.

[Illustration: Fig. 196.]

"Let it now be required to lay out 24 teeth upon this ellipse; that is,
6 in each quadrant; and for symmetry's sake we will suppose that the
centre of one tooth is to be at A, and that of another at C, Fig. 196.
We therefore divide L G into six equal parts at the points 1, 2, 3, &c.,
which will be the centres of the teeth upon the rectified ellipse. It is
practically necessary to make the spaces a little greater than the
teeth; but if the greatest attainable exactness in the operation of the
wheel is aimed at, it is important to observe that backlash, in
elliptical gearing, has an effect quite different from that resulting in
the case of circular wheels. When the pitch-curves are circles, they are
always in contact; and we may, if we choose, make the tooth only half
the breadth of the space, so long as its outline is correct. When the
motion of the driver is reversed, the follower will stand still until
the backlash is taken up, when the motion will go on with a perfectly
constant velocity ratio as before. But in the ease of two elliptical
wheels, if the follower stand still while the driver moves, which must
happen when the motion is reversed if backlash exists, the pitch-curves
are thrown out of contact, and, although the continuity of the motion
will not be interrupted, the velocity ratio will be affected. If the
motion is never to be reversed, the perfect law of the velocity ratio
due to the elliptical pitch-curve may be preserved by reducing the
thickness of the tooth, not equally on each side, as is done in circular
wheels, but wholly on the side not in action. But if the machine must be
capable of acting indifferently in both directions, the reduction must
be made on both sides of the tooth: evidently the action will be
slightly impaired, for which reason the backlash should be reduced to a
minimum. Precisely what _is_ the minimum is not so easy to say, as it
evidently depends much upon the excellence of the tools and the skill of
the workmen. In many treatises on constructive mechanism it is variously
stated that the backlash should be from one-fifteenth to one-eleventh of
the pitch, which would seem to be an ample allowance in reasonably good
castings not intended to be finished, and quite excessive if the teeth
are to be cut; nor is it very obvious that its amount should depend upon
the pitch any more than upon the precession of the equinoxes. On paper,
at any rate, we may reduce it to zero, and make the teeth and spaces
equal in breadth, as shown in the figure, the teeth being indicated by
the double lines. Those upon the portion L H are then laid off upon K I,
after which these divisions are transferred to curves. And since under
that condition the motion of this third line, relatively to each of the
others, is the same as though it rolled along each of them separately
while they remained fixed, the process of constructing the generated
curves becomes comparatively simple. For the describing line, we
naturally select a circle, which, in order to fulfil the condition, must
be small enough to roll within the pitch ellipse; its diameter is
determined by the consideration, that if it be equal to A P, the radius
of the arc A F, the flanks of the teeth in that region will be radial.
We have, therefore, chosen a circle whose diameter, A B, is
three-fourths of A P, as shown, so that the teeth, even at the ends of
the wheels, will be broader at the base than on the pitch line. This
circle ought strictly to roll upon the true elliptical curve, and
assuming as usual the tracing-point upon the circumference, the
generated curves would vary slightly from true epicycloids, and no two
of those used in the same quadrant of the ellipse would be exactly
alike. Were it possible to divide the ellipse accurately, there would be
no difficulty in laying out these curves; but having substituted the
circular arcs, we must now roll the generating circle upon these as
bases, thus forming true epicycloidal teeth, of which those lying upon
the same approximating arc will be exactly alike. Should the junction
of two of these arcs fall within the breadth of a tooth, as at D,
evidently both the face and the flank on one side of that tooth will be
different from those on the other side; should the junction coincide
with the edge of a tooth, which is very nearly the case at F, then the
face on that side will be the epicycloid belonging to one of the arcs,
its flank a hypocycloid belonging to the other; and it is possible that
either the face or the flank on one side should be generated by the
rolling of the describing circle partly on one arc, partly on the one
adjacent, which, upon a large scale and where the best results are aimed
at, may make a sensible change in the form of the curve.

[Illustration: Fig. 197.]

"The convenience of the constructions given in Fig. 194 is nowhere more
apparent than in the drawing of the epicycloids, when, as in the case
in hand, the base and generating circles may be of incommensurable
diameters; for which reason we have, in Fig. 197, shown its application
in connection with the most rapid and accurate mode yet known of
describing those curves. Let C be the centre of the base circle;
B that of the rolling one; A the point of contact. Divide the
semi-circumference of B into six equal parts at 1, 2, 3, &c.; draw the
common tangent at A, upon which rectify the arc A2 by process No. 1,
then by process No. 2 set out an equal arc A2 on the base circle, and
stepping it off three times to the right and left, bisect these spaces,
thus making subdivisions on the base circle equal in length to those on
the rolling one. Take in succession as radii the chords A1, A2, A3, &c.,
of the describing circle, and with centres 1, 2, 3, &c., on the base
circle, strike arcs either externally or internally, as shown
respectively on the right and left; the curve tangent to the external
arcs is the epicycloid, that tangent to the internal ones the
hypocycloid, forming the face and flank of a tooth for the base circle.

"In the diagram, Fig. 196, we have shown a part of an ellipse whose
length is 10 inches and breadth 6, the figure being half size. In order
to give an idea of the actual appearance of the combination when
complete, we show in Fig. 198 the pair in gear, on a scale of 3 inches
to the foot. The excessive eccentricity was selected merely for the
purpose of illustration. Fig. 198 will serve also to call attention to
another serious circumstance, which is that although the ellipses are
alike, the wheels are not; nor can they be made so if there be an even
number of teeth, for the obvious reason that a tooth upon one wheel must
fit into a space on the other; and since in the first wheel, Fig. 196,
we chose to place a tooth at the extremity of each axis, we must in the
second one place there a space instead; because at one time the major
axes must coincide, at another the minor axis, as in Fig. 191. If then
we use even numbers, the distribution and even the forms of the teeth
are not the same in the two wheels of the pair. But this complication
may be avoided by using an odd number of teeth, since, placing a tooth
at one extremity of the major axis, a space will come at the other.

"It is not, however, always necessary to cut teeth all round these
wheels, as will be seen by an examination of Fig. 199, C and D being the
fixed centres of the two ellipses in contact at P. Now P must be on the
line C D, whence, considering the free foci, we see P B is equal to P C,
and P A to P D; and the common tangent at P makes equal angles with C P
and P A, as is also with P B and P D; therefore, C D being a straight
line, A B is also a straight line and equal to C D. If then the wheels
be overhung, that is, fixed on the ends of the shafts outside the
bearings, leaving the outer faces free, the moving foci may be connected
by a rigid link A B, as shown.

[Illustration: Fig. 198.]

"This link will then communicate the same motion that would result from
the use of the complete elliptical wheels, and we may therefore dispense
with most of the teeth, retaining only those near the extremities of the
major axes which are necessary in order to assist and control the motion
of the link at and near the dead-points. The arc of the pitch-curves
through which the teeth must extend will vary with their eccentricity:
but in many cases it would not be greater than that which in the
approximation may be struck about one centre, so that, in fact, it would
not be necessary to go through the process of rectifying and subdividing
the quarter of the ellipse at all, as in this case it can make no
possible difference whether the spacing adopted for the teeth to be cut
would "come out even" or not if carried around the curve. By this
expedient, then, we may save not only the trouble of drawing, but a
great deal of labor in making, the teeth round the whole ellipse. We
might even omit the intermediate portions of the pitch ellipses
themselves; but as they move in rolling contact their retention can do
no harm, and in one part of the movement will be beneficial, as they
will do part of the work; for if, when turning, as shown by the arrows,
we consider the wheel whose axis is D as the driver, it will be noted
that its radius of contact, C P, is on the increase; and so long as this
is the case the other wheel will be compelled to move by contact of the
pitch lines, although the link be omitted. And even if teeth be cut all
round the wheels, this link is a comparatively inexpensive and a useful
addition to the combination, especially if the eccentricity be
considerable. Of course the wheels shown in Fig. 198 might also have
been made alike, by placing a tooth at one end of the major axis and a
space at the other, as above suggested. In regard to the variation in
the velocity ratio, it will be seen, by reference to Fig. 199, that if D
be the axis of the driver, the follower will in the position there shown
move faster, the ratio of the angular velocities being PD/PB; if the
driver turn uniformly the velocity of the follower will diminish, until
at the end of half a revolution, the velocity ratio will be PB/PD; in
the other half of the revolution these changes will occur in a reverse
order. But P D = L B; if then the centres B D are given in position, we
know L P, the major axis; and in order to produce any assumed maximum or
minimum velocity ratio, we have only to divide L P into segments whose
ratio is equal to that assumed value, which will give the foci of the
ellipse, whence the minor axis may be found and the curve described. For
instance, in Fig. 198 the velocity ratio being nine to one at the
maximum, the major axis is divided into two parts, of which one is nine
times as long as the other; in Fig. 199 the ratio is as one to three, so
that, the major axis being divided into four parts, the distance A C
between the foci is equal to two of them, and the distance of either
focus from the nearer extremity of the major axis equal to one, and from
the more remote extremity equal to three of these parts."

[Illustration: Fig. 199.]

Another example of obtaining a variable motion is given in Fig. 200. The
only condition necessary to the construction of wheels of this class is
that the sum of the radii of the pitch circles on the line of centres
shall equal the distance between the axes of the two wheels. The pitch
curves are to be considered the same as pitch circles, "so that," says
Willis, "if any given circle or curve be assumed as a describing (or
generating) curve, and if it be made to roll on the inside of one of
these pitch curves and on the outside of the corresponding portion of
the other pitch curve, then the motion communicated by the pressure and
sliding contact of one of the curved teeth so traced upon the other will
be exactly the same as that effected by the rolling contact (by
friction) of the original pitch curves."

[Illustration: Fig. 200.]

It is obvious that on B the corner sections are formed of simple
segments of a circle of which the centre is the axis of the shaft, and
that the sections between them are simply racks. The corners of A are
segments of a circle of which the axis of A is the centre, and the
sections between the corners curves meeting the pitch circles of the
rack at every point as it passes the line of centres.

Intermittent motion may also be obtained by means of a worm-wheel
constructed as in Fig. 201, the worm having its teeth at a right angle
to its axis for a distance around the circumference proportioned to the
required duration of the period of rest; or the motion may be made
variable by giving the worm teeth different degrees of inclination (to
the axis), on different portions of the circumference.

In addition to the simple operation of two or more wheels transmitting
motion by rotating about their fixed centres and in fixed positions, the
following examples of wheel motion may be given.

[Illustration: Fig. 201.]

[Illustration: Fig. 202.]

In Fig. 202 are two gear-wheels, A, which is fast upon its stationary
shaft, and B, which is free to rotate upon its shaft, the link C
affording journal bearing to the two shafts. Suppose that A has 40
teeth, while B has 20 teeth, and that the link C is rotated once around
the axis of A, how many revolutions will B make? By reason of there
being twice as many teeth in A as in B the latter will make two
rotations, and in addition to this it will, by reason of its connection
to the arm C, also make a revolution, these being two distinct motions,
one a rotation of B about the axis of A, and the other two rotations of
B upon its own axis.

A simple arrangement of gearing for reversing the direction of rotation
of a shaft is shown in Fig. 203. I and F are fast and loose pulleys for
the shaft D, A and C are gears free to rotate upon D, N is a clutch
driven by D; hence if N be moved so as to engage with C the latter will
act as a driver to rotate the shaft B, the wheel upon B rotating A in
an opposite direction to the rotation of D. But if N be moved to engage
with A the latter becomes the driving wheel, and B will be caused to
rotate in the opposite direction. Since, however, the engagement of the
clutch N with the clutch on the nut of the gear-wheels is accompanied
with a violent shock and with noise, a preferable arrangement is shown
in Fig. 204, in which the gears are all fast to their shafts, and the
driving shaft for C passes through the core or bore of that for A, which
is a sleeve, so that when the driving belt acts upon pulley F the shaft
B rotates in one direction, while when the belt acts upon E, B rotates
in the opposite direction, I being a loose pulley.

[Illustration: Fig. 203.]

If the speed of rotation of B require to be greater in one direction
than in the other, then the bevel-wheel on B is made a double one, that
is to say, it has two annular toothed surfaces on its radial face, one
of larger diameter than the other; A gearing with one of these toothed
surfaces, and C with the other. It is obvious that the pinions A C,
being of equal diameters, that gearing with the surface or gear of
largest diameter will give to B the slowest speed of rotation.

[Illustration: Fig. 204.]

Fig. 205 represents Watt's sun-and-planet motion for converting
reciprocating into rotary motion; B D is the working beam of the engine,
whose centre of motion is at D. The gear A is so connected to the
connecting rod that it cannot rotate, and is kept in gear with the wheel
C on the fly-wheel shaft by means of the link shown. The wheel A being
prevented from rotation on its axis causes rotary motion to the wheel C,
which makes two revolutions for one orbit of A.

[Illustration: Fig. 205.]

An arrangement for the rapid increase of motion by means of gears is
shown in Fig. 206, in which A is a stationary gear, B is free to rotate
upon its shaft, and being pivoted upon the shaft of A, at D, is capable
of rotation around A while remaining in gear with C. Suppose now that
the wheel A were absent, then if B were rotated around C with D as a
centre of motion, C and its shaft E would make a revolution even though
B would have no rotation upon its axis. But A will cause B to rotate
upon its axis and thus communicate a second degree of motion to C, with
the result that one revolution of B causes two rotations of C.

The relation of motion between B and C is in this case constant (2 to
1), but this relation may be made variable by a construction such as
shown in Fig. 207, in which the wheel B is carried in a gear-wheel H,
which rides upon the shaft D. Suppose now that H remains stationary
while A revolves, then motion will be transmitted through B to C, and
this motion will be constant and in proportion to the relative diameters
of A and C. But suppose by means of an independent pinion the wheel H be
rotated upon its axis, then increased motion will be imparted to C, and
the amount of the increase will be determined by the speed of rotation
of H, which may be made variable by means of cone pulleys or other
suitable mechanical devices.

[Illustration: Fig. 206.]

Fig. 208 represents an arrangement of gearing used upon steam
fire-engines and traction engines to enable them to turn easily in a
short radius, as in turning corners in narrow streets. The object is to
enable the driving wheel on either side of the engine to increase or
diminish its rotation to suit the conditions caused by the leading or
front pair of steering wheels.

[Illustration: Fig. 207.]

In the figures A is a plate wheel having the lugs L, by means of which
it may be rotated by a chain. A is a working fit on the shaft S, and
carries three pinions E pivoted upon their axes P. F is a bevel-gear, a
working fit on S, while C is a similar gear fast to S. The pinions B, D
are to drive gears on the wheels of the engine, the wheels being a
working fit on the axle. Let it now be noted that if S be rotated, C and
F will rotate in opposite directions and A will remain stationary. But
if A be rotated, then all the gears will rotate with it, but E will not
rotate upon P unless there be an unequal resistance to the motion of
pinions D and B. So soon, however, as there exists an inequality of
resistance between D and B then pinions E operate. For example, let B
have more resistance than D, and B will rotate more slowly, causing
pinion E to rotate and move C faster than is due to the motion of the
chain wheel A, thus causing the wheel on one side of the engine to
retard and the other to increase its motion, and thus enable the engine
to turn easily. From its action this arrangement is termed the
equalizing gear.

[Illustration: Fig. 208.]

In Figs. 209 to 214 are shown what are known as mangle-wheels from their
having been first used in clothes mangling machines.

[Illustration: Fig. 209.]

[Illustration: Fig. 210.]

The mangle-wheel[10] in its simplest form is a revolving disc of metal
with a centre of motion C (Fig. 209). Upon the face of the disc is fixed
a projecting annulus _a_ _m_, the outer and inner edges of which are cut
into teeth. This annulus is interrupted at _f_, and the teeth are
continued round the edges of the interrupted portion so as to form a
continued series passing from the outer to the inner edge and back
again.

  [10] From Willis's "Principles of Mechanism."

A pinion B, whose teeth are of the same pitch as those of the wheel, is
fixed to the end of an axis, and this axis is mounted so as to allow of
a short travelling motion in the direction B C. This may be effected by
supporting this end of it either in a swing-frame moving upon a centre
as at D, or in a sliding piece, according to the nature of the train
with which it is connected. A short pivot projects from the centre of
the pinion, and this rests in and is guided by a groove B S _f_ _t_ _b_
_h_ K, which is cut in the surface of the disc, and made concentric to
the pitch circles of the inner and outer rays of teeth, and at a normal
distance from them equal to the pitch radius of the pinion.

Now when the pinion revolves it will, if it be on the outside, as in
Fig. 209, act upon the spur teeth and turn the wheel in the opposite
direction to its own, but when the interrupted portion _f_ of the teeth
is thus brought to the pinion the groove will guide the pinion while it
passes from the outside to the inside, and thus bring its teeth into
action with the annular or internal teeth. The wheel will then receive
motion in the same direction as that of the pinion, and this will
continue until the gap _f_ is again brought to the pinion, when the
latter will be carried outwards and the motion again be reversed. The
_velocity ratio_ in either direction will remain constant, but the ratio
when the pinion is inside will differ slightly from the ratio when it is
outside, because the pitch radius of the annular or internal teeth is
necessarily somewhat less than that of the spur teeth. However, the
change of direction is not instantaneous, for the form of the groove S
_f_ _t_, which connects the inner and outer grooves, is a semicircle,
and when the axis of the pinion reaches S the velocity of the
mangle-wheel begins to diminish gradually until it is brought to rest at
_f_, and is again gradually set in motion from _f_ to _t_, when the
constant ratio begins; and this retardation will be increased by
increasing the difference between the radius of the inner and outer
pitch circles.

The teeth of a mangle-wheel are, however, most commonly formed by pins
projecting from the face of the disc as in Fig. 210. In this manner the
pitch circles for the inner and outer wheels coincide, and therefore the
velocity ratio is the same within and without, also the space through
which the pinion moves in shifting is reduced.

[Illustration: Fig. 211.]

[Illustration: Fig. 212.]

This space may be still further reduced by arranging the teeth as in
Fig. 211, that is, by placing the spur-wheel within the annular or
internal one; but at the same time the difference of the two velocity
ratios is increased.

If it be required that the velocity ratio vary, then the pitch lines of
the mangle-wheel must no longer be concentric.

Thus in Fig. 212 the groove _k_ _l_ is directed to the centre of the
mangle-wheel, and therefore the pinion will proceed during this portion
of its path without giving any motion to the wheel, and in the other
lines of teeth the pitch radius varies, hence the angular velocity ratio
will vary.

In Figs. 209, 210, and 211 the curves of the teeth are readily obtained
by employing the same describing circle for the whole of them. But when
the form Fig. 212 is adopted, the shape of the teeth requires some
consideration.

Every tooth of such a mangle-wheel may be considered as formed of two
ordinary teeth set back to back, the pitch line passing through the
middle. The outer half, therefore, appropriated to the action of the
pinion on the outside of the wheel, resembles that portion of an
ordinary spur-wheel tooth that lies beyond its pitch line, and the inner
half which receives the inside action of the pinion resembles the half
of an annular wheel that lies within the pitch circle. But the
consequence of this arrangement is, that in both positions the action of
the driving teeth must be confined to the approach of its teeth to the
line of centres, and consequently these teeth must be wholly within
their pitch line.

To obtain the forms of the teeth, therefore, take any convenient
describing circle, and employ it to describe the teeth of the pinion by
rolling within its pitch circle, and to describe the teeth of the wheel
by rolling within and without its pitch circle, and the pinion will then
work truly with the teeth of the wheel in both positions. The tooth at
each extremity of the series must be a circular one, whose centre lies
on the pitch line and whose diameter is equal to half the pitch.

[Illustration: Fig. 213.]

If the reciprocating piece move in a straight line, as it very often
does, then the mangle-_wheel_ is transformed into a _mangle-rack_ (Fig.
213) and its teeth may be simply made cylindrical pins, which those of
the mangle-wheel do not admit of on correct principle. B _b_ is the
sliding piece, and A the driving pinion, whose axis must have the power
of shifting from A to _a_ through a space equal to its own diameter, to
allow of the change from one side of the rack to the other at each
extremity of the motion. The teeth of the mangle-rack may receive any of
the forms which are given to common rack-teeth, if the arrangement be
derived from either Fig. 210 or Fig. 211.

But the mangle-rack admits of an arrangement by which the shifting
motion of the driving pinion, which is often inconvenient, may be
dispensed with.

[Illustration: Fig. 214.]

B _b_ Fig. 214, is the piece which receives the reciprocating motion,
and which may be either guided between rollers, as shown, or in any
other usual way; A the driving pinion, whose axis of motion is fixed;
the mangle rack C _c_ is formed upon a separate plate, and in this
example has the teeth upon the inside of the projecting ridge which
borders it, and the guide-groove formed within the ring of teeth,
similar to Fig. 211.

This rack is connected with the piece B _b_ in such a manner as to allow
of a short transverse motion with respect to that piece, by which the
pinion, when it arrives at either end of the course, is enabled by
shifting the rack to follow the course of the guide-groove, and thus to
reverse the motion by acting upon the opposite row of teeth.

The best mode of connecting the rack and its sliding piece is that
represented in the figure, and is the same which is adopted in the
well-known cylinder printing-engines of Mr. Cowper. Two guide-rods K C,
_k_ _c_ are jointed at one end K _k_ to the reciprocating piece B _b_,
and at the other end C _c_ to the shifting-rack; these rods are moreover
connected by a rod M _m_ which is jointed to each midway between their
extremities, so that the angular motion of these guide-rods round their
centres K _k_ will be the same; and as the angular motion is small and
the rods nearly parallel to the path of the slide, their extremities C
_c_ may be supposed to move at a right angle to that path, and
consequently the rack which is jointed to those extremities will also
move upon B _b_ in a direction at a right angle to its path, which is
the thing required, and admits of no other motion with respect to B _b_.

[Illustration: Fig. 215.]

To multiply plane motion the construction shown in Fig. 215 is
frequently employed. A and B are two racks, and C is a wheel between
them pivoted upon the rod R. A crank shaft or lever D is pivoted at E
and also (at P) to R. If D be operated C traverses along A and also
rotates upon its axis, thus giving to B a velocity equal to twice that
of the lateral motion of C.

The diameter of the wheel is immaterial, for the motion of B will always
be twice that of C.

Friction gearing-wheels which communicate motion one to the other by
simple contact of their surfaces are termed friction-wheels, or
friction-gearing. Thus in Fig. 216 let A and B be two wheels that touch
each other at C, each being suspended upon a central shaft; then if
either be made to revolve, it will cause the other to revolve also, by
the friction of the surfaces meeting at C. The degree of force which
will be thus conveyed from one to the other will depend upon the
character of the surface and the length of the line of contact at C.

[Illustration: Fig. 216.]

These surfaces should be made as concentric to the axis of the wheel and
as flat and smooth as possible in order to obtain a maximum power of
transmission. Mr. E. S. Wicklin states that under these conditions and
proper forms of construction as much as 300 horse-power may be (and is
in some of the Western States) transmitted.

In practice, small wheels of this class are often covered with some
softer material, as leather; sometimes one wheel only is so covered, and
it is preferred that the covered wheel drive the iron one, because, if a
slip takes place and the iron wheel was the driver, it would be apt to
wear a concave spot in the wood covered one, and the friction between
the two would be so greatly diminished that there would be difficulty in
starting them when the damaged spot was on the line of centre.

If, however, the iron wheel ceased motion, the wooden one continuing to
revolve, the damage would be spread over that part of the circumference
of the wooden one which continued while the iron one was at rest, and if
this occurred throughout a whole revolution of the wooden wheel its
roundness would not be apt to be impaired, except in so far as
differences in the hardness of the wood and similar causes might effect.

"To select the best material for driving pulleys in friction-gearing has
required considerable experience; nor is it certain that this object
has yet been attained. Few, if any, well-arranged and careful
experiments have been made with a view of determining the comparative
value of different materials as a frictional medium for driving iron
pulleys. The various theories and notions of builders have, however,
caused the application to this use of several varieties of wood, and
also of leather, india-rubber, and paper; and thus an opportunity has
been given to judge of their different degrees of efficiency. The
materials most easily obtained, and most used, are the different
varieties of wood, and of these several have given good results.

"For driving light machinery, running at high speed, as in sash, door,
and blind factories, basswood, the linden of the Southern and Middle
States (_Tilia Americana_) has been found to possess good qualities,
having considerable durability and being unsurpassed in the smoothness
and softness of its movement. Cotton wood (_Populus monilifera_) has
been tried for small machinery with results somewhat similar to those of
basswood, but is found to be more affected by atmospheric changes. And
even white pine makes a driving surface which is, considering the
softness of the wood, of astonishing efficiency and durability. But for
all heavy work, where from twenty to sixty horse-power is transmitted by
a single contact, soft maple (_Acer rubrum_) has, at present, no rival.
Driving pulleys of this wood, if correctly proportioned and well built,
will run for years with no perceptible wear.

"For very small pulleys, leather is an excellent driver and is very
durable; and rubber also possesses great adhesion as a driver; but a
surface of soft rubber undoubtedly requires more power than one of a
less elastic substance.

"Recently paper has been introduced as a driver for small machinery, and
has been applied in some situations where the test was most severe; and
the remarkable manner in which it has thus far withstood the severity of
these tests appears to point to it as the most efficient material yet
tried.

"The proportioning, however, of friction-pulleys to the work required
and their substantial and accurate construction are matters of perhaps
more importance than the selection of material.

"Friction-wheels must be most accurately and substantially made and kept
in perfect line so that the contact between the surfaces may not be
diminished. The bodies are usually of iron lagged or covered with wooden
segments.

"All large drivers, say from four to ten feet diameter and from twelve
to thirty inch face, should have rims of soft maple six or seven inches
deep. These should be made up of plank, one and a half or two inches
thick, cut into 'cants,' one-sixth, eighth, or tenth of the circle, so
as to place the grain of the wood as nearly as practicable in the
direction of the circumference. The cants should be closely fitted, and
put together with white lead or glue, strongly nailed and bolted. The
wooden rim, thus made up to within about three inches of the width
required for the finished pulley, is mounted upon one or two heavy iron
'spiders,' with six or eight radial arms. If the pulley is above six
feet in diameter, there should be eight arms, and two spiders when the
width of face is more than eighteen inches.

"Upon the ends of the arms are flat 'pads,' which should be of just
sufficient width to extend across the inner face of the wooden rim, as
described; that is, three inches less than the width of the finished
pulley. These pads are gained into the inner side of the rim; the gains
being cut large enough to admit keys under and beside the pads. When the
keys are well driven, strong 'lag' screws are put through the ends of
the arm into the rim. This done, an additional 'round' is put upon each
side of the rim to cover bolt heads and secure the keys from ever
working out. The pulley is now put to its place on the shaft and keyed,
the edges trued up, and the face turned off with the utmost exactness.

"For small drivers, the best construction is to make an iron pulley of
about eight inches less diameter and three inches less face than the
pulley required. Have four lugs, about an inch square, cast across the
face of this pulley. Make a wooden rim, four inches deep, with face
equal to that of the iron pulley, and the inside diameter equal to the
outer diameter of the iron. Drive this rim snugly on over the rim of the
iron pulley having cut gains to receive the lugs, together with a hard
wood key beside each. Now add a round of cants upon each side, with
their inner diameter less than the first, so as to cover the iron rim.
If the pulley is designed for heavy work, the wood should be maple, and
should be well fastened by lag screws put through the iron rim; but for
light work, it may be of basswood or pine, and the lag screws omitted.
But in all cases, the wood should be thoroughly seasoned.

"In the early use of friction-gearing, when it was used only as backing
gear in saw-mills, and for hoisting in grist-mills, the pulleys were
made so as to present the head of the wood to the surface; and we
occasionally yet meet with an instance where they are so made. But such
pulleys never run so smoothly nor drive so well as those made with the
fibre more nearly in a line with the work."[11]

  [11] By E. S. Wicklin.

[Illustration: Fig. 217.]

[Illustration: Fig. 218.]

The driving friction may be obtained from contact of the radial surfaces
in two ways: thus, Fig. 217 represents three discs, A, B, and C; the
edge of A being gripped by and between B and C, which must be held
together by a spiral spring S or other equivalent device. These wheels
may be made to give a variable speed of rotation by curving the surfaces
of the pair B C as in the figure. By means of suitable lever-motion A
may be made to advance towards or recede from the centre of B and C,
giving to their shaft an increased or diminished speed of revolution.

[Illustration: Fig. 219.]

A similar result may be obtained by the construction shown in Fig. 218,
in which D and E are two discs fast upon their respective shafts, and C
are discs of leather clamped in E. It is obvious that if D be the driver
the speed of revolution of E will be diminished in proportion as it is
moved nearer to the centre of D, and also that the direction of
revolution of D remaining constant, that of E will be in one direction
if on the side B of the centre of D, and in the other direction if it is
on the side A of the centre of D, thus affording means of reversing the
motion as well as of varying its speed. A similar arrangement is
sometimes employed to enable the direction of rotation of the driver
shaft to be reversed, or its motion to cease. Thus, in Fig. 219, R is a
driving rope driving the discs A, B, and _c_, _d_, _e_, _f_, _g_ are
discs of yellow pine clamped between the flanges _h_ _i_; when these
five discs are forced (by lifting shaft H), against the face of a motion
occurs in one direction, while if forced against B the direction of
motion of H is reversed.

For many purposes, such as hoisting, for example, where considerable
power requires to be transmitted, the form of friction wheels shown in
Fig. 220 is employed, the object being to increase the line of contact
between wheels of a given width of face. In this case the strain due to
the length of the line of contact partly counteracts itself, thus
relieving to that extent the journals from friction. Thus in Fig. 221 is
shown a single wedge and groove of a pair of wheels. The surface
pressure on each side will be at a right angle to the face, or in the
direction described by the arrows A and B. The surface contact acts to
thrust the bearings of the two shafts apart. The effective length of
surface acting to thrust the bearings apart being denoted by the dotted
line C. The relative efficiency of this class of wheel, however, is not
to be measured by the length of the line C, as compared to that of the
two contacting sides of the groove, because it is increased from the
wedge shape of the groove, and furthermore, no matter how solid the
wheels may be, there will be some elasticity which will operate to
increase the driving power due to the contact. It is to preserve the
wedge principle that the wedges are made flat at the top, so that they
shall not bottom in the grooves even after considerable wear has taken
place. The object of employing this class of gear is to avoid noise and
jar and to insure a uniform motion. The motion at the line of contact of
such wheels is not a rolling, but, in part, a sliding one, which may
readily be perceived from a consideration of the following. The
circumference of the top of each wedge is greater than that of the
bottom, and, in the case of the groove, the circumference of the top is
greater than that of the bottom; and since the top or largest
circumference of one contacts with the smallest circumference of the
other, it follows that the difference between the two represents the
amount of sliding motion that occurs in each revolution. Suppose, for
example, we take two of such wheels 10 inches in diameter, having wedges
and grooves 1/4 inch high and deep respectively; then the top of the
groove will travel 31.416 inches in a revolution, and it will contact
with the bottom of the wedge which travels (on account of its lesser
diameter) 29.845 inches per revolution.

[Illustration: Fig. 220.]

[Illustration: Fig. 221.]

Fig. 222 shows the construction for a pair of bevel wheels on the same
principle.

[Illustration: Fig. 222.]

[Illustration: Fig. 223.]

[Illustration: Fig. 224.]

A form of friction-gearing in which the journals are relieved of the
strain due to the pressure of contact, and in which slip is impossible,
is shown in Fig. 223. It consists of projections on one wheel and
corresponding depressions or cavities on the other. These projections
and cavities are at opposite angles on each half of each wheel, so as to
avoid the end pressure on the journals which would otherwise ensue.
Their shapes may be formed at will, providing that the tops of the
projections are narrower than their bases, which is necessary to enable
the projections to enter and leave the cavities. In this class of
positive gear great truth or exactness is possible, because both the
projections and cavities may be turned in a lathe. Fig. 224 represents a
similar kind of gear with the projections running lengthways of the
cylinder approaching more nearly in its action to toothed gearing, and
in this case the curves for the teeth and groves should be formed by the
rules already laid down for toothed gearing. The action of this latter
class may be made very smooth, because a continuous contact on the line
of centres may be maintained by reason of the longitudinal curve of the
teeth.

[Illustration: Fig. 225.]

Cams may be employed to impart either a uniform, an irregular, or an
intermittent motion, the principles involved in their construction being
as follows:--Let it be required to construct a cam that being revolved
at a uniform velocity shall impart a uniform reciprocating motion. First
draw an inner circle O, Fig. 225, whose radius must equal the radius of
the shaft that is to drive it, plus the depth of the cam at its
shallowest part, plus the radius of the roller the cam is to actuate.
Then from the same centre draw an outer circle S, the radius between
these two circles being equal to the amount the cam is to move the
roller. Draw a line O P, and divide it into any convenient numbers of
divisions (five being shown in the figure), and through these points
draw circles. Divide the outer circle S into twice as many equal
divisions as the line O P is divided into (as from 1 to 10 in the
figure), and where these lines pass through the circles will be points
through which the pitch line of the cam may be drawn.

[Illustration: Fig. 226.]

Thus where circle 1 meets line 1, or at point A, is one point in the
pitch line of the cam; where circle 2 meets line 2, or at B, is another
point in the pitch line of the cam, and so on until we reach the point
E, where circle 5 meets line 5. From this point we simply repeat the
process, the point E where line 6 cuts circle 4, being a point on the
pitch line, and so on throughout the whole 10 divisions, and through the
points so obtained we draw the pitch line.

[Illustration: Fig. 227.]

[Illustration: Fig. 228.]

If we were to cut out a cam to the outline thus obtained, and revolve it
at a uniform velocity, it would move a point held against its perimeter
at a uniform velocity throughout the whole of the cam revolution. But
such a point would rapidly become worn away and dulled, which would, as
the point broadened, vary the motion imparted to it, as will be seen
presently. To avoid this wear a roller is used in place of a point, and
the diameter of the roller affects the action of the cam, causing it to
accelerate the cam action at one and retard it at another part of the
cam revolution, hence the pitch line obtained by the process in Fig. 225
represents the path of the centre of the roller, and from this pitch
line we may mark out the actual cam by the construction shown in Fig.
226. A pair of compasses are set to the radius of the roller R, and from
points (such as at A, B, E, F), as the pitch line, arcs of circles are
struck, and a line drawn to just meet the crowns of these arcs will give
the outline of the actual cam. The motion of the roller, however, in
approaching and receding from the cam centre C, must be in a straight
line G G that passes through the centre C of the cam. Suppose, for
example, that instead of the roller lifting and falling in the line G G
its arm is horizontal, as in Fig. 227, and that this arm being pivoted
the roller moves in an arc of a circle as D D, and the motion imparted
to the arm will no longer be uniform. Furthermore, different diameters
of roller require different forms of cam to accomplish the same motion,
or, in other words, with a given cam the action will vary with different
diameters of roller. Suppose, for example, that in Fig. 228 we have a
cam that is to operate a roller along the line A A, and that B
represents a large and C a small roller, and with the cam in the
position shown in the figure, C will have contact with the cam edge at
point D, while B will have contact at the point E, and it follows that
on account of the enlarged diameter of roller B over roller C, its
action is at this point quicker under a given amount of cam motion,
which has occurred because the point of contact has advanced upon the
roller surface--rolling along it, as it were. In Fig. 229 we find that
as the cam moves forward this action continues on both the large and the
small roller, its effect being greater upon the large than upon the
small one, and as this rolling motion of the point of contact evidently
occurs easily, a quick roller motion is obtained without shock or
vibration. Continuing the cam motion, we find in Fig. 230 that the point
of contact is receding toward the line of motion on the large roller and
advancing upon the small one, while in Fig. 231 the two have contact at
about the same point, the forward motion being about completed.

[Illustration: Fig. 229.]

[Illustration: Fig. 230.]

[Illustration: Fig. 231.]

[Illustration: Fig. 232.]

To compare the motions of the respective rollers along the line of
motion A A we proceed as in Fig. 232, in which the two dots M and N are
the same distance apart as are the centres of the two rollers B and C
when in the positions they occupy in Fig. 228; hence a pair of compasses
set to the radius from the axis of the cam to that of roller B will, if
rested at N, strike the arc marked 1 above the line of motion A A, while
a pair of compasses set to the radius from the axis of the cam to that
of roller C in Fig. 228 will, if rested at M in Fig. 232, mark the arc 1
below the line of motion A A. Continuing this process, we set the
compasses to the radius from the axis of the cam to that of roller B in
Fig. 229, and mark this radius at arc 2 above the line A A in Fig. 232;
hence the distance apart of these two arcs is the amount the roller
travelled along the line A A while the cam moved from its position in
Fig. 228 to its position in Fig. 229. Next we set the compasses from the
axis of the cam to that of the large roller in Fig. 230, and then mark
arc 3 above the line in Fig. 232, and repeat the process for Fig. 233,
thus using the centre N for all the positions of the large roller and
marking its motion above the line A A. To get the motion of the small
roller C, we set the compasses to the radius from the axis of the cam to
the small roller in Fig. 228, and then resting one point of these
compasses on centre M in Fig. 232, we mark arc 1 below the line A A.
Turning to Fig. 229 we set the compasses from the cam axis to the centre
of roller C, and from centre N in Fig. 232 mark arc 2 below line A. From
Figs. 230 and 231 proceed in the same way to get lines 3 and 4 below
line A in Fig. 232, and we may at once compare the two motions. Thus we
find that while the cam moved from the position in Fig. 228 to that in
Fig. 229, the large roller moved twice as far as the small one, while at
230 the motions were rapidly equalizing again, the equalization being
completed at 231.

[Illustration: Fig. 233.]

[Illustration: Fig. 234.]

[Illustration: Fig. 235.]

We may now consider the return motion, and in Fig. 233 we find that the
order of things is reversed, for the small roller has contact at O,
while the large one has contact at P; hence the small one leads and
gives the most rapid motion, which it continues to do, as is shown in
Figs. 234, 235, and 236, and we may plot out the two motions as in Fig.
237--that for the large roller being above and that for the small one
below the line A A. First we set a pair of compasses to the radius from
the axis of the large and small roller when in the position shown in
Fig. 231 (which corresponds to the same radius in Fig. 228), and mark
two centres, M and N, as we did in Fig. 232. Of these N is the centre
for plotting the motion of the large roller and M the centre for
plotting the motion of the small one. We set a pair of compasses to the
radius from the axis of the cam and that of the large roller in Fig.
231, and then resting the compasses at N we mark arc 5 above the line A
A, Fig. 237. The compasses are then set from the cam to the roller axis
in Fig. 233, and arc 6 is marked above line A A. From Figs. 234, 235,
and 236 we get the radii to mark arcs 7, 8, 9 above A A, and the motion
of the large roller is plotted. We proceed in the same way for the small
one, but use the centre M, Fig. 237, to mark the arcs 5, 6, 7, 8, and 9
below the line A A, and find that the small roller has moved quickest
throughout. It appears, then, that the larger the roller the quicker the
forward motion and the slower the return one, which is advantageous,
because the object is to move the roller out quickly and close it
slowly, so that under a quick speed the cam shall not run away from the
roller as it is apt to do in the absence of a return or backing cam,
which consists of a separate cam for moving the roller on its return
stroke, thus dispensing with the use of springs or weights to keep the
roller upon the cam and making the motion positive.

[Illustration: Fig. 236.]

[Illustration: Fig. 237.]

[Illustration: Fig. 238.]

The return or backing cam obviously depends for its shape upon the
forward cam, and the latter having been determined, the requisite form
for the return cam may be found as follows. In Fig. 238 let A represent
the forward cam fastened in any suitable or convenient way to a disc of
paper, or, what is better, sheet zinc, B. The cam is pivoted by a pin
passing through it and the zinc, and driven into the drawing-board. A
frame F is made to carry two rollers R and R´, whose width apart exactly
equals the extreme length of the forward cam. The faces D D of the frame
F are in a line with a line passing through the centres of the rolls R
R´, and the cam is also pivoted on this line, so that when the four pins
P are driven into the drawing-board, the frame F will be guided by them
to move in a line that crosses the centre of the cam A. Suppose then
that, the pieces occupying the position shown in the engraving, we slide
F so that roller R touches the edge of cam A, and we may then take a
needle and mark an arc or line around the edge of R´. We then revolve
cam A a trifle, and, being fast to B, the two will move together, and
with R against A we mark a second arc, coincident with the edge of
roller R´. By continuing this process we mark the numerous short arcs
shown upon B, and the crowns of these arcs give us the outline of the
return cam. It is obvious that, while the edge of the cam A will not let
roller R (and therefore frame F) move to the right, roller R´ being
against the edge of the backing or return cam as marked upon B, prevents
the frame F from moving to the left; hence neither roll can leave its
cam.

[Illustration: Fig. 239.]

We have in this example supposed that the frame carrying the rollers is
guided to move in a straight line, and it remains to give an example in
which the rollers are carried on a pivoted shaft or rocking arm. In Fig.
239 we have the same cam A with a sheet of paper B fastened to it, the
rollers R R´ being carried in a rock shaft pivoted at X. It is essential
in this case that the rollers R and R´ and the centre upon which the cam
revolves shall all three be in the arc of a circle whose centre is the
axis of X, as is denoted by the arc D. The cam A is fastened to the
piece of stiff paper or of sheet zinc B, and the two are pivoted by a
pin passing through the axis E of the cam and into the drawing-board,
while the lever is pivoted at X by a pin passing into the drawing-board.
The backing or return cam is obviously marked out the same way as was
described with reference to Fig. 238.

[Illustration: Fig. 240.]

In Fig. 240 we have as an example the construction of a cam to operate
the slide valve of an engine which is to have the steam supply to the
cylinder cut off at one-half the piston stroke, and that will admit the
live steam as quickly as a valve having steam lap equal to, say,
three-fourths the width of the port. In Fig. 240 let the line A
represent a piston stroke of 24 inches, the outer circle B the path of
the outer edge of the cam, and the inner circle C the inner edge of the
cam, the radius between these circles representing the full width of the
steam port. Now, in a valve having lap equal to three-fourths the width
of the steam port, and travel enough to open both ports fully, the
piston of a 24-inch-stroke engine will have moved about 2 inches before
the steam port is fully opened, and to construct a cam that will effect
the same movement we mark a dot D, distant from the end E of piston
stroke 2/26 of the length of the line A, and by erecting the line F we
get at point G, the point at which the cam must attain its greatest
throw. It is obvious, therefore, that as the roller is at R the valve
will be in mid-position, as shown at the bottom of the figure, and that
when point G of the cam arrives at E the edge P of the valve will be
moved fair with edge S of the steam port T, which will therefore be full
open. To cut off at half stroke the valve must again be closed by the
time point N of the cam meets the roller R; hence we may mark point N.
We may then mark in the cam curve from N to M, making it as short as it
will work properly without causing the roller to fail to follow the
curve or strike a blow when reaching the circle C. To accomplish this
end in a single cam, it is essential to make the curve as gradual as
possible from point M to O, so as to start the roller motion easily. But
once having fairly started, its motion may be rapidly accelerated, the
descent from O to Q being rapid. To prevent the roller from meeting
circle C with a blow, the curve from Q to N is again made gradual, so as
to ease and retard the roller motion. The same remarks apply to the
curve from R to G, the object being to cause the roller to begin and end
its passage along the cam curve as slowly as the length of cam edge
occupied by the curve will permit. There is one objection to starting
the curve slowly at G, which is that the port S will be opened
correspondingly slowly for the live steam. This, however, may be
overcome by giving the valve an increased travel, as shown in Fig. 241,
which will simply cause the valve edge to travel to a corresponding
amount over the inside edge of the port. The increased travel is shown
by the circles Y and Z, and it is seen that the cam curve from W to R is
more gradual than in Fig. 240, while the roller R will be moved much
more quickly in the position shown in Fig. 241 than it will in that
shown in Fig. 240, both positions being that when the piston is at the
end of the stroke and the port about to open. While that part of the cam
curve from G to M in Fig. 241 is moving past the roller R, the valve
will be moving over the bridge, the steam port remaining wide open, and
therefore not affecting the steam distribution. After point M, Fig. 241,
has passed the roller, we have from M to T to start the roller
gradually, so that when it has arrived at T and the port begins to close
for the cut-off it may move rapidly, and continue to do so until the
point N reaches the roller and the cut-off has occurred, after which it
does not matter how slowly the valve moves; hence we may make the curve
from N to the circle Y as gradual as we like.

[Illustration: Fig. 241.]

[Illustration: Fig. 242.]

Fig. 242 represents a cam for a valve having the amount of lap
represented by the distance between circles C and Y, the cam occupying
the position it would do with the piston at one end of the stroke, as at
E. Obviously, a full port is obtained when point G reaches the roller,
and as point N is distant from E three-quarters of the diameter of the
outer circle, the cut-off occurs at three-quarter stroke, and we have
from N to Y to make the curve as gradual as we like, and from W to R in
moving the valve to open the port. We cannot, however, give more gradual
curves at G and at M without retarding the roller motion, and therefore
opening and closing the port slower, and it would simply be a matter of
increase of speed to cause the roller to fail to follow the cam surface
at these two points unless a return cam be employed.

We have in these engine cams considered the steam supply and point of
cut-off only, and it is obvious that a second and separate cam would be
required to operate the exhaust valves.

[Illustration: Fig. 243.]

Fig. 243 represents a groove-cam, and it is to be observed that the
roller cannot be maintained in a close fit in the groove, because the
friction on its two sides endeavours to drive it in opposite directions
at the same time, causing an abrasion that soon widens the groove and
reduces the roller diameter; furthermore, when the grooves are made of
equal width all the way down (and these cams are often made in this way)
the roller cannot have a rolling action only, but must have some sliding
motion. Thus, referring to Fig. 243, the amount of sliding motion will
be equal to the differences in the circumferences of the outer circle A
and the inner one B. To obviate this the groove and roller must be made
of such a taper that the axis of the cam and of the roller will meet on
the line of the cam axes and in the middle of the width, as is shown in
Fig. 244; but even in this case the cam will grind away the roller to
some extent, on account of rubbing its sides in opposite directions. To
obviate this, Mr. James Brady, of Brooklyn, N. Y., has patented the use
of two rollers, as in Fig. 245, one acting against one side and the
other against the other side of the groove, by which means lost motion
and rapid wear are successfully avoided.

[Illustration: Fig. 244.]

[Illustration: Fig. 245.]

In making a cam of this form, the body of the cam is covered by a
sleeve. The groove is cut through the sleeve and into the body, and is
made wider than the diameter of the roller. When the rollers are in
place on the spindle or journal, the sleeve is pushed forward, or rather
endways, and fastened by a set-screw. This gives the desired bearing on
both sides of the groove, while each roller touches one side only of the
groove. The edges of the sleeve are then faced off even with the cam
body, the whole appearing as in the figure.

[Illustration: _VOL. I._ =FORMS OF SCREW THREADS.= _PLATE II._


THE [V]-THREAD.

Fig. 246.

THE UNITED STATES STANDARD THREAD.

Fig. 247.

THE WHITWORTH, OR ENGLISH STANDARD THREAD.

Fig. 248.

THE SQUARE THREAD.

Fig. 249.

THE PITCH OF A THREAD.

Fig. 250.

A DOUBLE THREAD.

Fig. 251.

A RATCHET THREAD.

Fig. 252.

A "DRUNKEN" THREAD.

Fig. 253.

RIGHT AND LEFT HAND THREAD.

Fig. 254.]



CHAPTER IV.--SCREW THREAD.


Screw threads are employed for two principal purposes--for holding or
securing, and for transmitting motion. There are in use, in ordinary
machine shop practice, four forms of screw thread. There is, first, the
sharp [V]-thread shown in Fig. 246; second, the United States standard
thread, the Sellers thread, or the Franklin Institute thread, as it is
sometimes called--all three designations signifying the same form of
thread. This thread was originally proposed by William Sellers, and was
afterward recommended by the Franklin Institute. It was finally adopted
as a standard by the United States Navy Department. This form of thread
is shown in Fig. 247. The third form is the Whitworth or English
standard thread, shown in Fig. 248. It is sometimes termed the round top
and bottom thread. The fourth form is the square thread shown in Fig.
249, which is used for coarse pitches, and usually for the transmission
of motion.

The sharp [V]-thread, Fig. 246, has its sides at an angle of 60° one to
the other, as shown; or, in other words, each side of the thread is at
an angle of 60° to the axial line of the bolt. The United States
Standard, Fig. 247, is formed by dividing the depth of the sharp
[V]-thread into 8 equal divisions and taking off one of the divisions at
the top and filling in another at the bottom, so as to leave a flat
place at the top and bottom. The Whitworth thread, Fig. 248, has its
sides at an angle of 55° to each other, or to the axial line of the
bolt. In this the depth of the thread is divided into 6 equal parts, and
the sides of the thread are joined by arcs of circles that cut off one
of these parts at the top and another at the bottom of the thread. The
centres from which these arcs are struck are located on the second lines
of division, as denoted in the figure by the dots. Screw threads are
designated by their pitch or the distance between the threads. In Fig.
250 the pitch is 1/4 inch, but it is usual to take the number of threads
in an inch of length; hence the pitch in Fig. 250 would generally be
termed a pitch of 4, or 4 to the inch. The number of threads per inch of
length does not, however, govern the true pitch of the thread, unless it
be a "single" thread.

A single thread is composed of one spiral projection, whose advance upon
the bolt is equal in each revolution to the apparent pitch. In Fig. 251
is shown a double thread, which consists of two threads. In the figure,
A denotes one spiral or thread, and B the other, the latter being
carried as far as C only for the sake of illustration. The true pitch is
in this case twice that of the apparent pitch, being, as is always the
case, the number of revolutions the thread makes around the bolt (which
gives the pitch per inch), or the distance along the bolt length that
the nut or thread advances during one rotation. Threads may be made
double, treble, quadruple and so on, the object being to increase the
motion without the use of a coarser pitch single thread, whose increased
depth would weaken the body of the bolt.

The "ratchet" thread shown in Fig. 252 is sometimes used upon bolts for
ironwork, the object being to have the sides A A of the thread at a
right angle to the axis of the bolt, and therefore in the direct line of
the strain. Modifications of this form of thread are used in coarse
pitches for screws that are to thread direct into woodwork.

A waved or drunken thread is one in which the path around the bolt is
waved, as in Fig. 253, and not a continuous straight spiral, as it
should be. All threads may be either left hand or right, according to
their direction of inclination upon the bolt; thus, Fig. 254 is a
cylinder having a right-hand thread at A and a left-hand one at B. When
both ends of a piece have either right or left-hand threads, if the
piece be rotated and the nuts be prevented from rotating, they will move
in the same direction, and, if the pitches of the threads are alike, at
the same rate of motion; but if one thread be a right and the other a
left one, then, under the above conditions, the nuts will advance toward
or recede from each other according to the direction of rotation of the
male thread.

[Illustration: Fig. 255.]

In Fig. 255 is represented a form of thread designed to enable the nut
to fit the bolt, and the thread sides to have a bearing one upon the
other, notwithstanding that the diameter of the nut and bolt may differ.
The thread in the nut is what may be termed a reversed ratchet thread,
and that in the bolt an undercut ratchet thread, the amount of undercut
being about 2°. Where this form of thread is used, the diameter of the
bolt may vary as much as 1-32d of an inch in a bolt 3/4 inch in
diameter, and yet the nut will screw home and be a tight fit. The
difference in the thread fit that ordinarily arises from differences in
the standards of measurement from wear of the threading tools, does not
in this form affect the fit of the nut to the bolt. In screwing the nut
on, the threads conform one to the other, giving a bearing area
extending over the full sides of the thread. The undercutting on the
leading face of the bolt thread gives room for the metal to conform
itself to the nut thread, which it does very completely. The result is
that the nut may be passed up and down the bolt several times and still
remain too tight a fit to be worked by hand. Experiment has demonstrated
that it may be run up and down the bolt dozens of times without becoming
as loose as an ordinary bolt and nut. On account of this capacity of the
peculiar form of thread employed, to adapt itself, the threads may be
made a tight fit when the threading tools are new. The extra tightness
that arises from the wear of these tools is accommodated in the
undercutting, which gives room for the thread to adjust itself to the
opposite part or nut.

In a second form of self-locking thread, the thread on the bolt is made
of the usual [V]-shape United States standard. The thread in the nut,
however, is formed as illustrated in Fig. 256, which is a section of a
3/4-inch bolt, greatly enlarged for the sake of clearness of
illustration. The leading threads are of the same angle as the thread on
the bolt, but their diameters are 3/4 and 1-16th inch, which allows the
nut to pass easily upon the bolt. The angle of the next thread following
is 56°, the succeeding one 52°, and so on, each thread having 4° less
angle than the one preceding, while the pitch remains the same
throughout. As a result, the rear threads are deeper than the leading
ones. As the nut is screwed home, the bolt thread is forced out or up,
and fills the rear threads to a degree depending upon the diameter of
the bolt thread. For example, if the bolt is 3/4 inch, its leading or
end thread will simply change its angle from that of 60° to that of 44°,
or if the bolt thread is 3/4 and 1-64th inch in diameter, its leading
thread will change from an angle of 60° to one of 44°. It will almost
completely fill the loose thread in the nut. The areas of spaces between
the nut threads are very nearly equal, although slightly greater at the
back end of the nut, so that if the front end will enter at all, the nut
will screw home, while the thread fit will be tight, even under a
considerable variation in the bolt itself. From this description, it is
evident that the employment of nuts threaded in this manner is only
necessary in order to give to ordinary bolts all the advantages of
tightness due to this form of thread.

The term "diameter" of a thread is understood to mean its diameter at
the top of the thread and measured at a right angle to the axis of the
bolt. When the diameter of the bottom or root of the thread is referred
to it is usually specified as diameter at the bottom or at the root of
the thread.

The depth of a thread is the vertical height of the thread upon the
bolt, measured at a right angle to the bolt axis and not along the side
of the thread.

A true thread is one that winds around the bolt in a continuous and even
spiral and is not waved or drunken as is the thread in Fig. 253. An
outside or male thread is one upon an external surface as upon a bolt;
an internal or female thread is one produced in a bore or hole as in a
nut.

[Illustration: Fig. 256.]

The Whitworth or English standard thread, shown in Fig. 248, is that
employed in Great Britain and her colonies, and to a small extent in the
United States. The [V]-thread fig. 246 is that in most common use in the
United States, but it is being displaced by the United States standard
thread. The reasons for the adoption of the latter by the Franklin
Institute are set forth in the report of a committee appointed by that
Institute to consider the matter. From that report the following
extracts are made.

"That in the course of their investigations they have become more deeply
impressed with the necessity of some acknowledged standard, the
varieties of threads in use being much greater than they had supposed
possible; in fact, the difficulty of obtaining the exact pitch of a
thread not a multiple or sub-multiple of the inch measure is sometimes a
matter of extreme embarrassment.

"Such a state of things must evidently be prejudicial to the best
interests of the whole country; a great and unnecessary waste is its
certain consequence, for not only must the various parts of new
machinery be adjusted to each other, in place of being interchangeable,
but no adequate provision can be made for repairs, and a costly variety
of screwing apparatus becomes a necessity. It may reasonably be hoped
that should a uniformity of practice result from the efforts and
investigations now undertaken, the advantages flowing from it will be so
manifest, as to induce reform in other particulars of scarcely less
importance.

"Your committee have held numerous meetings for the purpose of
considering the various conditions required in any system which they
could recommend for adoption. Strength, durability, with reference to
wear from constant use, and ease of construction, would seem to be the
principal requisites in any general system; for although in many cases,
as, for instance, when a square thread is used, the strength of the
thread and bolt are both sacrificed for the sake of securing some other
advantage, yet all such have been considered as special cases, not
affecting the general inquiry. With this in view, your committee decided
that threads having their sides at an angle to each other must
necessarily more nearly fulfil the first condition than any other form;
but what this angle should be must be governed by a variety of
considerations, for it is clear that if the two sides start from the
same point at the top, the greater the angle contained between them, the
greater will be the strength of the bolt; on the other hand, the greater
this angle, supposing the apex of the thread to be over the centre of
its base, the greater will be the tendency to burst the nut, and the
greater the friction between the nut and the bolt, so that if carried to
excess the bolt would be broken by torsional strain rather than by a
strain in the direction of its length. If, however, we should make one
side of the thread perpendicular to the axis of the bolt, and the other
at an angle to the first, we should obtain the greatest amount of
strength, together with the least frictional resistance; but we should
have a thread only suitable for supporting strains in one direction, and
constant care would be requisite to cut the thread in the nut in the
proper direction to correspond with the bolt; we have consequently
classed this form as exceptional, and decided that the two sides should
be at an angle to each other and form equal angles with the base.

"The general form of the thread having been determined upon the above
considerations, the angle which the sides should bear to each other has
been fixed at 60°, not only because this seems to fulfil the conditions
of least frictional resistance combined with the greatest strength, but
because it is an angle more readily obtained than any other, and it is
also in more general use. As this form is in common use almost to the
exclusion of any other, your committee have carefully weighed its
advantages and disadvantages before deciding to recommend any
modification of it. It cannot be doubted that the sharp thread offers us
the simplest form, and that its general adoption would require no
special tools for its construction, but its liability to accident,
always great, becomes a serious matter upon large bolts, whilst the
small amount of strength at the sharp top is a strong inducement to
sacrifice some of it for the sake of better protection to the remainder;
when this conclusion is reached, it is at once evident a corresponding
space may be filled up in the bottom of the thread, and thus give an
increased strength to the bolt, which may compensate for the reduction
in strength and wearing surface upon the thread. It is also clear that
such a modification, by avoiding the fine points and angles in the tools
of construction, will increase their durability; all of which being
admitted, the question comes up, what form shall be given to the top and
bottom of the thread? for it is evident one should be the converse of
the other. It being admitted that the sharp thread can be made
interchangeable more readily than any other, it is clear that this
advantage would not be impaired if we should stop cutting out the space
before we had made the thread full or sharp; but to give the same shape
at the bottom of the threads would require that a similar quantity
should be taken off the point of the cutting tool, thus necessitating
the use of some instrument capable of measuring the required amount, but
when this is done the thread having a flat top and bottom can be quite
as readily formed as if it was sharp. A very slight examination sufficed
to satisfy us that in point of construction the rounded top and bottom
presents much greater difficulties--in fact, all taps and screws that
are chased or cut in a lathe require to be finished or rounded by a
second process. As the radius of the curve to form this must vary for
every thread, it will be impossible to make one gauge to answer for all
sizes, and very difficult, in fact impossible, without special tools, to
shape it correctly for one.

"Your committee are of opinion that the introduction of a uniform system
would be greatly facilitated by the adoption of such a form of thread as
would enable any intelligent mechanic to construct it without any
special tools, or if any are necessary, that they shall be as few and as
simple as possible, so that although the round top and bottom presents
some advantages when it is perfectly made, as increased strength to the
thread and the best form to the cutting tools, yet we have considered
that these are more than compensated by ease of construction, the
certainty of fit, and increased wearing surface offered by the flat top
and bottom, and therefore recommend its adoption. The amount of flat to
be taken off should be as small as possible, and only sufficient to
protect the thread; for this purpose one-eighth of the pitch would seem
to be ample, and this will leave three-fourths of the pitch for bearing
surface. The considerations governing the pitch are so various that
their discussion has consumed much time.

"As in every instance the threads now in use are stronger than their
bolts, it became a question whether a finer scale would not be an
advantage. It is possible that if the use of the screw thread was
confined to wrought iron or brass, such a conclusion might have been
reached, but as cast iron enters so largely into all engineering work,
it was believed finer threads than those in general use might not be
found an improvement; particularly when it was considered that so far as
the vertical height of thread and strength of bolt are concerned, the
adoption of a flat top and bottom thread was equivalent to decreasing
the pitch of a sharp thread 25 per cent., or what is the same thing,
increasing the number of threads per inch 33 per cent. If finer threads
were adopted they would require also greater exactitude than at present
exists in the machinery of construction, to avoid the liability of
overriding, and the wearing surface would be diminished; moreover, we
are of opinion that the average practice of the mechanical world would
probably be found better adapted to the general want than any
proportions founded upon theory alone."

       *       *       *       *       *

[Illustration: Fig. 257.]

[Illustration: Fig. 258.]

The principal requirements for a screw thread are as follows: 1. That it
shall possess a strength that, in the length or depth of a nut, shall be
equal to the strength of the weakest part of the bolt, which is at the
bottom of the bolt thread. 2. That the tools required to produce it
shall be easily made, and shall not alter their form by reason of wear.
3. That these tools shall (in the case of lathe work) be easily
sharpened, and set to correct position in the lathe. 4. That a minimum
of measuring and gauging shall be required to test the diameter and form
of the thread. 5. That the angles of the sides shall be as acute as is
consistent with the required strength. 6. That it shall not be unduly
liable to become loose in cases where the nut may require to be fastened
and loosened occasionally.

Referring to the first, by the term "the strength of a screw thread," is
not meant the strength of one thread, but of so many threads as are
contained in the nut. This obviously depends upon the depth or thickness
of the nut-piece. The standard thickness of nut, both in the United
States and Whitworth systems, as well as in general practice, or where
the common [V]-thread is used, is made equal to the diameter of the top
of the thread. Therefore, by the term "strength of thread" is meant the
combined strength of as many threads as are contained in a nut of the
above named depth. It is obvious, then, when it is advantageous to
increase the strength of a thread, that it may be done by increasing the
depth of the nut, or in other words, by increasing the number of threads
used in computing its strength. This is undesirable by reason of
increasing the cost and labor of producing the nuts, especially as the
threading tools used for nuts are the weakest, and are especially liable
to breakage, even with the present depth of nuts.

It has been found from experiments that have been made that our present
threads are stronger than their bolts, which is desirable, inasmuch as
it gives a margin for wear on the sides of the threads. But for threads
whose nuts are to remain permanently fastened and are not subject to
wear, it is questionable whether it were not better for the bolts to be
stronger than the threads. Suppose, for instance, that a thread strips,
and the bolt will remain in place because the nut will not come off the
bolt readily. Hence the pieces held by the bolt become loosened, but not
disconnected. If, on the other hand, the bolt breaks, it is very liable
to fall out, leaving the piece or pieces, as the case may be, to fall
apart, or at least become disconnected, so far as the bolt is concerned.
But since threads are used under conditions where the threads are liable
to wear, and since it is undesirable to have more than one standard
thread, it is better to have the threads, when new, stronger than the
bolts.

[Illustration: Fig. 259.]

Referring to the second requirement, screw threads or the tools that
produce them are originated in the lathe, and the difficulty with making
a round top and bottom thread lies in shaping the corner to cut the top
of the thread. This is shown in Fig. 257, where a Whitworth thread and a
single-toothed thread-cutting tool are represented. The rounded point A
of the tool will not be difficult to produce, but the hollow at B would
require special tools to cut it. This is, in fact, the plan pursued
under the Whitworth system, in which a hob or chaser-cutting tool is
used to produce all the thread-cutting tools. A chaser is simply a
toothed tool such as is shown in Fig. 258. Now, it would manifestly be
impracticable to produce a chaser having all the curves, A and B, at the
top and at the bottom of the teeth alike, by the grinding operations
usually employed in the workshop, and hence the employment of the hob.
Fig. 259 represents a hob, which is a threaded piece of steel with a
number of grooves such as shown at A, A, A, which divide the thread into
teeth, the edges of which will cut a chaser, of a form corresponding to
that of the thread upon the hob. The chaser is employed to produce taps
and secondary hobs to be used for cutting the threads in dies, &c., so
that the original hob is the source from which all the thread-cutting
tools are derived.

[Illustration: Fig. 260.]

For the United States standard or the common [V]-thread, however, no
standard hob is necessary, because a single-pointed tool can be ground
with the ordinary grinding appliances of the workshop. Thus, for the
United States standard, a flat-pointed tool, Fig. 260, and for the
common [V]-thread, a sharp-pointed tool, Fig. 260, may be used. So far
as the correctness of angle of pitch and of thread depth are concerned,
the United States standard and the common [V]-thread can both be
produced, under skilful operation, more correctly than is possible with
the Whitworth thread, for the following reasons:--

To enable a hob to cut, it must be hardened, and in the hardening
process the pitch of the thread alters, becoming, as a general rule
(although not always) finer. This alteration of pitch is not only
irregular in different threads, but also in different parts of the same
thread. Now, whatever error the hob thread receives from hardening it
transfers to the chaser it cuts. But the chaser also alters its form in
hardening, the pitch, as a general rule, becoming coarser. It may happen
that the error induced in the hob hardening is corrected by that induced
by hardening the chaser, but such is not necessarily the case.

[Illustration: Fig. 261.]

The single-pointed tool for the United States standard or for the common
[V]-thread is accurately ground to form after the hardening, and hence
need contain no error. On the other hand, however, the rounded top and
bottom thread preserves its form and diameter upon the thread-cutting
tools better than is the case with threads having sharp corners, for the
reason that a rounded point will not wear away so quickly as a sharp
point. To fully perceive the importance of this, it is necessary to
consider the action of a tool in cutting a thread. In Fig. 261 there is
shown a chaser, A, applied to a partly-formed thread, and it will be
observed that the projecting ends or points of the teeth are in
continuous action, cutting a groove deeper and deeper until a full
thread is developed, at which time the bottoms of the chaser teeth will
meet the perimeter of the work, but will perform no cutting duty upon
it. As a result, the chaser points wear off, which they will do more
quickly if they are pointed, and less quickly if they are rounded. This
causes the thread cut to be of increased and improper diameter at the
root.

[Illustration: Fig. 262.]

The same defect occurs on the tools for cutting internal threads, or
threads in holes or bores. In Fig. 262, for example, is shown a tool
cutting an internal thread, which tool may be taken to represent one
tooth of a tap. Here again the projecting point of the tool is in
continuous cutting action, while this, being a single-toothed tool, has
no bottom corners to suffer from wear. As a result of the wear upon the
tools for cutting internal threads, the thread grooves, when cut to
their full widths, will be too shallow in depth, or, more correctly
speaking, the full diameter of the thread will be too small to an amount
corresponding to twice the amount of wear that the tool point has
suffered. In single-pointed tools, such as are used upon lathe work,
this has but little significance, because it is the work of but a minute
or two to grind up the tool to a full point again, but in taps and solid
dies, or in chasers in heads (as in some bolt-cutting machines) it is
highly important, because it impairs the fit of the threads, and it is
difficult to bring the tools to shape after they are once worn.

[Illustration: Fig. 263.]

The internal threads for the nuts of bolts are produced by a tap formed
as at T in Fig. 263. It consists of a piece of steel having an external
thread and longitudinal flutes or grooves which cut the thread into
teeth. The end of the thread is tapered off as shown, to enable the end
of the tap to enter the hole, and as it is rotated and the nut N held
stationary, the teeth cut grooves as the tap winds through, thus forming
the thread.

[Illustration: Fig. 264.]

The threads upon bolts are usually produced either by a head containing
chasers or by a solid die such as shown at A in Fig. 264, B representing
a bolt being threaded. The bore of A is threaded and fluted to provide
cutting teeth, and the threads are chamfered off at the mouth to assist
the cutting by spreading it over several teeth, which enables the bolt
to enter the die more easily.

We may now consider the effect of continued use and its consequent wear
upon the threads or teeth of a tap and die or chaser.

The wear of the corners at the tops of the thread (as at A B in Fig.
265) of a tap is greater than the wear at the bottom corners at E F,
because the tops perform more cutting duty.

[Illustration: Fig. 265.]

First, the top has a larger circle of rotation than has the bottom, and,
therefore, its cutting speed is greater, to an amount equal to the
difference between the circumferences of the thread at the top and at
the bottom. Secondly, the tops of the teeth of tap perform nearly all
the cutting duty, because the thread in the nut is formed by the tops
and sides of the tap, which on entering cut a groove which they
gradually deepen, until a full thread is formed, while the bottoms of
the teeth (supposing the tapping hole to be of proper diameter and not
too small) simply meet the bore of the tapping hole as the thread is
finished. If, as in the case of hot punched nuts, the nut bore contains
scale, this scale is about removed by the time the bottoms of the top
teeth come into action, hence the teeth bottoms are less affected by the
hardness of the scale.

In the case of the teeth on dies and chasers, the wear at the corners C
D, in Fig. 266, is the greatest. Now, the tops of the teeth on the tap
(A B, in Fig. 265) cut the bottom or full diameter of the thread in the
nut, while the tops of the teeth (C D, in Fig. 266) in the die cut the
bottom of the thread on the bolt; hence the rounded corners cut on the
work by the tops of the teeth in the one case, meet the more square
corners left by the tops of the teeth in the other, and providing that
under these circumstances the thread in the nut were of equal diameter
to that on the bolt the latter would not enter the former.

If the bolt were made of a diameter to enable the nut to wind a close
fit upon the bolt, the corners only of the threads would fit, as shown
in Fig. 267, which represents at N a thread in a portion of a nut and at
S a portion of a thread upon a tap or bolt, the two threads being
magnified and shown slightly apart for clearness of illustration. The
corners A, B of the nut are then cut by the corners A B of the tap in
Fig. 265, and the corners C, C, D correspond to those cut by the corners
C, D of the die teeth in Fig. 266; corners E, F, Fig. 267, are cut by
corners C, D, in Fig. 266, and corners G, H are cut by corners G, H in
Fig. 266, and it is obvious that the roundness of the corners A, B, C,
and D in Fig. 267 will not permit the tops of the thread on the bolt to
meet the bottoms of the thread in the nut, but that the threads will
bear at the corners only.

[Illustration: Fig. 266.]

[Illustration: Fig. 267.]

So far, however, we have only considered the wear tending to round off
the sharp corners of the teeth, which wear is greater in proportion as
the corners are sharp, and less as they are rounded or flattened, and we
have to consider the wear as affecting the diameters of the male and
female thread at their tops and bottoms respectively.

Now, since the tops of the tap teeth wear the most, the diameter of the
thread decreases in depth, while, since the tops of the die teeth wear
most, the depth of the thread in the die also decreases. The tops of the
tap teeth cut the bottom of the thread in the nut and the tops of the
die teeth cut the bottoms of the thread upon the bolt.

Let it be supposed then that the points of the teeth of a tap have worn
off to a depth of the 1-2000th part of an inch, which they will by the
time they become sufficiently dulled to require resharpening, and that
the teeth of a die have become reduced by wear by the same amount, and
the result will be the production of threads such as shown in Fig. 268,
in which the diameter of the bolt is supposed to be an inch, and the
proper thread depth 1-10th inch. Now, the diameter at the root of the
thread on the bolt will be .802 inch in consequence of the wear, but the
smallest diameter of the nut thread is .800 inch, and hence too small to
admit the male or bolt thread. Again, the full diameter of the bolt
thread is 1 inch, whereas the full diameter of the nut thread is but
.998 inch, or, again, too small to admit the bolt thread. As a result,
it is found in practice that any standard form of thread that makes no
allowance for wear, cannot be rigidly adhered to, or if it is adhered
to, the tap must be made when new above the standard diameter, causing
the thread to be an easy fit, which fit will become closer as the
thread-cutting tools wear, until finally it becomes too tight
altogether. The fit, however, becomes too tight at the top and bottom,
where it is not required, instead of at the sides, where it should
occur. When this is the case, the nuts will soon wear loose because of
their small amount of bearing area.

[Illustration: Fig. 268.]

[Illustration: Fig. 269.]

It may be pointed out, however, that from the form in which the chasers
or solid dies for bolt machines, and also that in which taps are made,
the finishing points of the teeth are greatly relieved of cutting duty,
as is shown in Figs. 269 and 270. In the die the first two or three
threads are chamfered off, while in the tap the thread is tapered off
for a length usually equal to about two or three times the diameter for
taps to be used by hand, and six or seven times the diameter for taps to
be used in a machine. The wear of the die is, therefore, more than that
of the tap, because the amount of cutting duty to produce a given
length of thread is obviously the same, whether the thread be an
internal or an external one, and the die has less cutting edges to
perform this duty than the tap has. The main part of the cutting is, it
is true, in both cases borne by the beveled surfaces at the top of the
chamfered teeth of the cutting tools, but the fact remains that the
depth of the thread is finished by the extreme tops of the teeth, and
these, therefore, must in time suffer from the consequent wear, while
the bottoms of the teeth perform no cutting duty, providing that the
hole in the one case and the bolt in the other are of just sufficient
diameter to permit of a full thread being formed, as should be the case.
In threads cut by chasers the same thing occurs; thus in Fig. 271 is
shown at A a chaser having full teeth, as it must have when a full
thread is to pass up to a shoulder, as up to the head of a bolt. Here
the first tooth takes the whole depth of the cut, but if from wear this
point becomes rounded, the next tooth may remedy the defect. When,
however, a chaser is to be used upon a thread that terminates in a stem
of smaller diameter, as C in Fig. 271, then the chaser may have its
teeth bevelled off, as is shown on B.

[Illustration: Fig. 270.]

The evils thus pointed out as attending the wear of screw-cutting tools
for bolts and nuts, may be overcome by a slight variation in the form of
the thread. Thus in Fig. 272, at A is shown a form of thread for the
tools to cut internal threads, and at B a form of thread for dies to cut
external threads. The sides of the thread are in both cases at the same
angle, as say, 60°. The depth of the thread, supposing the angle of the
sides to meet in a point, is divided off into 11, or any number of equal
divisions. For a tap one of these divisions is taken off, forming a flat
top, while at the bottom two of these divisions are taken off, or if
desirable, 1-1/2 divisions may be taken off, since the exact amount is
not of primary importance. On the external thread cutting tool B, as say
a solid die, two divisions are taken off at the largest diameter, and
one at the smallest diameter, or, if any other proportion be selected
for the tap, the same proportion may be selected for the die, so long as
the least is taken off the largest diameter of the tap thread, and of
the smallest diameter of the die thread.

[Illustration: Fig. 271.]

The diameter of the tap may still be standard to ring or collar gauge,
as in the Franklin Institute thread, the angle at the sides being simply
carried in a less distance. In the die the largest diameter of the
thread has a flat equal to that on the bottom of the tap, while the
smallest diameter has a flat equal to that on the tops of the tap teeth,
the width or thickness of the threads remaining the same as in the
Franklin Institute thread at each corresponding diameter in its depth.

[Illustration: Fig. 272.]

The effect is to give to the threads on the work a certain amount of
clearance at the top and bottom of the thread, leaving the angles just
the same as before, and insuring that the contact shall be at the sides,
as shown in Fig. 273.

This form of thread retains the valuable features of the Franklin
Institute that it can be originated by any one, and that it can be
formed with a single-toothed or single-pointed tool. Furthermore, the
wear of the threading tools will not impair the diametral fit of the
work, while the permissible limit of error in diameter will be
increased.

By this means great accuracy in the diameters of the threads is rendered
unnecessary, and the wear of the screw-cutting tools at their corners is
rendered harmless, nor can any confusion occur, because the tools for
external threads cannot be employed upon internal ones. The sides only
of the thread will fit, and the whole contact and pressure of the fit
will be on those sides only.

[Illustration: Fig. 273.]

This is an important advantage, because if the tops of the thread are
from the wear of the dies and taps of too large or small diameter,
respectively, the threads cannot fit on the sides. Thus, suppose a bolt
thread to be loose at the sides, but to be 1-1000 of an inch larger in
diameter than the nut thread, then it cannot be screwed home until that
amount has been worn or forced off the thread diameter, or has been
bruised down by contact with the nut thread, and it would apparently be
a tight fit at the sides. Suppose a thread to have been cut in the
lathe to the correct diameter at the bottom of the thread, the sides of
the thread being at the correct angle, but let the diameter at the top
of the thread (a Franklin Institute thread is here referred to), be
1-1000 too large, then the nut cannot be forced on until that 1-1000 is
removed by some means or other, unless the nut thread be deepened to
correspond.

Now take this last bolt and turn the 1-1000 inch off, and it will fit,
turn off another 1-1000 or 1-64 inch, and it will still fit, and the fit
will remain so nearly the same with the 1-64 inch off that the
difference can scarcely be found. Furthermore, with a nut of a fit
requiring a given amount of force to screw it upon the bolt, the area of
contact will be much greater when that contact is on the sides than when
it is upon the tops and bottoms of the thread, while the contact will be
in a direction better to serve as an abutment to the thrust or strain.

In very fine pitches of thread such as are used in the manufacture of
watches, this plan of easing or keeping free the extremities of the
thread is found to be essential, and there appears every probability
that its adoption would obviate the necessity of using check nuts.

It has been observed that the threads upon tools alter in pitch from the
hardening operation, and this is an objection to the employment of
chasers cut from hobs.

Suppose, for instance, that a nut is produced having a thread of true
and uniform pitch, then after hardening, the pitch may be no longer
correct. The chasers cut from the hob will contain the error of pitch
existing in the hob, and upon being hardened may have added to it errors
of its own. If this chaser be used to produce a new hob, the latter will
contain the errors in the chaser added to whatever error it may itself
obtain in the hardening. The errors may not, it is true, all exist in
one direction, and those of one hardening may affect or correct those
caused by another hardening, but this is not necessarily the case, and
it is therefore preferable to employ a form of thread that can be cut by
a tool ground to correct shape after having been hardened, as is the
case with the [V]-thread and the United States standard.

[Illustration: Fig. 274.]

It is obvious that in originating either the sharp [V] or the United
States standard thread, the first requisite is to obtain a correct angle
of 60°, which has been done in a very ingenious manner by Mr. J. H.
Heyer for the Pratt and Whitney Company, the method being as follows.
Fig. 274 is a face and an end view of an equilateral triangle employed
as a guide in making standard triangles, and constructed as
follows:--Three bars, A, A, A, of steel were made parallel and of
exactly equal dimensions. Holes X were then pierced central in the width
of each bar and the same distance apart in each bar; the method of
insuring accuracy in this respect being shown in Figs. 275 and 276, in
which S represents the live spindle of a lathe with its face-plate on
and a plug, C, fitted into the live centre hole. The end of this plug is
turned cylindrically true, and upon it is closely fitted a bush, the
plug obviously holding the bush true by its hole. A rectangular piece
_e_ is provided with a slot closely fitting to the bush.

The rectangular piece _e_ is then bolted to the lathe face-plate and
pierced with a hole, which from this method of chucking will be exactly
central to its slot, and at a right angle to its base. The bush is now
dispensed with and the piece _e_ is chucked with its base against the
face-plate and the hole pierced as above, closely fitting to the pin on
the end of the plug _c_, which, therefore, holds _e_ true.

[Illustration: Fig. 275.]

The bars A are then chucked one at a time in the piece _e_ (the outer
end resting upon a parallel piece _f_), and a hole is pierced near one
end, this hole being from this method of chucking exactly central to the
width of the bar A, and at a right angle to its face.

[Illustration: Fig. 276.]

The parallel piece _f_ is then provided with a pin closely fitting the
hole thus pierced in the bar. The bars were turned end for end with the
hole enveloping the pin in _f_ (the latter being firmly fixed to the
face-plate), and the other end laid in the slot in _e_, while the second
hole was pierced. The holes (X, Fig. 274) must be, from this method of
chucking, exactly an equal distance apart on each bar. The bars were
then let together at their ends, each being cut half-way through and
closely fitting pins inserted in the holes X, thus producing an
equilateral triangle entirely by machine work, and therefore as correct
as it can possibly be made, and this triangle is kept as a standard
gauge whereby others for shop use may be made by the following
process:--

Into the interior walls of this triangle there is fitted a cylindrical
bush B, it being obvious that this bush is held axially true or central
to the triangle, and it is secured in place by screws _y_, _y_, _y_,
passing through its flange and into bars A.

At one end of the bush B, is a cylindrical part D, whose diameter is 2
inches or equal to the length of one side of an equilateral triangle
circumscribed about a circle whose diameter is 1.1547 inches, as shown
in Fig. 278 and through this bush B passes a pin P, having a nut N. A
small triangle is then roughed out, and its bore fitting to the stem of
pin P, and by means of nut N, the small triangle is gripped between the
under face of D and the head of P. The large triangle is then held to an
angle-plate upon a machine while resting upon the machine-table, and the
uppermost edge of the small triangle is dressed down level with the
cylindrical stem D, which thus serves as a gauge to determine how much
to take off each edge of the small triangle to bring it to correct
dimensions.

The truth of the angles of the small triangle depends, of course, also
upon the large one; thus with face H resting upon the machine-table,
face G is cut down level with stem D; with face F upon the table, face E
is cut down level with D; and with face L upon the table, face K is
dressed down level with D. And we have a true equilateral triangle
produced by a very ingenious system of chuckings, each of which may be
known to be true.

The next operation is to cut upon the small triangle the flat
representing the top and bottom of the United States standard thread,
which is done by cutting off one-eighth part of its vertical height, and
it then becomes a test piece or standard gauge of the form of thread.
The next step is to provide a micrometer by means of which tools for
various pitches may be tested both for angle and for width of flat, and
this is accomplished as follows:--

[Illustration: _VOL. I_ =MEASURING AND GAUGING SCREW THREADS.= _PLATE
III._

Fig. 279.

Fig. 280.

Fig. 281.

Fig. 282.

Fig. 285.

Fig. 286.

Fig. 283.

Fig. 284.

Fig. 287.]

In Fig. 278 F is a jaw fixed by a set screw to the bar of the
micrometer, and E is a sliding jaw; these two jaws being fitted to the
edges of the triangle or test piece T in the figure which has been made
as already described. To the sliding jaw E is attached the micrometer
screw C, which has a pitch of 40 threads per inch; the drum A upon the
screw has its circumference divided into 250 equidistant divisions,
hence if the drum be moved through a space equal to one of these
divisions the sliding jaw E will be moved the 1-250th part of 1-40th of
an inch, or in other words the 1-10,000th of an inch. To properly adjust
the position of the zero piece or pointer, the test piece T is placed in
the position shown in Fig. 278, and when the jaws were so adjusted that
light was excluded from the three edges of the test piece, the pointer
R, Fig. 277, was set opposite to the zero mark on the drum and fastened.

[Illustration: Fig. 277.]

To set the instrument for any required pitch of thread of the United
States standard form the micrometer is used to move the sliding jaw E
away from the fixed jaw F to an amount equal to the width of flat upon
the top and bottom, of the required thread, while for the sharp
[V]-thread the jaws are simply closed. The gauge being set the tool is
ground to the gauge.

[Illustration: Fig. 278.]

Referring to the third requirement, that the tools shall in the case of
lathe work be easily sharpened and set to correct position in the lathe,
it will be treated in connection with cutting screws in the lathe.
Referring to the fourth requirement, that a minimum of measuring and
gauging shall be required to test the diameter and form of thread, it is
to be observed that in a Whitworth thread the angle and depth of the
thread is determined by the chaser, which may be constantly ground to
resharpen without altering the angles or depth of the thread, hence in
cutting the tooth the full diameter of the thread is all that needs to
be gauged or measured. In cutting a sharp [V]-thread, however, the
thread top is apt to project (from the action of the single-pointed
tool) slightly above the natural diameter of the work, producing a
feather edge which it becomes necessary to file off to gauge the full
diameter of the thread. In originating a sharp [V]-thread it is
necessary first to grind the tool to correct angle; second, to set it at
the correct height in the latter, and with the tool angles at the proper
angle with the work (as is explained with reference to thread cutting in
the lathe) and to gauge the thread to the proper diameter. In the
absence of a standard cylindrical gauge or piece to measure from, a
sheet metal gauge, such as in Fig. 279, may be applied to the thread;
such gauges are, however, difficult to correctly produce.

So far as the diameter of a thread is concerned it may be measured by
calipers applied between the threads as in Figs. 280 and 281, a plan
that is commonly practised in the workshop when there is at hand a
standard thread or gauge known to be of proper diameter; and this method
of measuring may be used upon any form of thread, but if it is required
to test the form of the thread, as may occur when its form depends upon
the workman's accuracy in producing the single-pointed threading tools,
then, in the case of the United States standard thread, the top, the
bottom, and the angle must be tested. The top of the thread may (for all
threads) be readily measured, but the bottom is quite difficult to
measure unless there is some standard to refer it to, to obtain its
proper diameter, because the gauge or calipers applied to the bottom of
the thread do not stand at a right angle to the axis of the bolt on
which the thread is cut, but at an angle equal to the pitch of the
thread, as shown in Fig. 282.

Now, the same pitch of thread is necessarily used in mechanical
manipulation upon work of widely varying diameters, and as the angle of
the calipers upon the same pitch of thread would vary (decreasing as the
diameter of the thread increases), the diameter measured at the bottom
of the thread would bear a constantly varying proportion to the diameter
measured across the tops of the thread at a right angle to the axial
line of the work. Thus in Fig. 282, A A is the axial line of two
threaded pieces, B, C. D, D represents a gauge applied to B, its width
covering the tops of two threads and measuring the diameter at a right
angle to A A, as denoted by the dotted line E. The dotted line F
represents the measurement at the bottom of the thread standing at an
angle to E equal to half the pitch. The dotted line G is the measurement
of C at the bottom of the thread.

Now suppose the diameter of B to be 1-1/2 inches at the top of the
thread, and 1-1/8 inches at the bottom, while C is 1-1/8 inches on the
top and 3/4 at the bottom of the thread, the pitches of the two threads
being 1/4 inch; then the angle of F to E will be 1/8 inch (half the
pitch) in its length of 1-1/8 inches. The angle of G to E will be 1/8
inch (half the pitch) in 3/4 (the diameter at the bottom or root of the
thread).

It is obvious, then, that it is impracticable to gauge threads from
their diameters at the bottom, or root.

On account of the minute exactitude necessary to produce with lathe
tools threads of the sharp [V] and United States standard forms, the
Pratt and Whitney Company manufacture thread-cutting tools which are
made under a special system insuring accuracy, and provide standard
gauges whereby the finished threads may be tested, and since these tools
are more directly connected with the subject of lathe tools than with
that of screw thread, they are illustrated in connection with such
tools. It is upon the sides of threads that the contact should exist to
make a fit, and the best method of testing the fit of a male and female
thread is to try them together, winding them back and forth until the
bright marks of contact show. Giving the male thread a faint tint of
paint made of Venetian red mixed with lubricating oil, will cause the
bearing of the threads to show very plainly.

Figs. 283 and 284 represent standard reference gauges for the United
States standard thread. Fig. 283 is the plug or male gauge. The top of
the thread has, it will be observed, the standard flat, while the bottom
of the thread is sharp. In the collar, or female gauge, or the template,
as it may be termed, a side and a top view of which are shown in Fig.
284, and a sectional end view in Fig. 285, the flat is made on the
smallest diameter of the thread, while the largest diameter is left
sharp; hence, if we put the two together they will appear as in Fig.
286, there being clearance at both the tops and bottoms of the threads.
This enables the diameters of the threads to be in both cases tested by
standard cylindrical gauges, while it facilitates the making of the
screw gauges. The male or plug gauge is made with a plain part, A, whose
diameter is the standard size for the bottoms of the threads measured
at a right angle to the axis of the gauge and taking the flats into
account. The female gauge or template is constructed as follows:--A
rectangular piece of steel is pierced with a plain hole at B, and a
standard thread hole at A, and is split through at C. At D is a pin to
prevent the two jaws from springing, this being an important element of
the construction. E is a screw threaded through one jaw and abutting
against the face of the other, while at F is another screw passing
through one jaw and threaded into the other, and it is evident that
while by operating these two screws the size of the gauge bore A may be
adjusted, yet the screws will not move and destroy the adjustment,
because the pressure of one acts as a lock to the other. It is obvious
that in adjusting the female gauge to size, the thread of the male gauge
may be used as a standard to set it by.

To produce sheet metal templates such as was shown in Fig. 279, the
following method may be employed, it being assumed that we have a
threading tool correctly formed.

[Illustration: Fig. 288.]

[Illustration: Fig. 289.]

Suppose it is required to make a gauge for a pitch of 6 per inch, then a
piece of iron of any diameter may be put in the lathe and turned up to
the required diameter for the top of the thread. The end of this piece
should be turned up to the proper diameter for the bottom of the thread,
as at G, in Fig. 287. Now, it will be seen that the angle of the thread
to the axis A of the iron is that of line C to line A, and if we require
to find the angle the thread passes through in once winding around the
bolt, we proceed as in Fig. 288, in which D represents the circumference
of the thread measured at a right angle to the bolt axis, as denoted by
the line B in Fig. 287. F, Fig. 288 (at a right angle to D), is the
pitch of the thread, and line C therefore represents the angle of the
thread to the bolt axis, and corresponds to line C in Fig. 287. We now
take a piece of iron whose length when turned true will equal its
finished and threaded circumference, and after truing it up and leaving
it a little above its required finished diameter, we put a pointed tool
in the slide-rest and mark a line A A in Fig. 289, which will represent
its axis. At one end of this line we mark off below A A the pitch of the
thread, and then draw the line H J, its end H falling below A to an
amount equal to the pitch of the thread to be cut. The piece is then put
in a milling machine and a groove is cut along H J, this groove being to
receive a tightly-fitting piece of sheet metal of which a thread gauge
is to be made. This piece of sheet metal must be firmly secured in the
groove by set-screws. The piece of iron is then again put in the lathe
and its diameter finished to that of the required diameter of thread.
Its two ends are then turned down to the required diameter for the
bottom of the thread, leaving in the middle a section on which a full
thread can be cut, as in Fig. 290, in which F F represents the sheet
metal for the gauge. After the thread is cut, as in Fig. 290, we take
out the gauge and it will appear as in Fig. 291, and all that is
necessary is to file off the two outside teeth if only one tooth is
wanted.

[Illustration: Fig. 290.]

The philosophy of this process is that we have set the gauge at an angle
of 90°, or a right angle to the thread, as is shown in Fig. 289, the
line C representing the angle of the thread to the axis A A, and
therefore corresponding to the line C in Fig. 287. A gauge made in this
way will serve as a test of its own correctness for the following
reasons: Taking the middle tooth in Fig. 291, it is clear that one of
its sides was cut by one angle and the other by the other angle of the
tool that cut it, and as a correctly formed thread is of exactly the
same shape as the space between two threads, it follows that if the
gauge be applied to any part of the thread that was cut in forming it,
and if it fits properly when tried, and then turned end for end and
tried again, it is proof that the gauge and the thread are both correct.
Suppose, for example, that the tool was correct in its shape, but was
not set with its two angles equal to the line of lathe centres, and in
that case the two sides of the thread will not be alike and the gauge
will not reverse end for end and in both cases fit to the thread. Or
suppose the flat on the tool point was too narrow, and the flat at the
bottom of the thread will not be like that at the top, and the gauge
will show it.

[Illustration: Fig. 291.]

Referring to the fifth requirement, that the angles of the sides of the
threads shall be as acute as is consistent with the required strength,
it is obvious that the more acute the angles of the sides of the thread
one to the other the finer the pitch and the weaker the thread, but on
the other hand, the more acute the angle the better the sides of the
thread will conform one to the other. The importance of this arises from
the fact that on account of the alteration of pitch, already explained,
as accompanying the hardening of screw-cutting tools, the sides of
threads cut even by unworn tools rarely have full contact, and a nut
that is a tight fit on its first passage down its bolt may generally be
caused to become quite easy by running it up and down the bolt a few
times. Nuts that require a severe wrench force to wind them on the bolt,
may, even though they be as large as a two-inch bolt, often be made to
pass easily by hand, if while upon the bolt they are hammered on their
sides with a hand hammer. The action is in both cases to cause the sides
of the thread to conform one to the other, which they will the more
readily do in proportion as their sides are more acute. Furthermore, the
more acute the angles the less the importance of gauging the threads to
precise diameter, especially if the tops and bottoms of the male and
female thread are clear of one another, as in Fig. 273.

Referring to the sixth requirement, that the nut shall not be unduly
liable to become loose of itself in cases where it may require to be
fastened and loosened occasionally, it may be observed, that in such
cases the threads are apt from the wear to become a loose fit, and the
nuts, if under jar or vibration, are apt to turn back of themselves upon
the bolt. This is best obviated by insuring a full bearing upon the
whole area of the sides of the thread, and by the employment of as fine
pitches as is consistent with sufficient strength, since the finer the
pitch the nearer the thread stands at right angle to the bolt axis, and
the less the tendency to unscrew from the pressure on the nut face.

The pitches, diameters, and widths of flat of the United States
standard thread are as per the following table:--

UNITED STATES STANDARD SCREW THREADS.

  +-------------+-----------+-----------------+----------+
  | Diameter of |  Threads  |   Diameter at   | Width of |
  |   Screw.    | per inch. | root of Thread. |   Flat.  |
  +-------------+-----------+-----------------+----------+
  |    1/4      |    20     |      .1850      |  .0063   |
  |    5/16     |    18     |      .2403      |  .0069   |
  |    3/8      |    16     |      .2938      |  .0078   |
  |    7/16     |    14     |      .3447      |  .0089   |
  |    1/2      |    13     |      .4001      |  .0096   |
  |    9/16     |    12     |      .4542      |  .0104   |
  |    5/8      |    11     |      .5069      |  .0114   |
  |    3/4      |    10     |      .6201      |  .0125   |
  |    7/8      |     9     |      .7307      |  .0139   |
  |             |           |                 |          |
  |    1        |     8     |      .8376      |  .0156   |
  |    1-1/8    |     7     |      .9394      |  .0179   |
  |    1-1/4    |     7     |     1.0644      |  .0179   |
  |    1-3/8    |     6     |     1.1585      |  .0208   |
  |    1-1/2    |     6     |     1.2835      |  .0208   |
  |    1-5/8    |     5-1/2 |     1.3888      |  .0227   |
  |    1-3/4    |     5     |     1.4902      |  .0250   |
  |    1-7/8    |     5     |     1.6152      |  .0250   |
  |    2        |     4-1/2 |     1.7113      |  .0278   |
  +-------------+-----------+-----------------+----------+

The standard pitches for the sharp [V]-thread are as follows:--

                             SIZE OF BOLT.
  ---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---
  1/4| 5/|3/8| 7/|1/2|5/8|3/4|7/8| 1 | 1-| 1-| 1-| 1-| 1-| 1-| 1-| 2
     | 16|   | 16|   |   |   |   |   |1/8|1/4|3/8|1/2|5/8|3/4|7/8|
  ---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---
                       NUMBER OF THREADS TO INCH.
  ---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---
   20| 18| 16| 14| 12| 11| 10| 9 | 8 | 7 | 7 | 6 | 6 | 5 | 5 | 4-| 4-
     |   |   |   |   |   |   |   |   |   |   |   |   |   |   |1/2|1/2
  ---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---

The following table gives the threads per inch, pitches and diameters at
root of thread of the Whitworth thread. The table being arranged from
the diameter of the screw as a basis.

  +----------+---------+--------+----------------+
  | Diameter | Threads |        |  Diameter at   |
  |    of    |   per   | Pitch. | Root or Bottom |
  |  Screw.  |  Inch.  |        |   of Thread.   |
  +----------+---------+--------+----------------+
  |          |         |  Inch. |      Inch.     |
  |    1/8   | 40      |  .025  |      .0929     |
  |    3/16  | 24      |  .041  |      .1341     |
  |    1/4   | 20      |  .050  |      .1859     |
  |    5/16  | 18      |  .056  |      .2413     |
  |    3/8   | 16      |  .063  |      .2949     |
  |    7/16  | 14      |  .071  |      .346      |
  |    1/2   | 12      |  .083  |      .3932     |
  |    9/16  | 12      |  .083  |      .4557     |
  |    5/8   | 11      |  .091  |      .5085     |
  |    11/16 | 11      |  .095  |      .571      |
  |    3/4   | 10      |  .100  |      .6219     |
  |    13/16 | 10      |  .100  |      .6844     |
  |    7/8   |  9      |  .111  |      .7327     |
  |    15/16 |  9      |  .111  |      .7952     |
  |  1       |  8      |  .125  |      .8399     |
  |  1-1/8   |  7      |  .143  |      .942      |
  |  1-1/4   |  7      |  .143  |     1.067      |
  |  1-3/8   |  6      |  .167  |     1.1615     |
  |  1-1/2   |  6      |  .167  |     1.2865     |
  |  1-5/8   |  5      |  .200  |     1.3688     |
  |  1-3/4   |  5      |  .200  |     1.4938     |
  |  1-7/8   |  4-1/2  |  .222  |     1.5904     |
  |  2       |  4-1/2  |  .222  |     1.7154     |
  |  2-1/8   |  4-1/2  |  .222  |     1.8404     |
  |  2-1/4   |  4      |  .250  |     1.9298     |
  |  2-3/8   |  4      |  .250  |     2.0548     |
  |  2-1/2   |  4      |  .250  |     2.1798     |
  |  2-5/8   |  4      |  .250  |     2.3048     |
  |  2-3/4   |  3-1/2  |  .286  |     2.384      |
  |  2-7/8   |  3-1/2  |  .286  |     2.509      |
  |  3       |  3-1/2  |  .286  |     2.634      |
  |  3-1/4   |  3-1/4  |  .308  |     2.884      |
  |  3-1/2   |  3-1/4  |  .308  |     3.106      |
  |  3-3/4   |  3      |  .333  |     3.356      |
  |  4       |  3      |  .333  |     3.574      |
  |  4-1/4   |  2-7/8  |  .348  |     3.824      |
  |  4-1/2   |  2-7/8  |  .348  |     4.055      |
  |  4-3/4   |  2-3/4  |  .364  |     4.305      |
  |  5       |  2-3/4  |  .364  |     4.534      |
  |  5-1/4   |  2-5/8  |  .381  |     4.764      |
  |  5-1/2   |  2-5/8  |  .381  |     5.014      |
  |  5-3/4   |  2-1/2  |  .400  |     5.238      |
  |  6       |  2-1/2  |  .400  |     5.488      |
  +----------+---------+--------+----------------+

The standard degree of taper, both for the taps and the dies, is 1/16
inch per inch, or 3/4 inch per foot, for all sizes up to 10-inch bore.

The sockets or couplings, however, are ordinarily tapped parallel and
stretched to fit the pipe taper when forced on the pipe. For bores of
pipe over 10 inches diameter the taper is reduced to 3/8 inch per foot.
The pipes or casings for oil wells are given a taper of 3/8 inch per
foot, and their couplings are tapped taper from both ends. There is,
however, just enough difference made between the taper of the socket and
that of the pipe to give the pipe threads a bearing at the pipe end
first when tried with red marking, the threads increasing their bearing
as the pieces are screwed together.

The United States standard thread for steam, gas and water pipe is given
below, which is taken from the Report of the Committee on Standard Pipe
and Pipe Threads of The American Society of Mechanical Engineers,
submitted at the 8th Annual Meeting held in New York, November-December,
1886.

"A longitudinal section of the tapering tube end, with the screw-thread
as actually formed, is shown full size in Fig. 291_a_ for a nominal
2-1/2 inch tube, that is, a tube of about 2-1/2 inches internal
diameter, and 2-7/8 inches actual external diameter.

[Illustration: Fig. 291_a_.]

"The thread employed has an angle of 60°; it is slightly rounded off
both at the top and at the bottom, so that the height or depth of the
thread, instead of being exactly equal to the pitch, is only four fifths
of the pitch, or equal to 0.8 × 1/_n_ if _n_ be the number of threads
per inch. For the length of tube end throughout which the screw thread
continues perfect, the empirical formula used is (0.8_D_ + 4.8) × 1/_n_,
where _D_ is the actual external diameter of the tube throughout its
parallel length, and is expressed in inches. Further back, beyond the
perfect threads, come two having the same taper at the bottom, but
imperfect at the top. The remaining imperfect portion of the screw
thread, furthest back from the extremity of the tube, is not essential
in any way to this system of joint; and its imperfection is simply
incidental to the process of cutting the thread at a single operation.

The standard thicknesses of the pipes and pitches of thread are as
follows:--

STANDARD DIMENSIONS OF WROUGHT IRON WELDED TUBES.

  +-----------------------------+-----------+--------------------+
  |      DIAMETER OF TUBE.      |           |   SCREWED ENDS.    |
  +---------+---------+---------+ THICKNESS +----------+---------+
  | Nominal | Actual  |  Actual |    OF     |  Number  |Length of|
  | Inside. | Inside. | Outside.|  METAL.   |of Threads| Perfect |
  |         |         |         |           |per Inch. | Screw.  |
  +---------+---------+---------+-----------+----------+---------+
  | Inches. | Inches. | Inches. |   Inch.   |    No.   |  Inch.  |
  |    1/8  |  0.270  |  0.405  |   0.068   |  27      |  0.19   |
  |    1/4  |  0.364  |  0.540  |   0.088   |  18      |  0.29   |
  |    3/8  |  0.494  |  0.675  |   0.091   |  18      |  0.30   |
  |    1/2  |  0.623  |  0.840  |   0.109   |  14      |  0.39   |
  |    3/4  |  0.824  |  1.050  |   0.113   |  14      |  0.40   |
  |  1      |  1.048  |  1.315  |   0.134   |  11-1/2  |  0.51   |
  |  1-1/4  |  1.380  |  1.660  |   0.140   |  11-1/2  |  0.54   |
  |  1-1/2  |  1.610  |  1.900  |   0.145   |  11-1/2  |  0.55   |
  |  2      |  2.067  |  2.375  |   0.154   |  11-1/2  |  0.58   |
  |  2-1/2  |  2.468  |  2.875  |   0.204   |   8      |  0.89   |
  |  3      |  3.067  |  3.500  |   0.217   |   8      |  0.95   |
  |  3-1/2  |  3.548  |  4.000  |   0.226   |   8      |  1.00   |
  |  4      |  4.026  |  4.500  |   0.237   |   8      |  1.05   |
  |  4-1/2  |  4.508  |  5.000  |   0.246   |   8      |  1.10   |
  |  5      |  5.045  |  5.563  |   0.259   |   8      |  1.16   |
  |  6      |  6.065  |  6.625  |   0.280   |   8      |  1.26   |
  |  7      |  7.023  |  7.625  |   0.301   |   8      |  1.36   |
  |  8      |  8.982  |  8.625  |   0.322   |   8      |  1.46   |
  |  9      |  9.000  |  9.688  |   0.344   |   8      |  1.57   |
  | 10      | 10.019  | 10.750  |   0.366   |   8      |  1.68   |
  +---------+---------+---------+-----------+----------+---------+

The taper of the threads is 1/16 inch in diameter for each inch of
length or 3/4 inch per foot.

WHITWORTH'S SCREW THREADS FOR GAS, WATER, AND HYDRAULIC IRON PIPING.

NOTE.--The Internal and External diameters of Pipes, as given below, are
those adopted by the firm of Messrs. JAMES RUSSELL & SONS, in Pipes of
their manufacture.

  +---------------------------------------+
  |         GAS AND WATER PIPING.         |
  +-------------+-------------+-----------+
  |  Internal   |  External   |  No. of   |
  | Diameter of | Diameter of |  Threads  |
  |    Pipe.    |   Pipe.     | per Inch. |
  +-------------+-------------+-----------+
  |     1/8     |    .385     |    28     |
  |     1/4     |    .520     |    19     |
  |     3/8     |    .665     |    19     |
  |     1/2     |    .822     |    14     |
  |     3/4     |   1.034     |    14     |
  |   1         |   1.302 }   |           |
  |   1-1/8     |   1.492 }   |           |
  |   1-1/4     |   1.650 }   |           |
  |   1-3/8     |   1.745 }   |           |
  |   1-1/2     |   1.882 }   |           |
  |   1-5/8     |   2.021 }   |           |
  |   1-3/4     |   2.047 }   |           |
  |   1-7/8     |   2.245 }   |           |
  |   2         |   2.347 }   |           |
  |   2-1/8     |   2.467 }   |           |
  |   2-1/4     |   2.587 }   |    11     |
  |   2-3/8     |   2.794 }   |           |
  |   2-1/2     |   3.001 }   |           |
  |   2-5/8     |   3.124 }   |           |
  |   2-3/4     |   3.247 }   |           |
  |   2-7/8     |   3.367 }   |           |
  |   3         |   3.485 }   |           |
  |   3-1/4     |   3.698 }   |           |
  |   3-1/2     |   3.912 }   |           |
  |   3-3/4     |   4.125 }   |           |
  |   4         |   4.339 }   |           |
  +-------------+-------------+-----------+

  +----------------------------------------------------+
  |                 HYDRAULIC PIPING.                  |
  +----------+----------+------------------+-----------+
  | Internal | External | Pressure in lbs. | No. of    |
  | Diameter | Diameter |   per Square     | Threads   |
  | of Pipe. | of Pipe. |      Inch.       | per Inch. |
  +----------+----------+------------------+-----------+
  |          | {  5/8   |       4,000}     |           |
  |    1/4   | {  3/4   |       6,000}     |    14     |
  |          | {  7/8   |       8,000}     |           |
  |          | {1       |      10,000}     |           |
  |          |          |                  |           |
  |          | {  3/4   |       4,000}     |           |
  |    3/8   | {  7/8   |       6,000}     |    14     |
  |          | {1       |       8,000}     |           |
  |          | {1-1/8   |      10,000}     |           |
  |          |          |                  |           |
  |          | {1       |       4,000}     |    14     |
  |    1/2   | {1-1/8   |       6,000}     |           |
  |          | {1-1/4   |       8,000 }    |    11     |
  |          | {1-3/8   |      10,000 }    |           |
  |          |          |                  |           |
  |          | {1-1/8   |       4,000      |    14     |
  |    5/8   | {1-1/4   |       6,000}     |           |
  |          | {1-3/8   |       8,000}     |    11     |
  |          | {1-1/2   |      10,000}     |           |
  |          |          |                  |           |
  |          | {1-1/4   |       4,000}     |           |
  |    3/4   | {1-3/8   |       6,000}     |    11     |
  |          | {1-1/2   |       8,000}     |           |
  |          | {1-5/8   |      10,000}     |           |
  |          |          |                  |           |
  |          | {1-3/8   |       4,000}     |           |
  |    7/8   | {1-1/2   |       6,000}     |    11     |
  |          | {1-5/8   |       8,000}     |           |
  |          | {1-3/4   |      10,000}     |           |
  |          |          |                  |           |
  |          | {1-1/2   |       4,000}     |           |
  |  1       | {1-5/8   |       6,000}     |    11     |
  |          | {1-3/4   |       8,000}     |           |
  |          | {1-7/8   |      10,000}     |           |
  |          |          |                  |           |
  |          | {1-5/8   |       4,000}     |           |
  |  1-1/8   | {1-3/4   |       6,000}     |    11     |
  |          | {1-7/8   |       8,000}     |           |
  |          | {2       |      10,000}     |           |
  |          |          |                  |           |
  |          | {1-3/4   |       4,000}     |           |
  |  1-1/4   | {1-7/8   |       6,000}     |    11     |
  |          | {2       |       8,000}     |           |
  |          | {2-1/8   |      10,000}     |           |
  |          |          |                  |           |
  |          | {1-7/8   |       4,000}     |           |
  |  1-3/8   | {2       |       6,000}     |    11     |
  |          | {2-1/8   |       8,000}     |           |
  |          | {2-1/4   |      10,000}     |           |
  |          |          |                  |           |
  |          | {2       |       4,000}     |           |
  |          | {2-1/8   |       6,000}     |           |
  |  1-1/2   | {2-1/4   |       8,000}     |    11     |
  |          | {2-3/8   |      10,000}     |           |
  |          | {2-1/2   |      10,000}     |           |
  |          |          |                  |           |
  |          | {2-1/8   |       4,000}     |           |
  |  1-5/8   | {2-1/4   |       6,000}     |    11     |
  |          | {2-3/8   |       8,000}     |           |
  |          | {2-1/2   |      10,000}     |           |
  |          |          |                  |           |
  |          | {2-1/4   |       3,000}     |           |
  |          | {2-3/8   |       4,000}     |           |
  |  1-3/4   | {2-1/2   |       6,000}     |    11     |
  |          | {2-5/8   |       8,000}     |           |
  |          | {2-3/4   |      10,000}     |           |
  |          |          |                  |           |
  |          | {2-3/8   |       3,000}     |           |
  |          | {2-1/2   |       4,000}     |           |
  |  1-7/8   | {2-5/8   |       6,000}     |    11     |
  |          | {2-3/4   |       8,000}     |           |
  |          | {2-7/8   |      10,000}     |           |
  |          |          |                  |           |
  |          | {2-1/2   |       3,000}     |           |
  |          | {2-5/8   |       4,000}     |           |
  |  2       | {2-3/4   |       6,000}     |    11     |
  |          | {2-7/8   |       8,000}     |           |
  |          | {3       |      10,000}     |           |
  +----------+----------+------------------+-----------+

The English pipe thread is a sharp [V]-thread having its sides at an
angle of 60°, and therefore corresponds to the American pipe thread
except that the pitches are different.

The standard screw thread of The Royal Microscopical Society of London,
England, is employed for microscope objectives, and the nose pieces of
the microscope into which these objectives screw.

The thread is a Whitworth one, the original standard threading tools now
in the cabinet of the society having been made especially for the
society by Sir Joseph Whitworth. The pitch of the thread is 36 per inch.
The cylinder, or male gauge, is .7626 inch in diameter.

The following table gives the Whitworth standard of thread pitches and
diameters for watch and mathematical instrument makers.

WHITWORTH'S STANDARD GAUGES FOR WATCH AND INSTRUMENT MAKERS, WITH SCREW
THREADS FOR THE VARIOUS SIZES, 1881.

  +---------------------+-------------+-------------+
  |    No. of each      |   Size in   |  Number of  |
  | size in thousandths | decimals of | Threads per |
  |    of an inch.      |  an inch.   |    inch.    |
  +---------------------+-------------+-------------+
  |        10           |    .010     |     400     |
  |        11           |    .011     |      "      |
  |        12           |    .012     |     350     |
  |        13           |    .013     |      "      |
  |        14           |    .014     |     300     |
  |        15           |    .015     |      "      |
  |        16           |    .016     |      "      |
  |        17           |    .017     |     250     |
  |        18           |    .018     |      "      |
  |        19           |    .019     |      "      |
  |        20           |    .020     |     210     |
  |        22           |    .022     |      "      |
  |        24           |    .024     |      "      |
  |        26           |    .026     |     180     |
  |        28           |    .028     |      "      |
  |        30           |    .030     |      "      |
  |        32           |    .032     |     150     |
  |        34           |    .034     |      "      |
  |        36           |    .036     |      "      |
  |        38           |    .038     |     120     |
  |        40           |    .040     |      "      |
  |        45           |    .045     |      "      |
  |        50           |    .050     |     100     |
  |        55           |    .055     |      "      |
  |        60           |    .060     |      "      |
  |        65           |    .065     |      80     |
  |        70           |    .070     |      "      |
  |        75           |    .075     |      "      |
  |        80           |    .080     |      60     |
  |        85           |    .085     |      "      |
  |        90           |    .090     |      "      |
  |        95           |    .095     |      "      |
  |       100           |    .100     |      50     |
  +---------------------+-------------+-------------+

For the pitches of the threads of lag screws there is no standard, but
the following pitches are largely used.

  +-----------+-----------+
  |Diameter of|  Threads  |
  |  Screw.   | per Inch. |
  +-----------+-----------+
  |   Inch.   |           |
  |    1/4    |    10     |
  |   5/16    |     9     |
  |    3/8    |     8     |
  |   7/16    |     7     |
  |    1/2    |     6     |
  |   9/16    |     6     |
  |    5/8    |     5     |
  |  11/16    |     5     |
  |    3/4    |     5     |
  |    7/8    |     4     |
  |      1    |     4     |
  +-----------+-----------+


SCREW-CUTTING HAND TOOLS.

For cutting external or male threads by hand three classes of tools are
employed.

The first is the screw plate shown in Fig. 292. It consists of a
hardened steel plate containing holes of varying diameters and threaded
with screw threads of different pitches. These holes are provided with
two diametrically opposite notches or slots so as to form cutting edges.

[Illustration: Fig. 292.]

This tool is placed upon the end of the work and slowly rotated while
under a hand pressure tending to force it upon the work, the teeth
cutting grooves to form the thread and advancing along the bolt at a
rate determined by the pitch of the thread.

The screw plate is suitable for the softer metals and upon diameters of
1/8 inch and less, in which the cutting duty is light; hence the holes
do not so rapidly wear larger.

The second class consists of a stock and dies such as shown in Fig. 293.
For each stock there are provided a set of dies having different
diameters and pitches of thread.

In this class of tool the dies are opened out and placed upon the bolt.
The set screw is tightened up, forcing the dies to their cut, and the
stock is slowly rotated and a traverse taken down the work.

[Illustration: Fig. 293.]

In some cases the dies are then again forced to the work by the set
screw, and a cut taken by winding the stocks up the bolt, the operation
being continued until the thread is fully developed and cut to the
required diameter. In other cases the cut is carried down the bolt, only
the dies being wound back to the top of the bolt after each cut is
carried down. The difference between these two operations will be shown
presently.

[Illustration: Fig. 294.]

The thread in dies which take successive cuts to form a thread may be
left full clear through the die, and will thus cut a full thread close
up to the head collar or shoulder of the work. It is usual, however, to
chamfer off the half threads at the ends of the dies, because if left of
their full _height_ they are apt to break off when in use. It is
sometimes the practice, however, to chamfer off the first two threads on
one side of the dies, leaving the teeth on the other side full, and to
use the chamfered as the leading side in all cases in which the thread
on the work does not require to be cut up to a shoulder, but turning the
dies over with the full threaded teeth as the leading ones when the
thread _does_ require to be carried up to a head or shoulder on the
work.

To facilitate the insertion and extraction of the dies in and from their
places in the stock, the Morse Twist Drill Co. employ the following
construction. In Figs. 294 and 295 the pieces A, A´ which hold the dies
are pivoted in the stock at B, so as to swing outward as in Fig. 295,
and receive the dies which are slotted to fit them. These pieces are
then swung into position in the stock. The lower die is provided with a
hole to fit the pin C, hence when that die is placed home C acts as a
detaining piece locking the pieces A, A´ through the medium of the
bottom die.

[Illustration: Fig. 295.]

[Illustration: Fig. 296.]

In other dies of this class the two side pieces or levers which hold the
dies are pivoted at the corner of the angle, as in Fig. 296. In the
bottom of the stock is a sliding piece beveled at its top and meeting
the bottom face of the levers; hence, by pressing this piece inwards the
side pieces recede into a slot provided in the stock, and leave the
opening free for the dies to pass into their places, when the pin is
released and a spring brings the side pieces back. Now, since the bottom
die rests upon the bottom angle of the side pieces the pressure of the
set screw closes the side pieces to the dies holding them firmly.

[Illustration: Fig. 297.]

In Fig. 297 is shown Whitworth's stocks and dies, the cap that holds the
guide die _a_ and the two chasers _b_, _c_ in their seats or recesses in
the stock being removed to expose the interior parts. The ends of the
chasers _b_, _c_ are beveled and abut against correspondingly beveled
recesses in the key _d_, so that by operating the nut _e_ on the end of
the key the dies are caused to move longitudinally. The principles of
action are more clearly shown in Fig. 298. The two cutting chasers B and
C move in lines that would meet at D, and therefore at a point behind
the centre or axis of the bolt being threaded; this has the effect of
preserving their clearance. It is obvious, for example, that when these
chasers cut a thread on the work it will move over toward guide A on
account of the thread on the work sinking into the threads on A, and
this motion would prevent the chasers B, C from cutting if they moved in
a line pointing to the centre of the work. This is more clearly shown in
Fig. 299, in which the guide die A and one of the cutting dies or
chasers B is shown removed from the stock, while the bolt to be threaded
is shown in two positions--one when the first cut is taken, and the
other when the thread is finished. For the first cut the centre of the
work is at E, for the last one it is at G, and this movement would, were
the line of motion as denoted by the dotted lines, prevent the chaser
from cutting, because, while the line of chaser motion would remain at
J, pointing to the centre of work for the first cut, it would require a
line at K to point to that centre for the last one; hence, when
considered with relation to the work, the line of chaser motion has been
moved forward, presenting the cutting edges at an angle that would
prevent their cutting. By having their motion as shown in Fig. 299,
however, the clearance of the chasers is preserved.

[Illustration: Fig. 298.]

[Illustration: Fig. 299.]

[Illustration: Fig. 300.]

Referring now to the die A, it acts as a guide rather than as a cutting
chaser, because it has virtually no clearance and cannot cut so freely
as B and C; hence it offers a resistance to the moving of the bolt, or
of the dies upon the bolt, in a lateral direction when the chaser teeth
meet either a projection or a depression upon the work. The guide
principle is, however, much more fully carried out in a design by
Bodmer, which is shown in Fig. 300. Here there is but one cutting chaser
C, the bush G being a guide let into a recess in the stock and secured
thereon by a pin _p_. The chaser is set in a stock, D also let into a
recess in the stock, and this recess, being circular, permits of stock D
swinging. At S are two set-screws, which are employed to limit the
amount of motion permitted to D. the handle E screws through D, and acts
upon the edge of chaser C to put on the cut. The action of the tool is
shown in Fig. 301, where it is shown upon a piece of work. Pulling the
handle E causes D to swing in the stock, thus giving the chaser
clearance, as shown. When the cut is carried down, a new cut may be put
on by means of E, and on winding the stock in the opposite direction, D
will swing in its seat, and cant or tilt the chaser in the opposite
direction, giving it the necessary clearance to enable it to cut on the
upward or back traverse. Another point of advantage is that the cutting
edges are not rubbed by the work during the back stroke, and their
sharpness is, therefore, greatly preserved. A die of this kind will
produce work almost as true as the lathe, and, in the case of long,
slender work, more true than the lathe; but it is obvious that, on
account of the friction caused by the pressure of the work to the guide
G, the tool will require more power to operate than the ordinary stock
and die or the solid die.

[Illustration: Fig. 301.]

In adjustable dies which require to take more than one cut along the
bolt to produce a fully developed thread, there is always a certain
amount of friction between the sides of the thread in the die and the
grooves being cut, because the angle of the thread at the top of a
thread is less than the angle at the bottom. Thus in Fig. 302 the pitch
at the top of thread (at A, B) is the same as at the bottom (C, D). Now
suppose that in Fig. 303 _a_ _b_ represents the axial line of a bolt,
and _c_ _d_ a line at a right angle to _a_ _b_. The radius _e_ _f_ being
equal to the circumference of the top of the thread, the pitch being
represented by _b_; then _k_ represents the angle of the top of the
thread to the axial line _a_ _b_. Now suppose that the radius _e_ _g_
represents the circumference at the bottom of the thread and to the
pitch; then _l_ is the angle of the bottom of the thread to the axial
line of the work, and the difference in angle between _k_ and _l_ is the
difference in angle between the top and bottom of the thread in the dies
and the thread to be cut on the work.

[Illustration: Fig. 302.]

Now the tops of the teeth on the die stand at the greatest angle _l_, in
Fig. 303, when taking the first cut on the bolt, but the grooves they
cut will be on the full diameter of the bolt, and will, therefore, stand
at the angle _k_, hence the lengths of the teeth do not lie in the same
planes as the grooves which they cut.

[Illustration: Fig. 303.]

In cutting [V]-threads, however, the angle of the die threads gradually
right themselves with the plane of the grooves attaining their nearest
coincidence when closed to finish the thread.

Since, however, the full width of groove is in a square thread cut at
the first cut taken by the dies, it is obvious that a square thread
cannot be cut by this class of die, because the sides of the grooves
would be cut away each time the dies were closed to take another cut.

[Illustration: Fig. 304.]

[Illustration: Fig. 305.]

Dies of this class require to have the threaded hole made of a larger
diameter than is the diameter of the bolt they are intended to thread,
the reason being as follows:--

Suppose the threaded hole in the dies to be cut by a hob or master tap
of the same diameter as the thread to be cut by the dies; when the dies
are opened out and placed upon the work as in Fig. 304, the edges A, B
will meet the work, and there will be nothing to steady the dies, which
will, therefore, wobble and start a drunken thread, that is to say, a
thread such as was shown in Fig. 253.

[Illustration: Fig. 306.]

Instances have been known in the use of dies made in this manner,
wherein the workman using a right-hand single-threaded pair of dies has
cut a right or left-hand double or treble thread; the teeth of the dies
acting as chasers well canted over, as shown in Fig. 305. It is
necessary to this operation, however, that the diameter of the work be
larger than the size of hob the dies were threaded with.

In Fig. 306 is shown a single right-hand and a treble left-hand thread
cut by the author with the same pair of dies.

All that is necessary to perform this operation is to rotate the dies
from left to right to produce a right-hand thread, and from right to
left for a left-hand thread, exerting a pressure to cause the dies to
advance more rapidly along the bolt than is due to the pitch of the
thread. A double thread is produced when the dies traverse along the
work twice as fast as is due to the pitch of the thread in the dies, and
so on.

[Illustration: Fig. 307.]

It is obvious, also, that a piece of a cylindrical thread may be used to
cut a left-hand external thread. Thus in Fig. 307 is shown a square
piece of metal having a notch cut in on one side of it and a piece of an
external thread (as a tap inserted) in the notch. By forcing a piece of
cylindrical work through the hole while rotating it, the piece of tap
would cut upon the work a thread of the pitch of the tap, but a
left-handed thread, which occurs because, as shown by the dotted lines
of the figure, the thread on one side of a bolt slopes in opposite
directions to its direction on the other, and in the above operation the
thread on one side is taken to cut the thread on the other.

These methods of cutting left-hand threads with right-handed ones are
mentioned simply as curiosities of thread cutting, and not as being of
any practical value.

To proceed, then: to avoid these difficulties it is usual to thread the
dies with a hob or master tap of a diameter equal to twice the depth of
the thread, larger than the size of bolt the dies are to thread. In this
case the dies fit to the bolt at the first cut, as shown in Fig. 308, C,
D being the cutting edges. The relation of the circle of the thread in
the dies to that of the work during the final cut is shown in Fig. 309.

[Illustration: Fig. 308.]

[Illustration: Fig. 309.]

There is yet another objection to tapping the dies with a hob of the
diameter of the bolt to be threaded, in that the teeth fit perfectly to
the thread of the bolt when the latter is threaded to the proper
diameter, producing a great deal of friction, and being difficult to
make cut, especially when the cutting edges have become slightly dulled
from use.

Referring now to taking a cut up the bolt or work as well as down, it
will be noted that supposing the dies to have a right-hand thread, and
to be rotating from left to right, they will be passing down the bolt
and the edges C, D (Fig. 308) will be the cutting ones. But when the
dies are rotated from right to left to bring them to the end of the bolt
again, C, D will be rubbed by the thread, which tends to abrade them and
thus destroy their sharpness.

[Illustration: Fig. 310.]

In some cases two or more pairs of dies are fitted to the same stock, as
shown in Fig. 310, but this is objectionable, because it is always
desirable to have the hole in the dies central to the length of the
stock, so that when placed to the work the stock shall be balanced,
which will render it easier to start the thread true with the axial line
of the bolt.

[Illustration: Fig. 311.]

From what has been said with reference to Fig. 303, it is obvious that a
square thread cannot be cut by a die that opens and closes to take
successive cuts along the work, but such threads may be cut upon work
that is of sufficient strength to withstand the twisting pressure of the
dies, by making a solid die, and tapering off the threads for some
distance at the mouth of the die, so as to enable the die to take its
bite or grip upon the work, and start itself. It is necessary, however,
to give to the die as many flutes (and therefore cutting edges), as
possible, or else to make flutes wide and the teeth as short as will
leave them sufficiently strong, both these means serving to avoid
friction.

[Illustration: Fig. 312.]

The teeth for adjustable dies, such as shown in Fig. 293, are cut as
follows:--There is inserted between the two dies a piece of metal,
separating them when set together to a distance equal to twice the depth
of the thread, added to the distance the faces of the dies are to be
apart when the dies are set to cut to this designated or proper
diameter. The tapping hole is then drilled (with the pieces in place) to
the diameter of the bolt the die is for. The form of hob used by the
Morse Twist Drill & Machine Company, to cut the thread, is shown in Fig.
311. The unthreaded part at the entering end is made to a diameter equal
to that of the work the dies are to be used in; the thread at the
entering end is made sunk in one half the height of the full thread, and
is flattened off one half the height of a full thread, so that the top
of the thread is even with the diameter of the unthreaded part at the
entering end. The thread then runs a straight taper up the hob until a
distance equal to the diameter of the nut is reached, and the length of
hob equal to its diameter is made a full and parallel thread for
finishing the die teeth with. The thread on the taper part has more
taper at the root of the thread than it has at the top of the same, and
the diameter of the full and parallel part at the shank end of the
thread is made of a diameter equal to twice the height or depth of a
full thread, larger than the diameter at the entering end of the hob.
The hob thus becomes a taper and relieved tap cutting a full thread at
one passage through the dies. If the hob is made parallel and a full
thread from end to end, as in Fig. 312, the dies must traverse up and
down the hob, or the hob through the dies to form a full thread.

The third class of stock and die is intended to cut a full thread at one
passage along the work, while at the same time provision is made,
whereby, to take up the wear due to the abrasion of the cutting edges,
which wear would cause the diameter of thread cut to be above the
standard.

In Fig. 313 is shown the Grant adjustable die made by the Pratt &
Whitney Company. It consists of four chasers or toothed cutting tools,
inserted in radial recesses or slots in an iron disc or collet encircled
by an iron ring. Each chaser is beveled at its end to fit a
corresponding bevel in the ring, and is grooved on one of its side faces
to receive the hardened point of a screw that is inserted in the collet
to hold the chaser in its adjusted position. Four screws extend up
through the central flange or body of the collet, two of which serve to
draw down the ring, and by reason of the taper on the ring move the
chasers equally towards the centre and reduce the cutting diameter of
the die, while the other two hold the ring in the desired position, or
force it upward to enlarge the cutting diameter of the die. The range of
adjustment permitted by this arrangement is 1-32 inch. The dies may be
taken out and ground up to sharpen.

[Illustration: Fig. 313.]

The object of cutting grooves in the sides of the chasers is that the
fine burrs formed by the ends of the set screws do not prevent the
chasers from moving easily in the collet during the process of
adjustment; the groove also acts as a shoulder for the screw end to
press the chaser down to its seat. These chasers are marked to their
respective places in the collet, and are so made that if one chaser
should break, a new one can be supplied to fit to its place, the teeth
of the new one falling exactly in line with the teeth on the other
three, whereas under ordinary conditions if one chaser breaks, a full
set of four new ones must be obtained.

In this die, as in all others which cut a full thread at one passage
along the work, the front teeth of the chasers are beveled off as shown
in the cut; this is necessary to enable the dies to take hold of or
"bite" the work, the chamfer giving a relief to the cutting edge, while
at the same time forming to a certain extent a wedge facilitating the
entrance of the work into the die.

Fig. 314 represents J. J. Grant's patent die, termed by its makers
(Wiley and Russel) the "lightening die." In this, as in other similar
stocks, several collets with dies of various pitches and diameters of
thread, fit to one stock. The nut of the stock is split on one side, and
is provided with lugs on that side to receive a screw, which operates to
open and enlarge the bore to release a collet, or close thereon and
grip it, as may be required when inserting or extracting the same. The
dies are formed as shown in Fig. 315, in which A, A are the dies, and B
the collet. To open the dies within the collet, the screws E are
loosened and the screws D are tightened, while to close the dies D, D
are loosened and E are tightened; thus the adjustment to size is
effected by these four screws, while the screws D also serve to hold the
dies to the collet B. The collets are provided with a collar having a
bore F, through which the work passes, so that the dies may be guided
true when starting upon the work; but if it is required to cut a thread
close up to a head or shoulder, the stock is turned upside down, not
only to have the collet out of the way of the head or shoulder, but also
because the thread of the dies on the collet side are chamfered off (as
is necessary in all solid dies, or dies which cut a full thread at one
traverse down the work) so as to enable them to grip or bite the work,
and start the thread upon it as before stated.

[Illustration: Fig. 314.]

[Illustration: Fig. 315.]

In Fig. 316 is shown Stetson's die, which cuts a full thread at one
passage, is adjustable to take up its wear, and has a guide to steady it
upon the work and assist it in cutting a true thread. The guide piece
consists of a hub (through which the work passes) having a flange
fitting into the dies and being secured thereto by the two screws shown.
The holes in the flanges are slotted to permit of the dies being closed
(to take up wear) by means of the small screws shown at the end of the
die, which screws pass through one die in a plain hole and screw into
the other.

[Illustration: Fig. 316.]

In Fig. 317 is shown Everett's stocks and dies. In this tool the dies
are set up by a cam lever, the dies being set to standard size when the
lever arm stands parallel with the arm of the stock. By turning the
straight side of the cam lever opposite to the dies, the latter may be
instantly removed and another size of die inserted. The dies may be used
to cut on their passage up and down the bolt or by operating the cam.
When the dies are at the end of a cut the dies may be opened, lifted to
the top of the work and another cut taken, thus saving the time
necessary to wind the stock back. When the final cut is taken the dies
may be opened and lifted off the work.

[Illustration: Fig. 317.]

The hardening process usually increases the thickness of these dies,
making the pitch of the thread coarser. The amount of expansion due to
hardening is variable, but increases with the thickness of the die. The
hob as a rule shortens during the tempering, but the amount being
variable, no rule for its quantity can be given.[12]

  [12] See also page 108.

Stocks and dies for pipe work are made in the form shown in Fig. 318, in
which B is the stock having the detachable handles (for ease of
conveyance) A, H, the latter being shown detached. The solid
screw-cutting dies C are placed in the square recess at B, and are
secured in B by the cap D, which swings over (upon its pivoted end as a
centre) and is locked by the thumbscrew E. To guide the stocks and cause
them to cut a true thread, the bushes F are provided. These fit into the
lower end of B and are locked in position by four set screws G. The
bores of the bushes F are made an easy fit to the outside of the pipe to
be threaded, there being a separate bush for each size of pipe.

[Illustration: Fig. 318.]

The dies employed in stocks for threading steam and gas pipes by hand
are sometimes solid, as in Fig. 318 at C, and at others adjustable. In
Fig. 319 is shown Stetson's adjustable pipe die containing four chasers
or toothed thread-cutting tools. These are set to cut the required
diameter by means of a small screw in each corner of the die, while they
are locked in their adjusted position by four screws on the face.

The tap is a tool employed to cut screw threads in internal surfaces, as
holes or bores. A set of taps for hand use usually consist of three: the
taper tap, Fig. 320; plug tap, Fig. 321; and bottoming tap, Fig. 322.
(In England these taps are termed respectively the taper, second, and
plug tap.) The taper tap is the first to be inserted, and (when the
hole to be threaded passes entirely through the work) rotated until it
passes through the work, thus cutting a thread parallel in diameter
through the full length of the hole. If, however, the hole does not pass
through the work, the taper tap leaves a taper-threaded hole containing
more or less of a fully developed thread according to the distance the
tap has entered.

[Illustration: Fig. 319.]

[Illustration: Fig. 320.]

[Illustration: Fig. 321.]

[Illustration: Fig. 322.]

To further complete the thread the plug tap is inserted, it being
parallel from four or five threads from the entering end of the tap to
the other end. If the work will admit it, this tap is also passed
through, which not only saves time in many cases, by avoiding the
necessity to wind the tap back, but preserves the cutting edge which
suffers abrasion from being wound back. To cut a full thread as near as
possible to the bottom of a hole the bottoming tap is used, but when the
circumstances will admit, it is best to drill the hole rather deeper
than is actually necessary, to avoid the trouble incident to tapping a
hole clear to the bottom.

On wrought iron and steel, which are fibrous and tough, the tap, when
used by hand, will not (if the hole be deeper than the diameter of the
tap) readily operate by a continuous rotary motion, but requires to be
rotated about half a revolution back occasionally, which gives
opportunity for the oil to penetrate to the cutting edges of the tap,
frees the tap and considerably facilitates the tapping operation,
especially if the hole be a deep one.

[Illustration: Fig. 323.]

When the tap is intended to pass entirely through the work with a
continuous rotary motion, as is the case, for example, in tapping nuts
in a tapping machine, it is made of similar form to the taper hand tap,
but longer, as shown in Fig. 323, the thread being full and parallel at
the shank end for a distance at least equal to the full diameter of the
tap measured across the tops of the thread.

If the thread of a tap be in diametral section a full circle, the sides
of the thread rub against the grooves cut by the teeth, producing a
friction which augments as the sharp edge of the teeth become dulled
from use, but the tap cuts a thread of great diametral accuracy.

To reduce this friction to a minimum as much as is consistent with
maintaining the standard size of the tapped hole, taps are sometimes
given clearance in the thread, that is to say, the back of each tooth
recedes from a true circle, as shown in Fig. 324, in which A A
represents a washer, and B A tap in the same, the back of the teeth
receding at C, D, E, from the true circle of the bore of A A, the tap
cutting when revolved in the direction of the arrow. The objection to
this is that when the tap is revolved backwards, as it must be to
extract it unless the hole passes clear through the work, the cuttings
lodge between the teeth and the thread in the work, rendering the
extraction of the tap difficult, unless, indeed, the clearance be small
enough in amount to clear the sides of the thread in the work
sufficiently to avoid friction without leaving room for the cuttings to
enter. If an excess of clearance be allowed upon taps that require to be
used by hand, the tap will thread the hole taper, the diameter being
largest at the top of the hole. This occurs because the tap is not so
well steadied by its thread, which fails to act as a guide, and it is
impossible to revolve the tap steadily by hand. Taps that are revolved
by machine tools may be given clearance because both the taps and the
work are detained in line, hence the tap cannot wobble.

[Illustration: Fig. 324.]

[Illustration: Fig. 325.]

In some cases clearance is given by filing or cutting off the tops of
the threads along the middle of the teeth, as shown in Fig. 325 at A, B,
C, which considerably reduces the friction. If clearance were given to a
tap after this manner but extended to the sides and to the bottom of the
thread, it would produce the best of results (for all taps that do not
pass entirely through the hole), reducing the friction and leaving no
room for the cuttings to jam in the threads when the tap is being backed
out. The threads of Sir Joseph Whitworth's taper hand taps are made
parallel, measured at the bottom of the thread, and parallel at the tops
of the thread for a distance equal to the diameter of the tap at the
shank end; thence, to the entering end of the tap, the tops of the
thread are turned off a straight taper, the amount of taper being
slightly more than twice the depth of the thread: hence, the thread is
just turned out at the entering end of the tap, and that end is the
exact proper size for the tapping hole.

This enables the tap to enter the tapping hole for a distance enveloping
one or perhaps two of the tap threads, leaving the extreme end of the
tap with the thread just turned out. In the practice of some tap makers
the diameter of the thread at the top is made the same as in the
Whitworth system, but there is more depth at the root of the thread and
near the entering end of the tap, hence the bottoms of the thread at
that end perform no cutting duty. This is done to enable the tap to take
hold of, and start a thread in, the work more readily, which it does for
the following reasons. In Fig. 326 is a piece of work with a tap A,
having a tapered thread, and a tap B, in which the taper is given by
turning off the thread. In the case of A the teeth points cut a groove
that is gradually widened and deepened as the tap enters, until a full
thread is finally produced. In the case of B the teeth cut at first a
wide groove, leaving a small projection, that is a part of the actual
finished thread, and the groove gets narrower as the tap enters; so that
in the one case no part of the thread is finished until the tap has
entered to its full diameter, while in the other the thread is finished
as it is produced. On entering, therefore, more cutting duty is
performed by B than by A, because a greater length of cutting edge is in
operation and more metal is being removed, and as a result B requires
more power to start it, so that in practice it is necessary to exert a
pressure upon it, tending to force it into the hole while rotating it.
The cutting duty on B decreases as the tap enters, because it gets less
width and area of groove to cut, while the cutting duty on A increases
as the tap enters, because it gets a greater width and area of groove to
cut. In the latter case the maximum of pressure falls on the tap when it
has entered the hole deepest, and hence can be operated steadiest,
which, independent of its entering easiest, is an advantage. When,
however, the bottom of a thread is taper (as must be the case to enable
it to cut as at A), the cutting edge of each tooth does not cut a groove
sufficiently large in diameter to permit the tooth itself to pass
through. In Fig. 327, for example, is shown a tap which is taper and has
a full thread from end to end (as is necessary for pipe tapping). Its
diameter increases as the thread proceeds from the end towards the line
A B. Now take the tooth O P, which stands lengthwise, in the plane C D.
Its cutting edge is at P, but the diameter of the tap at P is less than
it is at O, while O has to pass through the groove that P cuts. To
obviate this difficulty the tap is given clearance, as shown in Fig.
324, the amount being slightly more than the difference in the diameter
of the tap at O and at P in that figure. It follows, therefore, that a
tap having taper from end to end and a full thread also, as shown in the
lower tap in Fig. 328, is wrong in principle, and from the unsteady
manner in which it operates is undesirable, even though its thread be
given clearance.

[Illustration: Fig. 326.]

[Illustration: Fig. 327.]

In some cases the thread is made parallel at the tops and turned taper
for a distance of 1/3 or 1/2 the length of the tap, the root of the
thread at the taper part being deepened and the tops being given a
slight clearance. This answers very well for shallow holes, because the
taper tap cuts more thread on entering a given depth so that the second
tap can follow more easily, but the tap will not operate so steadily as
when the taper part is longer.

[Illustration: Fig. 328.]

It is on account of the tops of the teeth performing the main part of
the cutting that a tap taper may be sharpened by simply grinding the
teeth tops. In the Pratt and Whitney taps, the hand taper tap is made
parallel at the shank end for a distance equal in length to the diameter
of the tap.

The entering end of the taper tap is made straight or parallel for a
distance equal in length to one half the diameter of the tap, the
diameter at this end being the exact proper size of tapping hole. The
parallel part serves as a guide, causing the tap to enter and keep
axially true with the hole to be tapped. The plug and bottoming taps are
made parallel in the thread, the former being tapered slightly at and
for two or three threads from the entering, as shown in Fig. 328. The
threads are made parallel at the roots.

The Pratt and Whitney taper taps for use in machines are of the
following form:--

The entering end of the tap is equal in diameter to the diameter of the
tapping hole into which the tap will enter for a distance of two or
three threads. The thread at the shank end is parallel both at the top
and at the root for a distance equal, in length, to twice the diameter
of the tap. The top of the thread has a straight taper running from the
parallel part at the shank to the point or entering end, while the roots
of the thread are made along this taper twice the taper that there is at
the top of the thread, which is done to make the tap enter and take hold
of the nut more easily.

[Illustration: Fig. 329.]

A form of tap that cuts very freely on account of the absence of
friction on the sides of the thread is shown in Fig. 329. The thread is
cut in parallel steps, increasing in size towards the shank, the last
step (from D to E in the figure) being the full size. The end of the tap
at A being the proper size for the tapping hole, and the flutes not
being carried through A, insures that the tap shall not be used in holes
too small for the size of the tap, and thus is prevented a great deal of
tap breakage. The bottom of the thread of the first parallel step (from
A to B) is below the diameter of A, so as to relieve the sides of the
thread of friction and cause the tap to enter easily. The first tooth
of each step does all the cutting, thus acting as a turning tool, while
the step within the work holds the tooth to its cut, as shown in Fig.
330, in which N represents a nut and T the tap, both in section. The
step C holds the tap to its work, and it is obvious that, as the tooth B
enters, it will cut the thread to its own diameter, the rest of the
teeth on that step merely following frictionless until the front tooth
on the next step takes hold. Thus, to sharpen the tap equal to new, all
that is required is to grind away the front tooth on each step, and it
becomes practicable to sharpen the tap a dozen times without softening
it at all. As a sample of duty, it may be mentioned that, at the
Harris-Corliss Works, a tap of this class, 2-7/8 inches diameter, with a
4 pitch, and 10 inches long, will tap a hole 5 inches deep, passing the
tap continuously through without any backing motion, two men performing
the duty with a wrench 4 feet long over all, the work being of cast
iron.

[Illustration: Fig. 330.]

[Illustration: Fig. 331.]

[Illustration: Fig. 332.]

Another form of free cutting tap especially applicable to taps of large
diameter has been designed by Professor Sweet. Its principles may be
explained as follows:--

In the ordinary tap, with the taper four or five diameters in length,
there are far more cutting-edges than are necessary to do the work; and
if the taper is made shorter, the difficulty of too little room for
chips presents itself. The evil results arising from the extra cutting
edges are that, if all cut, then it is cutting the metal uselessly
fine--consuming power for nothing; or if some of the cutting edges fail
to cut, they burnish down the metal, not only wasting power, but making
it all the harder for the following cutters. One plan to avoid this is
to file away a portion of the cutting edges; but the method adopted in
the Cornell University tap is still better. Assume that it is desired to
make three following cutters, to remove the stock down to the dotted
line in Fig. 331. Instead of each cutter taking off a layer one-third
the thickness and the full width, the first cutter is cut away on each
side to about one-third its full width, so that it cuts out the centre
to its full depth, as shown in Fig. 331, the next cutter cutting out the
metal at A, and so on. This is accomplished by filing, or in any other
way cutting away the sides of one row of the teeth all the way up; next
cutting away the upper sides of the next row and the lower sides of the
third, leaving the fourth row (if it be a four-fluted tap) as it is left
by the lathe, to insure a uniform pitch and a smooth thread.

[Illustration: Fig. 333.]

Figs. 333, 334 and 335 represent an adjustable tap designed by C. R.
French, of Providence, R. I., to thread holes accurate in diameter.

The plug tap, Fig. 333, has at its end a taper screw, and the tap is
split up as far as the flutes extend, a second screw binds the two sides
of the tap together, hence by means of the two screws the size of the
tap may be regulated at will. In the third or bottoming tap, Fig. 334,
the split extends farther up the shank, and four adjusting screws are
used as shown, hence the parallelism of the tap is maintained.

In the machine tap, Fig. 335, there are six adjusting screws, two of
those acting to close the tap being at the extreme ends so as to
strengthen it as much as possible.

[Illustration: Fig. 334.]

[Illustration: Fig. 335.]

In determining the number, the width, the depth, and the form of flutes
for a tap, we have the following considerations. In a tap to be used in
a machine and to pass entirely through the work, as in the case of
tapping nuts, the flute need not be deep, because the taper part of the
tap being long the cutting teeth extend farther along the tap; hence,
each tooth takes a less amount of cut, producing less cuttings, and
therefore less flute is required to hold them. In taps of this class,
the thread being given clearance, the length of the teeth may be a
maximum, because they are relieved of friction; on the other hand,
however, the shallower and narrower the flute the stronger the tap, so
long as there is room for the cuttings so that they shall not become
wedged in the flutes. Taps for general use by hand are frequently used
to tap holes that do not pass entirely through the work; hence, the
taper tap must have a short length of taper so that the second tap may
be enabled to carry a full thread as near as possible to the bottom of
the hole without carrying so heavy a cut as to render it liable to
breakage, and the second or plug tap must in turn have so short a length
of its end tapered that it will not throw too much duty upon the
bottoming tap. Now, according as the length of the taper on the taper
tap is reduced, the duty of the teeth is increased, and more room is
necessary in the flute to receive the cuttings, and supposing the tap
to be rotated continuously to its duty the flute must possess space
enough to contain all the cuttings produced by the teeth, but on account
of the cuttings filling the flutes and preventing the oil fed to the tap
from flowing down the flute to the teeth it is found necessary in hand
taps (when they cannot pass through the work, or when the depth of the
hole is equal to more than about the tap diameter), to withdraw the tap
and remove the cuttings. On account of the tap not being accurately
guided in hand-tapping it produces a hole that is largest at its mouth,
and it is found undesirable on this account to give any clearance to
hand taps, because such clearance gives more liberty to the tap to
wobble in the hole and to enlarge its diameter at the mouth. It is
obvious also, that the less of the tap circumference removed to form the
flutes the longer the tap-teeth and the more steadily the tap may be
operated. On the other hand, however, the longer the teeth the greater
the amount of friction between them and the thread in the hole and the
more work there is involved in the tapping, because the tap must
occasionally be rotated back a little to ease its cut, which it is found
to do.

[Illustration: Fig. 336.]

[Illustration: Fig. 337.]

Fig. 336 represents a form of flute recommended by Brown and Sharp. The
teeth are short, thus avoiding friction, and the flutes are shallow,
which leaves the tap strong. The inclination of the cutting edges, as A
B (the cutting direction of rotation being denoted by the arrow), is
shown by the dotted lines, being in a direction to curve the chip or
cutting somewhat upward and not throw them down upon the bottom of the
flute. A more common form, and one that perhaps represents average
American practice, is shown in Fig. 337, the cutting edges forming a
radial line as denoted by the dotted line. The flute is deeper, giving
more room for the chips, which is an advantage when the tap is required
to cut a thread continuously without being moved back at all, but the
tap is weaker on account of the increased flute depth, the teeth are
longer and produce more friction, and the flutes are deeper than
necessary for a tap having a long taper or that requires to be removed
to clear out the cuttings. Fig. 338 shows the form of flute in the Pratt
and Whitney Company's hand taps, the cutting edges forming radial lines
and the bottoms of the flutes being more rounded than is usual. It may
here be remarked that if the flutes have comparatively sharp corners, as
at C in Fig. 339, the tap will be liable to crack in the hardening
process. The form of flute employed in the Whitworth tap is shown in
Fig. 340; here there being but three flutes the teeth are comparatively
long, and on this account there is increased friction. But, on the other
hand, such a tap produces, when used by hand, more accurate work, the
threaded hole being more parallel and of a diameter more nearly equal to
that of the tap, it being observed that even though a hand tap have no
clearance it will usually tap a hole somewhat larger than itself so that
it will unwind easily. If a hand tap is given clearance not only will it
cut a hole widest at the mouth, but it will cut a thread larger than
itself in an increased degree, and, furthermore, when the tap requires
to be wound back to extract it the fine cuttings will become locked in
the threads and the points of the tap teeth are liable to become broken
off. To ease the friction of long teeth, therefore, it is preferable to
do so either as in Fig. 325 at A, B, C, or as in Fig. 341. In Fig. 325
the tops of the teeth are shown filed away, leaving each end full, so
that the cuttings cannot get in, no matter in which direction the tap is
rotated; but the clearance is not so complete as in Fig. 341, in which
the teeth are supposed to be eased away within the area enclosed by
dotted lines, which gives clearance to the bottom as well as to the tops
and sides of the thread and leaves the ends of each tooth a full thread.

[Illustration: Fig. 338.]

[Illustration: Fig. 339.]

[Illustration: Fig. 340.]

[Illustration: Fig. 341.]

Concerning the number of flutes in taps, it is to be observed that the
duty the tap is to be put to, has much influence in this respect. In
hand tapping the object is to tap as parallel and straight as possible
with the least expenditure of power. Now, the greater the number of
flutes the less the tap is guided, because more of the circumferential
guiding surface is cut away. But on the other hand, the less the number
of flutes, and therefore the less the number of cutting edges, the more
power it takes to operate the tap on account of the greater amount of
friction between the tap and the walls of the hole. In hand tapping on
what may be termed frame work (as distinguished from such loose work as
nuts, &c.), the object is to tap the holes as parallel as possible with
the least expenditure of power while avoiding having to remove the tap
from the hole to clear it of the cuttings. Obviously the more flutes and
cutting edges there are the more room there is for the cuttings and the
less frequent the tap requires to be cleaned. If the tapping hole is
round and straight the tapping may be made true and parallel if due care
is taken, whatever the number of flutes, but less care will be required
in proportion as there are less flutes, while, as before noted, more
power and more frequent tap removals will be necessary. But if the hole
is not round, other considerations intervene.

[Illustration: Fig. 342.]

Thus in Fig. 342 we have a three-flute tap in a hole out of round at A,
and it is obvious that when a cutting edge meets the recess at A, all
three teeth will cease to cut; hence there will be no inducement for the
tap to move over toward A. But in the case of the four-flute tap in Fig.
343, when the teeth come to A there will be a strain tending to force
the teeth over toward the depression A. How much a given tap would
actually move over would, of course, depend upon the amount of
clearance; but whether the tap has clearance or not, the three-flute tap
will not move over, while with four flutes the tap would certainly do
so. Again, with an equal width of flute there is more of the
circumference tending to guide and steady the three-flute than the
four-flute tap. If the hole has a projection instead of a depression, as
at B, Figs. 344 and 345, then the advantage still remains with the
three-flute tap, because in the case of the three flutes, any lateral
movement of the tap will be resisted at the two points _c_ and D,
neither of which are directly opposite to the location of the projection
B; hence, if the projection caused the tap to move laterally, say,
1-100th inch, the effect at _c_ and D would be very small, whereas in
the four-flute, Fig. 345, the effect at E would be equal to the full
amount of lateral motion of the tap.

[Illustration: Fig. 343.]

[Illustration: Fig. 344.]

[Illustration: Fig. 345.]

In hand taps the position of the square at the head of the tap with
relation to the cutting-edges is of consequence; thus, in Fig. 346,
there being a cutting-edge A opposite to the handle, any undue pressure
on that end of the handle would cause A to cut too freely and the tap to
enlarge the hole; whereas in Fig. 347 this tendency would be greatly
removed, because the cutting-edges are not in line with the handle. In a
three-flute tap it makes but little difference what are the relative
positions of the square to the flutes, as will be seen in Fig. 348,
where one handle of the wrench comes in the most favorable and the other
in the most unfavorable position. Taps for use by hand and not intended
to pass through the work are sometimes made with the shank and the
square end which receive the wrench of enlarged diameter. This is done
to avoid the twisting of the shank which sometimes occurs when the tap
is employed in deep holes, giving it much strain, and also to avoid as
much as possible the wearing and twisting of the square which occurs,
because in the course of time the square holes in solid wrenches enlarge
from wear, and the larger the square the less the wear under a given
amount of strain.

[Illustration: Fig. 346.]

Brass finishers frequently form the heads of their taps as in Fig. 349,
using a wrench with a slot in it that is longer than the flat of the tap
head.

[Illustration: Fig. 347.]

The thickness of the flat head at A is made equal for all the taps
intended to be used with the same wrench. By this means one wrench may
be used for many different diameters of taps.

[Illustration: Fig. 348.]

For gas, steam pipe, and other connections made by means of screw
threads, and which require to be without leak when under pressure, the
tap shown in Fig. 350 is employed. It is made taper and full threaded
from end to end, so that the fittings may be entered easily into their
places and screwed home sufficiently to form a tight joint.

[Illustration: Fig. 349.]

[Illustration: Fig. 350.]

The standard degree of taper for steam-pipe taps is 3/4 inch per foot of
length, the taper being the same in the dies as on the taps. The
threading tools for the pipes or casings for petroleum oil wells are
given a taper of 3/8 inch per foot, because it was not found practicable
to tap such large fittings with a quick taper, because of the excessive
strain upon the threading tools. Ordinary pipe couplings are, however,
tapped straight and stretch to fit when screwed home on the pipe.
Oil-well pipe couplings are tapped taper from both ends, and there is
just enough difference in the taper on the pipe and that in the socket
to show a bearing mark at the end only when the pipe and socket are
tested with red marking.

PITCHES OF TAP THREADS IN USE IN THE UNITED STATES.

  +-----------+---------+----------------+
  |           |         | No. of Threads |
  | Diameter. | Length. |   to Inch.     |
  +-----------+---------+----------------+
  |   1/4     | 2-3/4   |  16, 18 & 20   |
  |   5/16    | 2-7/8   |      16 & 18   |
  |   3/8     | 3-1/2   |      14 & 16   |
  |   7/16    | 3-13/16 |      14 & 16   |
  |   1/2     | 4-5/16  |  12, 13 & 14   |
  |   9/16    | 4-3/4   |      12 & 14   |
  |   5/8     | 5-1/8   |  10, 11 & 12   |
  |   11/16   | 5-3/8   |      11 & 12   |
  |   3/4     | 5-13/16 |  10, 11 & 12   |
  |   13/16   | 6       |      10        |
  |   7/8     | 6-1/8   |       9 & 10   |
  |   15/16   | 6-3/8   |       9        |
  | 1         | 6-13/16 |       8        |
  | 1-1/8     | 7-1/4   |       7 & 8    |
  | 1-1/4     | 8       |       7 & 8    |
  +-----------+---------+----------------+

Fig. 351 represents the form of tap employed by blacksmiths for rough
work, and for the axles of wagon wheels. These taps are given a taper of
1/2 inch per foot of length, and are made with right and left-hand
threads, so that the direction of rotation on both sides of a wagon
wheel shall be in a direction to screw up the nuts and not to unscrew
the nut, as would be the case if both ends of the axle were provided
with right-hand threads.

[Illustration: Fig. 351.]

Taps that are used in a machine are sometimes so constructed that upon
having tapped the holes to the required depth, the pieces containing the
tap teeth recede from the walls of the hole, so that the tap may be
instantly withdrawn from the hole instead of requiring to be rotated
backwards. This is an advantage, not only on account of the time saved,
but also because the cutting edges of the teeth are saved from the
abrasion and its consequent wear which occur in rotating a tap
backwards.

[Illustration: Fig. 352.]

Figs. 352 and 353 represent a collapsing tap that is much used in
manufactories of pipe fittings.

[Illustration: Fig. 353.]

A is driven by the spindle of the machine, and drives B through the
medium of the pin H. In B are three chasers C, fitting into the dovetail
and taper grooves D. These chasers are provided with lugs fitting into
an annular groove E sunk in A, so that if the piece H rises, the chasers
will not rise with it, but will simply close together by reason of the
lifting or rising of the core B, with its taper dovetail grooves; or, on
the other hand, if the core B descends, the taper grooves in B force the
chasers outward, increasing their cutting diameter.

When the tap is cutting, it is driven as denoted by the arrow, and the
pin H is driven by the ends of the grooves, of which there are two, one
diametrically opposite the other, inclined in the same direction. But
when the tap has cut a thread to the required depth on the work, the
handles H may be pulled or pushed the working way, passing along the
grooves I, and causing B to lift within A, and allowing the chasers to
close away from the thread just cut, and the tap may be instantly
withdrawn, and handles H pushed back to expand the chasers, ready for
the next piece of work.

[Illustration: Fig. 354.]

Fig. 354 represents a collapsing tap used in Boston, Massachusetts, at
the Hancock Inspirator Works, in a monitor or turret lathe. It consists
of an outer shell A carrying three chasers B, pivoted to A at C, having
a small lug E at one end, and being coned at the inner end D. The inner
shell F is reduced along part of its length to receive the lug E of the
chaser, and permit the chasers to open out full at their cutting end. F
has a cone at the end G, fitting to the internal cone on the chasers at
D. At the other end of F is a washer H, against which abuts the spiral
spring shown, the other end of this spring abutting against a shoulder
provided in A. The washer H is bevelled on its outer or end face to
correspond with the bevel on a notch provided in lever I, as is shown.
Within the inner tube F is the stem J, into the end of which is fitted
the piece K, and on which is fixed the cone L. Piece K, and therefore L,
is prevented from rotating by a spline in K, into which spline the pin M
projects.

The operation is as follows. In the position in which the parts are
shown in the engraving, F is pushed forward so that its coned end G has
opened out the chaser to its fullest extent, which opening is governed
by contact of the lug E with the reduced diameter of F. Suppose that the
tap is operating in the work, then, when the foot N of K meets with a
resistance (as the end of the hole being tapped), J, and therefore L,
will be gradually pushed to the right, until, finally, the cone on L
will raise the end of lever I until the notch on I is clear of H, when
the spiral spring, acting against H, will force F to the right, and the
shoulder on F, at X, will lift the end E of the chaser, causing the
cutting end to collapse within A, the pivot C being its centre of
motion. The whole device may then be withdrawn from the work. To open
the chasers out again the rod J is forced, by hand, to the left, the
cone-piece L meeting the face of H and pushing it to the left until cone
G meets cone D, when the chasers open until the end E meets the body of
F, as in the cut. The rod J is then pulled to the right until L again
meets the curved end of lever I and all the parts assume the positions
shown in the cut. To regulate the depth of thread the tap shall cut, the
body A is provided with a thread to receive the nut O, by means of which
the collar P may be moved along A. This collar carries the pivots Q for
levers I, so that, by shifting O, the position of I is varied, hence the
point at which L will act upon the end of I and lift it to release H is
adjustable.

When used upon steel, wrought iron, cast iron, copper, or brass, a tap
should be freely supplied with oil, which preserves its cutting edge as
well as causes it to cut more freely, but for cutting the soft metals
such as tin, lead, &c., oil is unnecessary.

The diameters of tapping holes should be equal to the diameter of the
thread at the root, but in the case of cast iron there is much
difference of opinion and practice. On the one hand, it is claimed that
the size of the tapping hole should be such as to permit of a full
thread when it is tapped; on the other hand, it is claimed that
two-thirds or even one-half of a full thread is all that is necessary in
holes in cast iron, because such a thread is, it is claimed, equally as
strong as a full one, and much easier to tap. In cases where it is not
necessary for the thread to be steamtight, and where the depth of the
thread is greater by at least 1/8 inch than the diameter of the bolt or
stud, three-quarters of a full thread is all that is necessary, and can
be tapped with much less labor than would be the case if the hole were
small enough to admit of a full thread, partly because of the diminished
duty performed by the tap, and partly because the oil (which should
always be freely supplied to a tap) obtains so much more free access to
the cutting edges of the tap. If a long tap is employed to cut a
three-quarter full thread, it may be wound continuously down the hole,
without requiring to be turned backwards at every revolution or so of
the tap, to free it from the tap cuttings or shavings, as would be
necessary in case a full thread were being cut. The saving of time in
consequence of this advantage is equal to at least 50 per cent. in favor
of the three-quarter full thread.

As round bar iron is usually rolled about 1/32 inch larger than its
designated diameter, a practice has arisen to cut the threads upon the
rough iron just sufficiently to produce a full thread, leaving the
latter 1/32 inch above the proper diameter, hence taps 1/32 inch above
size are required to thread nuts to fit the bolts. This practice
should be discountenanced as destroying in a great measure the
interchangeability of bolts and nuts, because 1/32 inch is too small a
measurement to be detected by the eye, and a measurement or trial of the
bolt and nut becomes necessary.

A defect in taps which it has been found so far impracticable to
eliminate is the alteration of pitch which takes place during the
hardening process. The direction as well as the amount of this variation
is variable even with the most uniform grades of steel, and under the
most careful manipulation. Mr. John J. Grant, in reply to a
communication upon this subject, informs me that, using Jones and
Colver's (Sheffield) steel, which is very uniform in grade, he finds
that of one hundred taps, about 5 per cent. will increase in length, the
pitch of the thread becoming coarser; 15 per cent. will suffer no
appreciable alteration of pitch, and 80 per cent. will shrink in length,
the pitch becoming finer, and these last not alike. But it must be borne
in mind that with different steel the results will be different, and the
greater the variation in the grade of the steel the greater the
difference in the alteration of pitch due to hardening.

It is further to be observed that the expansion or contraction of the
steel is not constant throughout the same tap; thus the pitches of three
or four consecutive teeth may measure correct to pitch, while the next
three or four may be of too coarse or too fine a pitch.

There is no general rule, even using the same grade of steel, for the
direction in which the size of a tap may alter in hardening, as is
attested by the following answers made by Mr. J. J. Grant to the
respective questions:--

"Do the taps that shorten most in length increase the most in diameter?"

Answer.--"Not always; sometimes a tap that shortens by hardening becomes
also smaller in diameter, while sometimes a tap will increase in length,
and also in diameter from hardening."

"Do taps that remain of true pitch after hardening remain true, or
increase or diminish in diameter?"

Answer.--"They will generally be of larger diameter."

"Do small taps alter more in diameter from hardening than large ones?"

Answer.--"No; the proportion is about the same, and is about .002 per
inch of diameter."

"What increase in diameter do you allow for shrinkage in hardening of
hob taps for tapping solid dies?"

Answer.--"As follows:--

  Diameter of   Shrinkage
    Hob Tap       about

   1/4 inch       .003
   1/2  "         .003
   3/4  "         .005
   1    "         .008"

"Suppose a tap that had been hardened and tempered to a straw color
contained an error 1/1000 inch both in diameter and in pitch, was
softened again, would it when soft retain the errors, or in what way
would softening affect the tap?"

Answer.--"We have repeatedly tried annealing or softening taps that were
of long or short pitch caused by tempering, and invariably found them
about the same as before the annealing. The second tempering will
generally shorten them more than the first. Sometimes, however, a second
tempering will bring a long pitch nearer correct."

"Do you soften your taps after roughing them out in the lathe?"

Answer.--"Never, if we can possibly avoid it. Sometimes it is necessary
because of improper annealing at first. The more times steel is annealed
the worse the results obtained in making the tool, and the less durable
the tool."

The following are answers to similar questions addressed to the Morse
Twist Drill and Machine Co.:--

"The expansion of taps during hardening varies with the diameter. A
1-inch tap would expand in diameter from 1/1000 to 3/1000 inch."

"Taps above 1/2 inch diameter expand in diameter to stop the gauge every
time."

"The great majority of taps contract in pitch during the hardening, they
seldom expand in length."

"The shortening of the pitch and the expansion in diameter have not much
connection necessarily, though steel that did not alter in one direction
would be more likely to remain correct in the other."

"There does not seem to be any change in the diameter or pitch of taps
if measured after hardening (and before tempering) and again after
tempering them."

"Taps once out in length seem to get worse at every heating, whether to
anneal or to harden."

[Illustration: Fig. 355.]

It will now be obvious to the reader that the diameter of a tap, to give
a standard sized bolt a required tightness of fit, will, as a general
rule, require to vary according to the depth of hole to be tapped,
because the greater that depth the greater the error in the pitch.
Suppose a tap, for example, to get of finer pitch to the amount of .002
per inch of length, then a hole an inch deep and tapped with that tap
would err .002 in its depth, while a hole two inches deep would err
twice as much in its depth.

[Illustration: Fig. 356.]

Therefore a bolt that would be a hand fit (that is, screw in under hand
pressure) in the hole an inch deep would require more force, and
probably the use of a wrench, to wind it through the hole 2 inches deep;
hence in cases where a definite degree of fit is essential, the
reduction in diameter of the male screw or thread necessary to
compensate for the error in the tap pitch must vary according to the
depth of the hole, and the degree of error in the tap.

[Illustration: Fig. 357.]

It is obvious that the longer a tap is the greater the error induced by
hardening, and it often becomes a consideration how to tap a long hole,
and obtain a thread true to pitch. This may be accomplished as follows.
Several taps are made of slightly different diameters, the largest being
of the required finished size. Each tap is made taper for a distance of
two or three threads only, and is hardened at this tapered end, but left
soft for the remainder of its length. The smallest tap is used first,
and when it has tapped a certain distance, a larger one is inserted, and
by continuing this interchange of taps and slightly varying the length
of the taper, the work may be satisfactorily done.

To test the accuracy, or rather the uniformity, of a thread that has
been hardened, a sheet metal gauge, such as at G or at G´ (Fig. 355),
may be used, there being at _a_ and _b_ teeth to fit the threads. If the
edge of the gauge meets the tops of the threads, then their depth is
correct. If it is desired to test only the pitch, then the gauge may be
made as at G´, where, as is shown in the figure, the edge of the gauge
clears the tops of the threads, and in this way may be tried at various
points along the thread length.

[Illustration: Fig. 358.]

A method of truing hardened threads proposed by the author of this work
in 1877, and since employed by the Pratt and Whitney Company to true
their hardened steel plug-thread gauges, is as follows:--A soft steel
wheel about 3-1/2 inches in diameter, whose circumference is turned off
to the shape of the thread, is mounted upon the slide rest of a lathe,
and driven by a separate belt after the manner of driving emery wheels;
this wheel is charged with diamond dust, which is pressed into its
surface by a roller, hence it grinds the thread true.

The amount allowed for grinding is 3/1000 inch measured in the angles of
the thread, as was shown in Figs. 280 and 281.

In charging the wheel with diamond dust it is necessary to use a roller
shaped as in Fig. 356, so that the axis of the roller R and wheel W
shall be at a right angle, as denoted by the dotted lines. If the roller
is not made to the correct cone its action will be partly a rolling and
partly a sliding one, and it will strip the diamond dust from the wheel
rather than force it in, the reasons for this being shown in Figs. 57
and 58 upon the subject of bevel-wheels.

[Illustration: Fig. 359.]

Taps for lead and similar soft metal are sometimes made with three flat
sides instead of grooves. The tapping holes may in this case be made of
larger diameter than the diameter of the end of the tap thread, because
the metal in the hole will compress into the tap thread, and so form a
full thread. Taps for other metal have also been made of half-round
section. Fig. 357 represents a tap of oval cross section, having two
flutes, as shown, but it may be observed that neither half-round nor
oval taps possess any points of advantage over the ordinary forms of
three or four fluted taps, while the former are more troublesome and
costly to manufacture.

[Illustration: Fig. 360.]

When it is required to tap a hole very straight and true, it is
sometimes the practice to provide a parallel stem to the tap, as shown
in figure at C. This stem is made a neat working fit to the tapping
hole, so that the latter serves as a guide to the tap, causing it to
enter and to operate truly.

TAP WRENCH.--Wrenches for rotating a tap are divided into two principal
classes, single and double wrenches. The former has the hole which
receives the squared end of the tap in the middle of its length, as
shown in Fig. 358 at E, there being a handle on each side to turn it by.

[Illustration: Fig. 361.]

The single wrench has its hole at one end, as shown in Fig. 359 at D,
and is employed for tapping holes in locations where the double wrench
could not be got in.

[Illustration: Fig. 362.]

[Illustration: Fig. 363.]

In some cases double tap wrenches are made with two or three sizes of
square holes to serve as many different sizes of taps, but this is
objectionable, because unless the handles of the wrench extend equally
on each side of the tap, the overhanging weight on one side of the tap
exerts an influence to pull the tap over to one side and tap the hole
out of straight. For taps that have square heads the wrench should be a
close but an easy fit to the tap head, otherwise the square corners of
the tap become rounded. For the smaller sizes of taps, adjustable
wrenches, such as shown in Fig. 360, are sometimes employed. These
contain two dies; the upper one, which meets the threaded end of C,
being a sliding fit, and the joint faces being formed as shown at A, B.
By rotating the handle C its end leaves the upper die, which may be
opened out, leaving the square hole between the dies large enough to
admit the squared tap end. After the wrench is placed on the tap, C is
rotated so as to close the dies upon the tap.

[Illustration: Fig. 364.]

[Illustration: Fig. 365.]

When the location of the tapping hole leaves room for the wrench to
rotate a full circle, C is screwed up so that the dies firmly grip the
tap head, which preserves the tap head; but when the wrench can only be
rotated a part of a revolution, C is adjusted to leave the dies an easy
fit to the tap head, so as to enable the wrench to be removed from the
tap head with facility and again placed upon the tap head. C is operated
by a round lever or pin introduced in a hole in the collar, or the
collar may be squared to receive a wrench.

To insure that a tap shall tap a hole straight, the machinist, in the
case of hand tapping, applies a square to the work and the tap, as shown
in Fig. 361, in which W represents a piece of work, T a tap, and S S two
squares. If the tap is a taper one the square is sighted with the shank
of the tap, as shown in position 1, but if the thread of the tap is
parallel, the square may be applied to the thread of the tap, as in
position 2. If the tap leans over to one side, as in Fig. 362, it is
brought upright by exerting a pressure on the tap wrench handle B (on
the high side) in the direction of the arrow A, while the wrench is
rotated; but if the tap leans much to one side it is necessary to rotate
the tap back and forth, exerting the pressure on the forward stroke
only.

[Illustration: Fig. 366.]

[Illustration: Fig. 367.]

It is necessary to correct the errors before the tap has entered the
hole deeply, because the deeper the tap has entered the greater the
difficulty in making the correction. If the pressure on the tap wrench
be made excessive, it is very liable to cause the tap to break,
especially in the case of small taps, that is to say, those of 5/8 inch
or less in diameter. The square should be applied as soon as the tap has
entered the hole sufficiently to operate steadily, and should be applied
several times during the tapping operation.

When the tap does not pass through the hole it may be employed with a
guide which will keep it true, as shown in Fig. 363, in which W is a
piece of work, T the tap, and S a guide, the latter being bolted or
clamped to the work at B. In this case the shank of the tap is made
fully as large in diameter as the thread. In cases where a number of
equidistant holes require tapping, as in the case of cylinder ends, this
device saves a great deal of time and insures that the tapping be
performed true, the hole to receive the bolt B and that to receive the
tap being distant apart to the same amount as are the holes in the work.

In shops where small work is made to standard gauge, and on the
interchangeable system, devices are employed, by means of which a piece
that has been threaded will screw firmly home to its place, and come to
some definite position, as in the following examples. In Fig. 364 let it
be required that the stud A shall screw in the slide S; the arm A to
stand vertical when collar B is firmly home, and a device such as in
Fig. 365 may be employed. P is a plate on which is fixed a chuck C to
receive the slide S. In plate P is a groove G to hold the head H at a
right angle to the slideway in C, there being a projection beneath H and
beneath C to fit into G. The tap T is threaded through H, but not fluted
at the part that winds through H when the tapping is being done, so as
not to cause the thread in H to wear. H acts as a guide to the tap and
causes it to start the thread at the same point in the bore of each
piece S, and the stem will be so threaded that the screw starts at the
same point in the circumference of each piece.

[Illustration: Fig. 368.]

A second example of uniform tapping is shown in Figs. 366, 367, and 368.
The piece, Fig. 366, is to have its bore A tapped in line with the slot
C, and the thread is to start at a certain point in its bore. In Fig.
367 this piece is shown chucked on a plate D. F is a chuck having a lug
E fitting into the slot (C, Fig. 366) of the work. This adjusts the work
in one direction. The face D of the plate adjusts the vertical height of
the work, and the alignment of the hole to the axis of the tap is
secured in the construction of the chuck, as is shown in Fig. 369. A lug
K is at a right angle to the face B of the chuck and stands in a line
with lug E, as denoted by the dotted line _g_ _g_, and as lug K fits
into the slot G, Fig. 367, the work will adjust itself true when bolted
to the plate.

[Illustration: Fig. 369.]

Fig. 368 shows a method of tapping or hobbing four chasers (as for a
bolt cutter), so that if the chasers are marked 1, 2, 3 and 4, as shown,
any chaser of No. 1 will work with the others, although not tapped at
the same operation. C is a chuck with four dies (A, B, C, D) placed
between the chasers. By tightening the set-screws S, the dies and
chasers are locked ready for the tapping. N is a hub to receive a
guide-pin P, which is passed through to hold the chasers true while
being set in the chuck, and it is withdrawn before the tapping
commences; _d_ _e_ _f_ are simply to take hold of when inserting and
removing the dies. It is obvious that a chuck such as this used upon a
plate, as in Fig. 365, with the hob guided in the head H there shown,
would tap each successive set of chasers alike as a set, and
individually alike, provided, of course, that the hob guide or head H is
at each setting placed the same distance from the face of the chuck, a
condition that applies to all this class of work. In the case of work
like chasers, where the tap or hob does not have much bearing to guide
it in the work, a three-flute hob should be used for four chasers, or a
four-flute hob for three chasers, which is necessary so that the hob may
work steadily and tap all to the same diameter.



CHAPTER V.--FASTENING DEVICES.


Bolts are usually designated for size by their diameters measured at the
cylindrical stem or body, and by their lengths measured from the inner
side of the head to the end of the thread, so that if a nut be used, the
length of the bolt, less the thickness of the nut and washer (if the
latter be used), is the thickness of work the bolt will hold. If the
work is tapped, and no nut is used, the full length of the bolt stem is
taken as the length of the bolt.

A _black_ bolt is one left as forged. A finished bolt has its body, and
usually its head also, machine finished, but a finished bolt sometimes
has a black head, the body only being turned.

A square-headed bolt usually has a square nut, but if the nut is in
a situation difficult of access for the wrench, or where the head
of the bolt is entirely out of sight (as secluded beneath a flange) the
nut is often made hexagon. A machine-finished bolt usually has a
machine-finished and hexagon nut. Square nuts are usually left black.

[Illustration: Fig. 370.]

The heads of bolts are designated by their shapes, irrespective of
whether they are left black or finished. Fig. 370 represents the various
forms: _a_, square head; _b_, hexagon head; _c_, capstan head; _d_,
cheese head; _e_, snap head; _f_, oval head, or button head; _g_,
conical head; _h_, pan head; _i_, countersink head.

The square heads _a_ are usually left black, though in exceptional cases
they are finished. Hexagon heads are left black or finished as
circumstances may require; when a bolt head is to receive a wrench and
is to be finished, it is usually made hexagon. Heads _c_ and _d_ are
almost invariably finished when used on operative parts of machines, as
are also _e_ and F. Heads _g_ are usually left black, while _h_ and _i_
are finished if used on machine work, and left black when used as rivets
or on rough unfinished work.

The heads from _e_ to _i_ assume various degrees of curve or angle to
suit the requirements, but when the other end of the bolt is threaded to
receive a nut, some means is necessary to prevent them from rotating in
their holes when the nut is screwed up, thus preventing the nut from
screwing up sufficiently tight. This is accomplished in woodwork by
forging either a square under the head, as in Fig. 371, or by forging
under the head a tit or stop, such as shown in Figs. 372 and 373 at P.
Since, however, forging such stops on the bolt would prevent the heads
from being turned up in the lathe, they are for lathe-turned bolts put
in after the bolts have been finished in the lathe, a hole being
subsequently drilled beneath the head to receive the pin or stop, P,
Fig. 372, which may be tightly driven in. A small slot is cut in the
edge of the hole to receive the stop.

[Illustration: Fig. 371.]

[Illustration: Fig. 372.]

[Illustration: Fig. 373.]

Bolts are designated for kinds, as in Fig. 374, in which _k_ is a
machine bolt; _l_ a collar bolt, from having a collar on it; _m_ a
cotter bolt, from having a cotter or key passing through it to serve in
place of a nut; _n_ a carriage bolt, from having a square part under the
head to sink in the wood and prevent the bolt from turning with the nut;
and _o_ a countersink bolt for cases where the head of the bolt comes
flush.

[Illustration: Fig. 374.]

The simple designation "machine bolt" is understood to mean a black or
unfinished bolt having a square head and nut, and threaded, when the
length of the bolt will admit it, and still leave an unthreaded part
under the bolt head, for a length equal to about four times the diameter
of the bolt head. If the bolt is to have other than a square head it is
still called a machine bolt, but the shape of the head or nut is
specially designated as "hexagon head machine bolt," this naturally
implying that a hexagon nut also is required.

In addition to these general names for bolts, there are others applied
to special cases. Thus Fig. 375 represents a patch bolt or a bolt for
fastening patches (as plate C to plate D), its peculiarity being that it
has a square stem A for the wrench to screw it in by. When the piece the
patch bolt screws into is thin, as in the case of patches on steam
boilers, the pitch of the thread may, to avoid leakage, be finer than
the usual standard.

In countersink head bolts, such as the patch bolt in Fig. 375, the head
is very liable to come off unless the countersink in the work (as in C)
is quite fair with the tapped hole (as in D) because the thread of the
bolt is made a tight fit to the hole, and all the bending that may take
place is in the neck beneath the head, where fracture usually occurs.
These bolts are provided with a square head A to screw them in by, and
are turned in as at B to a diameter less than that at the bottom of the
thread, so that if screwed up until they twist off, they will break in
the neck at B.

[Illustration: Fig. 375.]

[Illustration: Fig. 376.]

Instead of the hole being countersunk, however, it may be cupped or
counterbored, as in Fig. 376, in which the names of the various forms of
the enlargement of holes are given. The difference between a faced and a
counterbored hole is that in a counterbored hole the head or collar of
the pin passes within the counterbore, the use of the counterbore being
in this case to cause the pin to stand firmly and straight. The
difference between a dished and a cupped is merely that cupped is deeper
than dished, and that between grooved and recessed is that a recess is a
wide groove.

Eye bolts are those having an eye in place of a head, as in Fig. 377,
being secured by a pin passing through the eye, or by a second bolt, as
in the figure. When the bolt requires to pivot, that part that is
within the eye may be made of larger diameter than the thread, so as to
form a shoulder against which the bolt may be screwed firmly home to
secure it without gripping the eye bolt.

[Illustration: Fig. 377.]

[Illustration: Fig. 378.]

Fig. 378 represents a foundation bolt for holding frames to the stone
block of a foundation. The bolt head is coned and jagged with chisel
cuts. It is let into a conical hole (widest at the bottom) in the stone
block, and melted lead is poured around it to fill the hole and secure
the bolt head.

[Illustration: Fig. 379.]

[Illustration: Fig. 380.]

Another method of securing a foundation bolt head within a stone block
is shown in Fig. 379; a similar coned hole is cut in the block, and
besides the bolt head B a block W is inserted, the faces of the block
and bolt being taper to fit to a taper key K, so that driving K locks
both the bolt and the block in the stone. When the bolt can pass
entirely through the foundation (as when the latter is brickwork) it is
formed as in Fig. 380, in which B is a bolt threaded to receive a nut at
the top. At the bottom it has a keyway for a key K, which abuts against
the plate P. To prevent the key from slackening and coming out, it has a
recess as shown in the figure at the sectional view of the bolt on the
right of the illustration, the recess fitting down into the end of the
keyway as shown.

[Illustration: Fig. 381.]

Another method is to give the bolt head the form at B in Fig. 381, and
to cast a plate with a rectangular slot through, and with two lugs A C.
The plate is bricked in and a hole large enough to pass the bolt head
through is left in the brickwork. The bolt head is passed down through
the brickwork in the position shown at the top, and when it has passed
through the slot in the plate it is given a quarter turn, and then
occupies the position shown in the lower view, the lugs A C preventing
it from turning when the nut is screwed home. The objection to this is
that the hole through the brickwork must be large enough to admit the
bolt head. Obviously the bolt may have a solid square head, and a square
shoulder fitting into a square hole in the plate, the whole being
bricked in.

[Illustration: Fig. 382.]

[Illustration: Fig. 383.]

Figs. 382 and 383 represent two forms of hook bolt for use in cases
where it is not desired to have bolt holes through both pieces of the
work. In Fig. 382 the head projects under the work and for some distance
beneath and beyond the washer, as is denoted by the dotted line, hence
it would suspend piece A from B or piece B from A. But in Fig. 383 the
nut pressure is not beneath the part where the hook D grips the work,
hence the nut would exert a pressure to pull piece B in the direction of
the arrow; hence if B were a fixed piece the bolt would suspend A from
it, but it could not suspend B from A.

In woodwork the pressure of the nut is apt to compress the wood, causing
the bolt head and nut to sink into the wood, and to obviate this, anchor
plates are used to increase the area receiving the pressure; thus in
Fig. 384 a plate is tapped to serve instead of a nut, and a similar
plate may of course be placed under the bolt head.

[Illustration: Fig. 384.]

The Franklin Institute or United States Standard for the dimensions of
bolt heads and nuts is as follows. In Fig. 385, D represents the
diameter of the bolt, J represents the short diameter or width across
flats of the bolt head or of the nut, being equal to one and a half
times the diameter of the bolt, plus 1/16 inch for finished heads or
nuts, and plus 1/8 inch for rough or unfinished heads or nuts. K
represents the depth or thickness of the head or nut, which in finished
heads or nuts equals the diameter of the bolt minus 1/16 inch, and in
rough heads equals one half the distance between the parallel sides of
the head, or in other words one half the width across the flats of the
head.

H represents the thickness or depth of the nut, which for finished nuts
is made equal to the diameter of the bolt less 1/16 inch, and therefore
the same thickness as the finished bolt head, while for rough or
unfinished nuts it is made equal to the diameter of the bolt or the same
as the rough bolt head. I represents the long diameter or diameter
across corners, which, however, is a dimension not used to work to, and
is inserted in the following tables merely for reference:--

[Illustration: Fig. 385.]

TABLE OF THE FRANKLIN INSTITUTE STANDARD DIMENSIONS FOR THE HEADS OF
BOLTS AND FOR THEIR NUTS, WHEN BOTH HEADS AND NUTS ARE OF HEXAGON FORM,
AND ARE POLISHED OR FINISHED.

  +------------+-------------+-----------+---------------+-----------+
  |  Diameter  | Diameter at | Number of |    Diameter   | Thickness |
  |   at top   |  bottom of  |  Threads  | across Flats, |    or     |
  | of Thread. |   Thread.   | per inch. |    or short   |  Depth.   |
  |            |             |           |    diameter.  |           |
  +------------+-------------+-----------+---------------+-----------+
  |     1/4    |     .185    |   20      |       7/16    |    3/16   |
  |     5/16   |     .240    |   18      |       17/32   |    1/4    |
  |     3/8    |     .294    |   16      |       5/8     |    5/16   |
  |     7/16   |     .345    |   14      |       23/32   |    3/8    |
  |     1/2    |     .400    |   13      |       13/16   |    7/16   |
  |     9/16   |     .454    |   12      |       29/32   |    1/2    |
  |     5/8    |     .507    |   11      |     1         |    9/16   |
  |     3/4    |     .620    |   10      |     1-3/16    |   11/16   |
  |     7/8    |     .731    |    9      |     1-3/8     |   13/16   |
  |   1        |     .837    |    8      |     1-9/16    |   15/16   |
  |   1-1/8    |     .940    |    7      |     1-3/4     |  1-1/16   |
  |   1-1/4    |    1.065    |    7      |     1-15/16   |  1-3/16   |
  |   1-3/8    |    1.160    |    6      |     2-1/8     |  1-5/16   |
  |   1-1/2    |    1.284    |    6      |     2-5/16    |  1-7/16   |
  |   1-5/8    |    1.389    |    5-1/2  |     2-1/2     |  1-9/16   |
  |   1-3/4    |    1.491    |    5      |     2-11/16   |  1-11/16  |
  |   1-7/8    |    1.616    |    5      |     2-7/8     |  1-13/16  |
  |   2        |    1.712    |    4-1/2  |     3-1/16    |  1-15/16  |
  |   2-1/4    |    1.962    |    4-1/2  |     3-7/16    |  2-3/16   |
  |   2-1/2    |    2.176    |    4      |     3-13/16   |  2-7/16   |
  |   2-3/4    |    2.426    |    4      |     4-3/16    |  2-11/16  |
  |   3        |    2.629    |    3-1/2  |     4-9/16    |  2-15/16  |
  |  3-1/4     |    2.879    |    3-1/2  |     4-15/16   |  3-3/16   |
  |  3-1/2     |    3.100    |    3-1/4  |     5-5/16    |  3-7/16   |
  |  3-3/4     |    3.377    |    3      |     5-11/16   |  3-13/16  |
  |  4         |    3.567    |    3      |     6-1/16    |  3-15/16  |
  |  4-1/4     |    3.798    |    2-7/8  |     6-7/16    |  4-3/16   |
  |  4-1/2     |    4.028    |    2-7/8  |     6-13/16   |  4-7/16   |
  |  4-3/4     |    4.256    |    2-5/8  |     7-3/16    |  4-11/16  |
  |  5         |    4.480    |    2-1/2  |     7-9/16    |  4-15/16  |
  |  5-1/4     |    4.730    |    2-1/2  |     7-15/16   |  5-3/16   |
  |  5-1/2     |    4.953    |    2-3/8  |     8-5/16    |  5-7/16   |
  |  5-3/4     |    5.203    |    2-3/8  |     8-11/16   |  5-11/16  |
  |  6         |    5.423    |    2-1/4  |     9-1/16    |  5-15/16  |
  +------------+-------------+-----------+---------------+-----------+

Note that square heads are supposed to be always unfinished, hence there
is no standard for their sizes if finished.

The Franklin Institute standard dimensions for hexagon and square bolt
heads and nuts when the same are left unfinished or rough, as forged,
are as follows:--

  +----------+-------------+-------------+---------------+-----------+
  |          |   Diameter  |   Diameter  |      Short    | Thickness |
  |   Bolt   |    across   |    across   |    diameter,  |     or    |
  | Diameter | corners, or | corners or  |  or diameter  | depth for |
  |    in    |    long     |    long     |  across flats | square or |
  |  Inches. | diameter of | diameter of | for square or |  hexagon  |
  |          |   hexagon   |    square   | hexagon heads |   heads.  |
  |          |   heads.    |    heads.   |   and nuts.   |           |
  +----------+-------------+-------------+---------------+-----------+
  |          |    Inch.    |    Inch.    |     Inch.     |   Inch.   |
  |    1/4   |     37/64   |     7/10    |      1/2      |    1/4    |
  |    5/16  |     11/16   |     10/12   |      19/32    |    19/64  |
  |    3/8   |     51/64   |     63/64   |      11/16    |    11/32  |
  |    7/16  |     9/10    |   1-7/64    |      25/32    |    25/64  |
  |    1/2   |   1         |   1-15/64   |      7/8      |    7/16   |
  |    9/16  |   1-1/8     |   1-23/64   |      31/32    |    31/64  |
  |    5/8   |   1-7/32    |   1-1/2     |    1-1/16     |    17/32  |
  |    3/4   |   1-7/16    |   1-49/64   |    1-1/4      |    5/8    |
  |    7/8   |   1-21/32   |   2-1/32    |    1-7/16     |    23/32  |
  |  1       |   1-7/8     |   2-19/64   |    1-5/8      |    13/16  |
  |  1-1/8   |   2-2/32    |   2-9/16    |    1-13/16    |    29/32  |
  |  1-1/4   |   2-5/16    |   2-53/64   |    2          |  1        |
  |  1-3/8   |   2-17/32   |   3-3/32    |    2-3/16     |  1-3/32   |
  |  1-1/2   |   2-3/4     |   3-23/64   |    2-3/8      |  1-3/16   |
  |  1-5/8   |   2-31/32   |   3-5/8     |    2-9/16     |  1-9/32   |
  |  1-3/4   |   3-3/16    |   3-57/64   |    2-3/4      |  1-3/8    |
  |  1-7/8   |   3-13/32   |   4-5/32    |    2-15/16    |  1-15/32  |
  |  2       |   3-5/8     |   4-27/64   |    3-1/8      |  1-9/16   |
  |  2-1/4   |   4-1/16    |   4-61/64   |    3-1/2      |  1-3/4    |
  |  2-1/2   |   4-1/2     |   5-31/64   |    3-7/8      |  1-15/16  |
  |  2-3/4   |   4-29/32   |   6         |    4-1/4      |  2-1/8    |
  |  3       |   5-3/8     |   6-17/32   |    4-5/8      |  2-5/16   |
  |  3-1/4   |   5-13/16   |   7-1/16    |    5          |  2-1/2    |
  |  3-1/2   |   6-7/64    |   7-39/64   |    5-3/8      |  2-11/16  |
  |  3-3/4   |   6-21/32   |   8-1/8     |    5-3/4      |  2-7/8    |
  |  4       |   7-3/32    |   8-41/64   |    6-1/8      |  3-1/16   |
  |  4-1/4   |   7-9/16    |   9-3/16    |    6-1/2      |  3-1/4    |
  |  4-1/2   |   7-31/32   |   9-3/4     |    6-7/8      |  3-7/16   |
  |  4-3/4   |   8-13/32   |  10-1/4     |    7-1/4      |  3-5/8    |
  |  5       |   8-27/32   |  10-49/64   |    7-5/8      |  3-13/16  |
  |  5-1/4   |   9-9/32    |  11-23/64   |    8          |  4        |
  |  5-1/2   |   9-23/32   |  11-7/8     |    8-3/8      |  4-3/16   |
  |  5-3/4   |  10-5/32    |  12-3/8     |    8-3/4      |  4-3/8    |
  |  6       |  10-19/32   |  12-15/16   |    9-1/8      |  4-9/16   |
  +----------+-------------+-------------+---------------+-----------+

The depth or thickness of both the hexagon and square nuts when left
rough or unfinished is, according to the above standard, equal to the
diameter of the bolt.

The following are the sizes of finished bolts and nuts according to the
present Whitworth Standard. The exact sizes are given in decimals, and
the nearest approximate sizes in sixty-fourths of an inch:--

  +-------------+------------------------+----------------------+
  | Diameter of |  Width of nuts across  |    Height of bolt    |
  |    bolts.   |         flats.         |         heads.       |
  +-------------+----------+-------------+--------+-------------+
  |     1/8     |   .338   |   21/64 _f_ |  .1093 |   7/64      |
  |     3/16    |   .448   |   29/64 _b_ |  .1640 |   5/32      |
  |     1/4     |   .525   |   33/64 _f_ |  .2187 |   7/32      |
  |     5/16    |   .6014  |   19/32 _f_ |  .2734 |   17/64     |
  |     3/8     |   .7094  |   45/64 _f_ |  .3281 |   21/64     |
  |     7/16    |   .8204  |   53/64 _b_ |  .3828 |   3/8   _f_ |
  |     1/2     |   .9191  |   29/32 _b_ |  .4375 |   7/16      |
  |     9/16    |  1.011   | 1-1/64  _b_ |  .4921 |   31/64 _f_ |
  |     5/8     |  1.101   | 1-3/32  _f_ |  .5468 |   35/64     |
  |     11/16   |  1.2011  | 1-13/64 _b_ |  .6015 |   19/32 _f_ |
  |     3/4     |  1.3012  | 1-19/64 _f_ |  .6562 |   21/32     |
  |     13/16   |  1.39    | 1-25/64 _b_ |  .7109 |   45/64 _f_ |
  |     7/8     |  1.4788  | 1-31/64 _b_ |  .7656 |   49/64     |
  |     15/16   |  1.5745  | 1-37/64 _b_ |  .8203 |   13/16 _f_ |
  |   1         |  1.6701  | 1-43/64 _b_ |  .875  |   7/8       |
  |   1-1/8     |  1.8605  | 1-55/64 _f_ |  .9843 |   63/64     |
  |   1-1/4     |  2.0483  | 2-3/64  _f_ | 1.0937 | 1-3/32      |
  |   1-3/8     |  2.2146  | 2-7/32  _b_ | 1.2031 | 1-13/64     |
  |   1-1/2     |  2.4134  | 2-13/32 _f_ | 1.3125 | 1-5/16      |
  |   1-5/8     |  2.5763  | 2-37/64 _b_ | 1.4128 | 1-27/64     |
  |   1-3/4     |  2.7578  | 2-3/4   _f_ | 1.5312 | 1-17/32     |
  |   1-7/8     |  3.0183  | 3-1/16  _f_ | 1.6406 | 1-41/64     |
  |   2         |  3.1491  | 3-5/32  _b_ | 1.75   | 1-3/4       |
  |   2-1/8     |  3.337   | 3-11/32 _b_ | 1.8523 | 1-55/64     |
  |   2-1/4     |  3.546   | 3-35/64 _b_ | 1.9687 | 1-31/32     |
  |   2-3/8     |  3.75    | 3-3/4       | 2.0781 | 2-5/64      |
  |   2-1/2     |  3.894   | 3-57/64 _f_ | 2.1875 | 2-3/16      |
  |   2-5/8     |  4.049   | 4-3/64  _f_ | 2.2968 | 2-19/64     |
  |   2-3/4     |  4.181   | 4-3/16  _b_ | 2.4062 | 2-13/32     |
  |   2-7/8     |  4.3456  | 4-11/32 _f_ | 2.5156 | 2-33/64     |
  |   3         |  4.531   | 4-17/32 _b_ | 2.625  | 2-5/8       |
  +-------------+----------+-------------+--------+-------------+

The thickness of the nuts is in every case the same as the diameter of
the bolts: _f_ = full, _b_ = bare.

When bolts screw directly into the work instead of passing through it
and receiving a nut, they come under the head of either tap bolts, set
screws, cap screws, or machine screws. A tap bolt is one in which the
full length of the stem or body is threaded, and differs from a set
screw, which is similarly threaded, in the respect that in a set screw
the head is square and its diameter is the same as the square bar of
steel or iron (as the case may be) from which the screw was made, while
in the tap bolt the head is larger in diameter than the bar it was made
from. Furthermore a tap bolt may have a hexagon head, which is usually
left unfinished unless ordered to be finished, as is also the case with
set screws.

Cap screws are made with heads either hexagon, square, or round, and
also with a square head and round collar, as in Fig. 386, the square
heads being of larger diameter than the iron from which they were made.
When the heads of cap screws are finished they are designated as "milled
heads."

[Illustration: Fig. 386.]

[Illustration: Fig. 387.]

A machine screw is a small screw, such as in Fig. 387, the diameter of
the body being made to the Birmingham wire gauge, the heads being formed
by upsetting the wire of which they are made. They have saw slots S for
a screw driver, the threads having special pitches, which are given
hereafter. The forms of the heads are as in Fig. 387, A being termed a
Fillister, B a countersink, and C a round head. The difference between a
Fillister head of a machine screw and the same form of head in a cap
screw is that the former is upset cold, and the latter is either forged
or cut out of the solid metal.

When the end of a screw abuts against the work to secure it, it is
termed a set screw. The ordinary form of set screw is shown in Fig. 389,
the head being square and either black or polished as may be required.
The ends of the set screws of commerce, that is to say, that are kept on
sale, are usually either pointed as at A, Fig. 388, slightly bevelled as
at B, or cupped as at D. If left flat or only slightly bevelled as at B,
they are liable, if of steel and not hardened, or if of iron and
case-hardened only, to bulge out as at C. This prevents them from
slacking back easily or prevents removal if necessary, and even though
of hardened steel they do not grip very firmly. On this account their
points are sometimes made conical, as at A. This form, however,
possesses a disadvantage when applied to a piece of work that requires
accurate adjustment for position, inasmuch as it makes a conical
indentation in the work, and unless the point be moved sufficiently to
clear this indentation the point will fall back into it; hence the
conical point is not desirable when the piece may require temporary
fixture to find the adjustment before being finally screwed home. For
these reasons the best form of set screw end is shown at D, the outside
of the end being chamfered off and the inside being cupped, as denoted
by the dotted lines. This form cuts a ring in the work, but will hold
sufficiently for purposes of adjustment without being screwed home
firmly.

In some cases the end of the set screw is tapped through the enveloping
piece (as a hub) and its end projects into a plain hole in the internal
piece of the work, and in this case the end of the thread is turned off
for a distance of two or three threads, as at A in Fig. 390. Similarly,
when the head of the screw is to act or bear upon the work, the thread
may be turned off as at B in the figure.

When a bolt has no head, but is intended to screw into the work at one
end, and receive a nut at the other, it is termed a stud or standing
bolt. The simplest form of standing bolt is that in which it is parallel
from end to end with a thread at each end, and an unthreaded part in the
middle, but since standing bolts or studs require to remain fixed in the
work, it is necessary to screw them tightly into their places, and
therefore firmly home. This induces the difficulty that some studs may
screw a trifle farther into the work than others, so that some of the
stud ends may project farther through the nuts than others, giving an
appearance that the studs have been made of different lengths. The
causes of this may be slight variations in the tapping of the holes and
the threading of the studs. If those that appear longest are taken out
and reduced to the lengths of the others, it will be found sometimes
that the stud on the second insertion will pass farther into the work
than at the first, and the stud will project less through the nut than
the others. To avoid this those protruding most may be worked backward
and forward with the wrench and thus induced to screw home to the
required distance, but it is better to provide to the stud a shoulder
against which it may screw firmly home; thus in Fig. 391 is a stud,
whose end A is to screw into the work, part B is to enter the hole in
the work (the thread in the hole being cut away at the mouth to receive
B). In this case the shoulder between B and C screwing firmly against
the face of the work, all the studs being made of equal length from this
shoulder to end E, then the thickness of the flange or work secured by
the nut being equal, the nuts will pass an equal distance on end D, and
E will project equally through all the nuts. The length of the plain
part C is always made slightly less than the thickness of the flange or
foot of the work to be bolted up, so that the nut shall not meet C
before gripping the flange surface.

There are, however, other considerations in determining the shape and
size of the parts A and C of studs.

Thus, suppose a stud to have been in place some time, the nut on end E
being screwed firmly home on the work, and perhaps somewhat corroded on
E. Then the wrench pressure applied to the nut will be in a direction to
unscrew the stud out of the work, and if there be less friction between
A and the thread in the work than there is between D and the thread in
the nut, the stud and not the nut will unscrew. It is for this purpose
that the end A requires firmly screwing into the work. But in the case
of much corrosion this is not always sufficient, and the thread A is
therefore sometimes made of a larger diameter than the thread at D. In
this case the question at once arises, What shall be the diameter of the
plain part C?

[Illustration: Fig. 388.]

[Illustration: Fig. 389.]

[Illustration: Fig. 390.]

[Illustration: Fig. 391.]

If it be left slightly larger than D, but the depth of the thread less
than A, then it may be held sufficiently firmly by the fit of the
threads (without the aid of screwing against a shoulder) to prevent
unscrewing when releasing the nut, and may be screwed within the work
until its end projects the required distance; thus all the studs may
project an equal distance, but there will be the disadvantage that when
the studs require removing and are corroded the plain part is apt to
twist off, leaving the end A plugging the hole. The plain part C may be
left of same diameter as A, both being larger than D; but in this case
the difficulty of having all the studs project equally when screwed
home, as previously mentioned, is induced; hence C may be larger than A,
and a shoulder left at B, as in the figure; this would afford excellent
facility for unscrewing the stud to remove it, as well as insuring equal
projection of E. The best method of all is, so far as quality goes, to
make the plain part C square, as in Fig. 392, which is an English
practice, the square affording a shoulder to screw up against and secure
an equal projection while serving to receive a wrench to put in or
remove the stud. In this case the holes in the flange or piece bolted up
being squared, the stud cannot in any case unscrew with the nut. The
objection to this squared stud is that the studs cannot be made from
round bar iron, and are therefore not so easily made, and that the
squaring of the holes in the flange or part of the work supported by the
stud is again extra work, and for these reasons studs with square
instead of cylindrical mid-sections have not found favor in the United
States.

[Illustration: Fig. 392.]

[Illustration: Fig. 393.]

An excellent method of preventing the stud from unscrewing with the nut
is to make the end A longer than the nut end, as in Fig. 393, so that
its threads will have more friction; and this has the further advantage
that in cast iron it serves also to make the strength of the thread
equal to that of the stud. As the faces of the nuts are apt when screwed
home to score or mark the face of the work, it adds to the neatness of
the appearance to use a washer W beneath the nut, which distributes the
pressure over a greater area of work surface.

In some practice the ends A of studs are threaded taper, which insures
that they shall fit tight and enables their more easy extraction.

[Illustration: Fig. 394.]

An excellent tool for inserting studs of this kind to the proper
distance is shown in Fig. 394. It consists of a square body _a_ threaded
to receive the stud whose end is shown at _c_. The upper end is threaded
to receive an adjusting screw _b_, which is screwed in so that its end
_d_ meets the end _c_ of the stud. It is obvious that _b_ may be so
adjusted that when _a_ is operated by a wrench applied to its body until
its end face meets the work and the stud is inserted to the proper
depth, all subsequent studs may be put into the same depth.

[Illustration: Fig. 395.]

When the work pivots upon a stem, as in Fig. 395, the bolt is termed a
standing pin, and as in such cases the stem requires to stand firm and
true it is usual to provide the pin with a collar, as shown in the
figure, and to secure the pivoted piece in place with a washer and a
taper pin because nuts are liable to loosen back of themselves.
Furthermore, a pin and washer admit of more speedy disconnection than a
nut does, and also give a more delicate adjustment for end fit.

In drilling the tapping holes for standing bolts, it is the practice
with some to drill the holes in cast iron of such a size that the tap
will cut three-quarters only of a full thread, the claim being that it
is as strong as a full thread. The difference in strength between a
three-quarter and a full thread in cast iron is no doubt practically
very small indeed, while the process of tapping is very much easier for
the three-quarter full thread, because the tap may, in that case, be
wound continuously forward without backing it at every quarter or half
revolution, as would otherwise be necessary, in order to give the oil
access to the cutting edges of the tap--and oil should always be used in
the process of tapping (even though on cast iron it causes the cuttings
to clog in the flutes of the tap, necessitating in many cases that the
tap be once or twice during the operation taken out, and the cuttings
removed) because the oil preserves the cutting edges of the tap teeth
from undue abrasion, and, therefore, from unnecessarily rapid dulling.
With a tap having ordinarily wide and deep flutes, and used upon a hole
but little deeper than the diameter of the tap, the cuttings due to
making a three-quarter full thread will not more than fill the flutes of
the tap by the time its duty is performed. We have also to consider that
with a three-quarter full thread it is much easier to extract the
standing bolt when it is necessary to do so, so that all things
considered it is permissible to have such a thread, providing the
tapping hole does not pass through into a cylinder or chamber requiring
to be kept steam-tight, for in that case the bolt would be almost sure
to leak. As a preventive against such leakage, the threads are sometimes
cut upon the standing bolts without having a terminal groove, and are
then screwed in as far as they will go; the termination of the thread
upon the standing bolt at the standing or short end being relied upon to
jam into and close up the thread in the hole. A great objection to this,
however, is the fact that the bolts are liable to screw into the holes
to unequal depths, so that the outer ends will not project an equal
distance through the nuts, and this has a bad appearance upon fine work.
It is better, then, in such a case, to tap the holes a full thread, the
extra trouble involved in the tapping being to some extent compensated
for in the fact that a smaller hole, which can be more quickly drilled,
is required for the full than for the three-quarter thread.

The depth of the tapping hole should be made if possible equal to one
and a half times the diameter of the tap, so that in case the hole
bottoms and the tap cannot pass through, the taper, and what is called
in England the second, and in the United States the plug tap, will
finish the thread deep enough without employing a third tap, for the
labor employed in drilling the hole deeper is less than that necessary
to the employment of a third tap. If the hole passes through the work,
its depth need not, except for cast-iron holes, be greater than 1/8 inch
more than the diameter of the bolt thread, which amount of excess is
desirable so that in case the nut corrodes, the nut being as thick as
the diameter of the tap, and therefore an inch less than the depth of
the hole at the standing end, will be more likely to leave the stud
standing than to carry it with it when being unscrewed.

[Illustration: Fig. 396.]

When it is desirable to provide that bolts may be quickly removed, the
flanges may be furnished with slots, as in Fig. 396, so that the bolts
may be passed in from the outside, and in this case it is simply
necessary to slacken back the nut only. It is preferable, however, in
this case to have the bolt square under the head, as in Fig. 397, so as
to prevent the bolt from turning when screwing up or unscrewing the nut.
The bolt is squared at A, which fits easily into the flange. The
flanges, however, should in this case be of ample depth or thickness to
prevent their breakage, twice the depth of the nut being a common
proportion.

[Illustration: Fig. 397.]

[Illustration: Fig. 398.]

In cases where it is inconvenient for the bolt head to pass through the
work a [T] groove is employed, as in Fig. 398. In this case the bolt
head may fit easily at A B to the sides A B of the groove, so that while
the bolt head will slide freely along the groove, the head, being
square, cannot turn in the slot when the nut is screwed home. This,
however, is more efficiently attained when there is a square part
beneath the bolt head, as in Fig. 399, the square A of the bolt fitting
easily to the slot B of the groove.

[Illustration: Fig. 399.]

[Illustration: Fig. 400.]

When it is undesirable that the slots run out to the edge of the work
they may terminate in a recess, as at A in Fig. 400, which affords
ingress of the bolt head to the slot; or the bolt head may be formed as
in Fig. 401, the width A B of the bolt head passing easily through the
top A B of the slot, and the bolt head after its insertion being turned
in the direction of the arrow, which it is enabled to do by reason of
the rounded corners C D. In this case, also, there may be a square under
the head to prevent the bolt head from locking in the slot, but the
corners of the square must also be rounded as in Fig. 402.

[Illustration: Fig. 401.]

[Illustration: Fig. 402.]

The underneath or gripping surface of a bolt head should be hollow, as
at A in Fig. 403, rather than rounding as at B, because, if rounding,
the bolt will rotate with the nut when the latter grips the work
surface. It should also be true with the axial line of the bolt so as to
bear fairly upon the work without bending. The same remarks apply to the
bedding surface of the nut, because to whatever amount the face is out
of true it will bend the threaded end of the bolt, and this may be
sufficient to cause the bolt to break.

[Illustration: Fig. 403.]

In Fig. 404, for example, is shown a bolt and nut, neither of which bed
fair, being open at A and B respectively, and it is obvious that the
strain will tend to bend or break the bolt across the respective dotted
lines C, D. In the case of the nut there is sufficient elasticity in the
thread to allow of the nut forcing itself to a bed on the work, the bolt
bending; but in the case of the bolt head the bending is very apt to
break off the bolt short in the neck under the head. In a tap bolt where
the wrench is applied to the bolt head, the rotation, under severe
strain, of the head will usually cause it to break off in all cases
where the bolt is rigidly held, so that it cannot cant over and allow
the head to bed fair.

[Illustration: Fig. 404.]

A plain tap bolt should be turned up along its body, because if out of
true the hole it passes through must be made large enough to suit the
eccentricity of the bolt, or else a portion of the wrench pressure will
be expended in rotating the bolt in the hole instead of being expended
solely in screwing the bolt farther into the work.

It is obvious therefore, that if a tap bolt be left black the hole it
passes through must be sufficiently large to make full allowance for the
want of truth in the bolt. For the same reasons the holes for tapped
bolts require to be tapped very true.

Black studs possess an advantage (over tap bolts) in this respect,
inasmuch as that if the holes are not tapped quite straight the error
may be to some extent remedied by screwing them fully home and then
bending them by hammer blows.

Nuts are varied in form to suit the nature of the work. For ordinary
work, as upon bolts, their shape is usually made to conform to the shape
of the bolt head, but when the nut is exposed to view and the bolt head
hidden, the bolt end and the nut are (for finished work) finished while
the bolt heads are left black.

[Illustration: Fig. 405.]

[Illustration: Fig. 406.]

The most common form of hexagon nut is shown in Fig. 405, the upper edge
being chamfered off at an angle of about 40°. In some cases the lower
edge is cut away at the corners, as in Fig. 405 at A, the object being
to prevent the corners of the nut from leaving a circle of bearing marks
upon the work, but this gives an appearance at the corners that the nut
does not bed fair. Another shape used by some for the end faces of deep
nuts, that is to say, those whose depth exceeds the diameter of the
bolt, is shown in Fig. 406. Nuts of extra depth are used when, from the
nut being often tightened and released, the thread wear is increased,
and the extra thread length is to diminish the wear.

To avoid the difficulty of having some of the bolt ends project farther
through some nuts than others on a given piece of work, as is liable to
occur where the flanges to be bolted together are not turned on all four
radial faces, the form of nut shown in Fig. 407 is sometimes employed,
the thread in the nut extending beyond the bolt end.

[Illustration: Fig. 407.]

[Illustration: Fig. 408.]

As an example of the application of this nut, suppose a cylinder cover
to be held by bolts, then the cylinder flange not being turned on its
back face is usually of unequal thickness; hence to have the bolt ends
project equally through the nuts, each bolt would require to be made of
a length to suit a particular hole, and this would demand that each hole
and bolt be marked so that they may be replaced when taken out, without
trying them in their places. Another application of this nut is to make
a joint where the threads may be apt to leak. In this case the mouth of
the hole is recessed and coned at the edge; the nut is chamfered off
with a similar cone, and a washer W, Fig. 408, is placed beneath the nut
to compress and conform to the coned recess; thus with the aid of a
cement of some kind, as red or white lead (usually red lead), a tight
joint may be made independent of the fit of the threads.

When the hole through which the bolt passes is considerably larger in
diameter than the bolt, the flange nut shown in Fig. 409 is employed,
the flange covering the hole. A detached washer may be used for the same
purpose, providing that its hole fit the bolt and it be of a sufficient
thickness to withstand the pressure and not bend or sink into the hole.

[Illustration: Fig. 409.]

[Illustration: Fig. 410.]

Circular nuts are employed where, on account of their rotating at high
speed, it is necessary that they be balanced as nearly as possible so as
not to generate unbalanced centrifugal force. Fig. 410 represents a nut
of this kind: two diametrically opposite flat sides, as A, affording a
hold for the wrench. Other forms of circular nuts are shown in Figs. 411
and 412. These are employed where the nuts are not subject to great
strain, and where lightness is an object.

[Illustration: Fig. 411.]

[Illustration: Fig. 412.]

That in Fig. 411 is pierced around its circumference with cylindrical
holes, as A, B, C, to receive a round lever or rod or a wrench, such as
shown in Fig. 459.

That shown in Fig. 412 has slots instead of holes in its circumference,
and the form of its wrench is shown in Fig. 461.

When nuts are employed upon bolts in which the strain of the duty is
longitudinal to the bolt, and especially if the direction of motion is
periodically reversed, and also when a bolt is subject to shocks or
vibrations, a single nut is liable to become loose upon the bolt, and a
second nut, termed a check nut, jamb nut, or safety nut, becomes
necessary, because it is found that if two nuts be employed, as in Fig.
413, and the second nut be screwed firmly home against the first, they
are much less liable to come loose on the bolt.

Considerable difference of practice exists in relation to the thickness
of the two nuts when a check nut is employed. The first or ordinary nut
is screwed home, and the second or check nut is then screwed home. If
the second nut is screwed home as firmly as the first, it is obvious
that the strain will fall mainly on the second. If it be screwed home
more firmly than the first, the latter may be theoretically considered
to be relieved entirely of the strain, while if it be screwed less
firmly home, the first will be relieved to a proportionate degree of the
strain. It is usual to screw the second home with the same force as
applied to the first, and it would, therefore, appear that the first
nut, being relieved of strain, need not be so thick as the first, but it
is to be considered that, practically, the first nut will always have
some contact with the bolt threads, because from the imperfections in
the threads of ordinary bolts the area and the force of contact is not
usually the same nor in the same direction in both nuts, unless both
nuts were tapped with the same tap and at about the same time.

When, for example, a tap is put into the tapping machine, it is at its
normal temperature, and of a diameter due to that temperature, but as
its work proceeds its temperature increases, notwithstanding that it may
be freely supplied with oil, because the oil cannot, over the limited
area of the tap, carry off all the heat generated by the cutting of a
tap rotated at the speeds usually employed in practice. As a result of
this increase of temperature, we have a corresponding increase in the
diameter of the tap, and a variation in the diameter of the threads in
the nuts. The variation in the nuts, however, is less than that in the
tap diameter, because as the heated tap passes through the nut it
imparts some of its heat to the nut, causing it also to expand, and
hence to contract in cooling after it has been tapped, and, therefore,
when cold, to be of a diameter nearer to that of the tap.

Furthermore, as the tap becomes heated it expands in length, and its
pitch increases, hence here is another influence tending to cause the
pitches of the nut threads to vary, because although the temperature of
the tap when in constant use reaches a limit beyond which, so long as
its speed of rotation is constant, it never proceeds; yet, when the tap
is taken from the machine to remove the tapped nuts which have collected
on its shank, and it is cooled in the oil to prevent it from becoming
heated any more than necessary, the pitch as well as the diameter of the
tap is reduced nearer to its normal standard.

[Illustration: Fig. 413.]

[Illustration: Fig. 414.]

So far, then, as theoretical correctness, either of pitch or diameter in
nut threads, is concerned, it could only be attained (supposing that the
errors induced by hardening the tap could be eliminated) by employing
the taps at a speed of rotation sufficiently slow to give the oil time
to carry off all the heat generated by the cutting process. But this
would require a speed so comparatively slow as not to be commercially
practicable, unless followed by all manufacturers. Practically, however,
it may be considered that if two nuts be tapped by a tap that has become
warmed by use, they will be of the same diameter and pitch, and should,
therefore, have an equal area and nature of contact with the bolt
thread, supposing that the bolt thread itself is of equal and uniform
pitch. But the dies which cut the thread upon the bolt also become
heated and expanded in pitch. But if the temperature of the dies be the
same as that of the tap, the pitches on both the bolt and in the nut
will correspond, though neither may be theoretically true to the
designated standard.

In some machines for nut tapping the tap is submerged in oil, and thus
the error due to variations of temperature is practically eliminated,
though even in this case the temperature of the oil will gradually
increase, but not sufficiently to be of practical moment.

Let it now be noted that from the hardening process the taps shrink in
length and become of finer pitch, while the dies expand and become of
coarser pitch, and that this alone precludes the possibility of having
the nut threads fit perfectly to those on the bolt. It becomes apparent,
then, that only by cutting the threads in the lathe, and with a
single-toothed lathe tool that can be ground to correct angle after
hardening, can a bolt and nut be theoretically or accurately threaded.
Under skilful operation, however, both in the manufacture of the
screw-cutting tools and in their operation, a degree of accuracy can be
obtained in tapped nuts and die-threaded bolts that is sufficient with a
single nut for ordinary uses, but in situations in which the direction
of pressure on the nut is periodically reversed, or in which it is
subject to shocks or vibrations, the check nut becomes necessary, as
before stated.

An excellent method of preventing a nut from slackening back of itself
is shown in the safety nut in Fig. 414; it consists of a second nut
having a finer thread than the first one, so that the motion of the
first would in unscrewing exceed that of the second, hence the locking
is effectually secured.

[Illustration: Fig. 415.]

Work may be very securely fastened together by the employment of what
are called differential screws, the principle of whose action may be
explained with reference to Fig. 415, which is extracted from
"Mechanics." It represents a piston head and piston rod secured together
by means of a differential screw nut. The nut contains an internal
thread to screw on the rod, and an external one to screw into the piston
head, but the internal thread and that on the rod differ from the
external one, and that in the head by a certain amount, as say one tenth
of the pitch. The nut itself is furnished with a hexagonal head, and
when screwed into place draws the two parts together with the same power
as a screw having a pitch equal to the difference between the two
pitches.

[Illustration: Fig. 416.]

When putting the parts together the nut is first screwed upon the rod B.
The outside threads are then entered into the thread in the piston C,
and by means of a suitable wrench the nut is screwed into the proper
depth. As shown in the engraving, the nut goes on to the rod a couple of
threads before it is entered in the piston. The tightening then takes
place precisely as though the nut had a solid bearing on the piston and
a fine thread on the rod, the pitch of which is equal to the difference
between the pitches of the two threads. Fig. 416 shows its application
to the securing of a pump plunger upon the end of a piston-rod. In this
case, as the rod does not pass through the nut, the latter is provided
with a cap, which covers the end of the rod entirely.

[Illustration: Fig. 417.]

The principle of the differential screw may be employed to effect very
fine adjustments in place of using a very fine thread, which would soon
wear out or wear loose. Thus in Fig. 417 is shown the differential foot
screws employed to level astronomical instruments. C D is a foot of the
instrument to be levelled. It is threaded to receive screw A, which is
in turn threaded to receive the screw B, whose foot rests in the recess
or cup in E F. Suppose the pitch of screw A is 30 per inch, and that of
B is 40, and we have as follows. If A and B are turned together the foot
C D is moved the amount due to the pitch of A. If B is turned within a
the foot is moved the amount due to the pitch of B. If A is turned the
friction of the foot of B will hold B stationary, and the motion of C D
will equal the difference between the pitches of the threads of A and B.
Thus one revolution of A forward causes it to descend through C D 1/30
inch (its pitch), tending to raise C D 1/30 inch. But while doing this
it has screwed down upon the thread of B 1/40 inch (the pitch of B) and
this tends to lower C D, hence C D is moved 1/120 inch, because
1/30-1/40 = 1/120.

To cause a single nut to lock itself and dispense with the second or
jamb nut, various expedients have been employed. Thus in Fig. 418 is
shown a nut split on one side; after being threaded the split is closed
by hammer blows, appearing as shown in the detached nut. Upon screwing
the nut upon the bolt the latter forces the split nut open again by
thread pressure, and this pressure locks the nut. Now there will be
considerable elasticity in the nut, so that if the thread compresses on
its bearing area, this elasticity will take up the wear or compression
and still cause the threads to bind. Sometimes a set screw is added to
the split, as in Fig. 419, in which case the split need not be closed
with the hammer.

Another method is to split the nut across the end as shown in Fig. 420,
tapping the nut with the split open, then closing the split by hammer
blows. Here as before the nut would pass easily upon the bolt until the
bolt reached the split, when the subsequent threads would bind. In yet
another design, shown in Fig. 421, four splits are made across the end,
while the face of the nut is hollowed, so that a flat place near each
corner meets the work surface. The pressure induced on these corners by
screwing the nut home is relied on in this case to spring the nut,
causing the thread at the split end to close upon and grip the bolt
thread.

Check nuts are sometimes employed to lock in position a screw that is
screwed into the work, thus screws that require to be operated to effect
an adjustment of length (as in the case of eccentric rods and eccentric
straps) are supplied with a check nut, the object being to firmly lock
the screw in its adjusted position.

The following are forms of nuts employed to effect end adjustments of
length, or to prevent end motion in spindles or shafts that rotate in
bearings.

Fig. 422 shows two cylindrical check nuts, the inner one forming a
flange for the bearing. The objection to this is that in screwing up the
check nut the adjustment of the first nut is liable to become altered in
screwing up the second one, notwithstanding that the first be held by a
lever or wrench while the second is screwed home.

Another method is to insert a threaded feather in the adjustment nut and
having at its back a set screw to hold the nut in its adjusted position,
as in Fig. 423. In this case the protruding head of the set screw is
objectionable. In place of the feather the thread of the spindle may be
turned off and a simple set screw employed, as in Fig. 424; here again,
however, the projecting set screw head is objectionable. The grip of an
adjustment nut may be increased by splitting it and using a pinching or
binding screw, as in Fig. 425, in which case the bore of the thread is
closed by the screw, and the nut may be countersunk to obviate the
objection of a projecting head. For adjusting the length of rods or
spindles a split nut with binding screws, such as shown in Fig. 426, is
an excellent and substantial device. The bore is threaded with a
right-hand thread at one end and a left-hand one at the other, so that
by rotating the nut the rod is lengthened or shortened according to the
direction of rod rotation. Obviously a clamp nut of this class, but
intended to take up lost motion or effect end adjustment, may be formed
as in Fig. 427, but the projecting ears or screw are objectionable.

[Illustration: Fig. 429.]

Where there is sufficient length to admit it an adjustment nut, such as
in Fig. 428, is a substantial arrangement. The nut A is threaded on the
spindle and has a taper threaded split nut to receive the nut B. Nut A
effects the end adjustment by screwing upon the spindle, and is
additionally locked thereon by screwing B up the taper split nut,
causing it to close upon and grip the spindle.

[Illustration: Fig. 430.]

Lost motion in square threads and nuts may be taken up by forming the
nut in two halves, A and B, in Fig. 429 (A being shown in section) and
securing them together by the screws C C. The lost motion is taken up by
letting the two halves together by filing away the joint face D of
either half, causing the thread in the nut to bear against one side only
of the thread of the screw. The same end may be accomplished in nuts for
[V]-shaped threads by forming the nut either in two halves, as shown in
Fig. 430, in which A is a cap secured by screws B, the joint face C
being filed away to take up the lost motion. Or the nut may be in one
piece with the joint C left open, the screws B crossing the nut upon the
screw by pressure. In this case the nut closes upon the circumference of
the thread, taking up the wear by closing upon both sides of the thread
instead of on one side only as in the case of the square thread.

[Illustration: Fig. 431.]

[Illustration: Fig. 432.]

[Illustration: Fig. 433.]

In cases where nuts are placed under rapid vibration or motion they are
sometimes detained in their places by pins or cotters. The simplest form
of pin used for this purpose is the split pin, shown in Fig. 431. It is
made from half round wire and is parallel, and does not, therefore,
possess the capability of being tightened when the nut has become
loosened from wear. As the wire from which these pins are made is not
usually a full half circle the pins should, if the best results are to
be obtained, be filed to fit the hole, and in doing this, care should be
taken to have the pin bear fully in the direction of the split which is
longitudinal to the bolt, as shown in Fig. 432, where the pin is shown
with its ends opened out as is required to prevent the pin from coming
out. If the pin bears in a direction across the bolt as at A D, in Fig.
433, it will soon become loose.

[Illustration: _VOL. I._ =END-ADJUSTMENT AND LOCKING DEVICES.= _PLATE
IV_.

Fig. 418.

Fig. 419.

Fig. 420.

Fig. 421.

Fig. 422.

Fig. 423.

Fig. 424.

Fig. 425.

Fig. 426.

Fig. 427.

Fig. 428.]

Pins of this class are sometimes passed through the nut itself as well
as through the bolt; but when this is the case, there is the objection
that the nut cannot be screwed up to take up any wear, because in that
case the hole in the nut would not come fair with that in the bolt, and
the pin could not be inserted. When, therefore, such a pin passes
through the nut, lost motion must be taken up by placing an additional
or a thicker washer behind the nut. The efficiency of this pin as a
locking device is much increased by passing it through the nut, because
its bearing, and, therefore, wearing area, is increased, and the pin is
prevented from bending after the manner shown in Fig. 434, as it is apt
to do under excessive wear, with the result that the end pressure of the
nut almost shears or severs the pin close to the perimeter of the bolt.

[Illustration: Fig. 434.]

[Illustration: Fig. 435.]

To enable the pin to take up the wear, it is a good plan to file on it a
flat place, which must be parallel to the sides of the pin-head and
placed against the nut-face. The hole in the bolt is in this case made
to fall slightly under the nut, as in Fig. 435, so that the flat place
is necessary to enable the pin to enter. By filing the flat place taper,
the lost motion that may ensue from wear may be taken up by simply
driving the pin in farther.

[Illustration: Fig. 436.]

[Illustration: Fig. 437.]

In place of this class of split pin, solid taper pins are sometimes
used, but these, if employed in situations where they are subject to jar
and vibration, are apt sometimes to come loose, especially if they be
given much taper, because in that case they do not wedge so tightly in
the hole. But if a taper pin be made too nearly parallel, it will drive
through too easily, and has less capability to take up the play due to
wear. An ordinary degree of taper is about 5/8 inch per foot of length,
but in long pins having ample bearing area, 1/2 inch per foot of length
is ample. To prevent taper pins from coming loose from vibration, they
are sometimes forged split at the small end, as in Fig. 436, and opened
out at that end after the manner shown in Fig. 432. This forms a very
secure locking device, and one easily applied. The split ends are closed
by hammer blows to remove the pin, and it is found that such pins may be
opened and closed many times without breaking, even though made of cast
steel. The heads and ends are rounded so as to prevent them from
swelling from the hammer blows necessary to drive them in and out. When
a taper pin is passed through a nut and bolt, it simply serves as a
locking device to secure the nut in position, and the lost motion due to
wear must be taken up by the application of a washer beneath the nut, as
already described. If, however, the taper pin be applied outside the
nut, it may be made to take up the wear, by filing on it a flat place,
and locating the hole in the bolt so that it will fall partly beneath
the nut, as shown in Fig. 435. In this case, the nut may be screwed up
to take up the wear, and the pin by being driven farther in will still
bear against the nut and prevent its slacking back.

Another and excellent locking device for bolts or nuts, is the cotter
shown in Fig. 437, which is sometimes forged solid and sometimes split,
as in the figure. By being made taper from A to B, it will take up the
wear if driven farther in. Its width gives it strength in the direction
in which it acts to lock, the overhanging head is to drive it out by,
and the bevelled corner C is to enable its easy insertion, because if
left sharp it would be liable to catch against the edge of the
cotter-way and burr up. If made split, its ends are opened out after it
is inserted, as shown at D. When closing the ends of either split
cotters or split pins to extract them it is better to close one side
first and bend it over a trifle too much, so that, when closing the
other side, by the time the pin is straightened the two ends will be
closed together, and extraction becomes easy.

[Illustration: Fig. 438.]

A very safe method in the case of a single nut or bolt head is to
provide a separate plate, as in Fig. 438. The plate P is provided with
three sides, corresponding to the sides of the hexagon, as shown, and in
the middle of these sides are cut the notches A B C, so that by giving
the nut N one-twelfth of a turn its corners D E would be held by the
notches B C, S being a small screw to hold P. It is obvious that a
simple set screw passed through the walls of the nut would grip the bolt
thread and serve to hold the nut, but this would damage the bolt thread,
and, furthermore, that thread would under jar or vibration compress and
let the set screw come loose.

A better plan than this is to provide a thick washer beneath the nut and
let a set screw pass through the washer and grip the bolt, fastening or
setting up the set screw after the nut is screwed home. This, however,
makes the washer a gripping piece and in no wise serves to lock the nut.
In addition to the washer a pin may project through the radial face of
the washer and into the work surface, which will prevent, in connection
with the set screw, both the bolt and the washer from turning.

When a bolt has no thread but is secured by a taper pin, set screw,
cotter, or device other than a nut, it is termed a pin. So, likewise, a
cylindrical piece serving as a pivot, or to hold two pieces together and
having no head, is termed a pin.

The usual method of securing a pin is by a set screw or by a taper pin
and a washer; and since the term pin applying to both may lead to
misunderstanding, the term bolt will here be applied to the large and
the term pin to the small or securing pin only.

The object of pins and washers is to secure an exact degree of fit and
permit of rapid connection or disconnection. An application of a taper
pin and washer to a double eye is shown in Fig. 439. It is obvious, in
this case, the pin E will drive home until it fills the hole through the
bolt, and hence always to the same spot, so that the parts may be taken
apart and put together again rapidly, while the fit is self-adjusting,
providing that the pin fills the hole, bears upon the groove in the
washer, and is driven home, so that by first letting the pin bind the
washer W slightly too tight, and then filing the radial faces of the
joint to a proper fit (which will ease the bearing of the pin on the
washer), an exact degree of fit and great accuracy may be obtained,
whereas when a nut is used it is difficult to bring the nut to the exact
same position when screwing it home. When the joints are to be thus
fitted, it is a good plan to drill the pin-hole (through the bolt) so
that its centre falls coincident with the face of the washer; to then
file out the grooves in the washer not quite deep enough. The pin may
then be filed to fit the hole through the bolt, but left slightly too
large, so that it shall not pass quite far enough through the bolt. The
joint faces may then be filed true, and when finished, the parts may be
put together, and the groove through the washer and hole through the
bolt may be simultaneously finished by reaming with a taper reamer. This
will leave the job a good fit, with a full bearing, without much
trouble, the final reaming letting the taper pin pass to its proper
distance through the bolt.

[Illustration: Fig. 439.]

[Illustration: Fig. 440.]

Taper pins are sometimes employed to secure in position a bolt that
rotates, or one that requires locking in position, in situations in
which there is no room for the bolt end to project and receive a nut or
washer. Examples of these kinds are shown in section in Figs. 440 and
441. In 441, B is a stud pin, to rotate in the bore of A. C is a semi
circular groove in B, and P a taper pin entering one-half in the groove
C and one-half in B, thus preventing B from moving endwise in A, while
at the same time permitting its free rotation. In this case it is best
to fit B to its place, a fit tight enough to hold it firmly while the
pin-hole is drilled and reamed through A and B simultaneously, then B
can be put in the lathe, and the groove cut in to coincide with the
half-hole or groove caused in the pin by the drilling, and after the
groove is turned the stud pin may be eased to the required degree of
working fit. The process for Fig. 440 is precisely the same, except that
no groove turning or easing of the pin will be necessary, because the
pin being locked in position may be left a tight fit. If, however, it is
considered desirable to give the taper pin in Fig. 440 a little draft,
so that any looseness (that may occur to the pin or stud) from wear may
be taken up, then after the taper pin-hole has been drilled and reamed,
the pin or stud (D in the figure) may be taken out, and its taper
pin-hole _in the arm E_ may be filed out all the way through on one
side, as denoted by the dotted half-circle. This will give draft to the
pin and allow it to drive farther through and grip the pin as it wears
smaller.

If a bolt and nut fit too tightly in their threads the nut may be wound
back and forth upon the bolt under free lubrication, which will ease the
fit by wearing away or compressing that part of the thread surface that
is in contact. If this should not suffice we may generally ease a nut
that fits so tight that it cannot be screwed upon the bolt with an
ordinary wrench, by screwing the nut on a thread or two, then rest it on
an iron block, and lightly hammer its sides; it will loosen its fit, and
if continued, the nut may be made to pass down the bolt comparatively
easily. Now, in this operation, it is not that the nut has been
stretched, but that the points of contact on the threads have become
compressed and imbedded; we have, in other words, caused the shape of
each thread to conform nearer to that of the other than it is
practicable to make them, because of reasons explained in the remarks on
screw threads, and on taps.

[Illustration: Fig. 441.]

To remove nuts or bolts that have become corroded in their places, we
may adopt the following methods:--

If the nuts are so corroded that they will not unscrew with an ordinary
wrench, we may, if the standing bolts and the wrench are strong enough
to stand it, place a piece of gas or other pipe on the end of the
wrench, so as to get a longer leverage; and, while applying the power to
the wrench, we may strike the end face of the nut a few sharp blows with
the hammer, interposing a set chisel, if the nut is a small one, so as
to be sure to strike the nut in the proper place, and not rivet the
screw end. If the joint is made with tap bolts we may strike the bolt
heads with the hammer direct, using as before a light hammer and sharp
blows, which will, in a majority of cases, start the thread, after which
the wrench alone will usually suffice to unscrew it. If, however, this
is not effective, we should take a thick washer, large enough in its
bore to pass over the nut, and heat it to a yellow heat and place it
over the nut, and the nut heating more rapidly than the stud or standing
bolt, will be proportionately expanded and loosened; and, furthermore,
the iron becomes stronger by being heated, providing the temperature
does not exceed about 400°. If standing bolts or studs are employed on
the joint, the heating is still advantageous, for the increase of
strength more than compensates for the expansion. In this case the
heating, however, may be performed more slowly, so that the hole may
also become heated, and the bolt, therefore, not made a tighter fit by
its excessive expansion. So also, in taking out the standing bolts or
studs, heating them will often enable one to extract them without
breaking them off in the hole, which would necessitate drilling out the
broken piece or part. If, however, this should become necessary, we may
drill a hole a little smaller than the diameter of the bottom of the
bolt thread, and then drive into the hole a taper square reamer, as
shown in Fig. 442, in which W represents the work, R the square reamer,
and S the drilled screw end, and then, with a wrench applied to the
reamer, unscrew the bolt thread. If this plan fails there is no
alternative, after drilling the hole, but to take a round-nosed cape or
cross-cut chisel and cut out the screw as nearly as possible, then pick
out the thread at the entrance of the hole, and insert a plug tap to cut
out the remaining bolt thread.

To take out a standing bolt, take two nuts and screw them on the bolt
end; then hold the outer one still with a wrench and unscrew the inner
one tightly against it. We may then remove the wrench from the outer or
top nut, and unscrew the bolt by a wrench applied to the bottom or inner
one. If the thread of a standing bolt has become damaged or burred, we
can easily correct the evil by screwing a solid die or die nut down it,
applying a little oil to preserve the cutting edge of the nut. If it is
found impossible to take off a corroded nut without twisting off the
standing bolt, it is the better plan to sacrifice the nut in order to
save the bolt; and we may first hold a hammer beneath the nut, and take
a cold chisel, and holding it so that the cutting edge stands parallel
with the chamfered edge of the nut, and slanting it at an angle obtuse
to the direction in which the nut in unscrewing would travel, strike it
a few sharp blows, using a light hand-hammer; and this will often start
it, especially if the nut is heated as before directed. The hammer held
beneath the nut should be a heavy one, and should be pressed firmly
against the square or hexagon side of the nut, the object being to
support it, and thus prevent the standing bolt from bending or breaking,
as it would otherwise be very apt to do. If this plan succeeds, the nut
may, for rough work, be used over again, the burr raised by the chisel
head being hammered down to close it as much as possible before filing
it off. By holding the chisel precisely as directed, the seating of the
nut acts to support it, and thus aids the heavy hammer in its duty. If
this procedure fails we may cut the nut off, and thus preserve the bolt.

[Illustration: Fig. 442.]

To do this, we must use a cross-cut or cape chisel, and cut a groove
from the end face to the seating of the nut--a narrow groove will do,
and two may be cut if necessary; light cuts should be taken, and the
chisel should be ground at a keen angle, so that it will keep to its cut
when held at an angle, as nearly parallel to the centre line of the
length of the bolt as possible, in which case the force of the blows
delivered upon the chisel head will be in a direction not so liable to
bend the bolt. The groove or grooves should be cut down nearly to the
tops of the bolt threads, and then a wrench will unscrew the nut or else
cause it to open if one, and break in halves, if two grooves were cut.

After the nuts are all taken off, we may take a hammer and two or three
wedges, or chisels (according to the size of the joint), and drive them
an equal distance into the joint, striking one chisel first, and the
diametrically opposite one next, and going over all the wedges to keep
an equal strain upon each. If the joint resists this method, we may take
a hammer and strike blows between the standing bolts on the outside
face, interposing a block of hard wood to prevent damage to the face,
and holding the wood so that the hammer strikes it endwise of the grain;
and this will, in most cases, loosen the material of which the joint is
made, and break the joint. If, however, the joint, after repeated
trials, still resists, we may employ the hammer without the
interposition of the wood, using a copper or lead hammer, if one is at
hand, so as not to cause damage to the face of the work. To facilitate
the entrance of the wedges, grooves should be cut in the joint of one
face, their widths being about an inch, and their depth 1/16 inch.

WASHERS.--Washers are placed upon bolts for the following purposes.
First, to provide a smooth seating for the nut in the case of rough
castings. Second, to prevent the nut corners from marking and marring
the surface of finished work. Thirdly, to give a neat finish, and in
some cases to increase the bearing area of the nut and provide an
elastic cushion to prevent the nut from loosening. Washers are usually
of wrought iron, except in the case of brass nuts, when the washers also
are of brass. The standard sizes adopted by the manufacturers in the
United States for wrought iron washers is given in the following
table:--


MANUFACTURERS' STANDARD LIST.

Adopted by "The Association of Bolt and Nut Manufacturers of the United
States," at their meeting in New York, December 11th, 1872.

  +-----------+---------+-------------+---------+
  | Diameter. | Size of |  Thickness  | Size of |
  |           |  Hole.  | Wire Gauge. |  Bolt.  |
  +-----------+---------+-------------+---------+
  |    1/2    |   1/4   |   No. 18    |    3/16 |
  |    5/8    |   5/16  |    "  16    |    1/4  |
  |    3/4    |   5/16  |    "  16    |    1/4  |
  |    7/8    |   3/8   |    "  16    |    5/16 |
  |  1        |   7/16  |    "  14    |    3/8  |
  +-----------+---------+-------------+---------+
  |  1-1/4    |   1/2   |    "  14    |    7/16 |
  |  1-3/8    |   9/16  |    "  12    |    1/2  |
  |  1-1/2    |   5/8   |    "  12    |    9/16 |
  +-----------+---------+-------------+---------+
  |  1-3/4    |   11/16 |    "  10    |    5/8  |
  |  2        |   13/16 |    "  10    |    3/4  |
  +-----------+---------+-------------+---------+
  |  2-1/4    |   15/16 |    "   9    |    7/8  |
  |  2-1/2    | 1-1/16  |    "   9    |  1      |
  |  2-3/4    | 1-1/4   |    "   9    |  1-1/8  |
  |  3        | 1-3/8   |    "   9    |  1-1/4  |
  |  3-1/2    | 1-1/2   |    "   9    |  1-3/8  |
  +-----------+---------+-------------+---------+

[Illustration: Fig. 443.]

[Illustration: Fig. 444.]

The various forms of wrenches employed to screw nuts home or to remove
them are represented in the following figures. Fig. 443 represents what
is known as a solid wrench, the width between the jaws a being an easy
fit to the nuts across the flats. The opening between the jaws being at
an angle to the body enables the wrench to be employed in a corner which
would be too confined to receive a wrench in which the handle stood in a
line with the jaws, because in that common form of wrench the position
of the jaws relative to the handle would be the same whether the wrench
be turned over or not, whereas with the jaws at an angle as in the
figure, the wrench may be applied to the nut, rotating it a certain
distance until its handle meet an abutting piece, flange, or other
obstruction, and then turned over and the jaw embracing the same two
sides of the nut the handle will be out of the way and may again operate
the nut.

[Illustration: Fig. 445.]

In some cases each end of the wrench is provided with jaws, those at one
end standing at the same angle but being on the opposite side of the
wrench.

The proper angle of the jaws to the centre line of the jaws may be
determined as follows:--The most desirable angle is that which will
enable the wrench to operate the nut with the least amount of
wrench-motion, an object that is of great importance in cases where an
opening has to be provided to admit the wrench to the nut, it being
desirable to leave this opening as small as possible so as to impair the
solidity of the work as little as practicable. For a hexagon nut this
angle may be shown to be one of 15°, as in Fig. 444.

[Illustration: Fig. 446.]

[Illustration: Fig. 447.]

In Fig. 445, for example, the wrench is shown in the position in which
it will just engage the nut, and at the first movement it will move the
nut to the position shown in Fig. 446. The wrench is then turned upside
down and placed upon the nut as in Fig. 447, and moved to the position
shown in Fig. 448, thus moving the nut the sixth part of a revolution,
and bringing it to a position corresponding to that in Fig. 445, except
that it has moved the nut around to a distance equal to one of its
sides. Since the wrench has been moved twice to move the nut this
distance, and since there are six sides, it will take twelve movements
to give the nut a full revolution, and, there being 360° in the circle,
each movement will move the nut 30°, or one-twelfth of 360°, and
one-half of this must be the angle of the gripping faces of the jaws to
the body of the wrench. The width of the opening in the work to admit
the wrench in such a case as in Fig. 445 must be not less than 30°, plus
the width of the wrench handle, at the radius of the outer corner of the
opening.

[Illustration: Fig. 448.]

In the case of wrenches for square nuts it is similarly obvious that
when the nut makes one-eighth of a revolution its sides will stand in
the same position to receive the wrench that the nut started from, and
in one-eighth of a revolution there are 45°. As the wrench is applied
twice to the same side of the nut, its jaws must stand at one half this
angle (or 22-1/2°) to the handle.

[Illustration: Fig. 449.]

When a nut is in such a position that it can only be operated upon from
the direction of and in a line with the axis of the bolt, a box wrench
such as shown in Fig. 449, is employed, the cavity at B fitting over
the bolt head; but if there is no room to admit the cross handle, a hub
or boss is employed instead, and this hub is pierced with four radial
holes into which the point of a round lever may be inserted to turn the
wrench. Adjustable wrenches that may be opened and closed to suit the
varying sizes of nuts are represented in Figs. 450, 451, and 452. In
Fig. 450, A is the fixed jaw solid upon the square or rectangular bar E,
and passing through the wooden handle D. B is a sliding jaw embracing E,
and operated thereon by the screw C, whose head is serrated to afford a
good finger grip. Various modifications of this form of wrench are made;
thus, for example, in Fig. 451 A is the jaw, B a slotted shank, C the
handle, all made in one piece. D is the movable jaw having a sleeve
extension D´, and recesses which permit the jaw to slide on the shank
longitudinally, but which prevent it from turning. The movable jaw is
run to and from the nut or bolt head to be turned, by means of the screw
G.

[Illustration: Fig. 450.]

[Illustration: Fig. 451.]

In another class of adjustable wrench the jaws slide one within the
other; thus in Fig. 452, the fixed jaw of the wrench forms a part of the
handle, and is hollowed out and slotted to receive the stem of the loose
jaw, which plays therein, being guided by ribs in the slot, which take
into grooves in the stem of the loose jaw. A screw with a milled head
and a grooved neck serves to propel the loose jaw, being stopped from
moving longitudinally by a partly open fixed collar on the fixed jaw,
which admits the screw and engages the grooved neck of the same. The
threaded extremity of the screw engages a female screw in the loose jaw,
and while the same are engaged the screw cannot be released from the
embrace of the fixed collar, as it requires considerable lateral
movement to accomplish this.

[Illustration: Fig. 452.]

[Illustration: Fig. 453.]

Adjustable wrenches are not suited for heavy work because the jaws are
liable to spring open under heavy pressure and thus cause damage to the
edges of finished nuts, and indeed these wrenches are not suitable for
ordinary use on finely finished work unless the duty be light.
Furthermore, the jaws being of larger size than the jaws of solid
wrenches, will not pass so readily into corners, as may be seen from the
[S] wrench shown in Fig. 453. In the adjustable [S] wrench in Fig. 454,
each half is provided with a groove at one end and a tongue in the
other, so that when put together the tongues are detained in the
grooves. To open or close the wrench a right and left-hand screw is
tapped into the wrench as shown, the head being knurled or milled to
afford increased finger-grip.

[Illustration: Fig. 454.]

[Illustration: Fig. 455.]

In all wrenches the location of contact and of pressure on the nut is
mainly at the corners of the nut, and unless the wrench be a very close
fit, the nut corners become damaged. A common method of avoiding this is
to interpose between the wrench jaw and the nut a piece of soft metal,
as copper, sheet zinc, or even a piece of leather. The jaws of the
wrench are also formed to receive babbitt metal linings which may be
renewed as often as required. To save the trouble of adjusting an
accurately fitting wrench to the nut, Professor Sweet forms the jaws as
in Fig. 455, so that when moved in one direction the jaws will pass
around the nut without gripping it, but when moved in the opposite
direction the jaws will grip the nut but not damage the corners, while
to change the direction of a nut rotation it is simply necessary to turn
the wrench over.

[Illustration: Fig. 456.]

Fig. 456 represents a key wrench which is suitable for nuts of very
large size. The sliding jaw J is held by the key or wedge S, which is
operated by hammer blows. The projection at R is necessary to give
sufficient bearing to the sliding jaw.

[Illustration: Fig. 457.]

For use in confined places where but little handle-motion is obtainable,
the ratchet wrench is employed, consisting of a lever affording journal
bearing to a socket that fits the head of the bolt. The socket is
provided with a ratchet or toothed wheel in which a catch or pawl
engages. Fig. 457 represents the Lowell Wrench Company's ratchet wrench
in which a lag screw socket is shown affixed. The socket is removable so
that various sizes and shapes may be used with the same wrench. Each
socket takes two sizes of square and one of hexagon heads or nuts. So
long as the screw runs easily, it can be turned by the wooden handle
more conveniently and faster than by the fingers, and independently of
the ratchet motion. When this can no longer be done with ease, the
twelve-inch handle is brought into use to turn the screw home.

For carriage bolts used in woodwork that turn with the nut
notwithstanding the square under the head (as they are apt to do from
decay of the wood or from the bolt gradually working loose) the form of
wrench shown in Fig. 458 is exceedingly useful, it is driven into the
wood by hammer blows at A. The bevelled edges cause the jaws to close
upon the head in addition to the handle-pressure.

[Illustration: Fig. 458.]

For circular nuts such as was shown in Fig. 411, the pin wrench or
spanner wrench shown in Fig. 459 is employed, the pin P fitting into the
holes in the nut circumference. The pin P should be parallel and slope
very slightly in the direction of A, so that it may not meet and bruise
the mouths of the pin-holes, A, B, C. The pin must, of course, pass
easily into the pin-holes, and would, if vertical, therefore meet the
edge of the hole at the top, bruising it and causing the wrench to
spring or slip out, as would be the case if the pin stood in the
direction of B.

[Illustration: Fig. 459.]

[Illustration: Fig. 460.]

It is obvious that to reverse the motion of the nut it is necessary to
reverse the position of the wrench, because the handle end must, to
enable the wrench to grip the work, travel in advance of the pin end. To
avoid this necessity Professor Sweet forms the wrench as in Fig. 460, in
which case it can operate on the nut in either direction without being
reversed.

When a circular nut has its circumference provided with notches as was
shown in Fig. 412 the wrench is provided with a rectangular piece as
shown in Fig. 461. This piece should slope in the direction of a for the
reasons already explained with reference to the cylindrical pin in Fig.
459. It is obvious, however, that this wrench also may be made upon
Professor Sweet's plan, in which case the pin should be straight.

[Illustration: Fig. 461.]

KEYS AND KEYWAYS.--Keys and keyways are employed for two purposes--for
locking permanently in a fixed position, and for locking and adjusting
at the same time. Keys that simply permanently lock are usually simply
embedded in the work, while those that adjust the parts and secure them
in their adjusted position usually pass entirely through the work. The
first are termed sunk keys and keyways, the latter adjusting keys and
through keyways.

[Illustration: Fig. 462.]

The usual forms of sunk keyways are as follows:--Fig. 462 represents the
common sunk key, the head _h_ forming a gib for use in extracting the
key, which is done by driving a wedge between the head and the hub of
the work.

[Illustration: Fig. 463.]

The flat key, sunk key, and feather shown in Fig. 463, are alike of
rectangular form, their differences being in their respective
thicknesses, which is varied to meet the form of key way which receives
them. The flat key beds upon a flat place upon the shaft, the sunk key
beds in a recess provided in the shaft, and the feather is fastened
permanently in position in the shaft. The hollow key is employed in
places where the wheel or pulley may require moving occasionally on the
shaft, and it is undesirable that the latter have any flat place upon it
or recess cut in it. The flat key is used where it is necessary to
secure the wheel more firmly without weakening the shaft by cutting a
keyway in it. The sunk key is that most commonly used; it is employed in
all cases where the strain upon the parts is great. The feather is used
in cases where the keyway extends along the shaft beyond the pulley or
wheel, the feather being fast in the wheel, and its protruding part a
working fit in the shaft keyway. This permits the wheel to be moved
along the shaft while being driven through the medium of the feather
along the keyway or spline. The heads of the taper keys are sometimes
provided with a set screw as in Fig. 464, which may be screwed in to
assist in extracting the key.

[Illustration: Fig. 464.]

[Illustration: Fig. 465.]

Fig. 465 represents an application of keys to a square shaft that has
not been planed true. The wheel is hung upon the shaft and four
temporary gib-headed keys are inserted in the spaces _a_, _a_, _a_, _a_,
in Fig. 465. (It may be mentioned here that similar heads are generally
forged upon keys to facilitate their withdrawal while fitting them to
their seats, the heads being cut off after the key is finally driven
home.) These sustain the wheel while the permanent keys, eight in
number, as shown in the figure at _b_, _b_, _b_, _b_, _b_, _b_, _b_,
_b_, are fitted, the wheel being rotated and tested for truth from a
fixed point, the fitting of the keys being made subservient to making
the wheel run true.

The proportions of sunk keys are thus given by the Manchester (England)
rule. The key is square in cross section and its width or depth is
obtained by subtracting 1/2 from the diameter of the shaft and dividing
the sum thus obtained by 8, and then adding to the subtrahend 1/4.

Example.--A shaft is 6 inches in diameter, what should be the cross
section dimensions of its key diameter of shaft?

  6 - 1/2 = 5-1/2, 5-1/2 ÷ 8 = .687, and .687 + .25 = 937/1000 inch.

In general practice, however, the width of a key is made slightly
greater than its depth, and one-half its depth should be sunk in the
shaft.

[Illustration: Fig. 466.]

[Illustration: Fig. 467.]

Taper keys are tapered on their surfaces A and B in Fig. 466, and are
usually given 1/8-inch taper per foot of length. There is a tendency
either in a key or a set screw to force the hub out of true in the
direction of the arrow. It therefore causes the hub bore to grip the
shaft, and this gives a driving duty more efficient than the friction of
the key itself. But the sides also of the key being a sliding fit they
perform driving duty in the same manner as a feather which fits on the
sides A, D in Fig. 467, but are clear either top or bottom. In the
figure the feather is supposed to be fast in the hub and therefore free
at C, but were it fast in the shaft it would be free on the top face.

[Illustration: Fig. 468.]

[Illustration: Fig. 469.]

Fig. 468 represents a shaft held by a single set screw, the strain being
in the direction of the arrow, hence the driving duty is performed by
the end of the set screw and the opposite half circumference of the bore
and shaft. On account, however, of the small area of surface of the set
screw point the metal of the shaft is apt, under heavy duty and when the
direction of shaft rotation is periodically reversed, to compress (as
will also the set screw point unless it is of steel and hardened),
permitting the grip to become partly released no matter how tightly the
set screw be screwed home. On this account a taper key will under a
given amount of strain upon the hub perform more driving duty, because
the increased area of contact prevents compression. Furthermore, the
taper key will not become loose even though it suffer an equal amount of
compression. Suppose, for example, that a key be driven lightly to a
fair seating, then all the rest of the distance to which the key is
driven home causes the hub to stretch as it were, and even though the
metal of the key were to compress, the elasticity thus induced would
take up the compression, preventing the key from coming loose. It is
obvious, then, that set screws are suitable for light duty only, and
keys for either heavy or light duty. It is advanced by some authorities
that keys are more apt to cause a wheel or pulley to run out of true
than a set screw, but such is not the case, because, as shown in Figs.
466 and 468, both of them tend to throw the wheel out of true in one
direction; but a key may be made with proper fitting to cause a wheel to
run true that would not run true if held by a set screw, as is explained
in the directions for fitting keys given in examples in vice work.

If two set screws be used they should both be in the same line (parallel
to the shaft axis) or else at a right angle one to the other as in Fig.
469, so that the shaft and bore may drive by frictional contact on the
side opposite to the screws. Theoretically the contact of their surface
will be at a point only, but on account of the elasticity of the metal
the contact will spread around the bore in the arc of a circle, the
length of the arc depending upon the closeness of fit between the pulley
bore and the shaft. If the bore is a close fit to the shaft it is by
reason of the elasticity of the metal relieved of contact pressure on
the side on which the set screw or key is to an amount depending upon
the closeness of the bore fit, but this will not in a bore or driving
fit to the shaft be sufficient to set the wheel out of true.

If two set screws are placed diametrally opposite they will drive by the
contact of their ends only, and not by reason of their inducing
frictional contact between the bore and the shaft.

A very true method of securing a hub to a shaft is to bore it larger
than the shaft and to a taper of one inch to the foot. A bushing is then
bored to fit the shaft and turned to the same taper as the hub is
turned, but left, say, 1/100 inch larger in diameter and 1/4 or 3/8
longer. The bush is then cut into three pieces and these pieces are
driven in the same as keys, but care must be taken to drive them equally
to keep the hub true.

[Illustration: Fig. 470.]

[Illustration: Fig. 471.]

Feathers are used under the following conditions:--When the wheel driven
by a shaft requires to slide along the shaft during its rotation, in
which case the feather is fast in the wheel and the shaft is provided
with a keyway or spline (as it is termed when the sliding action takes
place), of the necessary length, the sides of the feather being a close
but sliding fit in the spline while fixed fast in the wheel.

It is obvious that the feather might extend along the shaft to the
requisite distance and the spline or keyway be made in the wheel: but in
this case the work is greater, because the shaft would still require
grooving to receive the feather, and the feather instead of being the
simple width of the wheel would require to be the width of the wheel
longer than the traverse of the wheel on the shaft. Nor would this
method be any more durable, because the keyway's bearing length would be
equal to the width of the wheel only.

When a feather is used to enable the easy movement of a wheel from one
position to another a set screw may be used to fix the wheel in position
through the medium of the feather as is shown in Fig. 470.

Through keys and keyways are employed to lock two pieces, and sometimes
to enable the taking up of the wear of the parts. Fig. 471 represents an
example in which the key is used to lock a taper shaft end into a socket
by means of a key passing through both of them. When the keyway is
completely filled by the key as in the figure it is termed a solid key
and keyway, indicating that there is no draft to the keyway. Fig. 472
represents a key and keyway having draft. One edge, A C, of the key
binds against the socket edges only, and the other edge E binds against
the edge B of the enveloped piece or plug, so that by driving in the key
with A hammer the two parts are forced together. The space or distance
between the edge D and the key, and between edges E and F, is termed the
draft. The amount of this draft is made equal to the taper of the key,
hence, when the key is driven in so that its head comes level with the
socket or work surface, the draft will be all taken up and the key will
fill the keyway.

[Illustration: Fig. 472.]

Draft is given to ensure all the strain of the key forcing the parts
together, to enable the key to be driven in to take up any wear and to
adjust movable parts, as straps, journal boxes or brasses, &c. When the
bore of the socket and the end of the rod are parallel, the end of the
rod F, Fig. 473, should key firmly against the end E of the socket,
while the end D of the socket should be clear of the shoulder on the
rod; otherwise instead of the key merely compressing the metal at F it
will exert a force tending to burst the end F from G of the rod,
furthermore, the area of contact at the shoulder D being small the metal
would be apt to compress and the key would soon come loose.

In some cases two keys are employed passing through a sleeve, the
arrangement being termed a coupling, or a butt coupling.

[Illustration: Fig. 473.]

The usual proportions for this class of key, when the rod ends and
socket boxes are parallel, is width of key equals diameter of socket
bore, thickness of key equals one-fourth its width, with a taper
edgeways of about 1/4 inch in 10 inches of length.

[Illustration: Fig. 474.]

[Illustration: Fig. 475.]

As the keys in through keyways often require to be driven in very tight,
and as the parts keyed together often remain a long time without being
taken apart and in some situations become rusted together, it is often a
difficult matter to get them apart. First, it is difficult to drive it
out because the blows swell the end of the key so that it cannot pass
through the keyway, and secondly, driving the socket off the plug of the
two parts keyed together often damages the socket and may bend the rod
to which it is keyed. Furthermore, as the diameter of the socket is
usually not more than half as much again as the diameter of the plug,
misdirected blows are apt to fall upon the rod instead of upon the
socket end and damage it. Hence, a piece of copper, of lead, or a block
of wood should always be placed against the socket end to receive the
hammer blows. To force a plug out of a socket, we may use reverse keys.
These are pieces formed as shown in Fig. 474. A, A and B, B are edge and
face views respectively of two pieces of metal, formed as shown, which
are inserted in the keyway as shown in Fig. 475, in which A is the plug
or taper end of a rod and B the socket, C is one and D the other of the
reverse keys, while E is a taper key inserted between them, B driving E
through the keyway, A and B are forced apart. The action of the reverse
keys is simply to reverse the direction of the draft in the keyway so
that the pressure due to driving E through the keyway is brought to bear
upon the rod end in the part that was previously the draft side of the
keyway, and in like manner upon the keyway in the socket on the side
that previously served as draft.

Reverse keys are especially serviceable to take off cross heads, piston
heads, keyed crank-pins, and parts that are keyed very firmly together.

[Illustration: Fig. 476.]

[Illustration: Fig. 477.]

Hubs are sometimes fastened to their shafts by pins passing through both
the hub and the shaft. These pieces may be made parallel or taper, but
the latter obviously secures the most firmly. If the pin is located as
in Fig. 476, its resisting strength is that due to its cross sectional
area at A and B. But if the pin be located as in Fig. 477 it secures the
hub more firmly, because it draws the bore (on the side opposite to the
pin) against the shaft, causing a certain amount of friction, and,
furthermore, the area resisting the pressure of the hub is increased,
and that pressure is to a certain degree in a crushing as well as a
shearing direction.

[Illustration: Fig. 478.]

If unturned pins are used and the holes are rough or drilled but not
reamed, it is better that two sides of the pin should be eased off with
a file or on the emery wheel, so that all the locking pressure of the
pin shall fall where it is the most important that it should--that is,
where it performs locking duty. This is shown in Fig. 478, the hole
being round and the pin being very slightly oval (not, of course, so
much as shown in the drawing), so that it will bind at A B, and just
escape touching at C, D, so that all the pressure of contact is in the
direction to bind the hub to the shaft.



CHAPTER VI.--THE LATHE.


The lathe may be justly termed the most important of all metal-cutting
machine tools. Not only on account of the rapidity of its execution
which is due to its cutting continuously while many others cut
intermittently, but also because of the great variety of the duty it
will perform to advantage. In the general operations of the lathe,
drilling, boring, reaming, and other processes corresponding to those
performed by the drilling machine, are executed, while many operations
usually performed by the planing machine, or planer as it is sometimes
termed, may be so efficiently performed by the lathe that it sometimes
becomes a matter of consideration whether the lathe or the planer is the
best machine to use for the purpose.

The forms of cutting tools employed in the planer, drilling machine,
shaping machine, and boring machine, are all to be found among lathe
tools, while the work-holding devices employed on lathe work include,
substantially, very nearly all those employed on all other machines and,
in addition, a great many that are peculiar to itself. In former times,
and in England even at the present day, an efficient turner (as a lathe
operator is termed), or lathe hand, is deemed capable of skilfully
operating a planer, boring machine, screw-cutting machine, drilling
machine, or any of the ordinary machine tools, whereas those who have
learned to operate any or all of those machine tools would prove
altogether inefficient if put to operate a lathe.

[Illustration: Fig. 479.]

In almost all the mechanic arts the lathe in some form or other is to be
found, varying in weight from the jewellers' lathe of a few pounds to
the pulley or fly-wheel lathe of the engine builder, weighing many tons.

The lathe is the oldest of machine tools and exists in a greater variety
of forms than any other machine tool. Fig. 479 represents a lathe of
primitive construction actually in use at the present day, and
concerning which the "Engineering" of London (England), says, "At the
Vienna Exhibition there were exhibited wood, glasses, bottles, vases,
&c., made by the Hucules, the remnant of an old Asiatic nation which had
settled at the time of the general migration of nations in the remotest
parts of Galicia, in the dense forests of the Carpathian Mountains. The
lathe they are using has been employed by them from time immemorial.
They make the cones _b_, _b_ (of maple) serve as centres, one being
fixed and the other movable (longitudinally). They rough out the work
with a hatchet, making one end _a_ cylindrical, to receive the rope for
giving rotary motion. The cross-bar _d_ is fastened to the trees so as
to form a rest for the cutting tool, which consists of a chisel." C, of
course, is the treadle, the lathe or pole being a sapling.

In other forms of ancient lathes a wooden frame was made to receive the
work-centres, and one of these centres was carried in a block capable of
adjustment along the frame to suit different lengths of work. In place
of a sapling a pole or lath was employed, and from this lath is probably
derived the term lathe.

It is obvious, however, that with such a lathe no cutting operation can
be performed while the work is rotating backwards, and further, that
during the period of rest of the cutting tool it is liable to move and
not meet the cut properly when the direction of work rotation is
reversed and cutting recommences, hence the operation is crude in the
extreme, being merely mentioned as a curiosity.

The various forms in which the lathe appears in ordinary machine shop
manipulation may be classified as follows:--

The _foot lathe_, signifying that the lathe is driven by foot.

The _hand lathe_, denoting that the cutting tools must be held in the
hands, there being no tool-carrying or feeding device on the lathe.

The _single-geared lathe_, signifying that it has no gear-wheels to
reduce the speed of rotation of the live spindle from that of the cone.

The _back-geared lathe_, in which gear-wheels at the back of the
headstock are employed to reduce the speed of the lathe.

The _self-acting lathe_, or _engine lathe_, implying that there is a
slide rest actuated automatically to traverse the tool to its cut or
feed.

The _screw-cutting lathe_, which is provided with a _lead_ screw, by
means of which other screws may be cut.

The _screw-cutting lathe with independent feed_, which denotes that the
lathe has two feed motions, one for cutting threads and another for
ordinary tool feeding; and

The _chucking lathe_, which implies that the lathe has a face plate of
larger diameter than usual, and that the bed is somewhat short, so as to
adapt it mainly to work held by being chucked, that is to say, held by
other means than between the lathe centres.

There are other special applications of the lathe, as the boring lathe,
the grinding lathe, the lathe for irregular forms, &c., &c.

This classification, however, merely indicates the nature of the lathe
with reference to the individual feature indicated in the title; thus,
although a foot lathe is one run by foot, yet it may be a single or
double gear (back-geared) lathe, or a hand or self-acting lathe, with
lead screw and independent feed motion.

Again, a hand lathe may have a hand slide rest, and in that case it may
also be a back-geared lathe, and a back-geared lathe may have a hand
slide rest or a self-acting feed motion or motions.

Fig. 480 represents a simple form of foot lathe. The office of the
shears or bed is to support the headstock and tailstock or tailblock,
and to hold them so that the axes of their respective spindles shall be
in line in whatever position the tailstock may be placed along the bed.
The duty of the headstock is to carry the live spindle, which is driven
by the cone, the latter being connected by the belt to the wheel upon
the crank shaft driven by the crank hook and the treadle, which are
pivoted by eyes W to the rod X, the operation of the treadle motion
being obvious. The work is shown to be carried between the live centre,
which is fitted to the live spindle, and the dead centre fitting into
the tail spindle, and as it has an arm at the end, it is shown to be
driven by a pin fixed in the face plate, this being the simplest method
of holding and driving work. The lathe is shown provided with a hand
tool rest, and in this case the cutting tools are supported upon the top
of the tool rest N, whose height may be adjusted to bring the tool edge
to the required height on the work by operating the set screw S, which
secures the stem of N in the bore of the rest.

To maintain the axes of the live and dead spindles in line, they are
fitted to a slide or guideway on the shears, the headstock being fixed
in position, while the tailstock is adjustable along the shears to suit
the length of the work.

To lock the tailstock in its adjusted position along the shears, it has
a bolt projecting down through the plate C, which bolt receives the hand
nut D. To secure the hand rest in position at any point along the
shears, it sets upon a plate A and receives a bolt whose head fits into
a [T]-shaped groove, and which, after passing through the plate P
receives the nut N, by which the rest is secured to the shears.

To adjust the end fit of the live spindle a bracket K receives an
adjusting screw L, whose coned end has a seat in the end J of the live
spindle, M being a check nut to secure L in its adjusted position.

[Illustration: Fig. 480.]

The sizes of lathes are designated in three ways, as follows:--First by
the _swing_ of the lathe and the total length of the bed, the term
_swing_ meaning the largest diameter of work that the lathe is capable
of revolving or swinging. The second is by the _height of the centres_
(from the nearest corner of the bed) and the length of the shears. The
height of the centres is obviously equal to half the swing of the lathe,
hence, for example, a lathe of 28-inch swing is the same size as one of
14-inch centres. The third method is by the swing or height of centres
and by the greatest length of work that can be held between the lathe
centres, which is equal to the length of the bed less the lengths of the
head and tailstock together.

The effective size of a lathe, however, may be measured in yet another
way, because since the hand rest or slide rest, as the case may be,
rests upon the shears or bed, therefore the full diameter of work that
the lathe will swing on the face plate cannot be held between the
centres on account of the height of the body of the hand rest or slide
rest above the shears.

Fig. 481 shows a hand lathe by F. E. Reed, of Worcester, Massachusetts,
the mechanism of the head and tail stock being shown by dotted lines.
The live spindle is hollow, so that if the work is to be made from a
piece of rod and held in any of the forms of chucks to be hereafter
described, it may be passed through the spindle, which saves cutting the
rod into short lengths. The front bearing of the headstock has two
brasses or boxes, A and B, set together by a cap C.

The rear bearing has also a bearing box, the lower half D being threaded
to receive an adjustment screw F and check nut G to adjust the end fit
of the spindle in its bearings. In place of grooved steps for the belt
the cone has flat ones to receive a flat belt.

The tail spindle is shown, in Fig. 482, to be operated by a screw H,
having journal bearing at I, and threaded into a nut fast in the tail
spindle at J. To hold the tail spindle firmly the end of the tail stock
is split, and the hand screw K may be screwed up to close the split and
cause the bore at L to clasp the tail spindle at that end.

To lock the tail stock to the shears the bolt M receives the lever N at
one end and at the other passes through the plate or clamp O, and
receives the nut P, so that the tail stock is gripped to or released
from the shears by operating N in the necessary direction. The hand
rest, Fig. 483, has a wheel W in place of a nut, which dispenses with
the use of a wrench.

What are termed bench lathes are those having very short legs, so that
they may for convenience be mounted on a bench or fastened to a second
frame, as shown in Fig. 484.

It is obvious that when work is turned by hand tools, the parallelism of
the work depends upon the amount of metal cut off at every part of its
length, which to obtain work of straight outline, whether parallel or
taper, involves a great deal of testing and considerable skill, and to
obviate these disadvantages various methods of carrying and accurately
guiding tools are employed. The simplest of these methods is by means of
a slide rest, such as shown in Fig. 485.

The tool T is carried in the tool post P, being secured therein by the
set screw shown, which at the same time locks the tool post to the upper
slider. This upper slider fits closely to the cross slide, and has a
nut projecting down into the slot shown in the same, and enveloping the
cross feed screw, whose handle is shown at C, so that operating C
traverses the upper slider on the cross slide and regulates the depth to
which the tool enters the work, or in other words, the depth of cut.

[Illustration: Fig. 481.]

The cross slide is formed on the top of the lower slider, which has
beneath a nut for the feed screw, whose handle is shown at A, hence
rotating A will cause the lower slider to traverse along the lower slide
and carry the tool along the work to its cut. To maintain the fit of the
sliders to the slides a slip of metal is inserted, as at _e_ and at _c_,
and these are set up by screws as at _f_, _f_ and _b_, _b_.

[Illustration: Fig. 482.]

The lower or feed traverse slide is pivoted to its base B, so that it
may be swung horizontally upon the same, and is provided with means to
secure it in its adjusted position, which is necessary to enable it to
turn taper as well as parallel work. To set this lower slide to a given
degree of angle it may be marked with a line and the edge of base B may
be divided into degrees as shown at D.

[Illustration: Fig. 483.]

When a piece of work is rotated between the lathe centres its axis of
rotation may be represented by an imaginary straight line and the lower
slides must, to obtain parallel work, be set parallel to this straight
line, while for taper work the slide rest must be set at an angle to it.
Now, in the form of slide rest shown in figure the cross slide is
carried by the lower or feed traverse slide, hence setting the lower
slide out of parallel with the work axis sets the cross slide out of a
right angle to the work axis, with the result that when a taper piece of
work is turned that has a collar or flange on it, the face of that
collar or flange will be turned not at a right angle to the work axis as
it should be, but at a right angle to the surface of the cone. Thus in
Fig. 486 A represents the axis of a piece of work, and the slide nut
having been set parallel to the work axis, the face C will be at a right
angle to the surface B or axis A, but with the slide nut set at an angle
to turn the cone D, the cross slide will be at an angle to A, hence the
face E will be undercut as shown, and at a right angle to the surface D
instead of to A A. This may be obviated by letting the cross slide be
the lower one as in the English form of slide rest shown in Fig. 487, in
which the upper slide is pivoted at its centre to the cross slide and
may be swung at an angle thereto and secured in its adjusted position by
the bolt at F. The projection at the bottom of the lower slider fits
between the shears of the lathe and holds the lower slider parallel with
the line of lathe centres, which causes the slide rest to cut all faces
at a sight angle to the work axis whether the feed traverse slide be set
to turn parallel or taper. In either case, however, there is nothing to
serve as a guide to set the feed traverse slide parallel to the work
axis, and this must, therefore, be done as near as may be by the eye and
by taking a cut and testing its parallelism.

[Illustration: Fig. 484.]

[Illustration: Fig. 485.]

[Illustration: Fig. 486.]

The rest may be set approximately true by bringing the operator's eye
into such a position that the edge _a_ _a_, Fig. 488, of the slide rest
come into line with the edge _b_ _b_ of the lathe shears, because that
edge is parallel to the line of lathe centres, and therefore to the work
axis.

Slide rests which have a slide for traversing the tool along the work to
its cut are but little used in the United States, being confined to very
small lathes, and then (except in the case of watchmakers' lathes whose
forms of slide rest will be shown hereafter), mainly as an expedient to
save expense in the cost of the lathe, it being preferred to feed the
tool for the feed traverse (as the motion of the cutting tool along the
work is termed) by mechanism operated from the live spindle and to be
hereafter described. In England, however, slide rests are much used, a
specimen construction being shown in Fig. 489. The end face A of the
rest comes flush so that the tool shall be carried firmly when taking
facing cuts in which solidity in the rest is of most importance. The
tool is held by two clamps instead of by single tool posts, because the
slide rest is employed to take heavy cuts, and when this is the case
with boring tools whose cutting edges stand far out from the slide rest,
a single tool post will not hold the tool sufficiently firm.

[Illustration: Fig. 487.]

[Illustration: Fig. 488.]

The gib _e_, Fig. 485, is sometimes placed on the front side of the
slider, as in the figure, and at others on the back; when it is placed
in the front the strain of the cut causes it to be compressed against
the slide, and there is a strain placed upon the screws _f_ which lifts
them up, whereas if placed on the other side the screws are relieved of
strain, save such as is caused by the setting of the gib up.

[Illustration: Fig. 489.]

On the other hand, the screws are easier to get at for adjustment if
placed in front. When the screws _b_ of the upper gib _c_, Fig. 485, are
on the right-hand side, as in that figure, there is considerable strain
on the screws when a boring tool is used to stand far out, as for boring
deep holes. On the other hand, however, the screws can be readily got at
in this position, and may therefore be screwed up tightly to lock the
upper slider firmly to the cross slide, which will be a great advantage
in boring and also in facing operations. But the screws must not in this
case have simple saw slot heads, such as shown on a larger scale in
Fig. 490, but should have square heads to receive a wrench, and if these
four screws are used, the two end ones may be set to adjust the slicing
fit of the slider, while the two middle ones may be used to set the
slider form on its slide when either facing or boring. The corners of
the gibs as well as those of the slider and slide may with advantage be
rounded so that they may not become bruised or burred, and, furthermore,
the slider is strengthened, and hence less liable to spring under the
pressure of a heavy cut.

[Illustration: Fig. 490.]

A slide rest for turning spherical work is shown in Fig. 491. A is the
lower slide way on which is traversed the slide B, upon which is fitted
the piece C, pivoted by the bolt D; there is provided upon C a
half-circle rack, shown at E, and into this rack gears a worm-wheel
having journal bearing on B, and operated by the handle F. As F is
rotated C would rotate on D as a centre of motion, hence the tool point
would move in an arc of a circle whose radius would depend upon the
distance of the tool point from D as denoted by J, which should be
coincident with the line of centres of the lathe.

[Illustration: Fig. 491.]

The slide G is constructed in the ordinary manner, but the way on which
it slides should be short, so as not to come into contact with the work.
If the base slide way A be capable of being traversed along the lathe
shears S S by a separate motion, then the upper slide way and slide may
be omitted, G and C being in one piece. It is to be noted in a rest of
this kind, however, that the tool must be for the roughing cut set too
far from D to an amount equal to about the depth of cut allowed to
finish with, and for the finishing cut to the radius of the finished
sphere in order to obtain a true sphere, because if B be operated so
that D does not stand directly coincident with the line of lathe
centres, the centre of motion, or of the circle described by the tool
point, will not be coincident with the centre on which the work rotates,
hence the work though running true would not be a true sphere but an
oval. This oval would be longest in the direction parallel with the line
of centres whenever the pivot D was past the line of centres, and an
oval of largest diameter at the middle or largest diameter turned by the
tool whenever the pivot D was on the handle H side of the line of
centres. To steady C it may be provided with a circular dovetail, as
shown at the end I, provision being made (by set screw or otherwise) for
locking C in a fixed position when using the rest for other than
spherical work.

To construct such a rest for turning curves or hollows whose outline
required to be an arc of a circle, the pivot D would require to be
directly beneath the tool post, which must in this case occupy a fixed
position. The radius of the arc would here again be determined by the
distance of the tool point from the centre of rotation of the pivot, or,
what would be the same thing, from that of the tool post.

Next to the hand slide rest lathe comes the self-acting or engine lathe.
These are usually provided with a feed motion for traversing the slide
rest in the direction of the length of the bed, and sometimes with a
self-acting cross feed, that is to say, a feed motion that will traverse
the tool to or from the line of centres and at a right angle to the
same.

In an engine lathe the parallelism or truth of the work depends upon the
parallelism of the line of centres with the shears of the lathe, and
therefore upon the truth of the shears or bed, and its alignment with
the cone spindle and tail spindle, while the truth of the radial faces
on the turned work depends upon the tool rest moving on the cross slide
at a true right angle to the line of centres.

[Illustration: Fig. 492.]

Fig. 492 represents an 18-inch engine (or self-acting) lathe designed by
and containing the patented improvements of S. W. Putnam, of the Putnam
Tool Company, of Fitchburg, Massachusetts. The lathe has an elevating
slide rest self-acting feed traverse and self-acting cross feed, both
feeds being operative in either direction. It has also a feed rod for
the ordinary tool feeding and a lead screw for screw-cutting purposes.

Fig. 493 represents a cross-sectional view of the shears beneath the
headstock; A A are the shears or bed having the raised [V]s marked V´
and V on which the headstock and tailstock rest, and V´´ and V´´´ on
which the carriage slides. A and A´ are the shears connected at
intervals by cross girts or webs B to stiffen them. C C are the bolts to
secure the headstock to the shears. D is a bracket bolted to A´ and
affording at E journal bearing for the spindle that operates the
independent feed spindle. E is split at _f_ and a piece of soft wood or
similar compressible material is inserted in the split. The bolt F is
operated to close the split, and, therefore, to adjust the bore E to
properly fit the journal of the feed spindle, and as similar means are
provided in various parts of the lathe to adjust the fits of journals
and bearings the advantages of the system may here be pointed out.
First, then, the fit of the bearing may be adjusted by simply operating
the screw, and, therefore, without either disconnecting the parts or
performing any fitting operation, as by filing. Secondly, the presence
of the wood prevents the ingress of dust, &c., which would cause the
bearings and journals to abrade; and, thirdly, the compression of the
wood causes a resistance and pressure on the adjusting screw thread,
which pressure serves to lock it and prevent it from loosening back of
itself, as such screws are otherwise apt to do.

[Illustration: Fig. 493.]

As the pressure of the tool cut falls mainly on the front side of the
carriage, and as the weight of the carriage itself is greatest on that
side, the wear is greatest; this is counteracted by forming the front
[V], marked V´´´ in figure, at a less acute angle, which gives it more
wearing area and causes the rest to lower less under a given amount of
wear.

The rib A´´ which is introduced to strengthen the shears against
torsional strains, extends the full length of the shears.

[Illustration: Fig. 494.]

Fig. 494 is a sectional side elevation of the headstock; A A´ represents
the headstock carrying the bearing boxes B and B´, which are capable of
bore closure so as to be made to accurately fit the spindle S by the
construction of the front bearing B, being more clearly shown in Fig.
495; B is of composition brass, its external diameter being coned to fit
the taper hole in the head; it is split through longitudinally, and is
threaded at each end to receive the ring nuts C and C´. If C be loosened
from contact with the radial face of A, then C´ may be screwed up,
drawing B through the coned hole in A, and, therefore, causing its bore
to close upon S.

At the other end of S, Fig. 496, C´´ is a ring nut for drawing the
journal box B´ through _a´_ to adjust the bore of B´ to fit the journal
of S, space to admit the passage of B´ being provided at _e_. D is a box
nut serving to withdraw B´ or to secure it firmly in its adjusted
position, and also to carry the end adjusting step E. F is a check nut
to lock E in its adjusted position.

The method of preventing end motion to S is more clearly shown in Fig.
496, in which _h_ is a steel washer enveloping S, having contact with
the radial face of B´ and secured in its adjusted position by the check
nuts _g_, hence it prevents S from moving forward to the right. _f_ is a
disk of raw hide let into E; the latter is threaded in D and is squared
at the end within F to admit of the application of a wrench, hence E may
be screwed in until it causes contact between the face of _f_ and the
end of S, thus preventing its motion to the left. By this construction
the whole adjustment laterally of S is made with the short length from
_h_ to _f_, hence any difference of expansion (under varying
temperature) between the spindle and the head A A´, or between the boxes
and the spindle S, has no effect towards impairing the end fit of S in
its bearings.

The method of adjusting the bearings to the spindle is as follows:--C´´
and C´ are slackened back by means of a "spanner wrench" inserted in the
holes provided for that purpose. C and D are then screwed up,
withdrawing B and B´ respectively, and leaving the journal fit too easy.
C´ is then screwed up until B is closed upon the spindle sufficiently
that the belt being loose on the cone pulley, the latter moved by the
hand placed upon the smallest step of the cone can just detect that
there is contact between the bore of B and the spindle, then, while
still moving the cone, turn C´ back very slowly and a very little, the
object being to relieve the bore of B from pressure against S. C may
then be screwed up, firmly locking B in its adjusted position. C´´ may
then be operated to adjust B´ in a similar manner, and D screwed up to
lock it in its adjusted position. Before, however, screwing up D it is
better to remove F and release E from pressure against _f_, adjusting
the end pressure of E after D has been screwed home against A´.

To prevent B and B´ from rotating in the head when the ring nuts are
operated, each is provided with a pin, _q_, grooves _c_ and _c´_
permitting of the lateral movement of B and B´ for adjustment. The boxes
B, B´ admit of being rotated in their sockets in A and A´ so as to
assume different positions, the pins _q_ and _q´_ being removable from
one to another of a series of holes in the boxes B, B´ when it is
desired to partly rotate those boxes. The tops of the boxes are provided
with oil holes, and the oil ways shown at _r_, _s_ being the oil groove
through the head and _a_ simply a stopper to prevent the ingress of
dust, &c.

[Illustration: Fig. 495.]

The thread on S at Z, Fig. 494, is to receive and drive the face plates,
chucks, &c., which are bored and threaded to fit over Z. To cause the
radial faces of such face plates or chucks to run true, there is
provided the plain cylindrical part _l_, to which the bore in the hub of
the face plate or chuck is an accurate fit when the radial face of that
hub meets the radial face _m_.

Referring again to Fig. 494, G´ is the pinion to drive the back gear
while G receives motion from the back-gear pinion. The object of the
back gear is to reduce the speed of rotation of S and to enable it to
drive a heavier cut, which is accomplished as follows:--G´ is secured
within the end K of the cone and is free to rotate with the cone upon S;
at the other end the cone is secured to M, which is free to rotate upon
S so far as its bore is concerned. G is fixed upon S and hence rotates
at all times with it; but G may be locked to or released from M as
follows:--

[Illustration: Fig. 496.]

In G is a radial slot through which passes a bolt I provided with a cap
nut H, in M is an annular groove J. When I is lifted its head passes
into a recess in M, then H is screwed up and G is locked to M. This is
the position of I when the back gear is not in use, the motion of the
cone being communicated to S through I. But if H be loosened and I be
moved inwards towards S, the head of I passes into the annular groove J,
and the cone is free to rotate upon S while the latter and G remain
stationary unless the back gear is put into operation. In this latter
case the pinion G´ rotating with the cone drives the large gear of the
back gear and the small pinion of the latter drives G, whose speed of
rotation is reduced by reason of the relative proportions of the gear
wheels.

In this case it is obvious that since the pulley rotates upon the
spindle it requires lubrication, which is accomplished through the oil
hole tubes L.

The means of giving motion to the feed spindle and lead screw are as
follows:--N, Fig. 494, is a pinion fast upon S and operating the gear O,
which is fast upon the spindle P, having journal bearing in a stem in A´
and also at G´´. P drives the three-stepped cone R, which is connected
by belt to a similar cone fast upon the independent feed spindle. The
seat for the driving gear of the change wheels for the lead screw is on
P at V. To provide ample bearing surface for P in A´ the bush or sleeve
shown is employed, but this sleeve also serves to pivot the swing frame
W which carries the studs for the change wheels that go between the
wheel on V and that on the lead screw; _x_ _y_ are simply oil holes to
lubricate P in its bearings.

To provide a wider range of tool feed than that obtainable by the steps
on the feed cones, as R, they are provided at their ends with seats for
change wheels, the swing frame W carrying the intermediate wheels for
transmitting motion from V to a similar seat on the cone on the feed
spindle.

Fig. 497 represents the tailstock (or tailblock as it is sometimes
termed), shown in section. A represents the base which slides upon the
raised [V]s on the bed and carries the upper part B, in which slides the
tail spindle C, which is operated longitudinally by the tail screw D,
having journal bearing in E, and threaded through the nut F which is
fast in C. The hand wheel G is for rotating D, whose thread operating in
the nut F, causes C to slide within B in a direction determined by the
direction of rotation of G. To lock C in its adjusted position the
handled nut H is employed in connection with the bolt I, which is shown
in dotted lines; C is split as shown by the dotted lines at _f_; J is
the dead centre fitting accurately into a conical hole in C. When it is
required to remove J from C the wheel G is operated to withdraw C
entirely within B, and the end _d_ of D meets the end _e_ of J and
forces J from the coned hole in C.

The method of securing the tailstock to the shears or releasing it from
the same is as follows. A vertical prolongation of B affords at B´´ a
bearing surface for the nut-handle L and washer M. K is a bolt threaded
into L passing through M, B´´ and N, the latter of which it carries. N
spans the shears beneath the two [V]s on which the tailstock slides.
Moving or rather partly rotating the handle L in the necessary direction
lifts K and causes N to rise, and grip the shears beneath, while the
pressure of M on B´´ causes B to grip A and the latter to grip the
raised [V]s on the shears. If L be rotated in the opposite direction it
will cause N to fall, leaving A free to slide along the shears. To
prevent N from partly rotating when free, its ends are shaped to fit
loosely between the shears as shown at _n_.

To give to N sufficient rise and fall to enable it to grip or fall
entirely free from the shears with the small amount of rotary motion
which the handle-lever L is enabled from its position to have, the
following device is provided. M is a washer interposed between L and
B´´. This washer has upon it steps of different thickness as shown at M
and _m_, the two thicknesses being formed by an incline as shown. The
face of L has, as shown, similar steps; now as shown in the cut the step
_l_ on lever L meets the steps _m_ of the washer, the handle having
receded to the limit of its motion. The bolt K then has fallen to the
amount due to unscrewing the threaded or nut end of L, and also to the
amount of the difference of thickness at M and at _m_ of the washer, the
plate N being clear of the lathe-shears. But suppose the handle L be
pulled towards the operator, then the surface _l_ passing from a thin
section on to a thick one as M of the washer, will lift the bolt K,
causing N to meet the under surface of the shears, and then the motion
of L continuing the pressure of the thread will bind or lock N to the
bed.

[Illustration: Fig. 497.]

The surface A´ in Fig. 497 affords a shelf or table whereon tools, &c.,
may be placed instead of lying on the lathe bed, where they may cause or
receive damage.

Fig. 498 represents an end view of the tailstock viewed from the dead
centre end, the same letters of reference applying to like parts that
are shown in Fig. 497. The split at _f_ is here shown to be filled with
a piece of soft wood which prevents the ingress of dust, &c. At _d_ is a
cup or receptacle for oil, _e_ being a stopper, having attached to it a
wire pin flattened and of barb shape at the end, the object being to
cause the wire to withdraw from the cup a drop of oil to lubricate the
dead centre and centre in the work. The proximity of _e_ to the dead
centre makes this a great convenience, while the device uses much less
oil than would be used by an oil can.

[Illustration: Fig. 498.]

The method of setting over the upper part B to enable the turning of the
diameter of work conical or taper instead of parallel is shown in Fig.
498: P and P´ are square-headed screws threaded into the walls of A and
meeting at their ends the surface of B´. In A there is at _a_ a wide
groove or way, and on B there is at _b_ a projection fitting into the
way _a_ so as to guide B when it slides across A, as it will when P is
unscrewed in A and P´ is screwed into A. This operation is termed
setting over the tailstock, and its effect is as follows:--Suppose it be
required to turn a piece of work of smaller diameter at the end which
runs on the dead centre, then, by operating the screw P towards the
front of the lathe (or to the left as shown in the cut) and screwing P´
farther into A, the end of P´ will meet the surface of B´, causing B´ to
move over, and the centre of the dead centre J (which is the axis of
rotation of the work at that end) will be nearer to the point of the
cutting tool. Or suppose the work requires to be turned a taper having
its largest diameter at the end running on the dead centre, then P´
would be unscrewed and P screwed farther into A, carrying B farther
towards the back of the lathe.

The [V] grooves Q and Q´ fit upon the inner raised [V]s shown at V, V´
in Fig. 499.

[Illustration: Fig. 499.]

Fig. 499 is a side view of the slide rest for holding and traversing the
cutting tool. A represents the carriage resting upon the raised [V]s
marked V´´ and V´´´ and prevented from lifting by its own weight, and in
front also by the gib _a_ secured to A by the bolt _b_ and having
contact at _c_ with the shears. A carries at _d_ a pivot for the cross
slide B and at _e_ a ball pivot for the cross slide elevating screw C.
This screw is threaded through the end of B so that by operating it that
end of B may be raised or lowered to adjust the height of the cutting
tool point to suit the work. To steady B there is provided (in addition
to the pivots at _d_) on A two lugs _f_, between the vertical surfaces
of which B is a close working fit. The upper surface of B is provided
with a [V]-slide-way _g_, to which is fitted the tool rest D (the
construction being more clearly shown in Fig. 500).

[Illustration: Fig. 500.]

The means for traversing D along the slide _g_ on B is as follows:--

A nut _i_ is secured to D by the screw bolt _j_, and threaded through
the nut _i_ is the cross-feed screw E, which has journal bearing in the
piece _k_, which is screwed into the end face of B; there is a collar on
E which meets the inner end of _k_, and the handle F being secured by
nut to that end of E its radial face forms a shoulder at _m_ which with
the collar prevents any end motion of E, so that when F is rotated E
rotates and winds through the nut _i_ which moves D along B.

An end view of A, B, and D is shown in Fig. 500, in which the letters of
reference correspond to those in Fig. 499. B´ and B´´ are the
projections that pass into A and receive the pivoting screws _d_ and
_d_. To adjust the fit and take up any wear that may ensue on the slide
_g_, on B and on the corresponding surface on D, the piece _n_ is
provided, being set up by the adjusting screws O.

To adjust the fit and take up the wear at the pivots _d_ they are made
slightly taper, fitting into correspondingly taper holes in B.

The dotted circle T´, represents a pinion fast upon the cross-feed screw
(E, Fig. 499); the similar circles T and S´´ also represent pinions, the
three composing a part of the method of providing an automatic or
self-acting cross feed or cross traverse to D by rotating it through a
gear-wheel motion derived from the rotation of the independent feed
spindle, as is described with reference to Fig. 501.

_m_ in Fig. 500 represents a cavity or pocket to receive wool, cotton or
other elastic or fibrous material to be saturated with oil and thus
lubricate the raised [V]s while keeping dirt from passing between the
rest and the [V]s. The shape of these pockets is such as to enable them
to hold the cotton with a slight degree of pressure against the slides,
thus insuring contact between them.

The mechanical devices for giving to the carriage a self-acting traverse
in either direction along the bed, so as to feed the tool automatically
to its cut, and for giving to the tool rest (D, Fig. 499) traverse
motion so as to feed the tool to or from the line of centres along the
cross slide, are shown in Fig. 501, which presents two views of the feed
table or apron. The lower view supposes the feed table to be detached
from the carriage and turned around so as to present a side elevation of
the mechanism. The upper view is a plan of the same with two pinions (N
and N´), omitted. A represents the part of the lathe carriage shown at A
in Fig. 500. It has two bolts _p_ and _p´_, which secure the apron G,
Fig. 501, to A. At H is the independent feed spindle or feed rod
operated by belt from the cone pulley R, Fig. 494, or by a gear on stud
P at V. H is carried in bearings fixed to each end of the lathe shears
or bed, both of these bearings being seen in Fig. 492. H is also
provided with a bearing fixed on the feed apron as seen in Fig. 501, and
is splined as shown at _h_. At I is a bracket fast upon the apron G and
affording journal bearing to J, which is a bevel pinion having a hub
which has journal bearing in the bracket I. The fit of the bearing to
the journal is here again adjusted by a split in the bearing with a
screw passing through the split and threaded in the lower half (similar
to the construction of D in Fig. 493); J is bored to receive H, and is
driven by means of a feather projecting into the spline _h_. When
therefore, the carriage A is moved it carries with it the apron G, and
this carries the bracket I holding the bevel pinion J, which is in gear
with the bevel-wheel K, and therefore operates it when H has rotary
motion. At the back of K, and in one piece with it, is a pinion K´, both
being carried upon the stud L; pivoted upon this same stud is a plate
lever M, carrying two pinions N and N´ in gear together, but N only is
in gear with K´, hence K´ drives N and N drives N´. Now in the position
shown neither N or N´ is in gear with the gear-wheel O, but either of
them may be placed in gear with it by means of the following
construction:--

At the upper end of M there is provided a handle stud M´ passing through
the slot M´´ in G. Screwing up this stud locks M fast by binding it
against the surface of G. Suppose, then, M´ to be unscrewed, then if it
be moved to the right in the slot M´´, N will be brought into gear with
O and the motion will be transmitted in the direction of the arrows, and
screwing up N would retain the gear in that position. But suppose that
instead of moving M´ to the right it be moved to the left, then N´ will
be brought into gear with O and the direction of rotation of O will be
reversed.

[Illustration: Fig. 501.]

Thus, then, O may be made to remain stationary or to rotate in either
direction according to the position of M´ in the slot M´´, and this
position may be regulated at will.

The gear O contains in its radial face a conical recess, and upon the
same stud or pin (P) upon which O is pivoted, there is fixed the disk
P´, which is in one piece with the pinion P´´; the edge of P´ is coned
to fit the recess in the wheel O, so that if the stud P is operated to
force the disk P´ into the coned recess in O the motion of wheel O will
be communicated to disk P´, by reason of the friction between their two
coned surfaces. Or if P be operated to force the coned edge of the disk
out of contact with the coned bore or recess in gear O, then O will
rotate while P´ and P´´ will remain stationary. Suppose the coned
surfaces to be brought (by operating _x_) into contact and P´ to rotate
with O, then P´´ being in gear with wheel Q will cause it to rotate. Now
Q is fast to the pinion Q´, hence it will also rotate, and being in
contact with the rack which is fixed along the shears of the lathe and a
section of which is shown in the cut, the whole feed table or apron will
be made to traverse along the lathe shears.

The direction in which this traverse will take place depends upon the
adjusted position of M´ in M´´, or in other words upon whether N or N´
be the pinion placed in gear with O. As shown in the cut neither of them
is in gear, and motion from H would be communicated to N and N´ and
would there cease; but if M´ be raised in the slot M´´, N would drive O,
and supposing P´ to be held to O, the motion of all the gears would be
as denoted by the arrows, and the lathe carriage A would traverse along
the lathe bed in the direction of arrow Q´´. But if N´ be made to drive
O all the motions would be in the opposite directions. The self-acting
feed motion thus described is obviously employed to feed the cutting
tool, being too slow in its operation for use to simply move the
carriage from one part of the lathe bed to another; means for this
purpose or for feeding the carriage and cutting tool by hand are
provided as follows:--R is a pinion in gear with Q and fast upon the
stud R´, which is operated by the handle R´´. The motion of R´´ passes
from R to Q and Q´ which is in gear with the rack. But Q´ being in gear
with P´´ the latter also rotates, motion ceasing at this point because
the cone on P´ is not in contact with the coned recess in O. When,
however, P´ and O are in contact and in motion, that motion is
transmitted to R´´, which cannot then be operated by hand.

It is often necessary when operating the cross feed to lock the carriage
upon the lathe bed so that it shall not move and alter the depth of the
tool-cut on the radial face of the work. One method of doing this is to
throw off the belt that operates the feed spindle H, place N in gear
with O and P´ in contact with O, so that the transverse feed motion will
be in action, and then pull by hand the cone pulley driving H, thus
feeding the tool to its necessary depth of cut. The objection to this
method, however, is that when the operator is at the end of the lathe,
operating the feed cone by hand he cannot see the tool and can but
guess how deep a cut he has put on. To overcome this difficulty a brake
is provided to the pinion R as follows:--

The brake whose handle is shown at V has a hub V´ enveloping the hub
R´´´ which affords journal bearing to the stud R´. In the bore of this
hub V´ is an eccentric groove, and in R´´´ is a pin projecting into the
eccentric groove and meeting at its other end the surface of the stud
R´. When, therefore, V is swung in the required direction (to the left
as presented in the cut), the cam groove in V´ forces _r_ inwards,
gripping it and preventing it from moving, and hence the movement of R
which also locks Q and Q´.

It remains now to describe the method of giving rotary motion to the
cross-feed screw E (Fig. 499) so as to enable it to self-act in either
direction. S is a lever pivoted upon the hub of O and carrying at one
end the pinion S´´, while at the other end is a stud S´ passing through
a slot in G. The pinion S´´ is in gear with O and would therefore
receive rotary motion from it and communicate such motion to pinion T,
which in turn imparts rotary motion to T´. Now T´ is fast upon the
cross-feed screw as shown in Fig. 499 and the cross-feed screw E in that
figure would by reason of the nut _i_ in figure cause the tool rest D to
traverse along the cross-slide in a direction depending upon the
direction of motion of T´, which may be governed as follows:--

If S´ be moved to the left S´´ will be out of gear with T and the
cross-feed screw may be operated by the handle (F, Fig. 499). If S´ be
in the position shown in cut and M´´ also in the position there shown
(Fig. 501), operating the feed screw by its handle would cause its
pinion T´ to operate T, S´´, and O; hence S´ should always be placed to
disconnect S´´ from T when the cross-feed screw is to be operated by
hand, and S´ operated to connect them only when the self-acting cross
feed is to operate. In this way when the cross feed is operated by hand
T´ and T will be the only gears having motion. It has been shown that
the direction of motion of O is governed by the position of M´, or in
other words, is governed by which of the two pinions N or N´ operates,
and as O drives S´´ its motion, and therefore that of T´, is reversible
by operating M´.

The construction of S´ is as follows:--Within the apron as shown in the
side elevation it consists of what may be described as a crank, its pin
being at _t_; in the feed table is a slot through which the shaft of the
crank passes; _s_ is a handle for operating the crank. By rotating _s_
the end S´ of S is caused to swing, the crank journal moving in the slot
to accommodate the motion and permit S to swing on its centre.

The device for forcing the cone disk P´ into contact with or releasing
it from O is as follows:--The stud P is fast at the other end in P´ and
has a collar at _b_; the face of this collar forms one radial face, and
the nut W affords the other radial face, preventing end motion to _x_
without moving P endwise. If _x_ be rotated its thread at _x´_ causes it
to move laterally, carrying P with it, and P being fast to P´ also moves
it laterally. P´ is maintained from end motion by a groove at O´ in
which the end of a screw _a_ projects, _a_ screwing through W and into
the groove O´.

[Illustration: Fig. 502.]

The lead screw of a lathe is a screw for operating the lathe carriage
when it is desired to cut threads upon the work. It is carried parallel
to the lathe shears after the same manner as the independent feed
spindle, and is operated by the change wheels shown in Fig. 492 at the
end of the lathe. These wheels are termed change wheels on account of
their requiring to be changed for every varying pitch of thread to be
cut, so that their relative diameters, or, what is the same thing, their
relative number of teeth, shall be such as to give to the lead screw the
speed of rotation per lathe revolution necessary to cut upon the work a
thread or screw of the required pitch.

The construction of the bearings which carry the lead screw in the S. W.
Putnam's improved lathe is shown in Fig. 502, in which A represents the
bearing box for the headstock end of the lathe, having the foot A´ as a
base to bolt it to the lathe shears. L represents the lead screw, having
on one side of A the collar L´ and on the other the nut and washer N and
N´. The seat for the change wheel that operates the lead screw is at
L´´, the stop pin _l_ fitting into a recess in the change wheel so as to
form a driving pin to the lead screw. The washer N´ is provided with a
feather fitting into a recess into L so that it shall rotate with L and
shall prevent the nut N from loosening back as it would be otherwise apt
to do. End motion to L is therefore prevented by the radial faces of L´
and N´.

[Illustration: Fig. 503.]

At the other end of the lathe there are no collars on the lead screw,
hence when it expands or contracts, which it will do throughout its
whole length under variations of atmospheric temperature, it is free to
pass through the bearing and will not be deflected, bent, or under any
tension, as would be the case if there were collars at the ends of both
bearings. The amount of this variation under given temperatures depends
upon the difference in the coefficients of expansion for the metal of
which the lead screw and the lathe shears are composed, the shears being
of cast iron while lead screws are sometimes of wrought iron and
sometimes of steel.

The bearings at both ends are split, with soft wood placed in the split
and a screw to close the split and adjust the bearing bore to fit the
journal, in the manner already described with reference to other parts
of this lathe.

The construction of the swing frame for carrying the change wheels that
go between the driving stud V, Fig. 494, and that on the seat L´´, Fig.
502, are as follows:--

Fig. 503 represents the change wheel swing frame, an edge view of which
is partly shown at W in Fig. 494. S is a slot narrower at _a_ than at
_b_. Into this slot fit the studs for carrying the change wheels.

By enabling a feed traverse in either direction the lathe carriage may
be traversed back (for screw-cutting operations) without the aid of an
extra overhead pulley to reverse the direction of rotation of the lathe,
but in long screws it is an advantage to have such extra overhead pulley
and to so proportion it as to make the lathe rotate quicker backwards
than forward, so as to save time in running the carriage back.

The mechanical devices for transmitting motion from the lead screw to
the carriage are shown in Fig. 504, representing a view from the end and
one from the back of the lathe. B is a frame or casting bolted by the
bolt _b_ to the carriage A of the lathe. C is a disk having a handle C´
and having rotary motion from its centre. Instead of being pivoted at
its centre, however, it is guided in its rotary motion by fitting at _d_
_d_ into a cylindrical recess provided in B to receive it. C contains
two slots D and D´ running entirely through it. These slots are not
concentric but eccentric to the centre of motion of C. Through these
slots there pass two stud bolts E and E´ shown by dotted lines in Fig.
504, and these bolts perform two services: first by reason of the nuts F
and F´ they hold C to its place in B, and next they screw into and
operate the two halves G and G´ of a nut.

[Illustration: Fig. 504.]

Suppose, now, that the handle C´ be operated or moved towards arrow _e_,
then the dot at _f_ being the centre of its motion and the slots D and
D´ gradually receding from _f_ as their ends _g_ are approached they
will cause E to move vertically upward and E´ to move vertically
downward, a slot in B (which slot is denoted by the dotted lines _h_)
guiding them and permitting this vertical movement.

Since E and E´ carry the two halves of the nut which envelops the lead
screw L it is obvious that operating C´ will either close or release the
half nuts from L according to which direction it (C´) is moved in.

The screws H and H´ screw tightly into B, and the radial faces of their
heads are made to have a fair and full bearing against the underside of
the shears, so that they serve as back gibs to hold the carriage to the
shears and may be operated to adjust the fit or to lock the carriage to
the bed if occasion may require. This lathe is made with a simple tool
rest as shown in the engravings or with a compound slide rest. In some
sizes the rest is held to the carriage by a weight upon a principle to
be hereafter described. The bed is made (as is usual) of any length to
suit the purposes for which the lathe is to be used.

The next addition to the lathe as it appears in the United States is
that of a compound slide rest.

[Illustration: Fig. 505.]

Fig. 505 represents a 28-inch swing lathe by the Ames Manufacturing
Company, of Chicopee, Massachusetts. It is provided with the usual
self-acting feed motion and also with a compound slide rest. The swing
frame for the studs carrying the change wheels for screw cutting here
swings upon the end of the lead screw, the same spindle that carries the
driving cone for the independent feed rod which is in front of the
lathe, also carries the driving gear for the change wheels used for
screw cutting.

The construction of the compound rest is shown in Figs. 506 and 507. N
is the nut for the cross-feed screw (not shown in the cut) and is
carried in the slide A. A and the piece L above it are virtually in one,
since the latter is made separate for convenience of construction and
then secured to it firmly by screws. B is made separate from C also for
convenience of construction and fixed to it by screws; L is provided
with a conical circular recess into which the foot B of C fits. E is a
segment of a circle operated by the set screw F to either grip or
release B. The bolt D simply serves as a pivot for piece B C; at its
foot C is circular and is divided off into the degrees of a circle to
facilitate setting it to any designated angle.

If, then, F be unscrewed, C may be rotated and set to the required
angle, in which position screwing up F will lock it through the medium
of E. G is the feed nut for the upper slider H, which operates along a
slide way provided on C, the upper feed screw having journal bearing at
C´. I is the tool post, having a stepped washer J, by means of which the
height of the tool K may be regulated to suit the work.

[Illustration: Fig. 506.]

[Illustration: Fig. 507.]

Suppose, now, that it be required to turn a shaft having a parallel and
a taper part; then the carriage may be traversed to turn the parallel
part, and the compound slide C may be set to turn the taper part, while
the lower feed screw operating in N may be used to turn radial faces.

[Illustration: Fig. 508.]

The object of making A and L in two pieces is to enable the boring and
insertion of B, which is done as follows:--The front end of L as L´ is
planed out, leaving in it a groove equal in diameter and depth to the
diameter and depth of B, so that B may be inserted laterally along this
groove to its place in L. The segment E is then inserted and a piece is
then fitted in at L´ and held fast to A by screws. It is into this piece
that the set screw F is threaded.

Various forms of construction are designed for compound rests, but the
object in all is to provide an upper sliding piece carrying the tool
holder, such sliding piece being capable of being so set and firmly
fixed that it will feed the tool at an angle to the line of the lathe
centres.

Another and valuable feature of the compound rest is that it affords an
excellent method of putting on a very fine cut or of accurately setting
the depth of cut to turn to an exact diameter; this is accomplished by
setting the upper slide at a slight angle to the line of centres and
feeding the tool to the depth of cut by means of the screw operating the
upper slide. In this way the amount of feed screw handle motion is
increased in proportion to the amount to which the tool point moves
towards the line of lathe centres, hence a delicate adjustment of depth
of cut may be more easily made.

Suppose, for example, that a cut be started and that it is not quite
sufficiently deep, then, while the carriage traverse is still
proceeding, the compound rest may be operated to increase the cut depth,
or if it be started to have too deep a cut the compound rest may be
operated to withdraw the tool and lessen its depth of cut. Or it may be
used to feed the tool in sharp corners when the feed traverse is thrown
out, or to turn the tops of collars or flanges when the tailstock is
set over to turn a taper.

It is obvious, however, that comparatively short tapers only can be
conveniently turned by a compound slide rest; but most tapers, however,
are short.

To turn long tapers the tailstock of the lathe is set over as described
with reference to the Putnam lathe, but for boring deep holes the slide
rest must either be a compound one or a taper turning former or
attachment must be employed.

[Illustration: Fig. 509.]

When, however, the tailstock is set over, the centres in the work are
apt to wear out of true and move their location (the causes of which
will be hereafter explained).

In addition to this, however, the employment of a taper turning
attachment enables the boring of taper holes without the use of a
compound slide rest, thus increasing the capacity of the lathe not
having a simple or single rest.

In Fig. 508 is shown a back view of a Pratt and Whitney weighted lathe
having a Slate's taper turning attachment, the construction of which is
as follows:--Upon the back of the lathe shears are three brackets having
their upper surfaces parallel with and in the same plane as the surface
of the lathe shears. Pivoted to the middle bracket is a bar which has at
each end a projection or lug fitting into grooves provided in the end
brackets, these grooves being arcs of a circle whose centre is the axis
of the pivot in the middle bracket.

The end brackets are provided with handled nuts upon bolts, by which
means the bar may be fixed at any adjusted angle to the lathe shears.
Upon the upper surface of the bar is a groove or way in which slides a
sliding block or die, so that this die in traversing the groove will
move in a straight line but at an angle to the lathe bed corresponding
to the angle at which the bar may be adjusted. The slide rest upon being
connected by a bar or rod to the die or sliding block is therefore made
to travel at the same angle to the lathe bed or line of centres as that
to which the bar is set. The method of accomplishing this in the lathe,
shown in Fig. 508, is as follows:--

In Fig. 509 A is the bar pivoted at C upon the centre bracket B; E is
the sliding block pivoted to the nut bar F. This nut bar carries the
cross-feed nut, which in turn carries the feed screw and hence the tool
rest. When the nut bar is attached to the sliding block to turn a taper
it is free to move endways upon the lower part of the carriage in which
it slides, but when the taper attachment is not in use the bar is
fastened to the lower part of the carriage by a set screw.

The screw at D is provided to enable an accurate adjustment for the
angle of the bar A. G and H are screws simply serving to adjust the
diameter to which the tool will turn after the manner shown in Fig. 588,
G being for external and H for internal work.

When the lathe has a bed of sufficient length to require it, a slide is
provided to receive the brackets, which may be adjusted to any required
position along the slide, as shown in Fig. 510. This is a gibbed instead
of a weighted lathe, and the method of attaching the sliding block to
the lathe rest is as follows:--

A separate rod is pivoted to the sliding block. This rod carries at its
other end a small cross head which affords general bearing to the end of
the cross-feed screw, which has a collar on one side of the cross head
and a fixed washer on the other, to prevent any end motion of the said
screw.

[Illustration: Fig. 510.]

The cross-feed nut is attached to the traversing cross slide. The other
or handle end of the cross-feed screw has simple journal bearing in the
slide rest, but no radial faces to prevent end motion, so that one may
from the rod attached to the sliding-block traverse the cross-feed
slide, which will carry with it the feed screw. As a result, the line of
motion of the tool rest is governed by the sliding die, but the diameter
to which the tool will turn is determined by the feed screw in the usual
manner. When it is not required to use the taper attachment, the rod or
spindle is detached from the sliding die and is locked by a clamp, when
the rest may be operated in the usual manner.

Fig. 511 represents a compound duplex lathe of a design constructed by
Sir Joseph Whitworth, of Manchester, England. The two rests are here
operated on the same cross slide by means of a right and left-hand
cross-feed screw.

The tool for the back rest is here obviously turned upside down.

The lead screw is engaged at two places by the feed nut, which is in two
pieces attached to levers; while at a third point in its circumference
it is supported by a bracket, bolted to the lathe bed.

[Illustration: Fig. 511.]

Fig. 512 represents the New Haven Manufacturing Company's three tool
slide rest, for turning shafting. It is provided with a follower rest,
in front of which are two cutting tools for the roughing cuts, and
behind which is a third tool for the finishing cut. The follower rest
receives bushes, bored to the requisite diameter, to leave a finishing
cut. The first tool takes the preliminary roughing cut; the second tool
turns the shaft down to fit the bush or collar in the follower rest;
and, as stated, the last tool finishes the work.

Fig. 513 represents a 44-inch swing lathe, showing an extra and
detachable slide rest, bolted on one side of the carriage and intended
for turning work of too large a diameter to swing over the slide rest.
By means of this extra rest the cutting tool can be held close in the
rest, instead of requiring to stand out from the tool-post to a distance
equal to the width of the work. The ordinary tool post is placed in this
extra rest.

[Illustration: Fig. 512.]

When it is desired to bolt work on the lathe carriage and rotate the
cutting tools, as in the case of using boring bars, the cross slide is
sunk into instead of standing above the top surface of the carriage so
as to leave a flat surface to bolt the work to, and [T]-shaped slots are
provided in the carriage, to receive bolts for fastening the work to the
carriage, an example of this kind being shown in Fig. 514.

[Illustration: Fig. 513.]

Fig. 515 represents a self-acting slide or engine lathe by William
Sellers and Co., of Philadelphia. These lathes are made in various sizes
from 12 inches up to 48 inches swing on the same general design,
possessing the following features:--The beds or shears are made with
flat tops, the carriage being gibbed to the edges of the shears, these
edges being at a right angle to the top face of the bed. The dead centre
spindle is locked at each end of its bearing in the tailstock, thus
securing it firmly in line with the live spindle. The ordinary tool feed
is operated by a feed rod in front of the lathe, and this rod is
operated by a disc feed, which may be altered without stopping the lathe
so as to vary the rate of tool feed; and an index is provided whereby
the operator may at once set the discs to give the required rate of
feed. The lead screw for screw cutting is placed in a trough running
inside the lathe bed, so that it is nearer to the cutting tool than if
placed outside that bed, while it is entirely protected from the lathe
cuttings and from dirt or dust; and the feed-driving mechanism is so
arranged that both may be in gear with the live spindle, and either the
rod feed or screw-cutting feed may be put into action instantly, while
putting one into action throws the other out, and thus avoid the
breakage that occurs when both may be put into action at the same time.
The direction of the turning feed is determined by the motion of a lever
conveniently placed on the lathe carriage, and the feed may be stopped
or started in either direction instantly. The mechanism for putting the
cross feed in action is so constructed (in those lathes having a
self-acting cross feed) that the cross feed cannot be in action at the
same time as the turning feed or carriage traverse by rod feed.

Lathes of 12 and 16 inches swing are back-geared, affording six changes
of speed, and the lathe tool has a vertical adjustment on a single slide
rest. Lathes of 20 inches swing are back-geared with eight changes of
speed. Lathes of 25 inches and up to 48 inches swing inclusive are
triple-geared, affording fifteen changes of speed, having a uniformly
progressive variation at each change.

The construction of the live head or headstock for a 36-inch lathe is
shown in the sectional side view in Fig. 516, and in the top view in
Fig. 517, and it will be seen that there are five changes of speed on
the cone, five with the ordinary back-gear, and five additional ones
obtained by means of an extra pinion on the end of the back-gear
spindle, and gearing with the teeth on the circumference of the face
plate, the ordinary pinion of the back-gear moving on the back-gear
spindle so as to be out of the way and clear the large gear on the cone
spindle when the wheel of the extra back-gear pinion is in use, as shown
in Fig. 517.

[Illustration: Fig. 514.]

The front bearing of the live spindle is made of large diameter to give
rigidity, and the usual collar for the face plate to screw against is
thus dispensed with. End motion to the live spindle is prevented by a
collar of hardened steel, this collar being fast on the live spindle and
abutting on one side against the end face of the back bearing and on the
other against a hardened steel thrust collar.

[Illustration: Fig. 515.]

All these parts are enclosed in a tight cast-iron tail-block, which
serves as an oil well to insure constant and perfect lubrication. The
surfaces which confine the revolving collar back and front are so
adjusted as to allow perfect freedom of rotary motion to the spindle and
collar, but no perceptible end motion. The securing of the live spindle
endwise is thus confined to the thickness of the steel collar only, and
this is so enclosed in a large mass of cast iron as to insure uniformity
of temperature in all its parts, hence there is no liability for the
live spindle to stick or jam in its bearings, while the expansion of the
live spindle endways from this collar (if it expands more than the lathe
head) is allowed for in freedom of end motion through the front journal,
which is a little longer than the bearing it runs in. In turning work
held between the lathe centres the end thrust is taken against the
hardened steel collar on the live spindle, and the hardened steel collar
at the back of it, while in turning work chucked to the face plate the
spindle is held in place endways by the confinement of the steel collar
on the spindle between the steel collar behind it and the back end of
the back bearing. With this arrangement of the spindle the change from
turning between the lathe centres and turning chucked work requires no
thought or attention to be given to any adjustment of the live spindle
to accommodate it for the changed condition of end pressure between
turning between the centres and turning chucked work, as is the case in
ordinary lathes.

The double-geared lathes, as those of 12, 16 and 20 inches swing, are
provided with face plates that unscrew from the live spindle to afford
convenience for changing from one size of face plate to another, and all
such lathes have their front live spindle journal made of sufficiently
enlarged diameter above that of the screw, to afford a shoulder for the
face plate to abut against. The nose of the live spindle is not threaded
along its entire length, but a portion next to the shoulder is made
truly cylindrical but without any thread upon it, and to this unthreaded
part the face plate accurately fits so that it is held true thereby, and
the screw may fit somewhat loosely so that all the friction acts to hold
the face plate true and hard up against the trued face of the spindle
journal. Face plates fitted in this way may be taken off and replaced as
often as need be, with the assurance that they will be true when in
place unless the surfaces have been abused in their fitting parts.

[Illustration: Fig. 516.]

The construction of the tailstock or poppet-head, as it is sometimes
termed, is shown in Figs. 518, 519, and 520. To hold it in line with the
live spindle it is fitted between the inner edges of the bed, and it
will be seen that one of the bed flanges (that on the left of the
figure) is provided on its under side with a [V], and the clamp is
provided with a corresponding [V], so that in tightening up the bolt
that secures the tailstock to the bed the tailstock is drawn up to the
edge of the shears, and therefore truly in line with the live spindle,
while when this bolt is released the tailstock is quite free to be moved
to its required position in the length of the bed. As a result of this
form of design there is no wear between the clamp and the underneath
[V], and the tailstock need not fit tightly between the edges of the
bed, hence wear between these surfaces is also avoided, while the
tailstock is firmly clamped against one edge of the bed as soon as the
clamp is tightened up by the bolt on that side.

[Illustration: Fig. 517.]

Fig. 520 shows the method of locking the tailstock spindle and of
preventing its lateral motion in the bearing in the tailstock. At the
front or dead centre end of this bearing there is between the spindle a
sleeve enveloping the spindle, and coned at its outer end, fitting into
a corresponding cone in the bore of the tailstock. Its bore is a fit to
the dead spindle, and it is split through on the lower side. Its inner
end is threaded to a sleeve that is within the headstock, and whose end
is coned to fit a corresponding cone at the inner end of the bore of the
tailstock.

[Illustration: Fig. 518.]

To this second sleeve the line shown standing vertically on the left of
the hand wheel is attached, so that operating this handle revolves the
second sleeve and the two sleeves screw together, their coned ends
abutting in their correspondingly coned seats in the tailstock bore, and
thus causing the first-mentioned and split sleeve to close upon the dead
centre spindle and yet be locked to the tailstock.

[Illustration: Fig. 519.]

As the bore of the tailstock is exactly in line with the live spindle,
it follows that the dead spindle will be locked also in line with it.

Figs. 521 and 522 represent sectional views of the carriage and slide
rest of these lathes of a size over 16 inches swing. On the feed rod
there are two bevel pinions P, one on each side of the bevel-wheel A,
and by a clutch movement either of these wheels may be placed in gear
with bevel-wheel A.

The clutch motion is operated by a lever which, when swung over to the
right, causes the bevel pinion on the right to engage with the
bevel-wheel A, and the carriage feeds to the right, while with the lever
swung over to the left the carriage feeds to the left.

On the inclined shaft is a worm, or, as the makers term it, a spiral
pinion of several teeth which gears into a straight toothed spur
gear-wheel, giving a smooth and rolling tooth contact, and therefore
producing an even and uniform feed motion.

This spur gear is fast on a shaft C, which is capable of end motion and
is provided on each of its side faces with an annular toothed clutch. On
each side of this spur-wheel is a clutch, one of which connects with the
train of gears for the turning feed, and the other with the cross-feed
gear B.

[Illustration: Fig. 520.]

When the shaft (whose end is shown at C, and to which the spur gear
referred to is fast) is pulled endways outwards from the lathe bed, its
front annular clutch engages with the clutch that sets the cross-feed
gear B in motion, and B engages with a pinion which forms the nut of the
cross-feed screw.

When shaft C is moved endways inwards its other annular clutch engages
the clutch on that side of it, and the turning feed is put into
operation. The method of operating shaft C endways is as follows:--

In a horizontal bearing D is a shaft at whose end is a weighted lever L,
and on the end of this shaft is a crank pin shown engaging a sleeve E
which affords journal bearing to the outer end of shaft C, so that
operating the weighted lever L operates E, and therefore shaft C with
the spur gear receiving motion from the worm. A simple catch confines
lever L to either of its required limits of motion, and allows the free
motion of the operating lever to start or stop either the longitudinal
or the cross feed, either of which is started or stopped by this lever,
but no mistake can occur as to which feed is operated, because the catch
above mentioned requires to be shifted to permit the feed to be
operated.

The lower end of the bell crank F engages with the sleeve E, so that
when the shaft C is operated outwards the horizontal arm of bell crank F
is depressed and the spur pinion of the cross-feed nut is free to
revolve, being driven by the cross-feed motion. When the lever F is
moved towards the lathe bed (which occurs when the stop or catch is set
to allow the longitudinal feed to be used) the nut of the cross feed is
locked fast by the horizontal arm of the bell crank F. This device makes
the whole action from one direction of feed to another automatic, and
the attention of the workman is not needed for any complicated
adjustment of parts preparatory to a change from one feed to the other.

At H is a hand wheel for hand feeding, the pinion R meshing into the
rack that extends along the front of the lathe bed; back of the hand
wheel and at H´ a clamp is provided whereby the saddle or carriage may
be locked to the lathe bed when the cross feed is being used, thus
obviating the use of a separate clamp on the bed.

The top slide of the compound rest is long and its guideway is short,
the nut being in the stationary piece G, and it will be observed that by
this arrangement at no time does the bearing surfaces of the slides
become exposed to the action of chips or dirt.

[Illustration: Fig. 521.]

Fig. 523 is a sectional view of the carriage and slide rest as arranged
for 12 and 16-inch lathes when not provided with a self-acting cross
feed. In this case end motion to shaft C is given by lever H, which is
held in its adjusted position by the tongue T. In this lathe the
screw-cutting and the turning feed cannot be put into gear at the same
time.

[Illustration: Fig. 522.]

The tool nut is arranged to enable the tool to be adjusted for height
after it is fastened in the tool post by pivoting it to the cross slide,
a spring S forcing it upwards at its outer end, thus holding the tool
point down and in the direction in which the pressure of the cut forces
it, thus preventing the wear of the pivot from letting the tool move
when it first meets the cut. The nut N is operated to adjust the tool
height, and at the same time enables the depth of cut to be adjusted
very minutely. A trough catches the water, cuttings, &c., and thus
protects the slides and slideways from undue wear.

In all these lathes the feeding mechanism is so arranged that there are
no overhanging or suspended shaft pins or spindles, each of such parts
having a bearing at each end and not depending on the face surface of a
collar or pin, as is common in many lathes. Furthermore, in these
lathes the handle for the hand carriage feed moves to the right when the
carriage moves to the right; the cross-feed screw (and the upper screw
also in compound slide rests) has a left-hand thread, so that the nut
being fixed the slides move in the same direction as though the nut
moved as in ordinary lathes. The tailstock or poppet-head screw is a
right hand because the nut moves in this case. The object of employing
right-hand screws in some cases, and left-hand ones in others, is that
it comes most natural in operating a screw to move it from right to left
to unscrew, and from left to right to screw up a piece, this being the
action of a right-hand screw, left-hand screws being comparatively
rarely used in mechanism, save when to attain the object above referred
to.

[Illustration: Fig. 523.]

Fig. 524 represents the Niles Tool Works car axle lathe, forming an
example in which the work is driven from the middle of its length,
leaving both ends free to be operated upon simultaneously by separate
slide rests.

[Illustration: Fig. 524.]

The work being driven from its centre enables it to rotate upon two dead
centres, possessing the advantage that both being locked fast there is
no liberty for the work to move, as is the case when an ordinary lathe
having one live or running spindle is used, because in that case the
live spindle must be held less firmly and rigidly than a dead centre, so
as to avoid undue wear in the live spindle bearings; furthermore, the
liability of the workman to neglect to properly adjust the bearings to
take up the wear is avoided in the case of two dead centres, and no
error can occur because of either of the centres running out of true, as
may be the case with a rotating centre.

The cone pulley and back gear are here placed at the head of the lathe
driving a shaft which runs between the lathe shears and drives a pinion
which gears with the gear on the work driving head shown to stand on the
middle of the shears. This head is hollow so that the axle passes
through it. On the face of this gear is a Clement's equalizing driver
constructed upon the principle of that shown hereafter in Fig. 756.

The means for giving motion to the feed screw and for enabling a quick
change from the coarse roughing feed to a finer finishing feed to the
cutting tool without requiring to change the gears or alter their
positions, is shown in Fig. 525. _a_ and _b_ are two separate pinions
bored a working fit to the end of the driving shaft S, but pierced in
the bore with a recess and having four notches or featherways _h_. The
end of the driving shaft S is pierced or bored to receive the handled
pin _i_, and contains four slots to receive the four feathers _j_ which
are fast in _i_. In the position shown in the figure these feathers
engage with neither _a_ nor _b_, hence the driving shaft would remain
motionless, but it is obvious that if pin _i_ be pushed in the feathers
would engage _b_ and therefore drive it; or if _i_ were pulled outwards
the feathers would engage _a_ and drive it, because _a_ and _b_ are
separate pinions with a space or annular recess between them sufficient
in dimensions to receive the feathers. The difference in the rate of
feed is obviously obtained through the difference in diameters of the
pair of wheels _a_, _c_ and the pair _d_, _b_, the lathe giving to the
lead screw the slowest motion and, therefore, the finest feed.

The means for throwing the carriage in and out of feed gear with the
feed screw and of providing a hand feed for operating the tool in
corners or for quickly traversing the carriage, is shown in Fig. 526, in
which S represents the feed screw and B a bracket or casting bolted to
the carriage and carrying the hand wheel and feed mechanism shown in the
general cut figure.

[Illustration: Fig. 525.]

B provides a slide way denoted by the dotted lines at _b_, for the two
halves N and N´ of the feed nut. It also carries a pivot pin shown at
_p_ in the front elevation, which screws into B as denoted by _p´_ in
the end view; upon this pivot operates the piece D, having the handle
_d_. In D are two cam grooves _a_ _a_; two pins _n_, which are fast in
the two half-nuts N N´, pass through slots _c_ _c_ in B, and into the
cam grooves _a_ _a_ respectively.

[Illustration: Fig. 526.]

As shown in the cut the handle _d_ of D is at its lowest point, and the
half-nuts N´ and N are in gear upon the feed screw; but suppose _d_ be
raised, then the grooves _a_ _a_ would force their respective pins _n_
up the slots _c_, and these pins _n_ being each fast to a half of the
nut, the two half-nuts would be opened clear of the feed screw, and the
carriage would cease to be fed.

The hand-feed or guide-carriage traverse motion is accomplished as
follows:--B provides at _e_ journal bearing to a stud on which is the
hand wheel shown in the general cut; attached to this hand wheel is a
pinion operating a large gear (also seen in general cut) whose pitch
line is seen at _g_, in figure. The stud carrying _g_ has journal
bearing at _f_, and carries a pinion whose pitch circle is at _h_ and
which gears with the rack.

Fig. 527, which is taken from _The American Machinist_, represents an
English self-acting lathe capable of swinging work of 12 inches diameter
over the top of the lathe shears, which are provided with a removable
piece beneath the live centre, which when removed leaves a gap,
increasing the capacity of the lathe swing. The gears for reversing the
direction of feed screw motion are here placed at the end of the live
head or headstock, the screw being used for feeding as well as for screw
cutting.

Fig. 528 represents a pattern-maker's lathe, by the Putnam Tool Co., of
Fitchburg, Massachusetts. This lathe is provided with convenient means
of feeding the tool to its cut by mechanism instead of by hand, as is
usually done by pattern-makers, and this improvement saves considerable
time, because the necessity of frequently testing the straightness of
the work is avoided.

It is provided with an iron extension shears, the upper shears sliding
in [V]-ways provided in the lower one. The hand-wheel is connected with
a shaft and pinion, which works in a rack, and is used for the purpose
of changing the position of the upper bed, which is secured in its
adjusted position by means of the tie bolts and nuts, as shown on the
front of the lower shears. This enables the gap in the lower shears to
be left open to receive work of large diameter, and has the advantage
that the gap need be opened no more than is necessary to receive the
required length of work. The slide-rest is operated by a worm set at an
angle, so as to operate with a rolling rather than a sliding motion of
the teeth, and the handle for operating the worm-shaft is balanced. The
carriage is gibbed to the bed. The largest and smallest steps of the
cone pulley are of iron, the intermediate steps being of wood, and a
brake is provided to enable the lathe to be stopped quickly. This is an
excellent improvement, because much time is often lost in stopping the
lathe while running at a high velocity, or when work of large diameter
is being turned. The lathe will swing work of 50 inches within the gap,
and the upper shears will move sufficiently to take in 4 additional feet
between the centres.

In the general view of the lathe, Fig. 528, the slide-rest is shown
provided with a [T]-rest for hand tools, but as this sets in a clip or
split bore, it may readily be removed and replaced by a screw tool,
poppet for holding a gauge, or other necessary tool. To enable the
facing of work when the gap is used, the extra attachment shown in Figs.
529 and 530 is employed. It consists of an arm or bar A, bolted to the
upper shears S by a bolt B, and clamp C, in the usual manner, and is
provided with the usual slideway and feed-screw _f_ for operating the
lower slide T, which carries a hollow stem D; over D fits a hub K, upon
the upper slide E, which hub is split and has a bolt at F, by means of
which the upper slide may be clamped to its adjusted angle or position.
The upper slider H receives the tool-post, which is parallel and fits in
a split hub, so that when relieved it may be rapidly raised or lowered
to adjust the height of the tool.

The construction of the brake for the cone pulley is shown in Figs. 531
and 532, in which P represents the pulley rim, L the brake lever, S a
wooden shoe, and W a counter-weight. The lever is pivoted at G to a lug
R, provided on the live headstock, and the brake obviously operates on
the lowest part of the cone flange; hence the lever handle is depressed
to put the brake in action.

[Illustration: _VOL. I._ =EXAMPLES IN LATHE CONSTRUCTION.= _PLATE V._

Fig. 527.

Fig. 528.

Fig. 529.]

The construction of the front and back bearings for the live spindle is
the same as that shown in Figs. 495 and 496.

[Illustration: Fig. 530.]

Wood turners sometimes have their lathes so made that the headstock can
be turned end for end on the lathe shears, so that the face plate may
project beyond the bed, enabling it to turn work of large diameter. A
better method than this is to provide the projecting end of the lathe
with a screw to receive the face plate as shown in Fig. 533, which
represents a lathe constructed by Walker Brothers of Philadelphia. At
the end of the lathe is shown a hand rest upon a frame that can be moved
about the floor to accommodate the location, requiring to be turned upon
the work.

[Illustration: Fig. 531.]

For very large work, wood-workers sometimes improvise a facing lathe, as
shown in Fig. 534, in which A is a headstock bolted to the upright B; C
is the cone pulley, and E a face plate built up of wood, and fastened to
an iron face plate by bolts. The legs A, of the tripod hand rest, Fig.
535, are weighted by means of the weights B.

[Illustration: Fig. 532.]

In Fig. 536 is shown a chucking lathe, especially adapted for boring and
facing discs, wheels, &c. The live spindle is driven by a worm-wheel,
provided around the circumference of the face plate. The driving worm
(which runs in a cup of oil) is on a driving shaft, running across the
lathe and standing parallel with the face of the face plate. This shaft
is driven by a pulley as shown, changes of speed being effected by
having a cone pulley on the counter-shaft and one on the line of
shafting.

[Illustration: Fig. 533.]

This lathe is provided with two compound slide rests. One of which may
be used for boring, while the other is employed for facing purposes.
These rests are adjustable for location across the bed of the lathe by
means of bolts in slots, running entirely across the lathe bed.

These slide rests are given a self-acting motion by the following
arrangement of parts: at the back of the live spindle is an eccentric
rod, operating a connecting rod, which is attached at its lower end to
the arm of a shaft running beneath the bed, and parallel to the lathe
spindle. This shaft passes beyond the bed where it carries a bevel
gear-wheel, which meshes with a bevel gear-wheel upon a cross shaft.
This cross shaft carries three arms, one at each end and inside its
journal bearings in the bed, and one beneath and at a right angle to the
other two. These receive oscillating motion by reason of the eccentric
connecting rod, &c.

For each compound rest there are provided two handles as usual, and in
addition an [L] lever, one arm of the latter being provided with a
series of holes, while the other carries a weight.

[Illustration: Fig. 534.]

The [L] lever carries a pawl which operates a ratchet wheel, placed on
the handle end of the slide rest cross feed screw. If then a chain be
attached to one of the holes of the [L] lever, and to the oscillating
arm, the motion in one direction of the latter will be imparted to the
[L] lever (when the chain is pulled). On the return motion of the
oscillating arm, the chain hangs loose, and the weight on the [L] lever
causes that lever arm to fall, taking up the slack of the chain, the
feed taking place (when the pawl is made to engage with the ratchet
wheel) during the motion of the oscillating arm from right to left, or
while pulling the chain.

The rate of feed is varied by attaching the chain to different holes in
the [L] lever.

To operate the rests in a line parallel to the lathe spindle, a similar
[L] lever is attached by chain to the third oscillating arm, which is
placed on the cross shaft, mid-way of the bed, or between the two slide
rests. It is obvious then that with an [L] lever attachment on each feed
screw, both slides of each rest may be simultaneously operated, while
either one may be stopped either by detaching the chain or removing the
[L] lever.

For operating the rests by hand, the usual feed-screw handles are used.

Fig. 537 represents a 90-inch swing lathe by the Ames Manufacturing
Company of Chicopee, Massachusetts.

[Illustration: Fig. 535.]

The distinguishing feature of this lathe is that the tailstock spindle
is made square, to better enable it to bear the strain due to carrying
cutting tools in place of the dead centre; and by means of a pulley
instead of a simple hand wheel for operating the tail spindle, that
spindle may be operated from an overhead countershaft, and a tool may be
put in to cut key-ways in pulleys, wheels, &c., chucked on the face
plate (which of course remains stationary during the operation), thus
dispensing with the necessity of cutting out such key-ways by hammer,
chisel, and file, in wheel bores too large and heavy to be operated upon
in a slotting machine.

[Illustration: Fig. 538.]

On account of the weight of the tailstock it is fitted with rollers,
which may be operated to lift it from the bed when it is to be moved
along the lathe bed.

[Illustration: _VOL. I._ =CHUCKING LATHES.= _PLATE VI._

Fig. 536.

Fig. 537.]

Fig. 538 represents a 50-inch swing lathe by the New Haven
Manufacturing Company of New Haven, Connecticut. The compound rest is
here provided with automatic feed so that it may be set at an angle to
bore tapers with a uniform feed. The tailstock is provided with a
bracket, carrying a pinion in gear with the hand-feed rack, so as to
move the tailstock along the bed by means of the pinion. The feed screw
is splined to give an independent feed, and the swing frame is operated
by a worm as shown.

[Illustration: Fig. 539.]


GAP LATHE OR BREAK LATHE.

The gap lathe is one in which the bed is provided with a gap beneath the
face plate, so as to enable that plate or the chucks to swing work of
larger diameter, an example being given in Fig. 539.

[Illustration: Fig. 540.]

It is obvious, however, that the existence of the gap deprives the slide
rest of support on one side, when it is used close to the face plate.
This is obviated in some forms of gap lathes by fitting into the gap a
short piece of bed that may be taken out when the use of the gap is
required.

The gap lathe has not found favor in the United States, the same result
being more frequently obtained by means of the extension lathe, which
possesses the advantages of the gap lathe, while at the same time
enabling the width of the gap to be varied to suit the length of the
work. Fig. 540 represents an extension lathe by Edwin Harrington and
Son, of Philadelphia. There are two beds A and B, the former sliding
upon the latter when operated by the hand-wheel E, which is upon the end
of a screw that passes between the two beds, has journal bearing in the
upper bed, and engages a nut in the lower one, so that as the screw is
operated the wheel moves longitudinally with the upper bed. C is the
feed rod which communicates motion to the feeding screw D, which has
journal bearing on the upper bed and therefore travels with it when it
is moved or adjusted longitudinally. The cross slide has sufficient
length to enable the slide rest to face work of the full diameter that
will swing in the gap, and to support the slide rest when moved outwards
to the full limit, it is provided with a piece F, which slides at its
base upon the guideway or slide G.

Fig. 541 represents a double face plate lathe such as is used for
turning the wheels for locomotives. The circumference of both the face
plates are provided with spur teeth, so that both are driven by pinions,
which by being capable of moving endways into or out of gear, enable
either face plate to be used singly, if required, as for boring
purposes.

The slide rests are operated by ratchet arms for the self feed, these
arms being operated by an overhead shaft, with arms and chains.

[Illustration: Fig. 541.]

Fig. 542 represents a chucking lathe adapted more especially for boring
purposes. Thus the cone pulley is of small diameter and the parts are
light, so that the lathe is more handy than would be the case with a
heavier built lathe, while at the same time it is sufficiently rigid for
large work that is comparatively light.

[Illustration: Fig. 542.]

The compound rest is upon a pedestal that can be bolted in any required
position on the lower cross slide, and is made self-acting for the feed
traverse by the change wheels and feed screw, while the self-acting
cross feed is operated by a ratchet handle, actuated by a chain from an
overhead reciprocating lever; the latter being actuated from the crank
pin at A, which is adjustable in a slot in the crank disk B. A lathe of
this kind is very suitable for brass work of unusually large diameter,
because in such work the cuts and feeds are light, and the cutting speed
is quick, hence a heavy construction is not essential.

Figs. 543 and 544 represent a large lathe built by Thomas Shanks and
Co., of Johnstone, near Glasgow, Scotland; all the figures of this lathe
being from _The American Machinist_.

Fig. 543 shows the headstock and two of the slide rests, while Fig. 544
represents the remainder of the bed, the tailstock, and two of the slide
rests.

It will be seen from the figures that there are a compound rest and a
column or pillar rest both at the front and at the back of the lathe,
and that there is an additional rest on the front end of the tailstock
which may be used for facing the ends of the work.

Fig. 545 represents a section through, and a partial plan of the
headstock, and it will be seen that the live spindle is free from the
cone pulley and from the gearing, the chuck plate being driven from a
pinion engaging an internal gear at the back of the chuck plate. By this
construction the balancing of such work as crank shafts is facilitated,
because the chuck plate is not affected by the friction of the driving
gears, and may therefore be easily revolved to test the balance of the
work.

Fig. 546 represents a cross section through the bed, and through one of
the compound rests, and one of the pillar rests, the latter rests being
made thin so that they may pass between the cheeks of crank shafts, to
turn their faces and the crank journals.

Fig. 547 represents a view from the back end of the headstock, and Fig.
548 a view of the lathe from the tailstock end.

Figs. 549 and 550 represent a plan and a side view of the headstock and
the two slide rests nearest to it. The lathe being shown at work on the
crank shaft of the steamship service, which is shown in dotted lines,
and it will be seen that for turning the stem of the shaft all the rests
can be used at once, those at the back of the lathe having their
cutting tools turned upside down (as will be more clearly seen in the
cross-sectional view of the rests in Fig. 546).

[Illustration: Fig. 543.]

Figs. 551 and 552 represent a plan and a side view of the other half of
the lathe in operation upon the same crank shaft, which is again shown
in dotted lines.

[Illustration: Fig. 544.]

Referring now to the general construction of the lathe, the headstock or
live spindle has a front journal bearing 18 inches diameter and 24
inches long, and a back bearing 12 inches diameter and 15 inches long,
the bearings being parallel. The driving cone has five changes of speed
for a 6-inch belt, and is carried on an independent spindle. The cone is
turned inside as well as outside, so as to be in balance at high speeds.

[Illustration: Fig. 545.]

The face plate is 12 feet diameter, cast with internal gear at the back.
It is provided with [T]-slots and square holes for fixing work. It is
bolted to a large flange in one piece with the spindle, and fitted with
four steel expanding gripping jaws worked with screws and toothed
blocks. These are for doing chuck work, or for gripping work to be
driven, as the collars of propeller or crank shafts, or work of a
similar character. By the system of gearing adopted, when desired, the
face plate can be revolved almost free, which facilitates balancing for
turning crank shafts, as well as other operations. The thrust against
the live spindle is taken by an adjustable steel tail piece.

[Illustration: Fig. 546.]

[Illustration: Fig. 547.]

[Illustration: Fig. 548.]

[Illustration: Fig. 549.]

The beds are double, 10 feet in width over all, the sections being
joined together by massive ground plates and bolts. They are made with
square lips to resist the upward strain of cutting. The front bed is
fitted with two saddles, each carrying a compound slide rest having the
following movements: First, screw-cutting, by means of a leading screw,
situated inside the bed, with a sliding disengaging nut and reversing
motion for right or left-hand threads, or for instantaneously stopping
the longitudinal movement of the saddle. This is accomplished by a set
of clutch mitres placed inside the bed at headstock end, and actuated by
a lever in front: Second, a self-acting surfacing motion to slide rest
by means of a longitudinal shaft at the front of the bed, and clutch
mitres for reversing the saddle screw.

[Illustration: Fig. 550.]

Third, power motion for moving the saddles quickly to position along the
bed. This is done through the fast and loose pulleys at the headstock
end of lathe.

Fourth, hand rack motion to saddle. The back bed is fitted with two
saddles, each carrying a pillar rest, fitted for all movements in plain
turning like the front rests, and also with swiveling motion for corner
turning.

[Illustration: Fig. 551.]

The tailstock has a spindle 9 inches diameter. It is fitted in [V]s on
the bed, and held down by three [T]-head bolts on each side. The top
section is adjustable for turning tapers. It is moved along the ways by
engaging a nut with the main screw. An end-cutting rest is fitted to the
tailstock, which is adapted for operating on flanged couplings and
similar work.

There is a separate set of change wheels for each saddle, so arranged as
to cut standard pitches up to 3-inch pitch, and for self-acting feeds
down to 50 per inch. By this means, when both tools are in operation on
a piece of work, one tool may be used with coarse feed for roughing out,
while the other may be taking a fine or finishing cut either on the same
or a different part of the piece; or one tool may be cutting towards and
the other from the face plate, always maintaining the balance of a front
and back cut.

[Illustration: Fig. 552.]

Complete counter driving motion, consisting of wall brackets, shaft,
cone, and sets of fast and loose pulleys for quick reversing motion in
screw cutting, also belt bar shipping motion, and full set of
case-hardened wrenches are provided.



CHAPTER VII.--DETAILS IN LATHE CONSTRUCTION.


Although in each class of lathe the requirements may be practically the
same, yet there is a variety of different details of construction by
means of which these requirements may be met or filled, and it may be
profitable to enter somewhat into these requirements and the different
constructions generally employed to meet them.

[Illustration: Fig. 553.]

The cone spindle or live spindle of a lathe should be a close working
fit to its boxes or bearings, so that it will not lift under a heavy
cut, or lift and fall under a cut of varying pressure. This lifting and
falling may occur even though the work be true, and the cut therefore of
even depth all around the work, because of hard seams or spots in the
metal.

It is obvious that the bearings should form a guide, compelling the live
spindle to revolve in a true circle and in a fixed plane, the axis of
revolution being in line with the centre line of the tail spindle and
that means should be provided to maintain this alignment while
preserving the fit, or in other words taking up the wear. The spindle
journals must, to produce truly cylindrical work, be cylindrically true,
or otherwise the axis of its revolution will change as it revolves, and
this change will be communicated through the live centre to the work, or
through the chuck plate to the work, as the case may be.

The construction of the bearings should be such, that end motion to the
spindle is prevented in as short a length of the spindle as possible,
the thrust in either direction being resisted by the mechanism contained
in one bearing.

In Fig. 553 is a form of construction for the front bearing (as that
nearest to the live centre is called), in which end motion to the
spindle is prevented at the same time as the diametral fit is adjusted.
The spindle is provided with a cone at C and is threaded at T to receive
two nuts N which draw the spindle cone within the bearing. In this case
the journal at the back end may be made parallel, so that if the spindle
either expands or contracts more under variations of temperature than
the frame or head carrying the bearings or bearing boxes, it will not
bind endwise, nor will the fit be impaired save inasmuch as there may be
an inequality of expansion in the length of the front journal and its
box. In this case, however, the end pressure caused by holding the work
between the lathe centres acts to force the spindle into its bearing and
increase the tightness of its fit, hence it is not unusual to provide at
the back bearing additional means to resist the thrust of the dead
centre.

[Illustration: Fig. 554.]

Fig. 554, which is taken from "Mechanics," represents Wohlemberg's
patent lathe spindle, in which both journals are coned, fitting into
bushes which can be replaced by new ones when worn; the end thrust is
here taken by a steel screw, while the end fit is adjusted by means of a
ring nut which binds the face of the large cone gear against the inside
face of the front bearing and by the face of the gear that drives the
change gears. It may be pointed out, however, that in this construction
the spindle must be drawn within to adjust the fit of the front bearing,
which can only be done by adjusting the pinion that drives the change
gears, or by screwing up the nut that is inside the cone, and therefore
cannot be got at. The back bearing can be adjusted by means of the ring
nuts provided at each of its ends.

[Illustration: Fig. 555.]

Fig. 555 represents another design of cone bearing, in which the spindle
is threaded to receive the nuts A which draw it within the front bearing
and thus adjust the fit, and at the same time prevent end motion. The
back bearing is provided with a bush parallel outside, and furnished
with a nut at B to adjust the fit of the end bearing. To prevent the end
pressure of the dead centre from forcing the spindle cones too tightly
within their bearings a cross piece P is employed (being supported by
two studs provided in the head), and through P passes an adjusting screw
D, having nuts N and C, one on each side of P. Between the end of D and
of the lathe spindle a washer of leather or of raw hide is placed to
prevent the end faces from abrading. A similar device for taking up the
end thrust is often provided to lathes in which the journals are both
parallel, fitting in ordinary boxes, a top view of the device being
illustrated in Fig. 556, in which B is the back bearing box, S S two
studs supporting cross-piece P, and N and C are adjusting nuts. G is the
gear for driving the change wheels for screw cutting or for ordinary
feeding as the case may be. In this design the gear wheel G remains
fixed and the combinations of gears necessary to cut various pitches of
thread must be made on the lead screw and on the swing frame, which must
be long enough to permit the change gear stud to pass up to permit the
smallest change wheel to gear with wheel G, and which is provided with
two grooves E and F, Fig. 557, for two studs to carry two compounded
pairs of change wheels. This compounding in two places on the swing
frame enables gear G to be comparatively large, and thus saves the teeth
from rapid wear, while it facilitates the cutting of left-hand threads,
because it affords more convenience for putting in a gear to change the
direction of feed screw revolution.

[Illustration: Fig. 556.]

In many lathes of American design the journals are made parallel, and
the end play is taken up at the back bearing, an example being given in
Fig. 558, in which the back bearing boxes are made in two halves A and
B, the latter having a set screw (with check nut) threaded through it
and bearing against a washer that meets the end of the spindle.

[Illustration: Fig. 557.]

A simple method of preventing end motion is shown in Fig. 559, a bracket
B affording a support for a threaded adjusting screw, which is sometimes
made pointed and at others flat. When pointed it acts to support the
spindle, but on the other hand it also acts to prevent the journal from
bedding fairly in the boxes. In some cases of small lathes the back
bearing is dispensed with, and a similar pointed adjusting screw takes
its place, which answers very well for very small work.

Since the strain of the cut carried by the cutting tool falls mainly
upon the live centre end of the cone spindle, it is obvious that the
bearing at that end has a greater tendency to wear.

[Illustration: Fig. 558.]

In addition to this the weight of the cone itself is greatest at that
end, and furthermore the weight of the face plate or chuck, and of the
work, is carried mainly at that end. If, however, one journal and
bearing wears more than the other, the spindle is thrown out of line
with the lathe shears, and with the tail block spindle. The usual method
of obviating this as far as possible is to give that end a larger
journal-bearing area.

[Illustration: Fig. 559.]

The direction in which this wear will take place depends in a great
measure upon the kind of work done in the lathe; thus in a lathe running
slowly and doing heavy work carried by chucks, or on the face plate, the
wear would be downwards and towards the operator, the weight of the
chuck, &c., causing the downward, and the resistance or work-lifting
tendency of the cut causing the lateral wear. As a general rule the wear
will be least in a lateral direction towards the back of the lathe, but
the direction of wear is so variable that provision for its special
prevention or adjustment is not usually made. In the S. W. Putnam lathe,
provision is made that the bearing boxes may be rotated in the head, so
that when the lathe is used on a class of work that caused the live
spindle to wear the bearing boxes on one side more than on another, the
boxes may be periodically partly rotated in the head so that further
wear will correct the evil.

The coned hole to receive the live centre should run quite true, so that
the live centre will run true without requiring, when inserted, to be
placed in exactly the same position it occupied when being turned up at
its conical point. But when this hole does not run true a centre punch
dot is made on the end of the spindle, and another on the centre, so
that by placing the two dots to coincide at all times, the centre will
run true.

The taper given to lathe centres varies from 9/16 per foot to 1 inch per
foot. In the practice of Pratt and Whitney a taper of 9/16 per foot is
given to all lathes, the lengths of the tapers for different sizes of
lathes being as follows:

                                   Length of Taper Socket
  Swing of Lathe.                  for Live Centre.

    13 inches                           5    inches.
    16   "                              3-3/4   "
    18 and 19 inches                    7-11/16 "
        "       "    with hollow spindle 5 inches long
       and 1-1/16 diameter at the small end.

The less the amount of taper the more firmly the centre is held, but the
more difficult it becomes to remove the centre when necessary.

[Illustration: Fig. 560.]

The principal methods of removing live centres are shown in Fig. 560, in
which is shown at B a square part to receive a wrench, it being found
that if not less than about 1/2-inch taper per foot of length be given
to the live spindle socket, then revolving the centre with a wrench will
cause it to release itself, enabling it to be removed by hand. Another
method employed on small lathes is to drill a hole through the live
spindle to receive a taper pin P, the live centre end being shown at C.

Another and excellent plan for large lathes, is to thread the centre and
provide it with a nut M, which on being screwed against the end face of
the live spindle will release the centre. The objection to the use of
the pin P is that it is apt to become mislaid, and it is not advisable
to use a hammer about the parts of the lathe, especially in such an
awkward place as between the journal bearing and the cone, which is
where the pin hole requires to be located. The square section is,
therefore, the best method for small lathes, and the nut for large ones.

In cases where the live spindle is made hollow a bar may be passed
through from the rear end to remove the centre; this also enables rods
of iron to be passed through the spindle, leaving the end projecting
through the chuck for any length necessary for the work to be turned out
of its exposed end.

The dead centre may be extracted from the tail spindle by a pin and hole
as in Fig. 560, or, what is better, by contact with the end of the tail
screw as described when referring to the tail stock of the S. W. Putnam
lathe.

The cone pulley should be perfectly balanced, otherwise at high speeds
the lathe will shake or tremble from the unbalanced centrifugal motion,
and the tremors will be produced to some extent on the work. The steps
of the cone should be amply wide, so that it may have sufficient power,
without overstraining the belt, to drive the heaviest cut the lathe is
supposed to take without the aid of the back gear.

In some cases, as in spinning lathes, the order of the steps is
reversed, the smallest step of the cone being nearest to the live
centre, the object being to have the largest step on the left, and
therefore more out of the way.

The steps of the cone should be so proportioned that the belt will shift
from one to the other, and have the same degree of tension, while at the
same time they should give a uniform graduation or variation of speed
throughout, whether the lathe runs in single gear or with the back gear
in. This is not usually quite the case although the graduation is
sufficiently accurate for practical purposes. The variation in the
diameter of the steps of a lathe cone varies from an inch for lathes of
about 12-inch swing, up to 2 inches for lathes of about 30-inch swing,
and 3 inches for lathes of 5 or more feet of swing.

To enable the graduation of speed of the cone to be uniform throughout,
while the tension of the belt is maintained the same on whatever step
the cone may be, the graduation of the steps may be varied, and this
graduation may be so proportioned as to answer all practical purposes if
the overhead or countershaft cone and that on the lathe are alike.

The following on this subject is from the pen of Professor D. E. Klein,
of Yale College.

"The numbers given in the following tables are the differences between
the diameters of the adjacent steps on either cone pulley, and are
accurate within half a hundredth of an inch, which is a degree of
accuracy sufficient for practical purposes.

By simply omitting a step at each end of the cone, the two tables given
will be found equally well adapted for determining the diameters of
cones having four and three steps respectively.

The following are examples in the use of the tables. Suppose the centres
of a pair of pulley shafts to be 60 inches apart, and that the
difference of diameter between the adjacent steps is to be as near to
2-1/2 inches as can be, to obtain a uniformity of speed graduation and
belt tension, also that each cone is to have six steps, the smallest of
which is to be of five inches diameter.

To find the diameters for the remaining steps, we look in Table I.
(corresponding to cone pulleys with six steps), under 60 in. and
opposite 2-1/2 in. and obtain the differences,

  2.37        2.43        2.50        2.57        2.63

Each of these differences is _subtracted_ from the _larger_ diameter of
the two adjacent steps to which it corresponds, thus:

                               17.50 = 1st step.
  Difference of 1st and 2nd =   2.37
                               -----
                               15.13 = 2nd  "
      "         2nd  "  3rd =   2.43
                               -----
                               12.70 = 3rd  "
      "         3rd  "  4th =   2.50
                               -----
                               10.20 = 4th  "
      "         4th  "  5th =   2.57
                               -----
                                7.63 = 5th  "
      "         5th  "  6th =   2.63
                               -----
                                5.00 = 6th "

EXAMPLE 2. If we suppose the same conditions as in Example 1, with the
exception that each cone is to have four steps instead of six, the
largest diameter will, in this case, equal 12-1/2 in. and we may obtain
the remaining diameters by omitting the end differences of the above
example, and then subtracting the remaining differences as follows:

                               12.50 = 2nd step.
  Difference of 2nd and 3rd =   2.43
                               -----
                               10.07 = 3rd   "
      "         3rd  "  4th =   2.50
                               -----
                                7.57 = 4th   "
      "         4th  "  5th =   2.57
                               -----
                                5.00 = 5th   "

The 2nd, 3rd, 4th, and 5th steps of the table correspond respectively to
the 1st, 2nd, 3rd, and 4th steps of the cone, having but four steps. If
the smallest diameter had not been assumed equal to 5 in. we might have
dropped a step at each end of the six-step cone of the preceding
example, and employed the remaining four diameters, 15.13 in. 12.70 in.
10.20 in. and 7.63 in. for one four-step cone.

The present and the previous examples show that we can assume the size
of the smallest step anything that we please, and, other things being
equal, can make the required cones large or small.

I.--TABLE FOR FINDING CONE PULLEY DIAMETERS WHEN THE TWO PULLEYS ARE
CONNECTED BY AN OPEN BELT, AND ARE EXACTLY ALIKE.

The numbers given in table are the differences between the diameters of
the adjacent steps on either cone pulley, and can be employed when there
are either six or four steps on a cone. When there are six steps, the
largest is the first, and the smallest the sixth step of the table. When
there are four steps, the largest is the second, and the smallest the
fifth step of the table.

  +-------------+-----------+------------------------------
  |   Average   |  Adjacent | DISTANCE BETWEEN THE CENTRES
  |  difference |   steps,  |       OF CONE PULLEYS.
  |    between  |   whose   +----+----+----+----+----+----+
  |      the    |   diffe-  |    |    |    |    |    |    |
  |   adjacent  |  rence is | 10 | 20 | 30 | 40 | 50 | 60 |
  |    steps.   |  given in |         i n c h e s.        |
  |             |   table.  |    |    |    |    |    |    |
  +-------------+-----------+----+----+----+----+----+----+
  |             |1st and 2nd|0.87|0.94|0.96|0.97|0.98|0.98|
  |             |2nd  "  3rd|0.94|0.97|0.98|0.98|0.99|0.99|
  | 1 inch      |3rd  "  4th|1.00|1.00|1.00|1.00|1.00|1.00|
  |             |4th  "  5th|1.06|1.03|1.02|1.02|1.01|1.01|
  |             |5th  "  6th|1.13|1.06|1.04|1.03|1.02|1.02|
  +-------------+-----------+----+----+----+----+----+----+
  |             |1st and 2nd|1.21|1.36|1.40|1.43|1.44|1.45|
  |             |2nd  "  3rd|1.36|1.43|1.45|1.46|1.47|1.48|
  | 1-1/2 inch  |3rd  "  4th|1.50|1.50|1.50|1.50|1.50|1.50|
  |             |4th  "  5th|1.64|1.57|1.55|1.54|1.53|1.52|
  |             |5th  "  6th|1.79|1.64|1.60|1.57|1.56|1.55|
  +-------------+-----------+----+----+----+----+----+----+
  |             |1st and 2nd|1.47|1.74|1.83|1.87|1.90|1.92|
  |             |2nd  "  3rd|1.74|1.87|1.92|1.93|1.95|1.96|
  | 2 inches    |3rd  "  4th|2.00|2.00|2.00|2.00|2.00|2.00|
  |             |4th  "  5th|2.26|2.13|2.08|2.07|2.05|2.04|
  |             |5th  "  6th|2.53|2.26|2.17|2.13|2.10|2.08|
  +-------------+-----------+----+----+----+----+----+----+
  |             |1st and 2nd|1.66|2.10|2.23|2.30|2.34|2.37|
  |             |2nd  "  3rd|2.10|2.30|2.37|2.40|2.42|2.43|
  |2-1/2 inches |3rd  "  4th|2.50|2.50|2.50|2.50|2.50|2.50|
  |             |4th  "  5th|2.90|2.70|2.63|2.60|2.58|2.57|
  |             |5th  "  6th|3.34|2.90|2.77|2.70|2.66|2.63|
  +-------------+-----------+----+----+----+----+----+----+
  |             |1st and 2nd|1.76|2.42|2.62|2.71|2.77|2.81|
  |             |2nd  "  3rd|2.42|2.71|2.81|2.86|2.88|2.90|
  | 3 inches    |3rd  "  4th|3.00|3.00|3.00|3.00|3.00|3.00|
  |             |4th  "  5th|3.58|3.29|3.19|3.14|3.12|3.10|
  |             |5th  "  6th|4.24|3.58|3.38|3.29|3.23|3.19|
  +-------------+-----------+----+----+----+----+----+----+
  |             |1st and 2nd|    |3.95|3.31|3.49|3.59|3.66|
  |             |2nd  "  3rd|2.94|3.49|3.66|3.75|3.80|3.83|
  | 4 inches    |3rd  "  4th|4.00|4.00|4.00|4.00|4.00|4.00|
  |             |4th  "  5th|5.06|4.51|4.34|4.25|4.20|4.17|
  |             |5th  "  6th|    |5.05|4.69|4.51|4.41|4.34|
  +-------------+-----------+----+----+----+----+----+----+
  |             |1st and 2nd|    |3.33|3.92|4.20|4.36|4.47|
  |             |2nd  "  3rd|3.31|4.19|4.47|4.60|4.68|4.74|
  | 5 inches    |3rd  "  4th|5.00|5.00|5.00|5.00|5.00|5.00|
  |             |4th  "  5th|6.69|5.81|5.53|5.40|5.32|5.26|
  |             |5th  "  6th|    |6.67|6.09|5.80|5.64|5.53|
  +-------------+-----------+----+----+----+----+----+----+
  |             |1st and 2nd|    |3.52|4.42|4.83|5.08|5.23|
  |             |2nd  "  3rd|    |4.83|5.23|5.42|5.54|5.62|
  | 6 inches    |3rd  "  4th|    |6.00|6.00|6.00|6.00|6.00|
  |             |4th  "  5th|    |7.17|6.77|6.58|6.46|6.38|
  |             |5th  "  6th|    |8.48|7.58|7.17|6.92|6.77|
  +-------------+-----------+----+----+----+----+----+----+

  +-------------+-----------+-----------------------------+
  |   Average   |  Adjacent | DISTANCE BETWEEN THE CENTRES|
  |  difference |   steps,  |       OF CONE PULLEYS.      |
  |    between  |   whose   +----+----+----+----+----+----+
  |      the    |   diffe-  |    |    |    |    |    |    |
  |   adjacent  |  rence is | 70 | 80 | 90 | 100| 120| 240|
  |    steps.   |  given in |         i n c h e s.        |
  |             |   table.  |    |    |    |    |    |    |
  +-------------+-----------+----+----+----+----+----+----+
  |             |1st and 2nd|0.98|0.98|0.99|0.99|0.99|1.00|
  |             |2nd  "  3rd|0.99|0.99|0.99|0.99|1.00|1.00|
  | 1 inch      |3rd  "  4th|1.00|1.00|1.00|1.00|1.00|1.00|
  |             |4th  "  5th|1.01|1.01|1.01|1.01|1.00|1.00|
  |             |5th  "  6th|1.02|1.02|1.01|1.01|1.01|1.00|
  +-------------+-----------+----+----+----+----+----+----+
  |             |1st and 2nd|1.46|1.46|1.47|1.47|1.48|1.49|
  |             |2nd  "  3rd|1.48|1.48|1.49|1.49|1.49|1.49|
  | 1-1/2 inch  |3rd  "  4th|1.50|1.50|1.50|1.50|1.50|1.50|
  |             |4th  "  5th|1.52|1.52|1.51|1.51|1.51|1.51|
  |             |5th  "  6th|1.54|1.54|1.53|1.53|1.52|1.51|
  +-------------+-----------+----+----+----+----+----+----+
  |             |1st and 2nd|1.93|1.93|1.94|1.95|1.96|1.98|
  |             |2nd  "  3rd|1.96|1.97|1.97|1.97|1.98|1.99|
  | 2 inches    |3rd  "  4th|2.00|2.00|2.00|2.00|2.00|2.00|
  |             |4th  "  5th|2.04|2.03|2.03|2.03|2.02|2.01|
  |             |5th  "  6th|2.07|2.07|2.06|2.05|2.04|2.02|
  +-------------+-----------+----+----+----+----+----+----+
  |             |1st and 2nd|2.39|2.40|2.41|2.42|2.43|2.47|
  |             |2nd  "  3rd|2.44|2.45|2.46|2.46|2.47|2.49|
  | 2-1/2 inches|3rd  "  4th|2.50|2.50|2.50|2.50|2.50|2.50|
  |             |4th  "  5th|2.56|2.55|2.54|2.54|2.53|2.51|
  |             |5th  "  6th|2.61|2.60|2.59|2.58|2.57|2.53|
  +-------------+-----------+----+----+----+----+----+----+
  |             |1st and 2nd|2.84|2.86|2.87|2.88|2.90|2.95|
  |             |2nd  "  3rd|2.92|2.93|2.94|2.94|2.95|2.98|
  | 3 inches    |3rd  "  4th|3.00|3.00|3.00|3.00|3.00|3.00|
  |             |4th  "  5th|3.08|3.07|3.06|2.06|3.05|3.02|
  |             |5th  "  6th|3.16|3.14|3.13|3.12|3.10|3.05|
  +-------------+-----------+----+----+----+----+----+----+
  |             |1st and 2nd|3.71|3.75|3.78|3.80|3.83|3.91|
  |             |2nd  "  3rd|3.85|3.87|3.88|3.89|3.91|3.96|
  | 4 inches    |3rd  "  4th|4.00|4.00|4.00|4.00|4.00|4.00|
  |             |4th  "  5th|4.15|4.13|4.12|4.11|4.09|4.04|
  |             |5th  "  6th|4.29|4.25|4.22|4.20|4.17|4.09|
  +-------------+-----------+----+----+----+----+----+----+
  |             |1st and 2nd|4.55|4.60|4.64|4.68|4.74|4.87|
  |             |2nd  "  3rd|4.77|4.80|4.82|4.84|4.86|4.93|
  | 5 inches    |3rd  "  4th|5.00|5.00|5.00|5.00|5.00|5.00|
  |             |4th  "  5th|5.23|5.20|5.18|5.16|5.14|5.07|
  |             |5th  "  6th|5.45|5.40|5.36|5.32|5.26|5.13|
  +-------------+-----------+----+----+----+----+----+----+
  |             |1st and 2nd|5.34|5.42|5.49|5.55|5.62|5.80|
  |             |2nd  "  3rd|5.67|5.71|5.75|5.77|5.81|5.90|
  | 6 inches    |3rd  "  4th|6.00|6.00|6.00|6.00|6.00|6.00|
  |             |4th  "  5th|6.33|6.29|6.25|6.23|6.19|6.10|
  |             |5th  "  6th|6.66|6.58|6.51|6.45|6.38|6.20|
  +-------------+-----------+----+----+----+----+----+----+

EXAMPLE 3. Let distance apart of the centres = 30 in. the average
difference between adjacent steps = 2 in. the diameter of the smallest
step = 4 in., and the number of steps on each of the cones = 5. The
largest step will then equal 12 in., and from Table II., under 30 in.
and opposite 2 in., we obtain the differences

  1.87         1.96          2.04          2.13

and then subtracting as before we get the required diameters

  12 in.    10.30 in.    8.17 in.    6.13 in.    4 in.

EXAMPLE 4. Let the conditions be as in the preceding example, the cone
pulley having, however, three steps instead of five, the largest
diameter will then equal 8 in.; and by dropping the end differences and
subtracting

                              8.00 = 2nd step.
  Difference of 2nd and 3rd = 1.96
                             -----
                              6.04 = 3rd  "
       "        3rd  "  4th = 2.04
                             -----
                              4.00 = 4th  "

we get the diameters 8 in., 6.04, and 4 in., which correspond
respectively to 2nd, 3rd, and 4th steps of the table, and to the 1st,
2nd, and 3rd steps of the three-step cone.

EXAMPLE 5. Let the distance apart of the centres be 60 in., the average
difference between the adjacent steps be 2-1/8 in., the smallest step 7
in. and the number of steps = 5. The largest step will then be 7 in. +
(4 × 2-1/8) = 15-1/2 inches.

Now an inspection of Table II. will show that it contains no horizontal
lines corresponding to the average difference 2-1/8 inches, we cannot,
therefore, as heretofore, obtain the required differences directly, but
must interpolate as follows: since 2-1/8 inches is quarter way between 2
inches and 2-1/2 inches, the numbers corresponding to 2-1/8 inches (for
any given distance apart of the centres), will be quarter way between
the numbers of the table corresponding to 2 inches and 2-1/2 inches.
Thus, in Table II., we have under 60 inches,

  and opposite 2-1/2 in.:  2.40    2.47    2.53    2.60
         "     2           1.93    1.98    2.02    2.07
                           ----    ----    ----    ----
                            .47     .49     .51     .53

Dividing these differences by 4, we get:

  .12       .12       .13       .13

to which we add,

  1.93      1.98      2.02      2.07

and get for the differences corresponding to 2-1/8 inches

  2.05      2.10      2.15      2.20

and subtracting as before,

                              15.5   1st step.
  difference of 1st and 2nd =  2.05
                              -----
                              13.45 = 2nd   "
       "        2nd  "  3rd =  2.10
                              -----
                              11.35 = 3rd   "
       "        3rd  "  4th =  2.15
                              -----
                               9.20 = 4th   "
       "        4th  "  5th =  2.20
                              -----
                               7.00 = 5th   "

Thus far, however, we have considered only the case where the two cone
pulleys were exactly alike. Now although this case occurs much more
frequently than the case in which the cone pulleys are unlike, it is
nevertheless true that unlike cone pulleys occur with sufficient
frequency to make it desirable that convenient means be established for
obtaining the diameters of their steps rapidly and accurately, and Table
III. was calculated by the writer for this purpose; its accuracy is more
than sufficient for the requirements of practice, the numbers in the
table being correct to within a unit of the fourth decimal place (_i.e._
within .0001). It should be noticed that the tabular quantities are not
the diameters of the steps, but these diameters divided by the distance
between the centres of the cone pulleys; in other words, the tabular
quantities are the effective diameters of the steps only when the
centres of the pulleys are a unit's distance apart. By thus expressing
the tabular quantities in terms of the distance apart of the axis, the
table becomes applicable to all cone pulleys whatever their distance
from each other, the effective diameters of the steps being obtained by
multiplying the proper tabular quantities by the distance between the
centres of the pulleys.

II.--TABLE FOR FINDING CONE PULLEY DIAMETERS WHEN THE TWO PULLEYS ARE
CONNECTED BY AN OPEN BELT, AND ARE EXACTLY ALIKE.

The numbers given in table are the differences between the diameters of
the adjacent steps on either cone pulley, and can be employed when there
are either five or three steps on a cone.

  +-------------+-----------+-----------------------------+
  |   Average   |  Adjacent | DISTANCE BETWEEN THE CENTRES|
  |  difference |   steps,  |       OF CONE PULLEYS.      |
  |    between  |   whose   +----+----+----+----+----+----+
  |     the     |   diffe-  |    |    |    |    |    |    |
  |   adjacent  |  rence is | 10 | 20 | 30 | 40 | 50 | 60 |
  |    steps.   |  given in |         i n c h e s.        |
  |             |   table.  |    |    |    |    |    |    |
  +-------------+-----------+----+----+----+----+----+----+
  |             |1st and 2nd|0.90|0.95|0.97|0.98|0.98|0.98|
  |             |2nd  "  3rd|0.97|0.98|0.99|0.99|0.99|0.99|
  | 1 inch      |3rd  "  4th|1.03|1.02|1.01|1.01|1.01|1.01|
  |             |4th  "  5th|1.10|1.05|1.03|1.02|1.02|1.02|
  +-------------+-----------+----+----+----+----+----+----+
  |             |1st and 2nd|1.28|1.39|1.43|1.45|1.46|1.46|
  |             |2nd  "  3rd|1.43|1.46|1.48|1.48|1.48|1.49|
  | 1-1/2 inch  |3rd  "  4th|1.57|1.54|1.52|1.52|1.52|1.51|
  |             |4th  "  5th|1.72|1.61|1.57|1.55|1.54|1.54|
  +-------------+-----------+----+----+----+----+----+----+
  |             |1st and 2nd|1.61|1.81|1.87|1.90|1.92|1.93|
  |             |2nd  "  3rd|1.87|1.94|1.96|1.97|1.97|1.98|
  | 2 inches    |3rd  "  4th|2.13|2.06|2.04|2.03|2.03|2.02|
  |             |4th  "  5th|2.39|2.19|2.13|2.10|2.08|2.07|
  +-------------+-----------+----+----+----+----+----+----+
  |             |1st and 2nd|1.89|2.20|2.30|2.35|1.38|2.40|
  |             |2nd  "  3rd|2.30|2.40|2.43|2.45|2.46|2.47|
  | 2-1/2 inches|3rd  "  4th|2.70|2.60|2.57|2.55|2.54|2.53|
  |             |4th  "  5th|3.11|2.80|2.70|2.65|2.62|2.60|
  +-------------+-----------+----+----+----+----+----+----+
  |             |1st and 2nd|2.10|2.57|2.71|2.78|2.83|2.86|
  |             |2nd  "  3rd|2.71|2.86|2.90|2.93|2.94|2.95|
  | 3 inches    |3rd  "  4th|3.29|3.14|3.10|3.07|3.06|3.05|
  |             |4th  "  5th|3.90|3.43|3.29|3.22|3.17|3.14|
  +-------------+-----------+----+----+----+----+----+----+
  |             |1st and 2nd|    |3.22|3.49|3.62|3.69|3.75|
  |             |2nd  "  3rd|3.48|3.74|3.83|3.87|3.90|3.91|
  | 4 inches    |3rd  "  4th|4.52|4.26|4.17|4.13|4.10|4.09|
  |             |4th  "  5th|    |4.78|4.51|4.38|4.31|4.25|
  +-------------+-----------+----+----+----+----+----+----+
  |             |1st and 2nd|    |3.77|4.20|4.40|4.52|4.60|
  |             |2nd  "  3rd|4.19|4.60|4.73|4.80|4.84|4.87|
  | 5 inches    |3rd  "  4th|5.81|5.40|5.27|5.20|5.16|5.13|
  |             |4th  "  5th|    |6.23|5.80|5.60|5.48|5.40|
  +-------------+-----------+----+----+----+----+----+----+
  |             |1st and 2nd|    |4.21|4.83|5.13|5.31|5.42|
  |             |2nd  "  3rd|4.82|5.42|5.62|5.71|5.77|5.81|
  | 6 inches    |3rd  "  4th|7.18|6.58|6.38|6.29|6.23|6.19|
  |             |4th  "  5th|    |7.79|7.17|6.87|6.69|6.58|
  +-------------+-----------+----+----+----+----+----+----+

  +-------------+-----------+-----------------------------+
  |   Average   |  Adjacent | DISTANCE BETWEEN THE CENTRES|
  |  difference |   steps,  |       OF CONE PULLEYS.      |
  |    between  |   whose   +----+----+----+----+----+----+
  |     the     |   diffe-  |    |    |    |    |    |    |
  |   adjacent  |  rence is | 70 | 80 | 90 | 100| 120| 240|
  |    steps.   |  given in |         i n c h e s.        |
  |             |   table.  |    |    |    |    |    |    |
  +-------------+-----------+----+----+----+----+----+----+
  |             |1st and 2nd|0.99|0.99|0.99|0.99|0.99|1.00|
  |             |2nd  "  3rd|0.99|1.00|1.00|1.00|1.00|1.00|
  | 1 inch      |3rd  "  4th|1.01|1.00|1.00|1.00|1.00|1.00|
  |             |4th  "  5th|1.01|1.01|1.01|1.01|1.01|1.00|
  +-------------+-----------+----+----+----+----+----+----+
  |             |1st and 2nd|1.47|1.47|1.48|1.48|1.48|1.49|
  |             |2nd  "  3rd|1.49|1.49|1.49|1.49|1.49|1.49|
  | 1-1/2 inch  |3rd  "  4th|1.51|1.51|1.51|1.51|1.51|1.51|
  |             |4th  "  5th|1.53|1.53|1.52|1.52|1.52|1.51|
  +-------------+-----------+----+----+----+----+----+----+
  |             |1st and 2nd|1.94|1.95|1.96|1.96|1.97|1.98|
  |             |2nd  "  3rd|1.98|1.98|1.99|1.99|1.99|1.99|
  | 2 inches    |3rd  "  4th|2.02|2.02|2.01|2.01|2.01|2.01|
  |             |4th  "  5th|2.06|2.05|2.04|2.04|2.03|2.02|
  +-------------+-----------+----+----+----+----+----+----+
  |             |1st and 2nd|2.41|2.42|2.43|2.44|2.45|2.47|
  |             |2nd  "  3rd|2.47|2.47|2.48|2.48|2.48|2.49|
  | 2-1/2 inches|3rd  "  4th|2.53|2.53|2.52|2.52|2.52|2.51|
  |             |4th  "  5th|2.59|2.58|2.57|2.56|2.55|2.53|
  +-------------+-----------+----+----+----+----+----+----+
  |             |1st and 2nd|2.87|2.89|2.90|2.91|2.93|2.96|
  |             |2nd  "  3rd|2.96|2.96|2.97|2.97|2.98|2.99|
  | 3 inches    |3rd  "  4th|3.04|3.04|3.03|3.03|3.02|3.01|
  |             |4th  "  5th|3.13|3.11|3.10|3.09|3.07|3.04|
  +-------------+-----------+----+----+----+----+----+----+
  |             |1st and 2nd|3.78|3.81|3.83|3.84|3.87|3.94|
  |             |2nd  "  3rd|3.92|3.94|3.94|3.95|3.96|3.98|
  | 4 inches    |3rd  "  4th|4.08|4.06|4.06|4.05|4.04|4.02|
  |             |4th  "  5th|4.22|4.19|4.17|4.16|4.13|4.06|
  +-------------+-----------+----+----+----+----+----+----+
  |             |1st and 2nd|4.66|4.71|4.73|4.76|4.80|4.90|
  |             |2nd  "  3rd|4.89|4.90|4.91|4.92|4.93|4.96|
  | 5 inches    |3rd  "  4th|5.11|5.10|5.09|5.08|5.07|5.04|
  |             |4th  "  5th|5.34|5.29|5.27|5.24|5.20|5.10|
  +-------------+-----------+----+----+----+----+----+----+
  |             |1st and 2nd|5.51|5.57|5.62|5.66|5.71|5.86|
  |             |2nd  "  3rd|5.83|5.86|5.87|5.88|5.90|5.95|
  | 6 inches    |3rd  "  4th|6.17|6.14|6.13|6.12|6.10|6.05|
  |             |4th  "  5th|6.49|6.43|6.38|6.34|6.29|6.14|
  +-------------+-----------+----+----+----+----+----+----+

Before describing and applying the table, we will call attention to the
term "effective" diameter. The effective radius--as is well
known--extends from the centre of the pulley to the centre of the belt;
the effective diameter, being twice this effective radius, must also
equal the actual diameter plus thickness of belt.

The table is so arranged that the diameter (divided by distance between
centres) of one step of a belted pair will always be found in the
extreme right-hand column; while its companion step will be found on the
same horizontal line, and in that vertical column of the table
corresponding to the length of belt employed. For example, if column 14
of the table corresponded to the length of belt employed, some of the
possible pairs of diameters would be as follows:

  .7118    .5813    .42    .2164    .0474
  .06      .24      .42    .60      .72

The upper row of this series of pairs being taken from column 14, and
the lower row from the extreme right-hand column, the numbers in each
pair being on the same horizontal line. If the distance between the
centers of the pulleys were 60 ins. the effective diameters of the steps
corresponding to the above pairs would be:

  42.71    34.88    25.2    12.98    2.84 ins.
   3.6     14.4     25.2    36.0    43.20

being obtained by multiplying the first series of pairs by 60; the
length of belt which would be equally tight on each of these pairs would
be 3.3195 × 60 ins. = 199.17 ins.

III.--TABLE FOR FINDING THE EFFECTIVE DIAMETERS OF THE STEPS OF CONE
PULLEYS, WHEN THE PULLEYS ARE CONNECTED BY AN OPEN BELT AND ARE UNLIKE.

Each vertical cone of the table corresponds to a given length of belt,
and the numbers in these columns are the required effective diameters of
the steps when the centres of the pulleys are a Unit's distance apart.

  +--------------------------------------------------------------+------+
  |  LENGTH OF BELT WHEN THE CENTRES OF THE CONE PULLEYS ARE A   |      |
  |                   UNIT'S DISTANCE APART.                     |      |
  +------+------+------+------+------+------+------+------+------+      |
  |2.0942|2.1885|2.2827|2.3770|2.4712|2.5655|2.6597|2.7540|2.8482|      |
  +------+------+------+------+------+------+------+------+------+  [A] |
  |  =1= |  =2= |  =3= |  =4= |  =5= |  =6= |  =7= |  =8= |  =9= |      |
  +------+------+------+------+------+------+------+------+------+------+
  | .0594| .1177| .1750| .2313| .2867| .3413| .3950| .4479| .5000| 0.00 |
  | .03  | .0894| .1477| .2050| .2613| .3167| .3713| .4250| .4779| 0.03 |
  |      | .06  | .1194| .1777| .2350| .2913| .3467| .4013| .4550| 0.06 |
  |      | .0294| .09  | .1494| .2077| .2650| .3213| .3767| .4313| 0.09 |
  |      |      | .0594| .12  | .1794| .2377| .2950| .3513| .4067| 0.12 |
  |      |      | .0275| .0894| .15  | .2094| .2677| .3250| .3813| 0.15 |
  |      |      |      | .0575| .1194| .18  | .2394| .2977| .3550| 0.18 |
  |      |      |      | .0244| .0875| .1494| .21  | .2694| .3277| 0.21 |
  |      |      |      |      | .0544| .1175| .1794| .24  | .2994| 0.24 |
  |      |      |      |      | .0200| .0844| .1475| .2094| .27  | 0.27 |
  |      |      |      |      |      | .0500| .1144| .1775| .2394| 0.30 |
  |      |      |      |      |      | .0140| .0800| .1444| .2075| 0.33 |
  |      |      |      |      |      |      | .0440| .1100| .1744| 0.36 |
  |      |      |      |      |      |      | .0064| .0740| .1400| 0.39 |
  |      |      |      |      |      |      |      | .0364| .1040| 0.42 |
  |      |      |      |      |      |      |      |      | .0664| 0.45 |
  |      |      |      |      |      |      |      |      | .0271| 0.48 |
  +------+------+------+------+------+------+------+------+------+------+

  +--------------------------------------------------------------+------+
  |  LENGTH OF BELT WHEN THE CENTRES OF THE CONE PULLEYS ARE A   |      |
  |                   UNIT'S DISTANCE APART.                     |      |
  +------+------+------+------+------+------+------+------+------+      |
  |2.9425|3.0367|3.1310|3.2252|3.3195|3.4137|3.5080|3.6022|3.6965|      |
  |------+------+------+------+------+------+------+------+------+  [A] |
  | =10= | =11= | =12= | =13= | =14= | =15= | =16= | =17= | =18= |      |
  +------+------+------+------+------+------+------+------+------+------+
  | .5514| .6020| .6518| .7010| .7495| .7974| .8447| .8913| .9373| 0.00 |
  | .5300| .5814| .6320| .6818| .7310| .7795| .8274| .8747| .9213| 0.03 |
  | .5079| .5600| .6114| .6620| .7118| .7610| .8095| .8574| .9047| 0.06 |
  | .4850| .5379| .5900| .6414| .6920| .7418| .7910| .8395| .8874| 0.09 |
  | .4613| .5150| .5679| .6200| .6714| .7220| .7718| .8210| .8695| 0.12 |
  | .4367| .4913| .5450| .5979| .6500| .7014| .7520| .8018| .8510| 0.15 |
  | .4113| .4667| .5213| .5750| .6279| .6800| .7314| .7820| .8318| 0.18 |
  | .3850| .4413| .4967| .5513| .6050| .6579| .7100| .7614| .8120| 0.21 |
  | .3577| .4150| .4713| .5267| .5813| .6350| .6879| .7400| .7914| 0.24 |
  | .3294| .3877| .4450| .5013| .5567| .6113| .6650| .7179| .7700| 0.27 |
  | .30  | .3594| .4177| .4750| .5313| .5867| .6413| .6950| .7479| 0.30 |
  | .2694| .33  | .3894| .4477| .5050| .5613| .6167| .6713| .7250| 0.33 |
  | .2375| .2994| .36  | .4194| .4777| .5350| .5913| .6467| .7013| 0.36 |
  | .2044| .2675| .3294| .39  | .4494| .5077| .5650| .6213| .6767| 0.39 |
  | .1700| .2344| .2975| .3594| .42  | .4794| .5377| .5950| .6513| 0.42 |
  | .1340| .2000| .2644| .3275| .3894| .45  | .5094| .5677| .6250| 0.45 |
  | .0964| .1640| .2300| .2944| .3575| .4194| .48  | .5394| .5977| 0.48 |
  | .0571| .1264| .1940| .2600| .3244| .3875| .4494| .51  | .5694| 0.51 |
  | .0160| .0871| .1564| .2240| .2900| .3544| .4175| .4794| .54  | 0.54 |
  |      | .0460| .1171| .1864| .2540| .3200| .3844| .4475| .5094| 0.57 |
  |      | .0029| .0760| .1471| .2164| .2840| .3500| .4144| .4775| 0.60 |
  |      |      | .0329| .1060| .1771| .2464| .3140| .3800| .4444| 0.63 |
  |      |      |      | .0629| .1360| .2071| .2764| .3440| .4100| 0.66 |
  |      |      |      | .0174| .0929| .1660| .2371| .3064| .3740| 0.69 |
  |      |      |      |      | .0474| .1229| .1960| .2671| .3364| 0.72 |
  |      |      |      |      |      | .0774| .1529| .2260| .2971| 0.75 |
  |      |      |      |      |      | .0292| .1074| .1829| .2560| 0.78 |
  |      |      |      |      |      |      | .0592| .1374| .2129| 0.81 |
  |      |      |      |      |      |      | .0081| .0892| .1674| 0.84 |
  |      |      |      |      |      |      |      | .0381| .1192| 0.87 |
  |      |      |      |      |      |      |      |      | .0681| 0.90 |
  |      |      |      |      |      |      |      |      | .0138| 0.93 |
  +------+------+------+------+------+------+------+------+------+------+

  +--------------------------------------------------------------+------+
  |  LENGTH OF BELT WHEN THE CENTRES OF THE CONE PULLEYS ARE A   |      |
  |                   UNIT'S DISTANCE APART.                     |      |
  +------+------+------+------+------+------+------+------+------+      |
  |3.7907|3.8850|3.9792|4.0735|4.1677|4.2620|4.3562|4.4504|4.5447|      |
  |------+------+------+------+------+------+------+------+------+  [A] |
  | =19= | =20= | =21= | =22= | =23= | =24= | =25= | =26= | =27= |      |
  +------+------+------+------+------+------+------+------+------+------+
  | .9828|1.0277|1.0721|1.1159|1.1593|1.2021|1.2444|1.2861|1.3274| 0.00 |
  | .9673|1.0128|1.0577|1.1021|1.1459|1.1893|1.2321|1.2744|1.3161| 0.03 |
  | .9513| .9973|1.0428|1.0877|1.1321|1.1759|1.2193|1.2621|1.3044| 0.06 |
  | .9347| .9813|1.0273|1.0728|1.1177|1.1621|1.2059|1.2493|1.2921| 0.09 |
  | .9174| .9647|1.0113|1.0573|1.1028|1.1477|1.1921|1.2359|1.2793| 0.12 |
  | .8995| .9474| .9947|1.0413|1.0873|1.1328|1.1777|1.2221|1.2659| 0.15 |
  | .8810| .9295| .9774|1.0247|1.0713|1.1173|1.1628|1.2077|1.2521| 0.18 |
  | .8618| .9110| .9595|1.0074|1.0547|1.1013|1.1473|1.1928|1.2377| 0.21 |
  | .8420| .8918| .9410| .9895|1.0374|1.0847|1.1313|1.1773|1.2228| 0.24 |
  | .8214| .8720| .9218| .9710|1.0195|1.0674|1.1147|1.1613|1.2073| 0.27 |
  | .8000| .8514| .9020| .9518|1.0010|1.0495|1.0974|1.1447|1.1913| 0.30 |
  | .7779| .8300| .8814| .9320| .9818|1.0310|1.0795|1.1274|1.1747| 0.33 |
  | .7550| .8079| .8600| .9114| .9620|1.0118|1.0610|1.1095|1.1574| 0.36 |
  | .7313| .7850| .8379| .8900| .9414| .9920|1.0418|1.0910|1.1395| 0.39 |
  | .7067| .7613| .8150| .8679| .9200| .9714|1.0220|1.0718|1.1210| 0.42 |
  | .6813| .7367| .7913| .8450| .8979| .9500|1.0014|1.0520|1.1018| 0.45 |
  | .6550| .7113| .7667| .8213| .8750| .9279| .9800|1.0314|1.0820| 0.48 |
  | .6277| .6850| .7413| .7967| .8513| .9050| .9579|1.0100|1.0614| 0.51 |
  | .5994| .6577| .7150| .7713| .8267| .8813| .9350| .9879|1.0400| 0.54 |
  | .57  | .6294| .6877| .7450| .8013| .8567| .9113| .9650|1.0179| 0.57 |
  | .5394| .60  | .6594| .7177| .7750| .8313| .8867| .9413| .9950| 0.60 |
  | .5075| .5694| .63  | .6894| .7477| .8050| .8613| .9167| .9713| 0.63 |
  | .4744| .5375| .5994| .66  | .7194| .7777| .8350| .8913| .9467| 0.66 |
  | .4400| .5044| .5675| .6294| .69  | .7494| .8077| .8650| .9213| 0.69 |
  | .4040| .4700| .5344| .5975| .6594| .72  | .7794| .8377| .8950| 0.72 |
  | .3664| .4340| .5000| .5644| .6275| .6894| .75  | .8094| .8677| 0.75 |
  | .3271| .3964| .4640| .5300| .5944| .6575| .7194| .78  | .8394| 0.78 |
  | .2860| .3571| .4264| .4940| .5600| .6244| .6875| .7494| .81  | 0.81 |
  | .2429| .3160| .3871| .4564| .5240| .5900| .6544| .7175| .7794| 0.84 |
  | .1974| .2729| .3460| .4171| .4864| .5540| .6200| .6844| .7475| 0.87 |
  | .1492| .2274| .3029| .3760| .4471| .5164| .5840| .6500| .7144| 0.90 |
  | .0981| .1792| .2574| .3329| .4060| .4771| .5464| .6140| .6800| 0.93 |
  | .0438| .1281| .2092| .2874| .3629| .4360| .5071| .5764| .6440| 0.96 |
  |      | .0738| .1581| .2392| .3174| .3929| .4660| .5371| .6064| 0.99 |
  |      | .0157| .1038| .1881| .2692| .3474| .4229| .4960| .5671| 1.02 |
  |      |      | .0457| .1338| .2181| .2992| .3774| .4529| .5260| 1.05 |
  |      |      |      | .0757| .1638| .2481| .3292| .4074| .4829| 1.08 |
  |      |      |      | .0131| .1057| .1938| .2781| .3592| .4374| 1.11 |
  |      |      |      |      | .0431| .1357| .2238| .3081| .3892| 1.14 |
  |      |      |      |      |      | .0731| .1657| .2538| .3381| 1.17 |
  |      |      |      |      |      | .0050| .1031| .1957| .2838| 1.20 |
  |      |      |      |      |      |      | .0350| .1331| .2257| 1.23 |
  |      |      |      |      |      |      |      | .0650| .1631| 1.26 |
  |      |      |      |      |      |      |      |      | .0950| 1.29 |
  |      |      |      |      |      |      |      |      | .0200| 1.32 |
  +------+------+------+------+------+------+------+------+------+------+

  +-----------------------------------------+------+
  | LENGTH OF BELT WHEN THE CENTRES OF THE  |      |
  |CONE PULLEYS ARE A UNIT'S DISTANCE APART.|      |
  +------+------+------+------+------+------+      |
  |4.6389|4.7332|4.8274|4.9217|5.0159|5.1102|      |
  |------+------+------+------+------+------+  [A] |
  | =28= | =29= | =30= | =31= | =32= | =33= |      |
  +------+------+------+------+------+------+------+
  |1.3682|1.4085|1.4484|1.4877|1.5266|1.5650| 0.00 |
  |1.3574|1.3982|1.4385|1.4784|1.5177|1.5566| 0.03 |
  |1.3461|1.3874|1.4282|1.4685|1.5084|1.5477| 0.06 |
  |1.3344|1.3761|1.4174|1.4582|1.4985|1.5384| 0.09 |
  |1.3221|1.3644|1.4061|1.4474|1.4882|1.5285| 0.12 |
  |1.3093|1.3521|1.3944|1.4361|1.4774|1.5182| 0.15 |
  |1.2959|1.3393|1.3821|1.4244|1.4661|1.5074| 0.18 |
  |1.2821|1.3259|1.3693|1.4121|1.4544|1.4961| 0.21 |
  |1.2677|1.3121|1.3559|1.3993|1.4421|1.4844| 0.24 |
  |1.2528|1.2977|1.3421|1.3859|1.4293|1.4721| 0.27 |
  |1.2373|1.2828|1.3277|1.3721|1.4159|1.4593| 0.30 |
  |1.2213|1.2673|1.3128|1.3577|1.4021|1.4459| 0.33 |
  |1.2047|1.2513|1.2973|1.3428|1.3877|1.4321| 0.36 |
  |1.1874|1.2347|1.2813|1.3273|1.3728|1.4177| 0.39 |
  |1.1695|1.2174|1.2647|1.3113|1.3573|1.4028| 0.42 |
  |1.1510|1.1995|1.2474|1.2947|1.3413|1.3873| 0.45 |
  |1.1318|1.1810|1.2295|1.2774|1.3247|1.3713| 0.48 |
  |1.1120|1.1618|1.2110|1.2595|1.3074|1.3547| 0.51 |
  |1.0914|1.1420|1.1918|1.2410|1.2895|1.3374| 0.54 |
  |1.0700|1.1214|1.1720|1.2218|1.2710|1.3195| 0.57 |
  |1.0479|1.1000|1.1514|1.2020|1.2518|1.3010| 0.60 |
  |1.0250|1.0779|1.1300|1.1814|1.2320|1.2818| 0.63 |
  |1.0013|1.0550|1.1079|1.1600|1.2114|1.2620| 0.66 |
  | .9767|1.0313|1.0850|1.1379|1.1900|1.2414| 0.69 |
  | .9513|1.0067|1.0613|1.1150|1.1679|1.2200| 0.72 |
  | .9250| .9813|1.0367|1.0913|1.1450|1.1979| 0.75 |
  | .8977| .9550|1.0113|1.0667|1.1213|1.1750| 0.78 |
  | .8694| .9277| .9850|1.0413|1.0967|1.1513| 0.81 |
  | .84  | .8994| .9577|1.0150|1.0713|1.1267| 0.84 |
  | .8094| .87  | .9294| .9877|1.0450|1.1013| 0.87 |
  | .7775| .8394| .90  | .9594|1.0177|1.0750| 0.90 |
  | .7444| .8075| .8694| .93  | .9894|1.0477| 0.93 |
  | .7100| .7744| .8375| .8994| .96  |1.0194| 0.96 |
  | .6740| .7400| .8044| .8675| .9294| .99  | 0.99 |
  | .6364| .7040| .7700| .8344| .8975| .9594| 1.02 |
  | .5971| .6664| .7340| .8000| .8644| .9275| 1.05 |
  | .5560| .6271| .6964| .7640| .8300| .8944| 1.08 |
  | .5129| .5860| .6571| .7264| .7940| .8600| 1.11 |
  | .4674| .5429| .6160| .6871| .7564| .8240| 1.14 |
  | .4192| .4974| .5729| .6460| .7171| .7864| 1.17 |
  | .3681| .4492| .5274| .6029| .6760| .7471| 1.20 |
  | .3138| .3981| .4792| .5574| .6329| .7060| 1.23 |
  | .2557| .3438| .4281| .5092| .5874| .6629| 1.26 |
  | .1931| .2857| .3738| .4581| .5392| .6174| 1.29 |
  | .1250| .2231| .3157| .4038| .4881| .5692| 1.32 |
  | .0500| .1550| .2531| .3457| .4338| .5181| 1.35 |
  |      | .0800| .1850| .2831| .3757| .4638| 1.38 |
  |      |      | .1100| .2150| .3131| .4057| 1.41 |
  |      |      | .0255| .1400| .2450| .3431| 1.44 |
  |      |      |      | .0555| .1700| .2750| 1.47 |
  |      |      |      |      | .0855| .2000| 1.50 |
  +------+------+------+------+------+------+------+

Legend: [A] = Assumed diameter of steps, divided by distance between the
centres of Cone Pulleys.

To get the actual diameters of these steps when thickness of belt = 7/32
= 0.22 in., we have simply to subtract 0.22 in. from the effective
diameters just given, thus:

  42.49    34.66    24.98    12.76    2.62 in.
   3.38    14.18    24.98    35.78   42.98

would be the series of pairs of actual diameters.

In solving problems relating to the diameters of cone pulleys by means
of the accompanying table, we must have, besides the distance between
centres, sufficient data to determine the column representing the length
of belt. The length of belt is seldom known because it is of small
practical importance to know its exact length; but it may be estimated
approximately, and then the determination of suitable diameters of the
steps becomes an extremely simple matter, as may be seen from what has
already preceded. When the length of the belt is not known, and has not
been assumed, we indirectly prescribe the length of belt by assuming the
effective diameters of the two steps of a belted pair; thus, in the
following Figure (561), the length of belt is prescribed when the
distance A B, and any one of the pairs of steps D_{1}_d__{1},
D_{2}_d__{2}, D_{3}_d__{3} and D_{4}_d__{4} are given. We will show in
the following examples how the length of belt and its corresponding
column of diameter may be found when a pair of steps (like
D_{1}_d__{1}), are given.

[Illustration: Fig. 561.]

EXAMPLE 1. Given the effective diameters

  4.5 in.   9 in.    15 in.    21 in. on cone A,
     --      --      15 in.      --     "     B,

and the distance between centres equal to 50 inches.

Required the remaining diameters on cone B.

Since in this example the steps of the given pair are equal, we look for
15/50 = 0.30, in the extreme right-hand column of table; we will find it
in the 11th line from the top; now looking along this line for the
diameter of the other step, = 15/50 = 0.30, we will find it in column
10; consequently the numbers of this column may be taken as the
diameters of the steps which are the companions or partners of those in
the extreme right-hand column.

We can now easily determine the remaining members of the pairs to which
4.5 in., 9 in., and 21 in. steps respectively belong. To find the
partner of the 4.5 step, we find 4.5/50 = 0.09 in the right-hand column,
and look along the horizontal line on which 0.09 is placed till we come
to column 10, in which we will find the number 0.4850; 0.4850 × 50 in. =
24.25 in. will be the effective diameter of the companion to the 4.5 in.
step.

To find the partner to the 9 in. step, we proceed as before, looking for
9/50 = 0.18 in the right-hand column, and then along the horizontal line
of 0.18 to column 10, then will 0.4113 × 50 in. = 20.57 in. be the
required companion to the 9 in. step of cone A.

In like manner for the partner of the 21 in. step we get 0.1700 × 50 in.
= 8.5 in. The effective diameter therefore will be,

   4.5 in.    9 in.    15 in.    21   in. on cone A,
  24.25      20.57     15 in.     8.5         "   B.

If the thickness of belt employed were 0.25 in. the _actual_ diameters
of steps would be,

   4.25     8.75    14.75   20.75 on cone A,
  24.00    20.32    14.75    8.25    "    B,

and the length of belt would be 2.9425 × 50 = 147.125 in.

EXAMPLE 2. Given the effective diameters

   6 in.    12 in.    18 in.    24 in. on cone A,
  30 in.     --        --        --        "   B,

and the distance between centres = 40 in.

Required the unknown diameters on cone B.

We must, as before, first find the vertical column corresponding to the
length of belt which joins the pair of steps 6 in/30 in. We find the
number 6/40 = .15 in the right-hand column, and then look along its
horizontal line for its partner 30/40 = 0.75. Since we do not find any
number exactly equal to .7500, we must interpolate. For the benefit of
those not familiar with the method of interpolation we will give in
detail the method of finding intermediate columns of the table. On the
aforesaid horizontal line we find in column 16 a number 0.7520, larger
than the required 0.7500, and in column 15 a number 0.7014, smaller than
0.7500; evidently the intermediate column, containing the required
0.7500, must lie between columns 16 and 15. To find how far the required
column is from column 16, we subtract as follows:

  0.7520   0.7520
  0.7500   0.7014
  ------   ------
   .0020   0.0506

then the fraction .0020/.00506 = 0.04 nearly will represent the position
of the required intermediate column; namely, that its distance from
column 16 is about 4/100 of the distance between the adjacent columns,
15 and 16.

To find other numbers in this intermediate column we have only to
multiply the difference between the adjacent numbers of columns 16 and
15 by 0.04, and subtract the product from the number in column 16. But
it is not necessary to find as many numbers of the intermediate columns
as are contained in either of the adjacent columns; it is only necessary
to find as many numbers as there are steps in each of the cone pulleys.
We will now illustrate what has preceded, by finding the partner to the
12 in. step of cone A. Find, as before, the horizontal line
corresponding to 12/40 = 0.30, then take the difference between the
numbers 0.6413 and 0.5867 of columns 16 and 15, and multiply this
difference, 0.0546, by 0.04; this product = 0.0022 subtracted from
0.6413, will give 0.6391, a number of the intermediate columns
corresponding to the length of belt of the present problem. Multiplying
by the distance between the axes = 40 in. we get 0.6391 × 40 = 25.56,
for the diameter of the step of cone B which is partner to the 12 in.
step of cone A.

To find the companion to the 18 in. step, we proceed in the same manner,
looking for the horizontal line 18/40 = 0.45, and interpolating as
follows:

  0.5094 - (0.5094 - 0.4500) × 0.04 = 0.5070.

Consequently, 0.5070 × 40 in. = 20.28 in. will be the required partner
of the 18 in. step.

In like manner, for the 24 in. step, we have

  0.3500 - (0.3500 - 0.2840) × 0.04 = 0.3474, and 0.3474 × 40 = 13.90.

The effective diameters are therefore

   6 in.   12 in.   18 in.   24   in. on cone A.
  30       25.56    20.28    13.9         "   B.

The actual diameters, when thickness of belt = 0.20 in., are:

   5.8    11.8     17.8     23.8 on cone A.
  29.8    25.36    20.08    13.7      "  B.

And the length of belt will be:

  [3.5080 - (3.5080 - 3.4137) × 0.04] × 40 in. = 140.17 in.

EXAMPLE 3. Given the effective diameters:

  12 in.    18 in.    24 in.    30 in. on cone A,
  33 in.     --        --        --        "   B,

and the distance between the centres = 60 in.

Required the remaining diameters on cone B.

The horizontal corresponding to 12/60 = 0.20 lies 2/3rd way between the
horizontal line, corresponding to 0.18 and 0.21; the number 33/60 =
0.5500, corresponding to the companion of the 12 in. step, will
therefore lie 2/3rd way between the horizontal lines 0.18 and 0.21. We
have now to find two numbers on this 2/3rd line, of which one will be
less and the other greater than 0.5500. An inspection of the table will
show that these greater and less numbers must lie in columns 13 and 12.
The numbers on the 2/3rd line itself may now be found as follows:

In column 13, 0.5750 - 2/3(0.5750 - 0.5513) = 0.5592.

In column 12, 0.5213 - 2/3(0.5213 - 0.4967) = 0.5049.

0.5592 will be the number on the 2/3rd line, which is greater than
0.5500, and 0.5049 will be the one which is less than 0.5500. The
position of the intermediate column, corresponding to the length of belt
of the present example, may now be found, as before, briefly. It is:

  0.5592 - 0.5500 = 0.0092
                           = 0.17.
  0.5592 - 0.5049 = 0.0543

Consequently the required column lies nearest column 13, 17/100th way
between columns 13 and 12. To find any other number in the required
column, we have only to multiply the difference between two adjacent
numbers of columns 13 and 12 by 17/100, and subtract the product from
the number in column 13. For example, to find the diameter of the
partner to the 18 in. step of cone A, we find the numbers 0.4750 and
0.4177 of columns 13 and 12, which lie on the horizontal line
corresponding to 18/60 = 0.30; the difference, 0.0573, between the two
numbers is multiplied by 0.17, and the product, 0.0573 × 0.17 = 0.0097,
subtracted from 0.4750. This last difference will equal 0.4653, and will
be the number sought. If we now multiply by 60, we will get 27.92 in. as
the effective diameter of that step on cone B which is the partner to
the 18 in. step of cone A.

To find the companion of the 24 in. step, we proceed after the same
fashion; the horizontal line 24/60 = 0.40 lies 1/3rd way between 0.39
and 0.42; hence,

In column 13, 0.3900 - 1/3(0.3900 - 0.3594) = 0.3798;

In column 12, 0.3294 - 1/3(0.3294 - 0.2975) = 0.3188;

And 0.3798 - (0.3798 - 0.3188) × 0.17 = 0.3694.

The required effective diameter of the step, which is partner to the 24
in. step, will therefore be 0.3694 × 60 = 22.16 in.

In like manner we obtain partner for the 30 in. step, thus:

In column 13, 0.2944 - 2/3(0.2944 - 0.2600) = 0.2715.

In column 12, 0.2300 - 2/3(0.2300 - 0.1940) = 0.2060.

Also 0.2715 - (0.2715 - 0.2060) × 0.17 = 0.2604, and 0.2604 × 60 in. =
15.62 in. = diam. of step belonging to the same belted pair as the 30
in. step of cone A.

The effective diameters will be:

  12 in.    18 in.    24 in.    30    in. on cone A,
  33        27.92     22.16     15.62        "    B,

and the actual diameters when belt is 0.22" thick:

  11.78      17.78      23.78      29.78 in.
  32.78      27.70      21.94      15.40

and the length of belt is found to be:

[3.2252 - (3.2252 - 3.1310) × 0.17] × 60 in. = 192.55 in.

In all the preceding problems it should be noticed that we arbitrarily
assumed _all_ the steps on one cone, and _one_ of the steps on the other
cone. It will be found that all of the practical problems relating to
cone-pulley diameters can finally be reduced to this form, and can
consequently be solved according to the methods just given.

For those who find difficulty in interpolating, the following procedure
will be found convenient: Estimate approximately the necessary length of
belt, then divide this length by the distance between the centres of the
cone pulleys; now find which one of the 33 lengths of belt (per unit's
distance apart of the centres) given in the table is most nearly equal
to the quotient just obtained, and then take the vertical column, at the
head of which it stands, for the companion to the right-hand column.
Those numbers of these companion columns which are on the same
horizontal line will be the companion steps of a belted pair. The table
is so large, that in the great majority of cases not only exact, but
otherwise satisfactory values can be obtained by this method, without
any interpolation whatever."

[Illustration: Fig. 562.]

The teeth of the back gear should be accurately cut so that there is no
lost motion between the teeth of one wheel, and the spaces of the other,
because on account of the work being of large diameter or of hard metal
(so as to require the slow speed), the strain of the cut is nearly
always heavy when the back gear is in use, and the strain on the teeth
is correspondingly great, causing a certain amount of spring or
deflection in the live spindle and back gear spindle. Suppose then, that
at certain parts of the work there is no cut, then when the tool again
meets the cut the work will meet the tool and stand still until the lost
motion in the gear teeth and the spring of the spindles is taken up,
when the cut will proceed with a jump that will leave a mark on the work
and very often break the tool. When the cut again leaves the tool a
second jump also leaving a mark on the work will be made. If the teeth
of the gears are cut at an angle to the axial line of the spindle, as is
sometimes the case, this jumping from the play between the teeth will be
magnified on account of a given amount of play, affording more back lash
in such gears.

The teeth of the wheels should always be of involute and not of
epicycloidal form, for the following reasons. The transmission of motion
by epicycloidal teeth is exactly uniform only when their pitch circles
exactly coincide, and this may not be the case in time because of wear
in the parts as in the live spindle journals and the bearings, and the
back gear spindle and its bearings, and _every variation of speed_ in
the cut, however slight it may be, produces a corresponding mark upon
the work. In involute teeth the motion transmitted will be smooth and
equal whether the pitch lines of the wheels coincide or not, hence the
wear of the journals and bearings does not impair their action.

The object of cutting the teeth at an angle is to have the point of
contact move or roll as it were from one end to the other of the teeth,
and thus preserve a more conterminous contact on the line of centres of
the two wheels, the supposition being that this would remove the marks
on the work produced by the tremor of the back gear. But such tremor is
due to errors in the form of the teeth, and also in the case of
epicycloidal teeth from the pitch lines of the teeth not exactly
coinciding when in gear.

The pitch of the teeth should be as fine as the requisite strength, with
the usual allowance of margin for wear and safety will allow, so as to
have as many teeth in continuous contact as possible.

Various methods of moving the back gear into and out of gear with the
cone spindle gears are employed. The object is to place the back gears
into gear to the exact proper depth to hold them securely in position,
and to enable the operator to operate the gears without passing to the
back of the lathe. Sometimes a sliding bearing box, such as shown in
Fig. 562, is employed; _a_ is the back gear spindle, _b_ its bearing
box, and _d_ a pin which when on the side shown holds _b_ in position,
when the back gear is in action. To throw it out of action _d_ is
removed, _b_ pushed back, and _d_ inserted in a hole on the right hand
of _b_; the objection is that there is no means of taking up the wear of
_b_, and it is necessary to pass to the back of the lathe to operate the
device.

[Illustration: Fig. 563.]

Another plan is to let the back gear move endwise and bush its bearing
holes with hardened steel bushes. This possesses the advantage that the
gear is sure, if made right, to keep so, but it has some decided
disadvantages: first, the pinion A, Fig. 563, must be enough larger than
the smallest cone-step B to give room between B and C for the belt, and
this necessitates that D also be larger than otherwise; secondly, the
gear-spindle F projects through the bearing at _f_, and this often comes
in the way of the bolt-heads used for chucking work to the face plate.
The method of securing the spindle from end motion is as follows: On the
back of the head is pivoted at _i_, a catch G, and on the gear shaft F
are two grooves. As shown in the sketch, G is in one of these grooves
while H is the other, but when the back gear is in, G would be in H.

[Illustration: Fig. 564.]

Sometimes a simple eccentric bush and pin is used as in Fig. 564, in
which _a_ is the spindle journal, _b_ a bush having bearing in the lathe
head, and _d_ a taper pin to secure _b_ in its adjusted position.

[Illustration: Fig. 565.]

In large heavy lathes having many changes of speed, there are various
other constructions, as will be seen upon the lathes themselves in the
various illustrations concerning the methods of throwing the back gear
in and out. The eccentric motion shown in Fig. 573 of the Putnam lathe,
is far preferable to any means in which the back-gear spindle moves
endways, because, as before stated, the end of the back-gear spindle
often comes in the way of the bolts used to fasten work to the large
face plate. This occurs mainly in chucked work of the largest diameter
within the capacity of the lathe.

[Illustration: Fig. 566.]

In many American lathes the construction of the gearing that conveys
motion from the live spindle is such that facility is afforded to throw
the change gears out of action when the lathe is running fast, as for
polishing purposes, so as to save the teeth from wear. Means are also
provided to reverse the direction of lead screw or feed screw
revolution. An example of a common construction of this kind is shown in
Fig. 565, in which the driving wheel A is on the inner side of the back
bearing as shown. It drives (when in gear) a pair of gears, one only of
which is seen in the figure at B, which drives C, and through R, D, I,
and S, the lead screw. A side view of the wheel A and the mechanism in
connection therewith is shown in Fig. 566, in which S represents the
live spindle and R is a spindle or shaft corresponding to R in Fig. 565.
L is a lever pivoted upon R and carrying two pinions B and E; pinion B
is of larger diameter than E, so that B gears with both C and E (C
corresponding to wheel C in Fig. 565), while E gears with B only.

[Illustration: Fig. 567.]

With the lever L in the position shown, neither B nor E engages with A,
hence they are at rest; but if lever L be raised as in Fig. 567, B will
gear with wheels A and C, and motion will be conveyed from A to C, wheel
E running as an idle wheel, thus C will revolve in the same direction as
the lathe spindle.

But if lever L be lowered as in fig. 568, then wheel E will gear with
and receive motion from A, which it will convey to B, and C will revolve
in the opposite direction to that in which the lathe spindle runs. To
secure lever L in position, a pin F passes through it and into holes as
I, J, provided in the lathe head. Lever L is sometimes placed inside the
head, and sometimes outside as in Fig. 569, and it will be obvious that
it may be used to cut left-hand threads without the use of an extra
intermediate change gear, which is necessary in the construction shown
in Fig. 570, in order to reverse the direction of lead screw revolution.

Sometimes the pin F is operated by a small spring lever attached
to L, so that the hand grasps the end of L and the spring lever
simultaneously, removing F from the hole in H, and therefore freeing L,
so as to permit its operation. By relaxing the pressure on the small
spring lever pin F finds its own way into the necessary hole in H, when
opposite to it, without requiring any hand manipulation.

In larger lathes the lever L is generally attached to its stud outside
the end bearing of the head H.

[Illustration: Fig. 568.]

It is preferable, however, that the device for changing the direction of
feed traverse be operative from the lathe carriage as in the Sellers
lathe, so that the operator need not leave it when it is necessary to
reverse the direction of traverse.

[Illustration: Fig. 569.]

The swing frame, when the driving gear D is outside of the back bearing
(as it is in Fig. 570), is swung from the axis of the lead screw as a
centre of motion, and has two slots for receiving studs for change
wheels. But when the driving gear is inside the back bearing as in Fig.
571, the swing frame may be suspended from the spindle (R, Fig. 565)
that passes through the lathe head, which may also carry the cone for
the independent feed as shown in Fig. 571, no matter on which side of
the lathe the lead screw and feed rod are. This affords the convenience
that when both lead screw and feed rod are in front of the lathe, the
feed may be changed from the screw cutting to the rod feed, or _vice
versâ_, by suitable mechanism in the apron, without requiring any change
to be made in the driving gears.

[Illustration: Fig. 570.]

In the lathe shown in Fig. 572, which is from the design of S. W.
Putnam, of the Putnam Tool Company of Fitchburg, Massachusetts, the cone
pinion for the back gear, and that for driving the feed motion, are of
the same diameter and pitch, so that the gear-wheel L in Fig. 573 may
(by means of a lever shown dotted in) be caused to engage with either of
them. When the latter is used in single gear it would obviously make no
difference which wheel drives L, but when the back gear is put in and L
is engaged with the cone pinion, its speed corresponds to that of the
cone, which being nine times faster than the live spindle, enables the
cutting of threads nine times as coarse as if the back gear was not in
use. This affords very great advantages for cutting worms and threads of
coarse pitches.

An excellent method of changing the direction of feed motion, and of
starting or stopping the same, is shown in Fig. 574, which represents
the design of the Ames Manufacturing Company's lathe.

[Illustration: Fig. 571.]

In the figure, A is the small step of the lathe cone, B the pinion to
drive the back gear, C a pinion to drive the feed gear, giving motion to
D, which drives E, the latter being fast to G and rotating freely upon
the shaft F, G drives H, which in turn drives I. The clutch J has a
featherway into which fits the feather _c_, on the shaft F, so that when
the clutch rotates it rotates J through the medium of _c_; K is a
circular fork in a groove in J, and operated by a lever operated by a
rod running along the front of the lathe bed. This rod is splined so
that a lever carried by the apron or feed-table, having a hub and
enveloping the rod, may by means of a feather filling into the spline
operate the rod by partly rotating it, and hence operate K. Suppose now
that this lever stands horizontal, then the clutch J would stand in the
position shown in the cut, and D, E, G, H, and I, would rotate, while F
would remain stationary. By lifting the lever, however, J would be moved
laterally on F (by means of K) and the lug _a_ on J would engage with
lug _b_ on G, and G would drive J, which through _c_ would drive F, on
which is placed a change gear at L, thus traversing the carriage
forward. To traverse it backward the lever would be lowered or depressed
below the horizontal level moving K, and therefore J, to the right, so
that lug _a_ would engage with lug _b_ on I, hence F would be driven by
I, whose motion is in an opposite direction to G, as is denoted by the
respective arrows.

[Illustration: Fig. 572.]

To throw all the feed motion out of gear, to run the lathe at its
quickest for polishing, &c., the operation is as follows.

[Illustration: Fig. 573.]

M is tubular and fast in N and affords journal bearing to wheel D.
Through M passes stud O, having a knob handle at P. At the end of the
hub of D is a cap fast in D, the latter being held endways between the
shoulder shown on O and the washer and nut T. If then P be pulled
outwards O will slide through M, and through the medium of T will cause
D to slide over M, in the direction of the arrow, and pass out of gear
from C, motion therefore ceasing at C.

Q is the swing frame for the studs to carry the change wheels, and R a
bolt for securing Q in its adjusted position. S is a journal and bearing
for H.

If it be considered sufficient the feed motion on small lathes (instead
of feeding in both directions on the lateral and cross feeds as in the
Putnam Lathe), may feed in the direction from the dead to the live
centre, and in one direction only on the cross slide.

[Illustration: Fig. 574.]

An example of a feed motion of this kind is shown in Figs. 575 and 576;
_f_ _f_ is the feed spindle splined and through the medium of a feather
driving the bevel pinion A having journal bearing in B. Pinion A drives
the bevel gear C, which is in one piece with pinion D. The latter drives
gear F, which drives pinion K, which is carried on a lever L, pivoted on
the stud which carries F, so that by operating L, pinion K is brought
into gear with pinion P, which is fast upon the cross-feed screw, and
therefore rotates it to effect the automatic cross feed.

[Illustration: Fig. 575.]

[Illustration: Fig. 576.]

As shown in the cut, the lever L is in such a position as to throw K out
of gear with P, and the cross feed screw is free to be operated by the
handle by hand. At M is a slot in L in which operates a cam or
eccentric, one end of which projects into L, while at the other end is
the round handle R, Fig. 575, which is rotated to raise or lower that
end of L so as to operate K. To operate the saddle or carriage the
motion is continued as follows:--at the centre of F is a pinion gear G
which operates a gear H, which is in one piece with the pinion I, and
the latter is in gear with the rack running along the lathe bed.

If the motion from A to I was continuous, the carriage feed or traverse
would be continuous, but means are provided whereby motion from F to I
may be discontinued, as follows:--A hand traverse or feed is provided.
J, Figs. 575 and 576, is carried by a stud having journal bearing in a
hub on X and receiving the handle Q; hence by operating Q, J is rotated,
operating the gear H, upon which is the pinion I, which is in gear with
the rack running along the lathe bed.

To lock the carriage in a fixed position, as is necessary when operating
the cross feed on large radial surfaces, the following device is
provided:--N is a stud fixed in a hub on X, and having a head which
overlaps the rim of H, as shown in figure. On the other side of that rim
is a washer Z on the same stud having a radial face also overlapping the
rim of H, but its back face is bevelled to a corresponding bevel on the
radial face on the hub of lever O (the hub of O being pivoted on the
same stud). When therefore O is depressed the two-bevel face of the hub
of O forces the washer Z against the face of the wheel H, whose radial
faces at the rim are therefore gripped between the face of the collar N
and that of the washer Z, hence H is locked fast. By raising the end of
lever O, Z is released and H is free to rotate.

Both the carriage feed and cross feed can only be traversed in one
direction so far as these gears and levers are concerned, but means are
provided on the lathe headstock for reversing the direction of motion of
the feed spindle _f_ so as to reverse the direction of the feeds. It
will be observed that so long as _f_ rotates, A, C, D, and F rotate, the
remaining motions only operating when S is screwed up.

In order to obtain a delicate tool motion from the handle Q it is
necessary to reduce the motion between J and I as much as possible, a
point in which a great many lathes as at present constructed are
deficient, because Q, although used to simply traverse the carriage
along the bed, in which case rapid motion of the latter is desirable, is
also used to feed the tool into corners when the lathe has no compound
rest to put on light cuts on radial faces, hence it should be capable of
giving a delicate tool motion.

[Illustration: Fig. 577.]

On account of the deficiency referred to it is often necessary to put on
a fine radial cut by putting the feed traverse in gear, and, throwing
the feed screw gear out of contact with the other change wheels, pull it
around by hand to put on the cut. In compound slide rests these remarks
do not apply, because the upper part of the rest may be used instead of
the handle Q.

Many small lathes are provided with a tool rest known as the _elevating
rest_, or weighted lathe.

An excellent example of an elevating rest for a weighted lathe is shown
in Figs. 577 and 578, which represent the construction in the Pratt and
Whitney lathe. A is the lathe shears upon which slides the carriage
provided with [V] slideways R for the sliding piece B, and provided at
the other end with the guides H. The cross slide S is pivoted upon B at
D, and fits at the other end between the guides H. At E is the elevating
screw which when operated raises or lowers that end of the elevating
rest to adjust the tool height. This also affords an excellent means of
making a minute adjustment for depth of tool cut. The tool rest F is
bolted to S.

[Illustration: Fig. 578.]

The weight W is suspended from S and, therefore, holds one end of S to
B, the lathe to C, and C to A; at the other end the weight holds S to C
(through the medium of the elevating screw E) and C to A. The cross feed
nut N is fast to S, the cross feed screw being operated by hand wheel G.
B is provided with the [V] slideways R, which slide upon corresponding
[V] slides R´ upon C; P is a lug cast upon C, and K is a screw threaded
in B. When the end of screw K abuts against P the motion of S, and,
therefore, of the cutting tool T, towards the work is arrested, hence
when the tool is adjusted to the proper depth of cut, K is operated to
abut firmly against P, and successive pieces may be turned to the same
diameter without requiring each piece to be measured for diameter. N is
the handle for opening and closing the nut for the feed screw Q, and Z
is the wheel for the hand feed traverse. The length of cross feed motion
is determined by the length of the cross [V] slides R´.

This class of rest possesses the advantage that no lost motion in the
slides occurs by reason of the wear, because the weight keeps the parts
in constant contact notwithstanding such wear; on the other hand,
however, the slide [V]s sustain the extra wear due to the weight W in
addition to the weight of the carriage. Lathes of this class are
intended for light work, and are less suited for boring than for plain
turning; they are, however, very convenient, and are preferred by many
to any other kind of lathe for short and light work.

[Illustration: Fig. 579.]

The tool rest being removable may be supplanted by other special forms
of rest. Thus Figs. 579 and 580 represent a special rest for carrying
two tools to cut pieces of work to the exact same length. Bolts D and E
are to secure the rest A to the elevating rest, and C C are the clamps
for the two tools B.

[Illustration: Fig. 580.]

Fig. 581 represents a cross sectional view of the Putnam Tool Company's
gibbed elevating rest, there being a gib on the underneath side of the
front shear. The elevating screw is pivoted by a ball joint. By
employing a gib instead of a weight, the bed may be provided with cross
girts or ribs joining the two sides of the shear, thus giving much
greater stiffness to it.

Figs. 582, 583, and 584 represent a lathe feed motion by William Munzer,
of New York. The object in this motion is to insure that no two feeds
can be put into operation simultaneously, because putting the feed in
motion in one direction throws it out of gear for either of the others.
Another object is to have the transmitting motion as direct as possible
so as to avoid the rotation of any wheels not actually necessary for the
transmission of the motion; and a third object is to enable the throwing
out of gear of all wheels (when no feed motion at all is required)
without leaving the apron.

The means employed to effect these objects are as follows:--

In Fig. 582 _f_ represents the independent feed spindle and S the
lead-screw: _f_ is splined to drive A, A´ and A´´, which is a sleeve in
one piece, and consists of a circular rack at A, a bevel pinion at A´,
and a second bevel pinion at A´´. This sleeve may be operated in either
direction along _f_ by rotating the pinion B. As shown in the cut A´ and
A´´ are both out of gear with the bevel-wheel C, but if B be rotated to
the right then A´ will be in gear with C, or if it be operated to the
left then A´´ will be in gear with C. Now the direction of rotation of C
will be governed by which pinion, A´ or A´´, drives it, and these are
the means by which the direction of the feed traverse and also of the
cross feed is determined.

If none of the feeds are required to operate, the sleeve occupies the
position shown in the cut, and the circular rack at A simply rotates
while B and all other parts remain at rest. On the same central pin as C
is the pinion D driving a spur gear E´´. On the same centre pin as E is
the gear F driving G, which is on the same central pin as C and D. The
gear H is fixed to and rotates with G and drives I; all these gears
serving to reduce the speed of motion when operating to feed the
carriage traverse in either direction.

A gear J is carried on the end of a lever K, being pivoted at L. In the
position shown J is out of gear with all gears, but it may be swung to
the right so as to engage with wheel I and wheel M, and convey the
motion of I to M. Upon the same spindle as M is the pinion N, engaging
with the rack O, which is fast on the lathe bed. This completes the
automatic feed traverse.

For a hand feed traverse, pinion P is employed to drive M, which is fast
to N. The cross feed is self-acted by moving lever K to the left,
causing it to engage with pinion Q as well as with T, Q being fast on
the cross feed screw. To lock J in either of its three positions there
is provided on lever K a spring locking pin R, shown clearly in Fig.
584, which represents an irregular section of the gearing viewed from
the headstock of the lathe. The pin R is pressed inward by the spiral
spring shown, and has a conical end fitting into holes provided in the
apron to receive it. There are three of these holes, shown in dotted
lines at _a_ _b_ _c_ in Fig. 582. When the pin is in _a_ the lever K,
and therefore wheel J, Fig. 582, is locked out of gear; when it is in
hole _b_ wheel J is locked in gear with I and M, and when it is in _c_
the wheel J is in gear with T and Q, and the cross feed is actuated.

[Illustration: Fig. 581.]

[Illustration: Fig. 582.]

A similar locking device is provided for the pinion B for actuating A;
thus in Fig. 582 B is the lever, the spring pin being at R´´; or
referring to Fig. 584, X is the lever fast at _x_ on the pin driving B,
and R´´ is the spring pin.

The nut for the lead screw is secured either in or out of gear with the
screw in the same manner, _x´_, Fig. 583, being the lever and R´ the
spring pin.

In screw cutting the cutting tool requires to be withdrawn from the
thread while the carriage traverses back, and it is somewhat difficult
to know just how far to move the tool in again in order to put on a
proper depth of cut. To facilitate this the device shown in Fig. 585
(which is taken from the "American Machinist,") is sometimes employed.

It consists of a ring C inserted between the cross slide D and the
handle hub B having journal bearing on and rotating with the latter.
When the first cut is put on, the mark on C is coincident with that on
D, and the ring is then, while the first cut is traversing, moved
(supposing the cross feed screw to have a right-hand thread) to the
left, as shown in the figure, to the amount the handle will be required
to move to the right to put on the next cut, and when the next cut is
put on the handle will be moved the distance it was moved to withdraw
the tool for the back traverse, and in addition enough to make the marks
coincide, then while the second cut is being taken the ring is again
moved to the left, as in the cut, to give the depth of cut for the next
traverse, and so on.

[Illustration: Fig. 583.]

If the cross feed screw has a left-hand thread, the mark on the ring
would require to be moved to the right instead of to the left of the
mark on D. It is obvious that this answers the same purpose, but is more
exact than the chalk mark before referred to, and, indeed, that chalk
mark could be used in the same way, leaving the chalk mark D and rubbing
out that on C while the cut is proceeding and making a new one for the
next cut.

[Illustration: Fig. 584.]

Another device for use on lathes specially designed for screw-cutting is
shown in Fig. 586, in which A represents the cross feed screw. It is
fast to the notched wheel B, and is operated by it in the usual way. C
is a short screw which provides journal bearing for the screw A by a
plain hole. It is screwed on the outside, and the plate in which it fits
acts as its nut. It is fast to the handle D, and is in fact operated by
it. The handle or lever is provided with a catch E, pivoted in the
enclosed box F, which also contains a means of detaining the catch in
the notches of the wheel, or of holding it free from the same when it is
placed clear. If, then, the lever D be moved back and forth the feed
screw A, and hence the slide rest, will be operated; while, if the catch
be placed in one of the notches on the wheel B, both the screws, A and
C, will act to operate the rests. When, therefore, the tool is set to
touch the diameter of the work, the catch E is lifted and the feed wheel
B rotated, putting on the cut until the catch E will fall into the next
notch in B, the lever D resting in the meantime on the stud G. When the
cut is carried along the work to the required distance the tool is
withdrawn by moving D over until it rests upon stud or stop H. While the
slide rest is traversing back E is lifted and B rotated so that E will
fall into the next notch, and when the tool starts forward again D is
moved over from H to G, as shown in the figure, and the tool cut is put
on.

[Illustration: Fig. 585.]

When the device is not required to be used E is thrown out, D rests on
E, and the feed is operated in the ordinary manner.

[Illustration: Fig. 586.]

A simple attachment for regulating on a slide rest the depth of tool cut
in screw cutting or for adjusting the cut to a requisite diameter when a
number of pieces are to be turned to diameter by a finishing cut, is
shown in Fig. 587, in which B represents the slide rest carriage, and E
the cross slide on which the slide rest A is traversed by means of the
cross feed screw _f_. A screw is screwed into the rest, as shown,
carrying the two circular milled edge nuts R P; the screw passes an easy
fit through the piece C, which is capable of being fixed in any position
along the slide E by means of the set screw S; the nut R is set in such
a position on the screw that it will abut against C when the tool is
clear of the work surface (for the back traverse) while P may be used in
two ways:--First it may be set so that when it comes against C the
thread is cut to the required depth, and thus act as stop to give the
thread depth without trying the gauge: or it may be used to answer the
same purpose and in the same way as the ring C in Fig. 585.

[Illustration: Fig. 587.]

The use of this device as a stop to gauge the thread depth is confined
to such lengths of work as enable the tool to cut several pieces without
requiring regrinding, because when the tool is removed to grind it, it
is impracticable to set it exactly the same distance out from the tool
post, hence the adjustment of P becomes destroyed. It is better,
therefore, in most cases where a number of threads of equal pitch and
diameter are to be cut, to rough them all out, cutting the threads a
little above the gauge diameter so as to leave a finishing cut to be
taken. In roughing out, however, the nut P may still be used to regulate
the depth.

For the finishing cut the tool may be ground and P adjusted to give the
requisite depth of cut, taking a single traverse over each thread to
finish it. This, of course, preserves the tool and enables it to finish
a larger number of threads without regrinding, and the consequent
readjustment of P.

It is obvious that the nut P may be employed in the same manner to turn
a number of plain pieces to an equal diameter.

[Illustration: Fig. 588.]

It is preferable in a device of this kind, however, to employ the two
adjusting nuts P and Q in Fig. 588, Q being a clamp nut that can be
closed by a screw so as to firmly grip the threaded stud. Q is adjusted
so that when P abuts against it the tool will cut to the correct
diameter when it is moved in as far as nuts P Q will permit. The use of
the second nut P is as follows:--Suppose a first cut has been taken and
P may be screwed up to just meet the face of clamp C. Then while the
carriage is traversing, P may be screwed back towards Q sufficiently to
put on the next cut, and so on, so that P is used to adjust the depths
of the roughing, and Q that of the finishing cut.

Sometimes a feed motion to a slide rest is improvised by what is known
as the _star feed_, the principle of action of which is as follows: Upon
the outer end of the feed screw of the boring bar or slide rest, as the
case may be, is fastened a piece of iron plate, which, from having the
form in which stars are usually represented, is called the star. If the
feed is for a slide rest a pin is fastened to the lathe face plate or
other revolving part, in such a position that during the portion of the
revolution in which it passes the star it will strike one of the star
wings, and move it around sufficiently to bring the next wing into
position to be struck by the pin during its succeeding revolution. When
the feed is applied to a revolving boring bar the construction is the
same, but in this case the pin is stationary and the star revolves with
the feed screw of the bar.

In Fig. 589 is shown a star feed applied to a slide rest. A is the slide
rest, upon the end of the feed screw of which the star, B, is fitted. C
is a pin attached to the face plate of the lathe, which, as it revolves,
strikes one of the star wings, causing it to partly rotate, and thus
move the feed screw. The amount of rotation of the feed screw will
depend upon the size of the star and how far the circle described by the
pin C intersects the circle described by the extreme points of the star
wings. Thus the circles denoted by D E show the path of the pin C; the
circle F H the path of the star points, and the distance from F to G the
amount which one intersects the other. It follows that at each
revolution of C an arm or wing of the star will be carried from the
point G to point F, which, in this case, is a sixth of a revolution. If
more feed is required, we may move the pin C, so that it may describe a
smaller circle than D E, and cause it to intersect F H to a greater
extent, in which case it will move the star through a greater portion of
its revolution, striking every other wing and doubling the amount of
feed.

It will be observed that the points F and G are both below the
horizontal level of the slide rest's feed screw, and therefore that the
sliding motion of the pin C upon the face of the star wings will be from
the centre towards the points. This is better, because the motion is
easier and involves less friction than would be the case if the pin
contact first approached and then receded from the centre, a remark
which applies equally to all forms of gearing, for a star feed is only a
form of gearing in which the star represents a tooth wheel, and the pin
a tooth in a wheel or a rack, according to whether its line of motion is
a circle or a straight line.

[Illustration: Fig. 589.]

It is obvious that in designing a star feed, the pitch of the feed screw
is of primary importance. Suppose, for example, that the pitch of a
slide rest feed screw is 4 to an inch, and we require to feed the tool
an inch to every 24 lathe revolutions; then the star must have 6 wings,
because each revolution of the screw will move the rest 1/6 in., while
each revolution of the pin C will move the star 1/6 of a revolution, and
4 × 6 = 24. To obtain a very coarse feed the star attachment would
require to have two multiplying cogs placed between it and the feed
screw, the smaller of the cogs being placed upon the feed screw.

In many lathes of European design, the feeds or some of them, are
actuated by ratchet handles, operated by an overhead shaft, having arms
which rock back and forth. Thus in Fig. 590 is a lathe on which there is
provided at A crank disc, carrying in a dovetail slot a pin P, for
rocking the overhead shaft from whose arms a chain is attached which may
be connected to the ratchet handle shown on the cross-feed screw, the
weight being for the purpose of carrying that handle down while the
chain pulls it up. To regulate the amount of feed the pin P is adjusted
in the slot in A, or the chain may be attached in different positions
along the length of the ratchet arm, the weight being provided with a
set screw so that it may be set in any required position along the
ratchet arm.

[Illustration: Fig. 590.]

TOOL-HOLDING DEVICES.--Perhaps no part of a lathe is found in American
practice with so many different forms of construction as the device for
holding the cutting tool. The requirements for a lathe to be used on
light work and where frequent changes in the position of the tool are
necessary, are quite different from those for a lathe intended to take
as heavy a cut as the lathe will properly drive, and wherein tools
having the cutting edge at times standing a long way out from the tool
post (as sometimes occurs in the use of boring tools). In the former
case a single holding screw will suffice, possessing the advantage that
the tool may be quickly inserted, adjusted for height and set to one
side or the other, with a range of motion which often permits of a tool
that has taken a parallel cut being moved in position to capacitate it
to take a facing one, which would not be the case were its capacity for
side adjustment limited.

In the case of the common American lathe having a self-acting feed and
no compound rest, the tool post is usually employed, the rest being
provided with a [T] slot such as shown in Fig. 577. This enables the
tool post to be moved from side to side of the tool rest, and swing
around in any required position. In connection with such tool posts
various contrivances are employed to enable the height of the cutting
edge of the tool to be readily adjusted. Thus in the Fig. 591, the tool
post is surrounded by a cupped washer W, and through the slot in the
tool post passes a gib G, which may be moved endways in the slot and
thus elevates or depresses the tool point.

The objection to this is that the tool is not lifted parallel, or in
other words is caused to stand out of its proper horizontal position
which alters the clearance of the tool, and by presenting the angles
forming the tool edge in an improper position, with relation to the
work, impair its cutting qualification, as will be shown hereafter when
treating of lathe cutting tools.

An improvement on this form has been pointed out by Professor John E.
Sweet, whose device is shown in Fig. 592. Here the washer or ring is
rounded and the bottom surface of the gib is hollowed, so that chips or
dirt will to a great extent fall off, and every time the tool post is
swung the gib acts to push off whatever dirt may lodge on the washer.

In the design shown in Fig. 593, the tool rests upon two washers W that
are tapered, and its height is adjusted by revolving one of these
washers, it being obvious that the limit of action to depress the tool
point is obtained when the two thin sides of the washers are placed
together, and on the same side of the tool post as the cutting edge of
the tool, while the limit of action to raise the tool point is obtained
when the washers have their thick sides together and nearest to the tool
point.

Here again the tool is thrown out of level, and to obviate this
difficulty the stepped washer shown in Fig. 594 may be used, the steps
on opposite sides of the washer being of an equal height. This enables
the tool to be raised or lowered without being set out of the horizontal
position; but it has the defect that the adjustment cannot be made any
finer than the height of the steps, and if the height is made to vary
but slightly, in order to refine, as it were, the adjustment, the range
of tool elevation or depression is correspondingly limited. Another form
of stepped washer is shown in Fig. 595, in which no two steps are of the
same height. This affords a wider range of adjustment, because the same
two steps will alter the height of the tool by simply turning the washer
one-half revolution. It has two defects, however; first, the least
amount of adjustment is that due to the difference in height of the
steps; and, second, when the tool is elevated it grips the washer at A,
so that the tool is not supported across the full width of face of the
washer, as it should be.

A defect common to all devices in which the tool is thrown out of level,
is that the binding screw does not bed fair upon the tool, and as a
result it is apt, if screwed home very firmly, as is necessary to hold
boring tools that stand far out from the tool post, to spread the screw
end as in Fig. 596, or to bend it.

A very convenient tool-adjusting device is shown in Fig. 597. It
consists of a threaded ring N receiving the threaded bush M, the tool
height being adjusted by screwing or unscrewing one within the other.

The objection to this is, that it occupies so much vertical height that
there is not always room to admit it, which occurs, for example, in
compound slide rests on small lathes.

On these rests, therefore, a single washer is more frequently used,
which answers very well when the tool post is in a slot, so that it can
be moved from side to side of the rest as occasion may require. When,
however, the position of the tool post is fixed it has the disadvantage
that the point P, Fig. 598, where the tool takes its leverage, is too
far removed, and the tool is therefore liable to bend or spring from the
pressure of the cut.

In Fig. 599 is an elevating device sometimes used on the compound rests
of large lathes. The top of the rest is provided with a hub H, threaded
externally to receive a ring nut R, around whose edge there are numerous
holes to receive a pin for operating the nut. The tool-post is situated
central in the hub. When the tool is loose the ring nut can be operated
by hand or the tool may be gripped lightly and the ring nut operated by
a pin. The level of the tool is here maintained; it is supported to
about the edge of the rest on account of the large diameter of the ring
nut, and a very delicate adjustment for height can be made, but such a
device is only suitable for large lathes on account of the depth of the
ring nut and hub.

[Illustration: Fig. 607.]

On small slide-rests the device shown in Fig. 600 is often found. It
consists of a holder H, in which is cut a seat for the tool, this seat
being inclined to give the piece of steel used as a tool a certain
constant degree of angle, and at the same time to permit of the tool
being moved endwise in the holder to set it for height; but, as the tool
requires to be pushed farther and farther through the holder to raise
it, it is not so well supported as is desirable when slight tools are
used, unless the holder is made long, so as to pass through the tool
post with the tool. Again, it does not support the tool sideways unless
the tool steel is dressed up and closely fits the groove in the holder.

In Fig. 601 W W are two inverted wedges which afford an accurate
adjustment, but the range is limited, because if the wedges have much
taper they are apt to move endways when the tool is fastened.

A convenient device for the compound rests of small lathes is shown in
Fig. 602. It consists of a holder pivoted upon a central post and
carrying two tool-binding screws, hence it can be revolved to set the
tool in any required position. A similar device is shown in Fig. 603, in
which the central post is slotted at A to receive the tool, and also
carries a plate C, held by the nut N, and provided with tool-holding
screws B and B´, which abut against the top of the rest, a top view of
the device being shown in Fig. 604. Plate C may thus be swung around to
set the tool in any required position on either side of the rest.

In Maudslay's slide rest, the tool clamp shown in Fig. 605 is employed.
Screws A are employed to grip the tool moderately firm, and a turn of
screws B (whose ends abut against the top of the slide rest) very firmly
secures the tool, since it moves the clamp C as a lever, whose fulcrum
is the screw A.

Figs. 606 and 607 represent the Whitworth tool clamp, the clamping
plates of which change about upon the four studs, and are supported at
their inner ends by a block equal in height to the height of the tool
steel.

Figs. 608, 609, 610, and 611 represent the "Lipe" tool post, so called
from the name of its inventor. The top of the cross slide is
cylindrical, and is bored to receive the tool post which has a
cylindrical stem. The cylindrical part of the tool post is split
vertically, and has two lips, the bolt D passing through one lip and
threading into the other, so that by operating bolt D the tool post may
be gripped very firmly or released, so that it may be revolved to bring
the tool into any required position after it is fastened in the tool
post, which is a great advantage because the tool is brought to a solid
seating in the post before its height is adjusted, and will not
therefore be altered in height by setting up the set screws as often
occurs in ordinary tool posts. From the shape of the tool post, the tool
may be gripped by one set screw only, when required for light duty, or
by two set screws for heavy duty or for boring, while in either case it
is supported clear to the edge of the rest.

[Illustration: Fig. 608.]

[Illustration: _VOL. I._ =TOOL-HOLDING AND ADJUSTING APPLIANCES.= _PLATE
VII._

Fig. 591.

Fig. 592.

Fig. 593.

Fig. 594.

Fig. 595.

Fig. 596.

Fig. 597.

Fig. 598.

Fig. 599.

Fig. 600.

Fig. 601.

Fig. 602.

Fig. 603.

Fig. 604.

Fig. 605.

Fig. 606.]

Fig. 608 shows the tool in position, held by a single screw, for work
requiring the tool to be close up to the work driver. In Fig. 609 a tool
is shown held as is required by work between centres, but both
set-screws are used. Fig. 610 shows a tool in position for boring, two
set-screws being used. Fig. 611 shows a tool being held for the same
purpose, but by a single screw, and it will be observed that the
advantage of the second set-screw is obtained without in any way
sacrificing the handiness of the post, when used with a single screw.
Whether one or two set-screws are used, the boring tool may be forged
from a single bar of octagon steel, which can be seated in a piece like
that shown at E in Fig. 610, which is grooved so as to receive and hold
the tool. As is well known, boring tools are the most troublesome both
to forge and to adjust in the lathe, and, as the result, a light tool is
often used because no other is at hand and it is costly to make a new
one. When, however, the tool can be forged from a plain piece of steel,
these objections are overcome, and a sufficient number of tools may be
had so that one can always be found suitable for any ordinary sized
hole, the object being to use as rigid a tool as can be got into the
hole bored. The feature of maintaining the tool level is of great
importance in boring work, because when the tool requires to be set out
of level to adjust its height, it will generally strike against the
mouth of the hole if the latter is of much depth. This annoyance is also
frequently met with in boring tools which are forged out of rectangular
steel, because the rounded stem is generally left taper. The largest end
of the taper is generally nearest the tool post. Hence the capacity to
use octagon steel and keep it level while adjusting its height, added to
the fact that the tool is supported clear to the edge of the tool rest,
and the tool post is so blocked as to virtually become a part of the
rest, constitute a very important advantage.

[Illustration: Fig. 609.]

[Illustration: Fig. 610.]

A common device on large lathes is shown in Fig. 612, the two clamps
being shown in position for outside turning, and being changed (so as to
stand at a right angle to the position they occupy in the figure) for
holding boring tools. The bolts are enveloped by spiral springs which
support the clamps.

Figs. 613 and 614 represent the tool holders employed in the Brown and
Sharpe small screw machines. In the front rest, Fig. 613, the piece R
receives two adjusting and tool-gripping screws S, upon which sits the
gib G, and upon this the tool is placed. The surface E at the top of the
tool post slot is curved so that it will bear upon the top of the tool
at a point only. The tool is here supported along the full length of the
gib, and there is no set-screw at the top of the tool post, which
enables a much more unobstructed view of the tool.

Fig. 614 is the tool post used at the back of the rest, the piece B
passing through the tool post slot. The tool rests upon the top of screw
E and upon the top of B at F, and is secured by set-screw S; its height
is therefore adjusted by means of screw E, which is threaded in B. The
set-screw S is not in this case objectionable, because it is at the back
of the rest, and therefore does not obstruct the view of the work, while
it is at the same time convenient to get at.

When the screw for traversing a lathe carriage is used for plain
feeding, it is termed the feed screw, but when it is used to cut threads
it is termed the lead screw.

A lead screw should be used for screw cutting only, so that it may be
preserved as much as possible from wear. As the greater portion of
threads cut in a lathe of a given size are short in comparison with the
length of the lathe, it follows that the part of the lead screw that is
in operation when the carriage or saddle is traversing over short work
is most worn, while the other end is least worn, hence it is not unusual
to so construct the screw and its bearings that it may be changed end
for end in the lathe, to equalize the wear. By turning a lead screw end
for end, therefore, to equalize the wear, the middle of the length of
the screw will become the least worn, and, therefore, the most true.
Hence it is better to use one end of the lead screw for general work,
and to reverse it and use the other end only for screws requiring to be
of very correct pitch.

[Illustration: Fig. 611.]

[Illustration: Fig. 612.]

[Illustration: Fig. 613.]

[Illustration: Fig. 614.]

To obviate the wear as much as possible the feed nut should embrace as
great a length of the screw as convenient, and should be of a material
that will suffer more from wear than the lead screw, or in other words
shall relieve the feed screw from wear as much as possible. The wear on
the nut being equal from end to end, the wearing away of one side of its
thread does not vary its pitch; hence the only consideration as to its
wearing qualification are the expense of its renewal and the length of
time that may occur between its being engaged with the lead screw and
giving motion to the lathe carriage, this time increasing in proportion
as the nut thread is worn. Under quick speeds or when the lathe is in
single gear, the rotation of the feed screw is so quick that not much
time is lost before the carriage feeds, but when the back gear is in
operation at the slowest speeds, the loss of time due to a nut much worn
is an item of importance.

In some lathes the feed screw is employed for screw cutting and for
operating an independent feed also. This is accomplished by cutting a
feather way or spline along it, so that a worm having journal bearing in
the apron of the rest carriage may envelop the lead screw and be driven
by it, through the medium of a feather fast into the worm gear. The
motion obtained from the worm gear is transferred through suitable
gearing to the rack pinion.

The spline is cut deeper than the thread, so as to prevent the latter as
far as possible from wear, by reason of the friction of the spline.

The lead screw if long should be supported, to prevent its sagging of
its own weight. In some cases the lead screw is supported in a trough
along its whole length, as is done in the Sellers lathe. In other cases,
bearings hanging from the [V]-slides, and movable along the bed, are
employed.

It is desirable that the feed screw and nut be as near the middle of the
carriage as possible, so that it shall pull the carriage at as short a
leverage as possible, thus avoiding the liability to tilt or twist the
carriage; but it is not practicable to place it midway between the lathe
shears, because in that case the cuttings, &c., from the work would fall
upon it, and cause excessive and rapid wear of the screw and nut.

In general the lead screw is located either in front, or at the back of
the lathe, and in considering the more desirable of the two locations,
we have as follows:

The feed nut should obviously remain axially true with the lead screw,
as by reason of the extra weight of the front of the carriage, both it
and the lathe shears wear most at the front, and the carriage,
therefore, falls to the amount of its own wear and the wear of the
shears. If the lead screw is used to feed with (as it should not be),
the nut wears coincidently with the carriage and the shears, and the
screw alignment is not impaired; but with an independent feed, only a
small portion of the carriage traversing is done with the lead screw,
hence the carriage lowers from the wear due to the independent feeding,
and when the lead screw comes to be used its nut is not in true
alignment with it. It is obviously preferable, then, to place the lead
screw at the back, where the carriage and shears wear the least.
Furthermore, this relieves the carriage front from the weight of the
nut, &c., tending to equalize the back and front wear, while removing
the nut-operating device from the front to the back of the shears, and
thus reducing the number of handles in front, and thus avoid
complication in small lathes.

LATHE LEAD SCREWS.--Lead screws have their pitches in terms of the inch
throughout all parts of the world; or, in other words, the lead screws
of all lathes contain so many full threads per inch of length.

Lead screws are usually provided with square threads of the usual form,
or with threads whose sides have about fifteen degrees of angle, so that
the two halves of the feed nut may be let together to take up the wear.
It is obvious that in a [V]-thread or in a thread whose sides are at an
angle, the feeding strain tends to force the two halves of the feed nut
apart, and therefore places a strain on the feed-nut operating mechanism
that does not exist in the case of a square thread. Furthermore it can
be shown that with a [V]-thread the opportunities to lock the carriage
on a wrong place, after traversing it back by hand in screw cutting, are
increased, thus augmenting the liability to cut intermediate and
improper threads.

[Illustration: Fig. 615.]

In Fig. 615, for example, we have a pitch of lead screw of three threads
per inch, and the gears arranged to cut six threads per inch on the
work. As the bottom wheel has twice as many teeth as the top one, it is
clear that, while the top one makes one, the bottom one will make half
revolution, and the lead screw will make half a turn for every turn the
work makes. Now, suppose the tool point to stand opposite to space A,
and the nut (supposing it to have but one thread only, which is all that
is required for our purpose), stand opposite to space D. Suppose,
further, that the lathe makes one revolution, and space B on the work
will have moved to occupy the position occupied by space A, or, rather,
there will still be a place at A fully in front of the tool, as should
be the case, but the lead screw will have made half revolution, the top
_e_ of the thread coming opposite to the feed nut, as in the position of
tool and nut shown in the figure at T and N; hence the nut would not
engage, without moving the lathe carriage sideways, and thus throwing
the tool to one side of the thread in the work. When, however, the work
had made another revolution, both the feed screw and the work would
again come into position for the tool and nut to engage properly, and it
follows that in this case the tool will always fall into proper position
for the nut to be locked.

It is obvious, however, that had the lead screw thread been a square
one, and the nut thread to accurately fit to the lead screw thread, so
as to completely fill it, then the nut could not engage with the lead
screw until the lathe had made a complete revolution, at which time the
work will have made two full or complete revolutions, and the tool
would, therefore, fall into proper position to follow in the groove or
part of a thread cut at the first tool traverse.

[Illustration: Fig. 616.]

In Fig. 616, we have the same lead screw geared to cut five or an odd
number of threads per inch. The tool and the nut are shown in position
to properly engage, but suppose, the nut being disengaged, that the work
makes one revolution, and during this period the lead screw will have
made 3/5ths of a revolution, hence the nut will not be in position to
engage properly, because, although space B will have travelled forward
so as to occupy the position of space A in the figure (that is, there
will be a space fairly in front of the tool point), yet the nut will not
engage properly, because the nut point will not be opposite to the
bottom of the lead screw thread. When the work has made its second
revolution, and space C moves to the position occupied by A, the lead
screw will have made 6/5 or 1-1/5 revolutions, and the nut cannot engage
properly; when the lathe has made its third revolution, the lead screw
will have made 1-4/5 revolutions and the nut will still fall to one side
of the thread space, and will not lock properly. The work having made
its fourth turn, the lead screw will have made 2-2/5 turns, and the nut
will not be in position to lock fairly. The work having made its fifth
turn, however, the lead screw will have made three turns, and the
threads will fall into the same position that they occupy in the figure,
and both tool and feed nut will fall into their proper positions in
their respective threads. It does not follow, however, that, the lead
screw having a [V]-shaped thread, the nut cannot be forced to engage but
once in every five turns of the lead screw, because, were this the case,
it would be impossible to lock the nut in an improper position.

[Illustration: Fig. 617.]

Suppose, for example, that we have in Fig. 617, the same piece of work
and lead screw as in Fig. 616, and that a first groove, A, has been cut
with the tool in the position shown, and the nut engaged in the position
marked 1. Now, suppose the nut be disengaged and the work allowed to
make one revolution, then the lead screw will, during this revolution,
revolve 3/5 of a revolution, and the position of the nut point with
relation to the lead screw will be as at position 2. If, then, the nut
was forced into the lead screw thread, it would, acting on the wedge
principle, move the carriage to the right sufficiently to permit the nut
to engage fully in thread G, and the tool would then cut a second groove
on thread B. If the nut then be withdrawn from thread G, and the work
allowed to make another revolution, the nut will stand in a precisely
similar position with relation to the lead screw thread as it did in
position 2, and by forcing it down into thread H the carriage would be
again forced to the right, causing a third thread, C, to be cut. By
repeating the operation of withdrawing the nut, letting the work make
another revolution and then engaging the nut again, it will seat in
thread K, and a fourth thread D will be cut. On again repeating the
operation, however, the nut will come into position 5, and, on being
drawn home into thread, or, rather, into space L, the tool will fall
into groove A again. Thus there will be four threads, each having a
pitch equal to that of the lead screw. The second (B) of these four will
fall to the left of thread A to an amount or distance equal to 2/5 of
the pitch of the lead screw, because, in forcing the nut from position 2
down into the lead screw, the slide rest, and therefore the tool, will
be moved to the right 2/5 of the pitch of the lead screw. The third
thread C will fall to the left of thread B also to an amount equal to
2/5 of the pitch of the lead screw, because, in forcing the thread to
seat itself into thread H from position 3, the slide rest was again
moved (to that amount) to the right. The fourth thread D will fall to
the left of thread C to the same amount and for the same reason.

But in this case, as before, if the lead screw had a square thread and
the nut threads completely filled the spaces between the lead screw
threads, then the nut could not engage at the 2nd, 3rd, or 4th work
revolution, hence the false threads B, C, and D, could not have been
cut, even though the feed nut was disengaged and the lathe carriage was
traversed back by hand.

Now, suppose that two threads on the work measure less than the amount
the lead screw advances during the time that the work makes a
revolution, and if the lead screw has a [V]-shaped thread, the case is
altered. We have, for example, in Fig. 618, a pitch of lead screw of 3
to cut 12 and 13 threads respectively. In the case of the 13 threads it
will be seen that, supposing there to have been a first cut taken on the
work, and the feed nut to be disengaged while the work makes a
revolution, then the lead screw will revolve 3/13 revolution and the
point A on the lead screw will have moved up to point B, and the nut
point remaining at N, seating it in the thread, would cause it to engage
with the same thread that it did before, and no second thread would be
cut. If the nut be then released, the work allowed to make another
revolution and the nut again closed, the operation would be the same as
before, and no error would be induced, and so on. Suppose, further, that
after the nut was disengaged the lathe was permitted to make two
revolutions, and the lead screw would make 6/13, or less than half a
turn, and closing it would still cause it to pass back into the same
thread on the lead screw and produce correct work. But if after the nut
was released the work made three turns, the lead screw would make 9/13
of a turn, and the nut would fall on the right-hand side of the lead
screw thread, and in closing would move the lathe carriage to the right,
causing the tool to cut a second thread. Now, the same operation that
occurred with the first thread would during the next three trials occur
with the second thread, and at the next or seventh trial a third thread
would be cut, which would be again operated upon during the next
succeeding three trials. At the eleventh trial a fourth thread would be
cut, but on the next three trials the tool would again fall into the
groove first cut and the work proceed correctly. In the case of the 12
threads, the thread cut at the first and second trials would be correct.
At the third trial the nut would seat itself in the groove C of the lead
screw, causing the carriage to move to the right to a distance equal to
twice the pitch of thread being cut, but the tool would still fall into
the same groove in the work, as it also would on the fourth. At the
fifth trial the process would be repeated, and so on, so that no second
thread would be cut.

[Illustration: Fig. 618.]

It may now be noted that if we draw the lead screw and the thread to be
cut as in the figure, and draw the dotted lines shown, then those that
meet the bottom of the thread on the lead screw, and also meet the
groove cut on the work, at the first trial, represent the cases in which
the nut will fall naturally into its proper position for the tool to
fall into the correct groove, while whenever the nut is being forced
home it seats in a groove in the lead screw, the bottom of which groove
meets a line drawn from the first thread cut; the results obtained will
be made correct by reason of the movement given to the slide nut when
artificially seating the nut. This is shown to be the case in Fig. 619,
which represents a lead screw having an even number of threads per inch,
and from which it appears that in cutting 12 threads (an even number
also) the nut cannot be engaged wrong, whereas in the case of 13 threads
it can be engaged right three times in 13 trials, and 10 times wrong,
the latter causing the tool to cut three wrong threads.

[Illustration: Fig. 619.]

To prevent end motion of a lead screw it should have collars on both
sides of one bearing, and not one at each bearing. By this means the
screw will be permitted to expand and contract under variations of
atmospheric temperature, without binding against the bearing faces.

When a lead screw is long it requires to be supported, otherwise, either
its weight will be supported or lifted by the feed nut in gear, or if
that nut does not lift the screw, the thread cut will be finer than that
due to the pitch of the lead screw, by reason of its deflection or sag.

A lead screw should preferably be as near as possible to the middle of
the lathe shears, and as close to the surface as possible, so as to
bring it as nearly in line with the strain on the tool as possible, but
on account of the cuttings, which falling upon the screw would cause it
to wear rapidly, it is usual to locate it on one side, so as to protect
it from the cuttings. It is better to locate it on the front side of the
lathe rather than on the back, because the strain of the cut falls
mainly on the front side (especially in work of large diameter when this
strain is usually greatest) and it is desirable to pull the carriage as
near in a line with the resistance of the cut as possible, because the
farther off the feed nut from the cutting tool point, the greater the
tendency to twist the carriage on the shears.

To preserve the nut from wear, it should be made as long as convenient,
as, say, five or six times the diameter of the lead screw; it is usually
made, however, three or four diameters.

It is obvious that the pitch of the thread should be as accurate as
possible, but it has not as yet been found practicable to produce a
screw so accurate that it would not show an error, if sufficient of its
length be tested, as, say, several feet.

If the error in a screw be equal, and in the same direction at all parts
of its length, various devices may be employed to correct it. Thus Fig.
620 represents a device employed by the Pratt and Whitney Co.

It was first ascertained by testing the lathe that its lead screw was
too short by 7/100ths of a revolution in a length of 2 feet, the pitch
of its thread being 6 to an inch. Now in 2 feet of the screw there would
be 144 threads, and since 7/100ths (the part of a revolution the thread
was too short) × 1/6 (the pitch of the thread) = 7/600ths (which was
called 1/85th), the error amounted to 1/85th inch in 144 turns of the
screw. The construction of the device employed to correct this error is
as follows: In Fig. 620, A represents the bearing of the feed screw of
the lathe, and B _b_ a sleeve, a sliding fit upon A, prevented from
revolving by the pin _h_, while still having liberty to move endways. C
represents a casing affording journal bearing to B _b_, having a fixed
gear-wheel at its end C´, and an external thread upon a hub at that end.
D is the flange of C to fasten the device to the shears of the latter,
being held by screws. E represents an arm fast upon the collar of the
feed screw, and carrying the pinion F, the latter being in gear with the
pinion C´, and also with G, which is a pinion containing two internal
threads, one fitting to B at _b_, and the other fitting to C at _c_, the
former having a pitch of 27 threads to an inch, the latter a pitch of 25
to an inch.

[Illustration: Fig. 620.]

The operation is as follows:--The ordinary change wheels are connected
to the feed screw, or lead screw, as it is sometimes termed, at J in the
usual manner. The arm E being fast to the feed screw will revolve with
it, and cause the pinion F to revolve around the stationary gear-wheel
C´. F also gears with G. Now, F is of 12 diametrical pitch and contains
26 teeth, C´ is of 12 diametrical pitch and contains 37 teeth, and G is
of 12 diametrical pitch and contains 36 teeth. It follows that the
pinion F, while moving around the fixed gear C´, will revolve the pinion
G (which acts as a nut), to an amount depending upon the difference in
the number of its teeth and those of fixed gear C´ (in this case as 36
is to 37), and upon the difference in the pitches of the two threads, so
that at each revolution G will move the feed screw ahead of the speed
imparted by the change gears, the end of the sleeve B abutting against
the collar of the feed screw to move it forward.

In this case there are 36 turns of the feed screw A for one turn of the
nut pinion G, the thread on sleeve B being 27, and that on the hub of C
being 25 to the inch; hence, 36 turns of the feed screw gives an end
motion to the sleeve B of 1/25 minus 1/27 = 2/675, and 1/36 of that =
1/12150 of an inch = the amount of sliding motion of the sleeve _b_, for
each revolution of the lathe feed screw. By varying the proportions
between the number of teeth in C´ and G and the pitches of the two
threads in a proper and suitable ratio, the device enables the cutting
of a true thread from any untrue one in which the variation is regular.

It is usual to fasten to the side of the lathe head stock a brass plate,
giving a table of threads, and the wheels that will cut them, and
obviously such tables vary according to the pitch of the lead screw, but
a universal table may be constructed, such as the following table
(prepared by the author) that will serve for any lathe.

At the top of the table is the number of teeth in wheels, advancing by
four from 12 to 80 teeth, but it may be carried as much beyond 80 as
desired. On the left hand of the table is a column of the same wheels.
At the bottom of the scale are pitches of lead screw from 3 up to 20
threads per inch. Over each lead screw pitch are thread pitches, thus on
lead screw pitch 4 we have 20, 19, 18, and so on.

The use of the table is as follows:--

Find the pitch of the lead screw, and at the head of that column is the
number of teeth for the lathe stud or mandril. Then find in that column
the number of threads to be cut, and on the same line, but at the left
hand, will be found the number of teeth for the lead screw.

NUMBERS OF TEETH FOR WHEEL TO GO ON LATHE SPINDLE, LATHE STUD, OR
MANDRIL.

  ------+--+--+--+--+--+--+--+---+---+---+---+---+---+---+---+---+---+---
   Lead |12|16|20|24|28|32|36| 40| 44| 48| 52| 56| 60| 64| 68| 72| 76| 80
  Screw.|  | *|  |  |  |  |  |   |   |   |   |   |   |   |   |   |   |
  ------+--+--+--+--+--+--+--+---+---+---+---+---+---+---+---+---+---+---
   12   | 3| 3| 3| 3| 3| 3| 3|  3|  3|  3|  3|  3|  3|  3|  3|  3|  3|  3
  ------+--+--+--+--+--+--+--+---+---+---+---+---+---+---+---+---+---+---
   16   | 4| 4| 4| 4| 4| 4| 4|  4|  4|  4|  4|  4|  4|  4|  4|  4|  4|  4
  ------+--+--+--+--+--+--+--+---+---+---+---+---+---+---+---+---+---+---
   20   | 5| 5| 5| 5| 5| 5| 5|  5|  5|  5|  5|  5|  5|  5|  5|  5|  5|  5
  ------+--+--+--+--+--+--+--+---+---+---+---+---+---+---+---+---+---+---
   24   | 6| 6| 6| 6| 6| 6| 6|  6|  6|  6|  6|  6|  6|  6|  6|  6|  6|  6
  ------+--+--+--+--+--+--+--+---+---+---+---+---+---+---+---+---+---+---
   28   | 7| 7| 7| 7| 7| 7| 7|  7|  7|  7|  7|  7|  7|  7|  7|  7|  7|  7
  ------+--+--+--+--+--+--+--+---+---+---+---+---+---+---+---+---+---+---
   32   | 8| 8| 8| 8| 8| 8| 8|  8|  8|  8|  8|  8|  8|  8|  8|  8|  8|  8
  ------+--+--+--+--+--+--+--+---+---+---+---+---+---+---+---+---+---+---
   36   | 9| 9| 9| 9| 9| 9| 9|  9|  9|  9|  9|  9|  9|  9|  9|  9|  9|  9
  ------+--+--+--+--+--+--+--+---+---+---+---+---+---+---+---+---+---+---
   40   |10|10|10|10|10|10|10| 10| 10| 10| 10| 10| 10| 10| 10| 10| 10| 10
  ------+--+--+--+--+--+--+--+---+---+---+---+---+---+---+---+---+---+---
   44   |11|11|11|11|11|11|11| 11| 11| 11| 11| 11| 11| 11| 11| 11| 11| 11
  ------+--+--+--+--+--+--+--+---+---+---+---+---+---+---+---+---+---+---
   48   |12|12|12|12|12|12|12| 12| 12| 12| 12| 12| 12| 12| 12| 12| 12| 12
  ------+--+--+--+--+--+--+--+---+---+---+---+---+---+---+---+---+---+---
  *52   |13|13|13|13|13|13|13| 13| 13| 13| 13| 13| 13| 13| 13| 13| 13| 13
  ------+--+--+--+--+--+--+--+---+---+---+---+---+---+---+---+---+---+---
   56   |14|14|14|14|14|14|14| 14| 14| 14| 14| 14| 14| 14| 14| 14| 14| 14
  ------+--+--+--+--+--+--+--+---+---+---+---+---+---+---+---+---+---+---
   60   |15|15|15|15|15|15|15| 15| 15| 15| 15| 15| 15| 15| 15| 15| 15| 15
  ------+--+--+--+--+--+--+--+---+---+---+---+---+---+---+---+---+---+---
   64   |16|16|16|16|16|16|16| 16| 16| 16| 16| 16| 16| 16| 16| 16| 16| 16
  ------+--+--+--+--+--+--+--+---+---+---+---+---+---+---+---+---+---+---
   68   |17|17|17|17|17|17|17| 17| 17| 17| 17| 17| 17| 17| 17| 17| 17| 17
  ------+--+--+--+--+--+--+--+---+---+---+---+---+---+---+---+---+---+---
   72   |18|18|18|18|18|18|18| 18| 18| 18| 18| 18| 18| 18| 18| 18| 18| 18
  ------+--+--+--+--+--+--+--+---+---+---+---+---+---+---+---+---+---+---
   76   |19|19|19|19|19|19|19| 19| 19| 19| 19| 19| 19| 19| 19| 19| 19| 19
  ------+--+--+--+--+--+--+--+---+---+---+---+---+---+---+---+---+---+---
   80   |20|20|20|20|20|20|20| 20| 20| 20| 20| 20| 20| 20| 20| 20| 20| 20
  ------+--+--+--+--+--+--+--+---+---+---+---+---+---+---+---+---+---+---
  Lead  |  |  |  |  |  |  |  |   |   |   |   |   |   |   |   |   |   |
  Screw |3.|4.|5.|6.|7.|8.|9.|10.|11.|12.|13.|14.|15.|16.|17.|18.|19.|20.
  Pitch |  |  |  |  |  |  |  |   |   |   |   |   |   |   |   |   |   |
  ------+--+--+--+--+--+--+--+---+---+---+---+---+---+---+---+---+---+---

EXAMPLE.--The lead screw has a pitch of 4, and I require to cut 13
threads per inch. At the head of the column is 16, and on a line with
the 13 of the column, but on the left is 52, each number being marked by
a * hence the 16 and 52 are the wheels; if we have not those wheels,
multiply both by 2 and 32, and 104 will answer.

If the pitch of the lead screw is 2 threads per inch, the wheels must
advance by 6 teeth, as indicated below:--

NUMBERS OF TEETH FOR WHEEL TO GO ON LATHE STUD, LATHE SPINDLE OR
MANDRIL.

  +-----+--------+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
  |     |Lead    |12|18|24|30|36|42|48|54|60|66|72|78|84|90|96|
  |     |Screw.  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
  |NUM- +--------+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
  |BER  | 12     | 2| 2| 2| 2| 2| 2| 2| 2| 2| 2| 2| 2| 2| 2| 2|
  |OF   | 18     | 3| 3| 3| 3| 3| 3| 3| 3| 3| 3| 3| 3| 3| 3| 3|
  |TEETH| 24     | 4| 4| 4| 4| 4| 4| 4| 4| 4| 4| 4| 4| 4| 4| 4|
  |FOR  | 30     | 5| 5| 5| 5| 5| 5| 5| 5| 5| 5| 5| 5| 5| 5| 5|
  |WHEEL| 36     | 6| 6| 6| 6| 6| 6| 6| 6| 6| 6| 6| 6| 6| 6| 6|
  |TO   | 42     | 7| 7| 7| 7| 7| 7| 7| 7| 7| 7| 7| 7| 7| 7| 7|
  |GO   | 48     | 8| 8| 8| 8| 8| 8| 8| 8| 8| 8| 8| 8| 8| 8| 8|
  |ON   | 54     | 9| 9| 9| 9| 9| 9| 9| 9| 9| 9| 9| 9| 9| 9| 9|
  |LEAD | 60     |10|10|10|10|10|10|10|10|10|10|10|10|10|10|10|
  |SCREW+--------+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
  |     |Pitch of|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
  |     |Lead    | 2| 3| 4| 5| 6| 7| 8| 9|10|11|12|13|14|15|16|
  |     |Screw.  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
  +-----+--------+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+

This table may be used for compound lathes by simply dividing the pitch
of the lead screw by the ratio of the compounded pair of wheels. For
example, for the wheels to cut 8 threads per inch, the pitch of lead
screw being 4 and the compounded gears 2 to 1, as the ratio of the
compounded pair is 2 to 1, we divide the pitch of lead screw by 2, which
gives us 2, and we thus find the wheels in the column of pitch of lead
screw 2, getting 12 and 48 as the required wheels, the 12 going on top
of the lathe because it is at the top of the table, and the 48 on the
lead screw because it is at the left-hand end of the table, and the lead
screw gear is at the left-hand end of the lathe.

The table may be made for half threads as well as whole ones by simply
advancing the left-hand column by two teeth, instead of by four, thus:--

  +------+--------------------------------------------------------------+
  |Teeth |                   Teeth for Wheel on Stud.                   |
  | for  |------+------+------+------+------+------+------+------+------+
  |Wheel |      |      |      |      |      |      |      |      |      |
  |  on  | 12   | 16   | 20   | 24   | 28   | 32   | 36   | 40   | 44   |
  | Lead |      |      |      |      |      |      |      |      |      |
  |Screw.|      |      |      |      |      |      |      |      |      |
  +------+------+------+------+------+------+------+------+------+------+
  |  12  | 3    | 3    | 3    | 3    | 3    | 3    | 3    | 3    | 3    |
  +------+------+------+------+------+------+------+------+------+------+
  |  14  | 3-1/2| 3-1/2| 3-1/2| 3-1/2| 3-1/2| 3-1/2| 3-1/2| 3-1/2| 3-1/2|
  +------+------+------+------+------+------+------+------+------+------+
  |  16  | 4    | 4    | 4    | 4    | 4    | 4    | 4    | 4    | 4    |
  +------+------+------+------+------+------+------+------+------+------+
  |  18  | 4-1/2| 4-1/2| 4-1/2| 4-1/2| 4-1/2| 4-1/2| 4-1/2| 4-1/2| 4-1/2|
  +------+------+------+------+------+------+------+------+------+------+
  |  20  | 5    | 5    | 5    | 5    | 5    | 5    | 5    | 5    | 5    |
  +------+------+------+------+------+------+------+------+------+------+
  |  22  | 5-1/2| 5-1/2| 5-1/2| 5-1/2| 5-1/2| 5-1/2| 5-1/2| 5-1/2| 5-1/2|
  +------+------+------+------+------+------+------+------+------+------+
  |  24  | 6    | 6    | 6    | 6    | 6    | 6    | 6    | 6    | 6    |
  +------+------+------+------+------+------+------+------+------+------+
  |  26  | 6-1/2| 6-1/2| 6-1/2| 6-1/2| 6-1/2| 6-1/2| 6-1/2| 6-1/2| 6-1/2|
  +------+------+------+------+------+------+------+------+------+------+
  |  28  | 7    | 7    | 7    | 7    | 7    | 7    | 7    | 7    | 7    |
  +------+------+------+------+------+------+------+------+------+------+
  |  30  | 7-1/2| 7-1/2| 7-1/2| 7-1/2| 7-1/2| 7-1/2| 7-1/2| 7-1/2| 7-1/2|
  +------+------+------+------+------+------+------+------+------+------+
  |  32  | 8    | 8    | 8    | 8    | 8    | 8    | 8    | 8    | 8    |
  +------+------+------+------+------+------+------+------+------+------+
  |  34  | 8-1/2| 8-1/2| 8-1/2| 8-1/2| 8-1/2| 8-1/2| 8-1/2| 8-1/2| 8-1/2|
  +------+------+------+------+------+------+------+------+------+------+
  |  36  | 9    | 9    | 9    | 9    | 9    | 9    | 9    | 9    | 9    |
  +------+------+------+------+------+------+------+------+------+------+
  |  38  | 9-1/2| 9-1/2| 9-1/2| 9-1/2| 9-1/2| 9-1/2| 9-1/2| 9-1/2| 9-1/2|
  +------+------+------+------+------+------+------+------+------+------+
  |  40  |10    |10    |10    |10    |10    |10    |10    |10    |10    |
  +------+------+------+------+------+------+------+------+------+------+
  |  42  |10-1/2|10-1/2|10-1/2|10-1/2|10-1/2|10-1/2|10-1/2|10-1/2|10-1/2|
  +------+------+------+------+------+------+------+------+------+------+
  |Pitch |      |      |      |      |      |      |      |      |      |
  |  of  | 3    | 4    | 5    | 6    | 7    | 8    | 9    |10    |11    |
  | Lead |      |      |      |      |      |      |      |      |      |
  |Screw.|      |      |      |      |      |      |      |      |      |
  +------+------+------+------+------+------+------+------+------+------+

For quarter threads we advance the left-hand column by one tooth, or for
thirds of threads by three teeth, and so on.

If we require to find what wheels to provide for a lathe, we take the
pitch of the lead screw for the numerator, and the pitch required for
the denominator, and multiply them first by 2, then by 3, then by 4, and
so on, continuing until the numerator or denominator is as large as it
can be to give the required proportion of teeth, and not exceed the
greatest number that the largest wheel can contain.

For example: A lathe has single gear, and its lead screw pitch is 8 per
inch, what wheels will cut 18, 17, 16, 15, 14, or 13 threads per inch?

                                 Wheels.

  Pitch of lead screw  8         16   24   32
                      --  × 2 =  --   --   --
  Pitch required      18         36   54   72

  Pitch of lead screw  8         16   24   32
                      --    "    --   --   --
  Pitch required      17         34   51   68

  Pitch of lead screw  8         16   24   32
                      --    "    --   --   --
  Pitch required      16         32   48   64

  Pitch of lead screw  8         16   24   32
                      --    "    --   --   --
  Pitch required      15         30   45   60

  Pitch of lead screw  8         16   24   32
                      --    "    --   --   --
  Pitch required      14         28   42   56

  Pitch of lead screw  8         16   24   32   40
                      --    "    --   --   --   --
  Pitch required      13         26   39   52   65

If we suppose that the greatest number of teeth permissible in one wheel
is not to exceed 100, then in this table we have all the combinations of
wheels that can be used to cut the given pitches; and having made out
such a table, comprising all the pitches to be cut, we may select
therefrom the least number of wheels that will cut those pitches. The
whole table being made out it will be found, of course, that the
numerators of the fractions are the same in each case; that is, in this
case, 16, 18, 24, 32, and so on as far as we choose to carry the
multiplication of the numerator. We shall also find that the
denominators diminish in a regular order: thus taking the fractions
whose numerators are in each case 16, we find their denominators are, as
we pass down the column, 36, 34, 32, 30, 28, and 26, respectively, thus
decreasing by 2, which is the number we multiplied the left-hand column
by to obtain them. Similarly in the fractions whose numerators are 24,
the denominators diminish by 3, being respectively 54, 51, 48, 45, 42,
and 39; hence the construction of such a table is a very simple matter
so far as whole numbered threads are concerned, as no multiplication is
necessary save for the first line representing the finest pitch to be
cut.

For fractional threads, however, instead of using the pitch of the lead
screw for the numerator, we must reduce it to terms of the fraction it
is required to cut. For example, for 5-1/2 threads we proceed as
follows. The pitch of the lead screw is 8, and in 8 there are 16 halves,
hence we use 16 instead of 8, and as in the 5-1/2 there are 11 halves we
use the fraction 16/11 and multiply it first by 2, then by 3, and then
by 4, and so on, obtaining as follows: 16/11, 32/22, 48/33, 64/44,
obtaining as before three sets of wheels either of which will cut the
required pitch. In selecting from such a table the wheels to cut any
required number of pitches, the set must, in order to cut a thread of
the same pitch as the lead screw, contain two wheels having the same
number of teeth.

Now, suppose that the pitch of the lead screw was 6 instead of 8 threads
per inch, and the table will be as follows:--

   6           12           18           24
  --           --           --           --
  18           36           54           72

   6           12           18           24
  --           --           --           --
  17           34           51           68

   6           12           18           24
  --           --           --           --
  16           32           48           64

   6           12           18           24
  --           --           --           --
  15           30           45           60

   6           12           18           24
  --           --           --           --
  14           28           42           56

   6           12           18           24
  --           --           --           --
  13           26           39           52

Here, again, we find that in the first vertical column the denominators
decrease by two for each thread less per inch, in the second column they
decrease by three, and in the third by four; this decrease equalling the
number the first fraction was multiplied by.

But suppose the lead screw pitch is an odd one, as, say, 3 threads per
inch, and we construct the table as before, thus--

  Pitch of lead screw   3      6      9      12      15
                       --     --     --      --      --
  Pitch to be cut      18     32     54      72      90

Now it is useless to multiply by 2 or by 3, because they give a less
number of teeth than the smallest wheel should have, hence the first
multiplier should be 4, giving the following table:--

   3      12      15      18
  --      --      --     ---
  18      72      90     108

   3      12      15      18
  --      --      --     ---
  17      68      85     102

   3      12      15      18
  --      --      --      --
  16      64      80      96

   3      12      15      18
  --      --      --      --
  15      60      75      90

By continuing the table for other pitches we shall find that in the
first vertical column the denominators diminish by 4, the second column
by 5, and the third by 6; and it is seen that by diminishing the pitch
of the lead screw, we have rendered necessary one of two things, which
is, that either larger wheels containing more teeth must be used, or the
change gears must be compounded.

Assuming that the pitch of the lead screw was 5 per inch, the table
would be as follows:--

   5        15      20      25
  -- × 3 =  --      --      --
  18        54      72      90

   5        15      20      25
  --   "    --      --      --
  17        51      68      85

   5        15      20      25
  --   "    --      --      --
  16        48      64      80

The wheels in the first column here decrease by 3, the second by 4, and
the third by 5.

In nearly all lathes the advance or decrease is by 4 or by 6. In
determining this rate of advance or decrease, there are several
elements, among which are the following. Suppose the lathe to be geared
without compounding, then the distance between the lathe spindle and the
lead screw will determine what shall be the diameters of the largest and
of the smallest wheel in the set, it being understood that the smallest
wheel must not contain less than 12 teeth. Assume that in a given case
the distance is 10 inches, and it is obvious that the pitch of the teeth
at once commands consideration, because the finer the pitch the smaller
the wheel that will contain 12 teeth, and the larger the wheel on the
lead screw may be made. Of course the pitch must be coarse enough to
give the required tooth strength.

Let it be supposed that the arc pitch is 3/4-inch, then the pitch
circumference of a 12-toothed wheel would be 9 inches and its radius
1.432 in.; this subtracted from the 10 leaves 8.568 in. as the radius,
or 17.136 in. as the largest diameter of wheel that can be used on the
lead screw, supposing there to be no intermediate gears. Now a wheel of
this diameter would be capable of containing more than 75 teeth, but
less than 76. But from the foregoing tables it will be seen that it
should contain a number of teeth divisible either by 4 or by 6 without
leaving a remainder, and what that number should be is easily determined
by means of a table constructed as before explained. Thus from the
tables it would be found that 72 teeth would be best for a lead screw
having a pitch of either 8, 6, 5, or 3 threads per inch, and the
screw-cutting capacity of the lathe would (unless compounded) be
confined to such pitches as may be cut with wheels containing between 12
and 72 teeth both inclusive.

But assume that an arc pitch of 3/8-inch be used for the wheel teeth,
and we have as follows: A wheel of this pitch and containing 12 teeth
will have a radius of 7-16/1000 inches, leaving 9.284 in. as the radius
of the largest wheel, assuming it to gear direct with the 12-tooth
pinion. With this radius it would contain 155 teeth and a fraction of a
tooth; we must, therefore, take some less number, and from what has been
said, it will be obvious that this lesser number should be one divisible
by either 4 or 6. If made divisible by 6, the number will be 150,
because that is the highest number less than 155 that is divisible by 6
without leaving a remainder. But if made divisible by 4, it may contain
152 teeth, because that number is divisible by 4 without leaving a
remainder. With 150 teeth the latter could cut a thread 12-1/2 times as
fine as the lead screw, because the largest wheel contains 12-1/2 times
as many teeth as the smallest one; or it would cut a thread 12-1/2 times
as coarse as the lead screw, if the largest wheel be placed on the
mandril and the smallest on the lead screw. With 152 teeth the lathe
would be able to cut a thread 12-84/100 times as fine or as coarse as
the lead screw. Unless, however, the lathe be required to cut fractional
pitches, it is unnecessary that the largest wheel have more teeth than
divisible, without leaving a remainder, by the number of teeth in the
smallest wheel, which being 12 we have 144 as the number of teeth for
the largest wheel. In the United States standard pitches of thread,
however, there are several pitches in fractions of an inch, hence it is
desirable to have wheels that will cut these pitches.

LATHE SHEARS OR BEDS.--The forms of the shears and beds may be
classified as follows.

The term shear is generally applied when the lathe is provided with
legs, while the term bed is used when there are no legs; it may be
noted, however, that by some workmen the two terms of _shear_ and _bed_
are used indiscriminately.

The forms of shears in use on common lathes are, in the United States,
the raised [V], the flat shear and the shear, with the edge at an angle
of 90° or with parallel edges. In England and on the continent of
Europe, the flat shear is almost exclusively employed.

Referring to the raised [V] it possesses an important advantage in that,
first, the slide rest does not get loosely guided from the wear; and
second, the wear is in the direction that least affects the diameter of
the work.

[Illustration: Fig. 621.]

In Fig. 621, for example, is a section of a lathe shear, with a slide
rest shown in place, and it will be observed that the wear of the [V]
upon the lathe bed, and of the [V]-groove in the slide rest, will cause
the rest to fall in the direction of arrow A, and that a given amount of
motion in that direction will have less effect in altering the diameter
than it would in any other direction. This is shown on the right hand of
the figure as follows: Suppose the cutting point of the tool is at _a_,
and the work will be of the diameter shown by the full circle in the
figure. If we suppose the tool point to drop down to _f_, the work would
be turned to the diameter denoted by dotted arc _g_, while if the tool
were moved outwards from _a_ to _c_ the work would be turned to the
diameter _e_. Now since _f_ and _c_ are equidistant from the point _a_,
therefore the difference in the diameters of _e_ and _g_ represents the
difference of effect between the wear letting the rest merely fall, or
moving it outwards, and it follows that, as already stated, the diameter
of the work is less affected by a given amount of wear, when this wear
is in the direction of A, than when it is in the direction of B. When
the carriage is held down by a weight as is shown in Figs. 577 and 578,
there is therefore no lost motion or play in the carriage, which
therefore moves steadily upon the shears, unless the pressure of the cut
is sufficient in amount, and also in a direction to lift the carriage
(as it is in the case of boring with boring tools); but to enable the
carriage to remain firm upon the shears under all conditions, it is
necessary to provide means to hold it down upon the [V]s, which is done
by means of gibs G, G, which are secured to the carriage, and fit
against the bottom of the bed flange as shown.

Now since lathes are generally used much more frequently on short than
on long work, therefore the carriage traverses one part of the shears
more than another, and the [V]s wear more at the part most traversed,
and it follows that if gibs G are set to slide properly at some parts
they will not be properly set at another or other parts of the length of
the shears; hence the carriage will in some parts have liberty to move
from the bed, there being nothing but the weight of the carriage, &c.,
to hold it down to the [V]s. Now, the wear in the direction of A acts
directly to cause this inequality of gib fit, whereas that in the
direction of B does so to a less extent, as will appear hereafter.

Meantime it may be noted that when the carriage is held down by a
suspended weight the shears cannot be provided with cross girts, and are
therefore less rigid and more subject to torsion under the strain of the
cut; furthermore the amount of the weight must be sufficient to hold the
carriage down under the maximum of cut, and this weight acts
continuously to wear the [V]s, whether the carriage is under cutting
duty or not, but the advantage of keeping the carriage firmly down upon
the [V]s is sufficiently great to cause many to prefer the weighted
carriage for light work driven between the lathe centres.

[Illustration: Fig. 622.]

Fig. 622 represents the flat shear, the edges being at an angle and the
fit of the carriage to the shears being adjusted by the gibs at _a_ _a_,
which are set up by bolts _c_ _c_ and _d_ _d_. In this case there is a
large amount of wearing surface at _b_ _b_, to prevent the fall of the
carriage _c_, but the amount of end motion (in the direction of B, Fig.
621), permitted to the carriage by reason of the wear of the gibs and
shear edges, is greater than the amount of the wear because of the edges
being at an angle. It is true that the amount of fall of the carriage on
the raised [V] is also (on account of the angle of the [V]) greater than
the actual amount of the wear, but the effect upon the work diameter is
in this case much greater, as will be readily understood from what has
already been said. The wearing surface of the raised [V] may obviously
be increased by providing broader [V]s, or two [V]s instead of having
four. This would tend to keep the lathe in line, because the wear due to
moving the tailblock would act upon those parts of the shear length that
are less acted upon by the carriage, and since the front journal and
bearing of the live spindle wear the most, the alignment of the lathe
centres would be more nearly preserved.

[Illustration: Fig. 623.]

Fig. 623 represents another form of parallel edged shears in which the
fit of the carriage to the shears is effected at the front end only, the
other or back edge being clear of contact with the carriage, but
provided with a gib to prevent the carriage from lifting. This allows
for any difference in expansion and contraction between the carriage and
the shears, while maintaining the fit of the carriage to the bed.

[Illustration: Fig. 624.]

A modification of this form (both these forms being taken from
"Mechanics") is shown in Fig. 624, in which the underneath side of the
front edge is beveled so that but one row of screws is required to
effect the adjustment.

[Illustration: Fig. 625.]

Fig. 625 represents a form of bed in which the fit adjustment is also
made at the front end only of the bed, and there is a flange or slip at
_a_, which receives the thrust outwards of the carriage; and a similar
design, but with a bevelled edge, is shown in Fig. 626.

[Illustration: Fig. 626.]

[Illustration: Fig. 627.]

In Fig. 627 is shown a lathe shear with parallel edges, the fit being
adjusted by a single gib D, set up by set-screws S. In this case the
carriage will fall or move endwise, to an amount equal to whatever the
amount of the wear may be, and no more, but it may be observed that in
all the forms that admit of wear endways (that is to say in the
direction of B in Fig. 621), the straightness of the shears is impaired
in proportion as its edges are more worn at one part than at another.

[Illustration: Fig. 628.]

A compromise between the flat and the raised [V]-shear is shown in Fig.
628, there being a [V]-guide on one side only, as at J. When the
carriage is moved by mechanism on the front side of the lathe, and close
to the [V], this plan may be used, but if the feed screw or other
mechanism for traversing the carriage is within the two shears, the
carriage should be guided at each end, or if the operating mechanism is
at the back of the lathe, the carriage should be guided at the back end,
if not at both ends.

In flat shear lathes the tailstock is fitted between the inside edges of
the two shears, and the alignment of the tailstock depends upon
maintaining a proper fit notwithstanding the wear that will naturally
take place in time. The inside edges of the shears are sometimes
tapered; this taper makes it much easier to obtain a correct fit of the
tailstock to the shears, but at the same time more hard to move the
tailstock along the bed. To remedy this difficulty, rollers are
sometimes mounted upon eccentrics having journal bearing in the
tailstock, so that by operating these eccentrics one half a turn, the
rollers will be brought down upon the upper face of the shears, lifting
the tailstock and enabling it to be easily moved along the bed to its
required position.

In many of the watchmakers' lathes the outer edges are beveled off as in
Fig. 629, the bearing surfaces being on the faces _b_ as well as on the
edges _a_. As a result, edges _a_ are relieved of weight, and therefore
to some extent of wear also, and whatever wear faces _b_ have helps the
fit at _a_ _a_.

In the Barnes lathe, as in several other forms in which the lathe is
made (as, for example, in screw-making lathes) the form of bed in Fig.
630 is employed. The tailblock may rest on the surfaces A, A´, B, C, D,
and E, or as in the Barnes lathe the tailstock may fit to angles A B,
but not to E D, while the carriage fits to B E, and C D, but not to A,
the intention being to equalize the wear as much as possible.

[Illustration: Fig. 629.]

The shears of lathes require to be as rigid as possible, because the
pressure of the cut, as well as the weight of the carriage, slide rest,
and tailstock, and of the work, tends to bend and twist them.

[Illustration: Fig. 630.]

The pressure of the dead centre against the end of the work considered
individually, is in a direction to bend the lathe shears upward, but the
weight of the work itself acts in an opposite direction.

The strain due to the cut falls in a direction variable with the shape
of the cutting tool, but mainly in a direction towards the operator,
and, therefore, tending to twist the shears. To resist these strains,
lathe shears are usually given the [I] form shown in the cuts.

[Illustration: Fig. 631.]

[Illustration: Fig. 632.]

Figs. 631 and 632 represent the ribbing in the Putnam Tool Company's
lathe; a middle rib running the entire length, which greatly stiffens
it.

The legs supporting lathe shears are, in lathes of ordinary length,
placed at each end of the bed, so that the weight of the two heads, that
of the work, and that of the carriage and slide rest, as well as the
downward pressure of the cut, act combined to cause it to deflect or
bend. It is necessary, therefore, in long beds to provide intermediate
resting or supporting points to prevent this deflection.

[Illustration: Fig. 633.]

Professor Sweet has pointed out that a lathe shears will be more truly
supported on three than on four resting points, if the foundation on
which the legs rest do not remain permanently level, and in lathes
designed by him has given the right-hand end of the shears a single
supporting point, as shown at _a_ in Fig. 633.

[Illustration: Fig. 634.]

J. Richards in an article in "Engineering," has pointed out also that,
when the lathe legs rest upon a floor that is liable from moving loads
upon it to move its level, it is preferable that the legs be shaped as
in Fig. 634, being narrowest at the foot, whereas when upon a permanent
foundation, in which the foundation is intended to impart rigidity to
the legs, they should be broader at the base, as in Fig. 635.

[Illustration: Fig. 635.]

The rack on a lathe bed should be a cut one, and not simply a cast one,
because when a cutting tool is running up to a corner as against a
radial face, the self-acting motion must be stopped and the tool fed
into the corner by hand. As a very delicate tool movement is required to
cut the corner out just square, it should be capable of easy and steady
movement, but in the case of cast racks, the rest will, from defects in
the rack teeth, move in little jumps, especially if the pitch of the
teeth be coarse. On the other hand it is difficult to cast fine pitches
of teeth perfectly, hence the racks as well as the gear teeth should be
cut gear and of fine pitch.

The tailblock of a lathe should be capable of easy motion for adjustment
along the shears, or bed of the lathe, and readily fixable in its
adjusted position. The design should be such as to hold the axial line
of its spindle true with the axial line of the live spindle. If the
lathe bed has raised [V]s there are usually provided two special [V]s
for the tailblock to slide on, the slide rest carriage sliding on two
separate ones. In this case the truth of the axial line of the tail
spindle depends upon the truth of the [V]s.

If the lathe bed is provided with ways having a flat surface, as was
shown in Fig. 622, the surfaces of the edges and of the projection are
apt in time to wear, permitting an amount of play which gives room for
the tailblock to move out of line. To obviate this, various methods are
resorted to, an example being given in the Sellers lathe, Fig. 518.

In wood turners' lathes, where tools are often used in place of the dead
centre, and in which a good deal of boring is done by such use of the
tail spindle, it is not unusual to provide a device for the rapid motion
of that spindle. Such a device is shown in Fig. 636; it consists of an
arm A to receive the end C of the lever B, C being pivoted to A. The
spindle is provided with an eye at E, the wheel W is removed and a pin
passed through D and E, so that by operating the handle the spindle can
be traversed in and out without any rotary motion of the screw.

When the tailblock of a lathe fits between the edges of the shears,
instead of upon raised [V]s, it is sometimes the practice to give them
a slight taper fitting accurately a corresponding taper on the edges of
the shears. This enables the obtenance of a very good fit between the
surfaces, giving an increased area of contact, because the surfaces can
be filed on their bearing marks to fit them together; but this taper is
apt to cause the tailstock to fit so tightly between the shears as to
render it difficult to move it along them, and in any event the friction
is apt to cause the fit to be destroyed from the wear. An excellent
method of obviating these difficulties is by the employment of rollers,
such as shown at R in Figs. 637 and 638, which represent the tailstock
of the Putnam Tool Company's lathe. In some cases such rollers are
carried on eccentric shafts so that they may be operated to lift the
tailstock from the bed when moving it.

[Illustration: Fig. 636.]

[Illustration: Fig. 637.]

[Illustration: Fig. 638.]

[Illustration: Fig. 639.]

A very ready method of securing or releasing a small tailstock to a
lathe shears is shown applied to a wood turner's hand rest in Fig. 639,
in which A A represents the lathe shears, B the hand rest, C the
fastening bolt, D a piece hinged at each end and having through its
centre a hole to receive the fastening bolt, and a counter-sink or
recess to receive the nut and prevent it unscrewing. E represents a
hinged plate, and F a lever, having a cam at its pivoted end. A slot for
the fastening bolt to pass through is provided in the plate E. In this
arrangement a very moderate amount of force applied to bring up the cam
lever will cause the plate D to be pressed down, carrying with it the
nut, and binding the tailstock or the tool rest, as the case may be,
with sufficient force for a small lathe.

When a piece of work is driven between the lathe centres, the weight of
the work tends to deflect or bend down the tail spindle. The pressure of
the cut has also to be resisted by the tail spindle, but this pressure
is variable in direction, according to the shape of the tool and the
direction of the feed; usually it is laterally towards the operator and
upwards. In any event, however, the spindle requires locking in its
adjusted position, so as to keep it steady. The pressure on the conical
point of the dead centre is in a direction to cause the tail screw to
unwind, unless it be a left-hand thread, as is sometimes the case.

If the spindle and the bore in which it operates have worn, the
resulting looseness affords facility for the spindle to move in the bore
as the pressure of the cut varies, especially when the spindle is far
out from the tailstock.

Now, in locking the tail spindle to obviate these difficulties, it is
desirable that the locking device shall hold that spindle axially true
with the live spindle of the lathe, notwithstanding any wear that may
have taken place. The spindle is released from the pressure of the
locking device whenever it is adjusted to the work, whether the cut be
proceeding or not. Hence, the wear takes place on the bottom of the
spindle and of the hole, wear only ensuing on the top of the spindle and
bore when the spindle is operated under a