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Title: On the History of Gunter's Scale and the Slide Rule during the Seventeenth Century
Author: Cajori, Florian
Language: English
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            Vol. 1, No. 9, pp. 187-209     February 17, 1920

                        THE SLIDE RULE DURING THE
                           SEVENTEENTH CENTURY

                             FLORIAN CAJORI


                           TABLE OF CONTENTS

  I. Introduction                                                     187
  II. Innovations in Gunter’s Scale                                   188
    Changes introduced by Edmund Wingate                              188
    Changes introduced by Milbourn                                    189
    Changes introduced by Thomas Brown and John Brown                 190
    Changes introduced by William Leybourn                            192
  III. Richard Delamain’s “Grammelogia”                               192
    Different editions or impressions                                 194
    Description of Delamain’s instrument of 1630                      195
    Delamain’s later designs, and directions for using his
          instruments                                                 197
  IV. Controversy between Oughtred and Delamain on the invention of
          the circular slide rule                                     199
  V. Independence and priority of invention                           203
  VI. Oughtred’s “Gauging Line,” 1633                                 206
  VII. Other seventeenth century slide rules                          207

                              I. INTRODUCTION

  In my history of the slide rule[1], and my article on its invention[2] it
is shewn that William Oughtred and not Edmund Wingate is the inventor, that
Oughtred’s circular rule was described in print in 1632, his rectilinear
rule in 1633. Richard Delamain is referred to as having tried to
appropriate the invention to himself[3] and as having written a scurrilous
pamphlet against Oughtred. All our information about Delamain was taken
from De Morgan,[4] who, however, gives no evidence of having read any of
Delamain’s writings on the slide rule. Through Dr. Arthur Hutchinson of
Pembroke College, Cambridge, I learned that Delamain’s writings on the
slide rule were available. In this article will be given: First, some
details of the changes introduced during the seventeenth century in the
design of Gunter’s scale by Edmund Wingate, Milbourn, Thomas Brown, John
Brown and William Leybourn; second, an account of Delamain’s book of 1630
on the slide rule which antedates Oughtred’s first publication (though
Oughtred’s date of invention is earlier than the date of Delamain’s alleged
invention) and of Delamain’s later designs of slide rules; third, an
account of the controversy between Delamain and Oughtred; fourth, an
account of a later book on the slide rule written by William Oughtred, and
of other seventeenth century books on the slide rule.

                     II. INNOVATIONS IN GUNTER’S SCALE

                       Changes introduced by Wingate

  We begin with Anthony Wood’s account of Wingate’s introduction of
Gunter’s scale into France.[5]

  In 1624 he transported into France the rule of proportion, having a
  little before been invented by Edm. Gunter of Gresham Coll. and
  communicated it to most of the chiefest mathematicians then residing in
  Paris: who apprehending the great benefit that might accrue thereby,
  importun’d him to express the use thereof in the French tongue. Which
  being performed accordingly, he was advised by monsieur Alleawne the
  King’s chief engineer to dedicate his book to monsieur the King’s only
  brother, since duke of Orleans. Nevertheless the said work coming forth
  as an abortive (the publishing thereof being somewhat hastened, by reason
  an advocate of Dijon in Burgundy began to print some uses thereof, which
  Wingate had in a friendly way communicated to him) especially in regard
  Gunter himself had learnedly explained its use in a far larger volume.[6]

Gunter’s scale, which Wingate calls the “rule of proportion,” contained, as
described in the French edition of 1624, four lines: (1) A single line of
numbers; (2) a line of tangents; (3) a line of sines; (4) a line, one foot
in length, divided into 12 inches and tenths of inches, also a line, one
foot in length, divided into tenths and hundredths.

The English editions of this book which appeared in 1623 and 1628 are
devoid of interest. The editions of 1645 and 1658 contain an important
innovation.[7] In the preface the reasons why this instrument has not been
used more are stated to be: (1) the difficulty of drawing the lines with
exactness, (2) the trouble of working thereupon by reason (sometimes) of
too large an extent of the compasses, (3) the fact that the instrument is
not readily portable. The drawing of Wingate’s arrangement of the scale in
the editions of 1645 and 1658 is about 66 cm. (26.5 in.) long. It contains
five parallel lines, about 66 cm. long, each having the divisions of one
line marked on one side and of another line on the other side. Thus each
line carries two graduations: (1) A single logarithmic line of numbers; (2)
a logarithmic line of numbers thrice repeated; (3) the first scale
repeated, but beginning with the graduations which are near the middle of
the first scale, so that its graduation reads 4, 5, 6, 7, 8, 9, 1, 2, 3;
(4) a logarithmic line of numbers twice repeated; (5) a logarithmic line of
tangents; (6) a logarithmic line of sines; (7) the rule divided into 1000
equal parts; (8) the scale of latitudes; (9) a line of inches and tenths of
inches; (10) a scale consisting of three kinds, viz., a gauge line, a line
of chords, and a foot measure, divided into 1000 equal parts.

Important are the first and second scales, by which cube root extraction
was possible “by inspection only, without the aid of pen or compass;”
similarly the third and fourth scales, for square roots. This innovation is
due to Wingate. The 1645 edition announces that the instrument was made in
brass by Elias Allen, and in wood by John Thompson and Anthony Thompson in
Hosier Lane.

                      Changes introduced by Milbourn

William Leybourn, in his The Line of Proportion or Numbers, Commonly called
Gunter’s Line, Made Easie, London, 1673, says in his preface “To the

  The Line of Proportion or Numbers, commonly called (by Artificers)
  Gunter’s Line, hath been discoursed of by several persons, and variously
  applied to divers uses; for when Mr. Gunter had brought it from the
  Tables to a Line, and written some Uses thereof, Mr. Wingate added divers
  Lines of several lengths, thereby to extract the Square or Cube Roots,
  without doubling or trebling the distance of the Compasses: After him Mr.
  Milbourn, a Yorkshire Gentleman, disposed it in a Serpentine or Spiral
  Line, thereby enlarging the divisions of the Line.

On pages 127 and 128 Leybourn adds:

  Again, One T. Browne, a Maker of Mathematical Instruments, made it in a
  Serpentine or Spiral Line, composed of divers Concentrick Circles,
  thereby to enlarg the divisions, which was the contrivance of one Mr.
  Milburn a Yorkshire Gentleman, who writ thereof, and communicated his
  Uses to the aforesaid Brown, who (since his death) attributed it to
  himself: But whoever was the contriver of it, it is not without
  inconvenience; for it can in no wise be made portable; and besides
  (instead of compasses) an opening Joynt with thirds [threads] must be
  placed to move upon the Centre of the Instrument, without which no
  proportion can be wrought.

This Mr. Milburn is probably the person named in the diary of the
antiquarian, Elias Ashmole, on August 13 [1646?]; “I bought of Mr. Milbourn
all his Books and Mathematical Instruments.”[8] Charles Hutton[9] says that
Milburne of Yorkshire designed the spiral form about 1650. This date is
doubtless wrong, for Thomas Browne who, according to Leybourn, got the
spiral form of line from Milbourn, is repeatedly mentioned by William
Oughtred in his Epistle[10] printed some time in 1632 or 1633. Oughtred
does not mention Milbourn, and says (page 4) that the spiral form “was
first hit upon by one Thomas Browne a Joyner, . . . the serpentine
revolution being but two true semicircles described on severall

            Changes introduced by Thomas Brown and John Brown

Thomas Brown did not publish any description of his instrument, but his
son, John Brown, published in 1661 a small book,[12] in which he says
(preface) that he had done “as Mr. Oughtred with Gunter’s Rule, to a
sliding and circular form; and as my father Thomas Brown into a Serpentine
form; or as Mr. Windgate in his Rule of Proportion.” He says also that
“this brief touch of the Serpentine-line I made bold to assert, to see if I
could draw out a performance of that promise, that hath been so long
unperformed by the promisers thereof.” Accordingly in Chapter XX he gives a
description of the serpentine line, “contrived in five (or rather 15)
turn.” Whether this description, printed in 1661, exactly fits the
instrument as it was developed in 1632, we have no means of knowing. John
Brown says:

  1. First next the center is two circles divided one into 60, the other
  into 100 parts, for the reducing of minutes to 100 parts, and the

  2. You have in seven turnes two inpricks, and five in divisions, the
  first Radius of the sines (or Tangents being neer the matter, alike to
  the first three degrees,) ending at 5 degrees and 44 minutes.

  3. Thirdly, you have in 5 turns the lines of numbers, sines, Tangents, in
  three margents in divisions, and the line of versed sines in pricks,
  under the line of Tangents, according to Mr. Gunter’s cross-staff: the
  sines and Tangents beginning at 5 degrees, and 44 minutes where the other
  ended, and proceeding to 90 in the sines, and 45 in the Tangents. And the
  line of numbers beginning at 10, and proceeding to 100, being one entire
  Radius, and graduated into as many divisions as the largeness of the
  instrument will admit, being 10 to 10 50 into 50 parts, and from 50 to
  100 into 20 parts in one unit of increase, but the Tangents are divided
  into single minutes from the beginning to the end, both in the first,
  second and third Radiusses, and the sines into minutes; also from 30
  minutes to 40 degrees, and from 40 to 60, into every two minutes, and
  from 60 to 80 in every 5th minute, and from 80 to 85 every 10th, and the
  rest as many as can be well discovered.

  The versed sines are set after the manner of Mr. Gunter’s Cross-staff,
  and divided into every 10th minutes beginning at 0, and proceeding to 156
  going backwards under the line of Tangents.

  4. Fourthly, beyond the Tangent of 45 in one single line, for one Turn is
  the secants to 51 degrees, being nothing else but the sines reitterated
  beyond 90.

  5. Fifthly, you have the line of Tangents beyond 45, in 5 turnes to 85
  degrees, whereby all trouble of backward working is avoided.

  6. Sixthly, you have in one circle the 180 degrees of a Semicircle, and
  also a line of natural sines, for finding of differences in sines, for
  finding hour and Azimuth.

  7. Seventhly, next the verge or outermost edge is a line of equal parts
  to get the Logarithm of any number, or the Logarithm sine and Tangent of
  any ark or angle to four figures besides the carracteristick.

  8. Eightly and lastly, in the space place between the ending of the
  middle five turnes, and one half of the circle are three prickt lines
  fitted for reduction. The uppermost being for shillings, pence and
  farthings. The next for pounds, and ounces, and quarters of small
  Averdupoies weight. The last for pounds, shillings and pence, and to be
  used thus: If you would reduce 16s. 3d. 2q. to a decimal fraction, lay
  the hair or edge of one of the legs of the index on 16. 3½ in the line of
  1. s. d. and the hair shall cut on the equal parts 81 16; and the
  contrary, if you have a decimal fraction, and would reduce it to a proper
  fraction, the like may you do for shillings, and pence, and pounds, and

                      The uses of the lines follow.

  As to the use of these lines, I shall in this place say but little, and
  that for two reasons. First, because this instrument is so contrived,
  that the use is sooner learned then any other, I speak as to the manner,
  and way of using it, because by means of first second and third
  radiusses, in sines and Tangents, the work is always right on, one way or
  other, according to the Canon whatsoever it be, in any book that treats
  of the Logarithms, as Gunter, Wells, Oughtred, Norwood, or others, as in
  Oughtred from page 64 to 107.

  Secondly, and more especially, because the more accurate, and large
  handling thereof is more then promised, if not already performed by more
  abler pens, and a large manuscript thereof by my Sires meanes, provided
  many years ago, though to this day not extant in print; so for his sake I
  claiming my interest therein, make bold to present you with these few
  lines, in order to the use of them: And first note,

  1. Which soever of the two legs is set to the first term in the question,
  that I call the first leg always, and the other being set to the second
  term, I call the second leg . . .

The exact nature of the contrivance with the “two legs” is not described,
but it was probably a flat pair of compasses, attached to the metallic
surface on which the serpentine line was drawn. In that case the instrument
was a slide rule, rather than a form of Gunter’s line. In his publication
of 1661, as also in later publications,[13] John Brown devoted more space
to Gunter’s scales, requiring the use of a separate pair of compasses, than
to slide rules.

                  Changes introduced by William Leybourn

The same remark applies to William Leybourn who, after speaking of Seth
Partridge’s slide rule, returns to forms of Gunter’s scale, saying:[14]

  There is yet another way of disposing of this Line of Proportion, by
  having one Line of the full length of the Ruler, and another Line of the
  same Radius broken in two parts between 3 and 4; so that in working your
  Compasses never go off of the Line: This is one of the best contrivances,
  but here Compasses must be used. These are all the Contrivances that I
  have hitherto seen of these Lines: That which I here speak of, and will
  shew how to use, is only two Lines of one and the same Radius, being set
  upon a plain Ruler of any length (the larger the better) having the
  beginning of one Line, at the end of the other, the divisions of each
  Line being set so close together, that if you find any number upon one of
  the Lines, you may easily see what number stands against it on the other
  Line. This is all the Variation. . . .

  Example 1. If a Board be 1 Foot 64 parts broad, how much in length of
  that Board will make a Foot Square? Look upon one of your Lines (it
  matters not which) for 1 Foot 64 parts, and right against it on the other
  Line you shall find 61; and so many parts of a Foot will make a Foot
  square of that Board.

This contrivance solves the equation 1.64x=1, yielding centesimal parts of
a foot.

James Atkinson[15] speaks of “Gunter’s scale” as “usually of Boxwood . . .
commonly 2 ft. long, 1½ inch broad” and “of two kinds: long Gunter or
single Gunter, and the sliding Gunter. It appears that during the
seventeenth century (and long after) the Gunter’s scale was a rival of the
slide rule.


We begin with a brief statement of the relations between Oughtred and
Delamain. At one time Delamain, a teacher of mathematics in London, was
assisted by Oughtred in his mathematical studies. In 1630 Delamain
published the Grammelogia, a pamphlet describing a circular slide rule and
its use. In 1631 he published another tract, on the Horizontall
Quadrant.[16] In 1632 appeared Oughtred’s Circles of Proportion[17]
translated into English from Oughtred’s Latin manuscript by another pupil,
William Forster, in the preface of which Forster makes the charge (without
naming Delamain) that “another . . . went about to pre-ocupate” the new
invention. This led to verbal disputes and to the publication by Delamain
of several additions to the Grammelogia, describing further designs of
circular slide rules and also stating his side of the bitter controversy,
but without giving the name of his antagonist. Oughtred’s Epistle was
published as a reply. Each combatant accuses the other of stealing the
invention of the circular slide rule and the horizontal quadrant.

The two title-pages of the edition of the Grammelogia in the British Museum
             in London which we have called “Grammelogia IV.”

                    Different editions or impressions

There are at least five different editions, or impressions, of the
Grammelogia which we designate, for convenience, as follows:

  Grammelogia I, 1630. One copy in the Cambridge University Library.[18]

  Grammelogia II, I have not seen a copy of this.

  Grammelogia III, One copy in the Cambridge University Library.[19]

  Grammelogia IV, One copy in the British Museum, another in the Bodleian
  Library, Oxford.[20]

  Grammelogia V, One copy in the British Museum.

In Grammelogia I the first three leaves and the last leaf are without
pagination. The first leaf contains the title-page; the second leaf, the
dedication to the King and the preface “To the Reader;” the third leaf, the
description of the Mathematical Ring. Then follow 22 numbered pages.
Counting the unnumbered pages, there are altogether 30 pages in the
pamphlet. Only the first three leaves of this pamphlet are omitted in
Grammelogia IV and V.

In Grammelogia III the Appendix begins with a page numbered 52 and bears
the heading “Conclusion;” it ends with page 68, which contains the same two
poems on the mathematical ring that are given on the last page of
Grammelogia I but differs slightly in the spelling of some of the words.
The 51 pages which must originally have preceded page 52, we have not seen.
The edition containing these we have designated Grammelogia II. The reason
for the omission of these 51 pages can only be conjectured. In Oughtred’s
Epistle (p. 24), it is stated that Delamain had given a copy of the
Grammelogia to Thomas Brown, and that two days later Delamain asked for the
return of the copy, “because he had found some things to be altered
therein” and “rent out all the middle part.” Delamain labored “to recall
all the bookes he had given forth, (which were many) before the sight of
Brownes Lines.” These spiral lines Oughtred claimed that Delamain had
stolen from Brown. The title-page and page 52 are the only parts of the
Appendix, as given in Grammelogia III, that are missing in the Grammelogia
IV and V.

Grammelogia IV answers fully to the description of Delamain’s pamphlet
contained in Oughtred’s Epistle. It was brought out in 1632 or 1633, for
what appears to be the latest part of it contains a reference (page 99) to
the Grammelogia I (1630) as “being now more then two yeares past.”
Moreover, it refers to Oughtred’s Circles of Proportion, 1632, and
Oughtred’s reply in the Epistle was bound in the Circles of Proportion
having the Addition of 1633. For convenience of reference we number the two
title-pages of Grammelogia IV, “page (1)” and “page (2),” as is done by
Oughtred in his Epistle. Grammelogia IV contains, then, 113 pages. The page
numbers which we assign will be placed in parentheses, to distinguish them
from the page numbers which are printed in Grammelogia IV. The pages
(44)-(65) are the same as the pages 1-22, and the pages (68)-(83) are the
same as the pages 53-68. Thus only thirty-eight pages have page numbers
printed on them. The pages (67) and (83) are identical in wording, except
for some printer’s errors; they contain verses in praise of the Ring, and
have near the bottom the word “Finis.” Also, pages (22) and (23) are
together identical in wording with page (113), which is set up in finer
type, containing an advertisement of a part of Grammelogia IV explaining
the mode of graduating the circular rules. There are altogether six parts
of Grammelogia IV which begin or end by an address to the reader, thus: “To
the Reader,” “Courteous Reader,” or “To the courteous and benevolent Reader
. . .,” namely the pages (8), (22), (68), (89), (90), (108). In his Epistle
(page 2), Oughtred characterizes the make up of the book in the following

  In reading it . . . I met with such a patchery and confusion of
  disjoynted stuffe, that I was striken with a new wonder, that any man
  should be so simple, as to shame himselfe to the world with such a

Grammelogia V differs from Grammelogia IV in having only the second
title-page. The first title-page may have been torn off from the copy I
have seen. A second difference is that the page with the printed numeral 22
in Grammelogia IV has after the word “Finis” the following notice:

  This instrument is made in Silver, or Brasse for the Pocket, or at any
  other bignesse, over against Saint Clements Church without Temple Barre,
  by Elias Allen.

This notice occurs also on page 22 of Grammelogia I and III, but is omitted
from page 22 of Grammelogia V.

               Description of Delamain’s instrument of 1630

In his address to King Charles I, in his Grammelogia I, Delamain emphasizes
the ease of operating with his slide rule by stating that it is “fit for
use . . . as well on Horse backe as on Foot.” Speaking “To the Reader,” he
states that he has “for many yeares taught the Mathematicks in this Towne,”
and made efforts to improve Gunter’s scale “by some Motion, so that the
whole body of Logarithmes might move proportionally the one to the other,
as occasion required. This conceit in February last [1629] I struke upon,
and so composed my Grammelogia or Mathematicall Ring; by which only with an
ocular inspection, there is had at one instant all proportionalls through
the said body of Numbers.” He dates his preface “first of January, 1630.”
The fifth and sixth pages contain his “Description of the Grammelogia,” the
term Grammelogia being applied to the instrument, as well as to the book.
His description is as follows:

  The parts of the Instrument are two Circles, the one moveable, and the
  other fixed; The moveable is that unto which is fastened a small pin to
  move it by; the other Circle may be conceived to be fixed; The
  circumference of the moveable Circle is divided into unequall parts,
  charactered with figures thus, 1. 2. 3. 4. 5. 6. 7. 8. 9. these figures
  doe represent themselves, or such numbers unto which a Cipher or Ciphers
  are added, and are varied as the occasion falls out in the speech of
  Numbers, so 1. stands for 1. or 10. or 100., &c. the 2. stands for 2. or
  20. or 200. or 2000., &c. the 3. stands for 30. or 300. or 3000., &c.

After elaborating this last point and explaining the decimal subdivisions
on the scales of the movable circle, he says that “the numbers and
divisions on the fixed Circle, are the very same that the moveable are, .
.” There is no drawing of the slide rule in this publication. The
twenty-two numbered pages give explanations of the various uses to which
the instrument can be put: “How to performe the Golden Rule” (pp. 1-3),
“Further uses of the Golden Rule” (pp. 4-6), “Notions or Principles
touching the disposing or ordering of the Numbers in the Golden Rule in
their true places upon the Grammelogia” (pp. 7-11), “How to divide one
number by another” (pp. 12, 13), “to multiply one Number by another” (pp.
14, 15), “To find Numbers in continuall proportion” (pp. 16, 17), “How to
extract the Square Root,” “How to extract the Cubicke Root” (pp. 18-21),
“How to performe the Golden Rule” (the rule of proportion) is explained

  Seeke the first number in the moveable, and bring it to the second number
  in the fixed, so right against the third number in the moveable, is the
  answer in the fixed.

  If the Interest of 100. li. be 8. li. in the yeare, what is the Interest
  of 65. li. for the same time.

  Bring 100. in the moveable to 8. in the fixed, so right against 65. in
  the moveable is 5.2. in the fixed, and so much is the Interest of 65. li.
  for the yeare at 8. li. for 100. li. per annum.

  The Instrument not removed, you may at one instant right against any
  summe of money in the moveable, see the Interest thereof in the fixed:
  the reason of this is from the Definition of Logarithmes.

These are the earliest known printed instructions on the use of a slide
rule. It will be noticed that the description of the instrument at the
opening makes no references to logarithmic lines for the trigonometric
functions; only the line of numbers is given. Yet the title-page promised
the “resolution of Plaine and Sphericall Triangles.” Page 22 throws light
upon this matter:

  If there be composed three Circles of equal thicknesse, A.B.C. so that
  the inner edge of D [should be B] and the outward edge of A bee
  answerably graduated with Logarithmall signes [sines], and the outward
  edge of B and the inner edge of A with Logarithmes; and then on the
  backside be graduated the Logarithmall Tangents, and againe the
  Logarithmall signes oppositly to the former graduations, it shall be
  fitted for the resolution of Plaine and Sphericall Triangles.

After twelve lines of further remarks on this point he adds:

  Hence from the forme, I have called it a Ring, and Grammelogia by
  annoligie of a Lineary speech; which Ring, if it were projected in the
  convex unto two yards Diameter, or thereabouts, and the line Decupled, it
  would worke Trigonometrie unto seconds, and give proportionall numbers
  unto six places only by an ocular inspection, which would compendiate
  Astronomicall calculations, and be sufficient for the Prosthaphaeresis of
  the Motions: But of this as God shall give life and ability to health and

The unnumbered page following page 22 contains the patent and copyright on
the instrument and book:

  Whereas Richard Delamain, Teacher of Mathematicks, hath presented vnto Vs
  an Instrument called Grammelogia, or The Mathematicall Ring, together
  with a Booke so intituled, expressing the use thereof, being his owne
  Invention; we of our Gracious and Princely favour have granted unto the
  said Richard Delamain and his Assignes, Privilege, Licence, and
  Authority, for the sole Making, Printing and Selling of the said
  Instrument and Booke: straightly forbidding any other to Make, Imprint,
  or Sell, or cause to be Made, or Imprinted, or Sold, the said Instrument
  or Booke within any our Dominions, during the space of ten yeares next
  ensuing the date hereof, upon paine of Our high displeasure. Given under
  our hand and Signet at our Palace of Westminster, the fourth day of
  January, in the sixth yeare of our Raigne.

    Delamain’s later designs, and directions for using his instruments

In the Appendix of Grammelogia III, on page 52 is given a description of an
instrument promised near the end of Grammelogia I:

  That which I have formerly delivered hath been onely upon one of the
  Circles of my Ring, simply concerning Arithmeticall Proportions, I will
  by way of Conclusion touch upon some uses of the Circles, of Logarithmall
  Sines, and Tangents, which are placed on the edge of both the moveable
  and fixed Circles of the Ring in respect of Geometricall Proportions, but
  first of the description of these Circles.

  First, upon the side that the Circle of Numbers is one, are graduated on
  the edge of the moveable, and also on the edge of the fixed the
  Logarithmall Sines, for if you bring 1. in the moveable amongst the
  Numbers to 1. in the fixed, you may on the other edge of the moveable and
  fixed see the sines noted thus 90. 90. 80. 80. 70. 70. 60. 60. &c. unto
  6.6. and each degree subdivided, and then over the former divisions and
  figures 90. 90. 80. 80. 70. 70. &c. you have the other degrees, viz. 5.
  4. 3. 2. 1. each of those divided by small points.

  Secondly, (if the Ring is great) neere the outward edge of this side of
  the fixed against the Numbers, are the usuall divisions of a Circle, and
  the points of the Compasse: serving for observation in Astronomy, or
  Geometry, and the sights belonging to those divisions, may be placed on
  the moveable Circle.

  Thirdly, opposite to those Sines on the other side are the Logarithmall
  Tangents, noted alike both in the moveable and fixed thus &c. unto 45.45. which numbers or
  divisions serve also for their Complements to 90. so 40 gr. stands for
  50. gr. 30. gr. for 60 gr. 20. gr. for 70. gr. &c. each degree here both
  in the moveable and fixed is also divided into parts. As for the degrees
  which are under 6. viz. they are noted with small figures over
  this divided Circle from &c. and each of those degrees
  divided into parts by small points both in the moveable and fixed.

  Fourthly, on the other edge of the moveable on the same side is another
  graduation of Tangents, like that formerly described. And opposite unto
  it, in the fixed is a Graduation of Logarithmall sines in every thing
  answerable to the first descrition of Sines on the other side.

  Fifthly, on the edge of the Ring is graduated a parte of the Æquator,
  numbered thus 10 20. 30. unto 100. and there unto is adjoyned the degrees
  of the Meridian inlarged, and numbered thus 10 20.30 unto 70. each degree
  both of the Æquator, and Meridian are subdivided into parts; these two
  graduated Circles serve to resolve such Questions which concerne
  Latitude, Longitude, Rumb, and Distance, in Nauticall operations.

  Sixthly, to the concave of the Ring may be added a Circle to be elevated
  or depressed for any Latitude, representing the Æquator, and so divided
  into houres and parts with an Axis, to shew both the houre, and Azimuth,
  and within this Circle may be hanged a Box, and Needle with a Socket for
  a staffe to slide into it, and this accommodated with scrue pines to
  fasten it to the Ring and staffe, or to take it off at pleasure.

The pages bearing the printed numbers 53-68 in the Grammelogia III, IV and
V make no reference to the dispute with Oughtred and may, therefore, be
assumed to have been published before the appearance of Oughtred’s Circles
of Proportion. On page 53, “To the Reader,” he says:

  . . . you may make use of the Projection of the Circles of the Ring upon
  a Plaine, having the feet of a paire of compasses (but so that they be
  flat) to move on the Center of that Plaine, and those feet to open and
  shut as a paire of Compasses . . . now if the feet bee opened to any two
  termes or numbers in that Projection, then may you move the first foot to
  the third number, and the other foot shall give the Answer; . . . it hath
  pleased some to make use of this way. But in this there is a double
  labour in respect to that of the Ring, the one in fitting those feet unto
  the numbers assigned, and the other by moving them about, in which a man
  can hardly accommodate the Instrument with one hand, and expresse the
  Proportionals in writing with the other. By the Ring you need not but
  bring one number to another, and right against any other number is the
  Answer without any such motion. . . . upon that [the Ring] I write,
  shewing some uses of those Circles amongst themselves, and conjoyned with
  others . . . in Astronomy, Horolographie, in plaine Triangles applyed to
  Dimensions, Navigation, Fortification, etc. . . . But before I come to
  Construction, I have thought it convenient by way introduction, to
  examine the truth of the graduation of those Circles . . .

These are the words of a practical man, interested in the mechanical
development of his instrument. He considers not only questions of
convenience but also of accuracy. The instrument has, or may have now, also
lines of sines and tangents. To test the accuracy of the circles of
Numbers, “bring any number in the moveable to halfe of that number in the
fixed: so any number or part in the fixed shall give his double in the
moveable, and so may you trie of the thirds, fourths &c. of numbers, vel
contra,” (p. 54). On page 55 are given two small drawings, labelled, “A
Type of the Ringe and Scheme of this Logarithmicall projection, the use
followeth. These Instruments are made in Silver or Brasse by John Allen
neare the Sauoy in the Strand.”

                            CIRCULAR SLIDE RULE

Delamain’s publication of 1630 on the ‘Mathematicall Ring’ does not appear
at that time to have caused a rupture between him and Oughtred. When in
1631 Delamain brought out his Horizontall Quadrant, the invention of which
Delamain was afterwards charged to have stolen from Oughtred, Delamain was
still in close touch with Oughtred and was sending Oughtred in the Arundell
House, London, the sheets as they were printed. Oughtred’s reference to
this in his Epistle (p. 20) written after the friendship was broken, is as

  While he was printing his tractate of the Horizontall quadrant, although
  he could not but know that it was injurious to me in respect of my free
  gift to Master Allen, and of William Forster, whose translation of my
  rules was then about to come forth: yet such was my good nature, and his
  shamelessnesse, that every day, as any sheet was printed, hee sent, or
  brought the same to mee at my chamber in Arundell house to peruse which I
  lovingly and ingenuously did, and gave him my judgment of it.

Even after Forster’s publication of Oughtred’s Circles of Proportion, 1632,
Oughtred had a book, A canon of Sines Tangents and Secants, which he had
borrowed from Delamain and was then returning (Epistle, page (5)). The
attacks which Forster, in the preface to the Circles of Proportion, made
upon Delamain (though not naming Delamain) started the quarrel. Except for
Forster and other pupils of Oughtred who urged him on to castigate
Delamain, the controversy might never have arisen. Forster expressed
himself in part as follows:

  . . . being in the time of the long vacation 1630, in the Country, at the
  house of the Reverend, and my most worthy friend, and Teacher, Mr.
  William Oughtred (to whose instruction I owe both my initiation, and
  whole progresse in these Sciences.) I vpon occasion of speech told him of
  a Ruler of Numbers, Sines, & Tangents, which one had be-spoken to be made
  (such as it vsually called Mr. Gunter’s Ruler) 6 feet long, to be vsed
  with a payre of beame-compasses. “He answered that was a poore invention,
  and the performance very troublesome: But, said he, seeing you are taken
  with such mechanicall wayes of Instruments, I will shew you what deuises
  I have had by mee these many yeares.” And first, hee brought to mee two
  Rulers of that sort, to be vsed by applying one to the other, without any
  compasses: and after that hee shewed mee those lines cast into a circle
  or Ring, with another moueable circle vpon it. I seeing the great
  expeditenesse of both those wayes; but especially, of the latter, wherein
  it farre excelleth any other Instrument which hath bin knowne; told him,
  I wondered that hee could so many yeares conceale such vseful inuentions,
  not onely from the world, but from my selfe, to whom in other parts and
  mysteries of Art, he had bin so liberall. He answered, “That the true way
  of Art is not by Instruments, but by Demonstration: and that it is a
  preposterous course of vulgar Teachers, to begin with Instruments, and
  not with the Sciences, and so in-stead of Artists, to make their
  Schollers only doers of tricks, and as it were Iuglers: to the despite of
  Art, losse of precious time, and betraying of willing and industrious
  wits, vnto ignorance and idlenesse. That the vse of Instruments is indeed
  excellent, if a man be an Artist: but contemptible, being set and opposed
  to Art. And lastly, that he meant to commend to me, the skill of
  Instruments, but first he would haue me well instructed in the Sciences.
  He also shewed me many notes, and Rules for the vse of those circles, and
  of his Horizontall Instrument, (which he had proiected about 30 yeares
  before) the most part written in Latine. All which I obtained of him
  leaue to translate into English, and make publique, for the vse, and
  benefit of such as were studious, and louers of these excellent Sciences.

  Which thing while I with mature, and diligent care (as my occasions would
  give me leaue) went about to doe: another to whom the Author in a louing
  confidence discouered this intent, using more hast then good speed, went
  about to preocupate; of which vntimely birth, and preuenting (if not
  circumuenting) forwardnesse, I say no more: but aduise the studious
  Reader, onely so farre to trust, as he shal be sure doth agree to truth &

While in this dedication reference is made to a slide rule or “ring” with a
“moveable circle,” the instrument actually described in the Circles of
Proportion consists of fixed circles “with an index to be opened after the
manner of a paire of Compasses.” Delamain, as we have seen, had decided
preference for the moveable circle. To Oughtred, on the other hand, one
design was about as good as the other; he was more of a theorist and
repeatedly expressed his contempt for mathematical instruments. In his
Epistle (page (25)), he says he had not “the one halfe of my intentions
upon it” (the rule in his book), nor one with a “moveable circle and a
thread, but with an opening Index at the centre (if so be that bee cause
enough to make it to bee not the same, but another Instrument) for my part
I disclaime it: it may go seeke another Master: which for ought I know,
will prove to be Elias Allen himselfe: for at his request only I altered a
little my rules from the use of the moveable circle and the thread, to the
two armes of an Index.”

All parts of Delamain’s Grammelogia IV, except pages 1-22 and 53-68
considered above, were published after the Circles of Proportion, for they
contain references to the ill treatment that Delamain felt or made believe
that he felt, that he had received in the book published by Oughtred and
Forster. Oughtred’s reference to teachers whose scholars are “doers of
tricks,” “Iuglers,” and Forster’s allusion to “another to whom the Author
in a loving confidence” explained the instrument and who “went about to
preocupate” it, are repeatedly mentioned. Delamain says, (page (89)) that
at first he did not intend to express himself in print, “but sought peace
and my right by a private and friendly way.” Oughtred’s account of
Delamain’s course is that of an “ill-natured man” with a “virulent tongue,”
“sardonical laughter” and “malapert sawsiness.” Contrasting Forster and
Delamain, he says that, of the former he “had the very first moulding” and
made him feel that “the way of Art” is “by demonstration.” But Delamain was
“already corrupted with doing upon Instruments, and quite lost from ever
being made an Artist.” (Epistle page (27)). Repeatedly does Oughtred assert
Delamain’s ignorance of mathematics. The two men were evidently of wholly
different intellectual predilections. That Delamain loved instruments is
quite evident, and we proceed to describe his efforts to improve the
circular slide rule.

The Grammelogia IV is dedicated to King Charles I. Delamain says:

  . . . Everything hath his beginning, and curious Arts seldome come to the
  height at the first; It was my promise then to enlarge the invention by a
  way of decuplating the Circles, which I now present unto your sacred
  Majestie as the quintessence and excellencie there of . . .

His enlarged circular rules are illustrated in the Bodleian Library copy of
Grammelogia IV by four diagrams, two of them being the two drawings on the
two title-pages at the beginning of the Grammelogia IV, 4 inches in
external diameter, and exhibiting eleven concentric circular lines carrying
graduations of different sorts. In the second of these designs all circles
are fixed. The other two drawings are each 10¾ inches in external diameter
and exhibit 18 concentric circular lines; the folded sheet of the first of
these drawings is inserted between pages (23) and (24), the second folded
sheet between pages (83) and (84). All circles of this second instrument
are fixed. Counting in the two small drawings in Grammelogia III, there are
in all six drawings of slide rules in the Bodleian Grammelogia IV. On pages
(24) to (43) Delamain explains the graduation of slide rules. He takes
first a rule which has one circle of equal parts, divided into 1000 equal
divisions. From a table of logarithms he gets log 2 = 0.301; from the
number 301 in the circle of equal parts he draws a line to the center of
the circle and marks the intersection with the circles of numbers by the
figure 2. Thus he proceeds with log 3, log 4, and so on; also with log sin
x and log tan x. For log sin x he uses two circles, the first (see page
(27)) for angles from 34′ 24″ to 5° 44′ 22″, the second circle from 5° 44′
22″ to 90°. The drawings do not show the seconds. He suggests many
different designs of rules. On page (29) he says:

  For the single projection of the Circles of my Ring, and the dividing and
  graduating of them: which may bee so inserted upon the edges of Circles
  of mettle turned in the forme of a Ring, so that one Circle may moove
  betweene two fixed, by helpe of two stayes, then may there be graduated
  on the face of the Ring, upon the outer edge of the mooveable and inner
  edge of the fixed, the Circle of Numbers, then upon the inner edge of
  that mooveable Circle, and the outward edge of that inner fixed Circle
  may be inserted the Circle of Sines, and so according to the description
  of those that are usually made.

In addition to these lines he proceeds to mention the circle giving the
ordinary division into degrees and minutes, and two circles of tangents on
the other side of the rule.

Next Delamain explains an arrangement of all the graduation on one side of
the rule by means of “a small channell in the innermost fixed Circle, in
which may be placed a small single Index, which may have sufficient length
to reach from the innermost edge of the Mooveable Circle, unto the outmost
edge of the fixed Circle, which may be mooved to and fro at pleasure, in
the channell, which Index may serve to shew the opposition of Numbers” (p.
(31)). From this it is clear that the invention of the “runner” goes back
to the very first writers on the slide rule.

After describing a modification of the above arrangement, he adds, “many
other formes might be deliverd, about this single projection” (p. (32)).

Proceeding to the “enlarging” of the circles in the Ring, to, say, the
“Quadruple to that which is single, that is, foure times greater,” the
“equall parts” are distributed over four circles instead of only one
circle, but the general method of graduation is the same as before (p.
(33)); there being now four circles carrying the logarithms of numbers, and
so on. Next he points out “severall wayes how the Circles of the
Mathematicall Ring (being inlarged) may be accommodated for practicall
use:” (1) The Circles are all fixed in a plain and movable flat compasses
(or better, a movable semicircle) are used for fixing any two positions;
(2) There is a “double projection” of each logarithmic line “inlarged on a
Plaine,” one fixed, the other movable, as shown in his first figure on the
title-page, a single index only being used; (3) use of “my great Cylinder
which I have long proposed (in which all the Circles are of equall
greatnesse,) and it may be made of any magnitude or capacity, but for a
study (hee that will be at the charge) it may be of a yard diameter and of
such an indifferent length that it may containe 100 or more Circles fixed
parallel one to the other on the Cylinder, having a space betweene each of
them, so that there may bee as many mooveable Circles, as there are fixed
ones, and these of the mooveable linked, or fastened together, so that they
may all moove together by the fixed ones in these spaces, whose edges both
of the fixed, and mooveable being graduated by helpe of a single Index will
shew the proportionalls by opposition in this double Projection, or by a
double Index in a single Projection” (p. (36)).

Next follows the detailed description of his Ring “on a Plaine, according
to the diagramme that was given the King (for a view of that projection)
and afterwards the Ring it selve.” The diagram is the large one which we
mentioned as inserted between pages (23) and (24). The instrument has two
circles, one moveable, upon each of which are described 13 distinct
circular graduations. The lines on the fixed circle are: “The Circle of
degrees and calendar,” E. “Circle of equall parts, and part of the Equator,
and Meridian,” TT. “The Circle of Tangents,” S. “The Circle of Sines,” D.
“The Circle of Decimals,” N. “The Circle of Numbers.” The lines on the
movable circle are: N. “The Circle of Numbers,” E. “The Circle of equated
figures, and bodies,” S. “The Circle of Sines,” TT. “The Circle of
Tangents,” Y. “The Circle of time, yeares, and monethes.”

On pages (84)-(88) Delamain explains an enlargement of his Ring for
computations involving the sines of angles near to 90°. On page (86) he

  I have continued the Sines of the Projection unto two severall
  revolutions, the one beginning at 77.gr. 45.m. 6.s. and ends at 90.gr.
  (being the last revolution of the decuplation of the former, or the
  hundred part of that Projection) the other beginning at 86.gr. 6.m. 48.s.
  and ends at 90.gr. (being the last of a ternary of decuplated
  revolutions, or the thousand part of that Projection) and may bee thus

He explains the manner of using these extra graduations. Thus he claims to
have attained degrees of accuracy which enabled him to do what “some one”
had declared “could not bee done.” It is hardly necessary to point out that
Delamain’s Grammelogia IV suggests designs of slide rules which inventors
two hundred or more years later were endeavouring to produce. Which of
Delamain’s designs of rules were actually made and used, he does not state
explicitly. He refers to a rule 18 inches in diameter as if it had been
actually constructed (pages (86), (88)). Oughtred showed no appreciation of
such study in designing and ridiculed Delamain’s efforts, in his Epistle.

Additional elucidations of his designs of rules, along with explanations of
the relations of his work to that of Gunter and Napier, and sallies
directed against Oughtred and Forster, are contained on pages (8)-(21) of
his Grammelogia IV.


The question of independence and priority of invention is discussed by
Delamain more specifically on pages (89)-(113); Oughtred devotes his entire
Epistle to it. It is difficult to determine definitely which publication is
the later, Delamain’s Grammelogia IV or Oughtred’s Epistle. Each seems to
quote from the other. Probably the explanation is that the two publications
contain arguments which were previously passed from one antagonist to the
other by word of mouth or by private letter. Oughtred refers in his Epistle
(p. (12)) to a letter from Delamain. We believe that the Epistle came after
Delamain’s Grammelogia IV. Delamain claims for himself the invention of the
circular slide rule. He says in his Grammelogia IV. (p. (99)), “when I had
a sight of it, which was in February, 1629 (as I specified in my Epistle) I
could not conceale it longer, envying my selfe, that others did not tast of
that which I found to carry with it so delightfull and pleasant a goate
[taste] . . .” Delamain asserts (without proof) that Oughtred “never saw it
as he now challengeth it to be his invention, untill it was so fitted to
his hand, and that he made all his practise on it after the publishing of
my Booke upon my Ring, and not before; so it was easie for him or some
other to write some uses of it in Latin after Christmas, 1630 and not the
Sommer before, as is falsely alledged by some one . . .” (p. (91)).
Delamain’s accusation of theft on the part of Oughtred cannot be seriously
considered. Oughtred’s reputation as a mathematician and his standing in
his community go against such a supposition. Moreover, William Forster is a
witness for Oughtred. The fact that Oughtred had the mastery of the
rectilinear slide rule as well, while Delamain in 1630 speaks only of the
circular rule, weighs in Oughtred’s favour.

Oughtred says he invented the slide rule “above twelve yeares agoe,” that
is, about 1621, and “I with mine owne hand made me two such Circles, which
I have used ever since, as my occasions required,” (Epistle p. (22)). On
the same page, he describes his mode of discovery thus:

  I found that it required many times too great a paire of Compasses [in
  using Gunter’s line], which would bee hard to open, apt to slip, and
  troublesome for use. I therefore first devised to have another Ruler with
  the former: and so by setting and applying one to the other, I did not
  onely take away the use of Compasses, but also make the worke much more
  easy and expedite: when I should not at all need the motion of my hand,
  but onely the glancing at my sight: and with one position of the Rulers,
  and view of mine eye, see not one onely, but the manifold proportions
  incident unto the question intended. But yet this facility also wanted
  not some difficulty especially in the line of tangents, when one arch was
  in the former mediety of the quadrant, and the other in the latter: for
  in this case it was needful that either one Ruler must bee as long againe
  as the other; or else that I must use an inversion of the Ruler, and
  regression. By this consideration I first of all saw that if those lines
  upon both Rulers were inflected into two circles, that of the tangents
  being in both doubled, and that those two Circles should move one upon
  another; they with a small thread in the center to direct the sight,
  would bee sufficient with incredible and wonderfull facility to worke all
  questions of Trigonometry . . .

Oughtred said that he had no desire to publish his invention, but in the
vacation of 1630 finally promised William Forster to let him bring out a
translation. Oughtred claims that Delamain got the invention from him at
Alhallontide [November 1], 1630, when they met in London. The accounts of
that meeting we proceed to give in double column.

                            Delamain’s Statement
                         Grammelogia IV, page (98)

  “. . . about Alhalontide 1630. (as our Authors reporteth) was the time he
  was circumvented, and then his intent in a loving manner (as before) he
  opened unto me, which particularly I will dismantle in the very naked
  truth: for, wee being walking together some few weekes before Christmas,
  upon Fishstreet hill, we discoursed upon sundry things Mathematicall,
  both Theoreticall and Practicall, and of the excellent inventions and
  helpes that in these dayes were produced, amongst which I was not a
  little taken with that of the Logarythmes, commending greatly the
  ingenuitie of Mr. Gunter in the Projection, and inventing of his Ruler,
  in the lines of proportion, extracted from these Logarythmes for ordinary
  Practicall uses; He replyed unto me (in these very words) What will yov
  say to an Invention that I have, which in a lesse extent of the Compasses
  shall worke truer then that of Mr. Gunters Ruler, I asked him then of
  what forme it was, he answered with some pause (which no doubt argued his
  suspition of mee that I might conceive it) that it was Arching-wise, but
  now hee sayes that hee told mee then, it was Circular (but were I put to
  my oath to avoid the guilt of Conscience I would conclude in the former.)
  At which immediately I answered, I had the like my selfe, and so we
  discoursed not a word more touching that subject . . . Then after my
  coming home I sent him a sight of my Projection drawne in Pastboard: Now
  admit I had not the Invention of my Ring before I discoursed . . . it was
  not so facil for mee . . . to raise and compose so complete, and absolute
  an Instrument from so small a principle, or glimpse of light . . .”

                            Oughtred’s Statement
                             Epistle, page (23)

  “Shortly after my gift to Elias Allen, I chanced to meet with Richard
  Delamain in the street (it was at Alhallontide) and as we walked together
  I told him what an Instrument I had given to Master Allen, both of the
  Logarithmes projected into circles, which being lesse then one foot
  diameter would performe as much as one of Master Gunters Rulers of sixe
  feet long: and also of the Prostaphaereses of the Plannets and second
  motions. Such an invention have I said he: for now his intentions (that
  is his ambition) beganne to worke: . . . But he saith, Then after my
  comming home I sent him a sight of my projection drawne in past-board.
  See how notoriously he jugleth without an Instrument. Then after: how
  long after? a sight of my projection: of how much? More then seven weekes
  after on December 23, he sent to mee the line of numbers onely set upon a
  circle: . . . and so much onely he presented to his Majesty: but as for
  Sine or tangent of his, there was not the least shew of any. Neither
  could he give to Master Allen any direction for the composure of the
  circles of his Ring, or for the division of them: as upon his oath Master
  Allen will testify how hee misled him, and made him labour in vain above
  three weeks together, until Master Allen himselfe found out his ignorance
  and mistaking, which is more cleare then is possible with any impudence
  to be outfaced.”

Oughtred makes a further statement (Epistle, p. (24)) as follows:

  Delamain hearing that Brown with his Serpentine had another line by which
  he could worke to minutes in the 90 degree of sines . . . gave the [his]
  booke to Browne: who in thankfulnesse could not but gratify Delamain with
  his Lines also: and teach him the use of them, but especially of the
  great Line: with this caution on both sides, that one should not meddle
  with the others invention. Two dayes after Delamain . . . because he had
  found some things to be altered therin, . . . asked for the booke . . .
  but as soone as he had got it in his hands he rent out all the middle
  part with the two Schemes & put them up in his pocket & went his way . .
  . and . . . laboureth to recall all the bookes he had given forth . . .
  And shortly after this he got a new Printer (who was ignorant of his
  former Schemes) to print him new: giving him an especiall charge of the
  outermost line newly graven in the Plate, which indeed is Brownes very
  line: and then altering his book . . .

This and other statements made by Oughtred seem damaging to Delamain’s
reputation. But it is quite possible that Oughtred’s guesses as to
Delamain’s motives are wrong. Moreover, some of Oughtred’s statements are
not first hand knowledge with him, but mere hearsay. One may accept his
first hand facts and still clear Delamain of wrong doing. There is always
danger that rival claimants of an invention or discovery will proceed on
the assumption that no one else could possibly have come independently upon
the same devices that they themselves did; the history of science proves
the opposite. Seldom is an invention of any note made by only one man. We
do not feel competent to judge Delamain’s case. We know too little about
him as a man. We incline to the opinion that the hypothesis of independent
invention is the most plausible. At any rate, Delamain figures in the
history of the slide rule as the publisher of the earliest book thereon and
as an enthusiastic and skillful designer of slide rules.

The effect of this controversy upon interested friends was probably small.
Doubtless few people read both sides. Oughtred says:[21] “this scandall . .
. hath with them, to whom I am not knowne, wrought me much prejudice and
disadvantage . .” Aubrey,[22] a friend of Oughtred, refers to Delamain “who
was so sawcy to write against him” and remembers having seen “many yeares
since, twenty or more good verses made” against Delamain. Another friend of
Oughtred, William Robinson, who had seen some of Delamain’s publications,
but not his Grammelogia IV, wrote in a letter to Oughtred, shortly before
the appearance of the latter’s Epistle:

  I cannot but wonder at the indiscretion of Rich. Delamain, who being
  conscious to himself that he is but the pickpurse of another man’s wit,
  would thus inconsiderately provoke and awake a sleeping lion . . . he
  hath so weakly (though in my judgment, vaingloriously enough) commended
  his own labour . . .[23]

Delamain presented King Charles I with one of his sun-dials, also with a
manuscript and, later, with a printed copy of his book of 1630. A drawing
of his improved slide rule was sent to the King and the Grammelogia IV is
dedicated to him. The King must have been favorably impressed, for Delamain
was appointed tutor to the King in mathematics. His widow petitioned the
House of Lords in 1645 for relief; he had ten children.[24]

Anthony Wood states that Charles I, on the day of his execution, commanded
his friend Thomas Herbert “to give his son the duke of York his large
ring-sundial of silver, a jewel his maj. much valued.” Anthony Wood adds,
“it was invented and made by Rich. Delamaine a very able mathematician, who
projected it, and in a little printed book did shew its excellent use in
resolving many questions in arithmetic and other rare operations to be
wrought by it in the mathematics.”[25]

                    VI. OUGHTRED’S GAUGING LINE, 1633

It has not been generally known, hitherto, that Oughtred designed a
rectilinear slide rule for gauging and published a description thereof in
1633.[26] In his Circles of Proportion, chapter IX, Oughtred had offered a
closer approximation than that of Gunter for the capacity of casks. The
Gauger of London expostulated with Oughtred for presuming to question
anything that Gunter had written. The ensuing discussion led to an
invitation extended by the Company of Vintners to the instrument maker
Elias Allen to request Oughtred to design a gauging rod.[27] This he did,
and Allen received an order for “threescore” instruments. On page 19
Oughtred describes his ‘Gauging Rod:’

  It consisteth of two rulers of brasse about 32 ynches of length, which
  also are halfe an ynch broad, and a quarter of an ynch thick . . . At one
  end of both those rulers are two little sockets of brasse fastened on
  strongly: by which the rulers are held together, and made to move one
  upon another, and to bee drawne out unto any length, as occasion shall
  require: and when you have them at the just length, there is upon one of
  the sockets a long Scrue-pin to scrue them fast.

There are graduations on three sides of the rulers, one graduation being
the logarithmic line of numbers. He says (p. 39), “the maner of computing
the Gauge-divisions I have concealed.” W. Robinson, who was a friend of
Oughtred, wrote him as follows:[28]

  I have light upon your little book of artificial gauging, wherewith I am
  much taken, but I want the rod, neither could I get a sight of one of
  them at the time, because Mr. Allen had none left . . . I forgot to ask
  Mr. Allen the price of one of them, which if not much I would have one of
  them.” Oughtred annotated this passage thus: “Or in wood, if any be made
  in wood by Thompson or any other.”

Another of Oughtred’s admirers, Sir Charles Cavendish, wrote, on February
11, 1635 thus:[29]

  I thank you for your little book, but especially for the way of
  calculating the divisions of your gauging rod. I wish, both for their own
  sakes and yours, that the citizens were as capable of the acuteness of
  this invention, as they are commonly greedy of gain, and then I doubt not
  but they would give you a better recompense than I doubt now they will.

On April 20, 1638, we find Oughtred giving Elias Allen directions[30]
“about the making of the two rulers.” As in 1633,[31] so now, Oughtred
takes one ruler longer than the other. This 1633 instrument was used also
as “a crosse-staffe to take the height of the Sunne, or any Starre above
the Horizon, and also their distances.” The longer ruler was called staffe,
the shorter transversarie. While in 1633 he took the lengths of the two in
the ratio “almost 3 to 2,” in 1638, he took “the transversary three
quarters of the staff’s length, . . . that the divisions may be larger.”


In my History of the Slide Rule I treat of Seth Partridge, Thomas Everard,
Henry Coggeshall, W. Hunt and Sir Isaac Newton.[32] Of Partridge’s Double
Scale of Proportion, London, I have examined a copy dated 1661, which is
the earliest date for this book that I have seen. As far as we know, 1661
is the earliest date of publications on the slide rule, since Oughtred and
Delamain. But it would not be surprising if the intervening 28 years were
found not so barren as they seem at present. The 1661 and 1662 impressions
of Partridge are identical, except for the date on the title-page. William
Leybourn, who printed Partridge’s book, speaks in high appreciation of it
in his own book.[33]

In 1661 was published also John Brown’s first book, Description and Use of
a Joynt-Rule, previously mentioned. In Chapter XVIII he describes the use
of “Mr. Whites rule” for the measuring of board and timber, round and
square. He calls this a “sliding rule.” The existence, in 1661, of a
“Whites rule” indicates activities in designing of which we know as yet
very little. In his book of 1761, previously quoted, Brown gives a drawing
of “White’s sliding rule” (p. 193); also a special contrivance of his own,
as indicated by him in these words:

  A further improvement of the Triangular Quadrant, as I have made it
  several times, with a sliding Cover on the in-side, when made hollow, to
  carry Ink, Pens, and Compasses; then on the sliding Cover, and Edges, is
  put the Line of Numbers, according to Mr. White’s first Contrivance for
  manner of operation; but much augmented, and made easie, by John Brown.

He gives no drawing of his “triangular quadrant,” hence his account of it
is unsatisfactory. He explains the use of “gage-points.” His placing
logarithmic lines on the edges of instrument boxes was outdone in oddity
later by Everard who placed them on tobacco-boxes.[34] In Brown’s
publication of 1704 the White slide rule is given again, “being as neat and
ready a way as ever was used.” He tells also of a “glasier’s sliding rule.”
William Leybourn explains in 1673 how Wingate’s double and triple lines for
squaring and cubing, or square and cube root, can be used on slide

Beginning early in the history of the slide rule, when Oughtred designed
his “gauging rod,” we notice the designing of rules intended for very
special purposes. Another such contrivance, which enjoyed long popularity,
was the Timber Measure by a Line, by Hen. Coggeshall, Gent., London, 1677,
a booklet of 35 pages. Coggeshall says in his preface:

  For what can be more ready and easie, then having set twelve to the
  length, to see the Content exactly against the Girt or Side of the
  Square. Whereas on Mr. Partridge’s Scale the Content is the Sixth Number,
  which is far more troublesome then [even] with Compasses.

One line on Coggeshall’s rule begins with 4 and extends to 40, these
numbers being the “Girt” (a quarter of the circumference), which in
ordinary practice of measuring round timber lies between 4 inches and 40
inches. This “Girt line” slides “against the line of Numbers in two
Lengths, to which it is exactly equal.” A second edition, 1682, shows some
changes in the rule, as well as an enlargement and change of title of the
book itself: A Treatise of Measures, by a Two-foot Rule, by H. C. Gent,
London, 1682. In this, the description of the rule is given thus:

  There are four Lines on each flat of this Rule; two next the outward
  edges, which are Lines of Measure; and two next the inward edges, which
  are Lines of Proportion. On one flat, next the inward edges, is the
  Square-line [Girt-line in round timber measurement] with the Line of
  Numbers his fellow. Next the outward, a Line of Inches divided into
  Halfs, Quarters, and Half-Quarters; from 1 to 12 on one Rule; and from 12
  to 24 on the other. On the other flat, next the inward edges, is the
  double Scale of Numbers [for solving proportions]. Next the outward on
  one Rule a Line of Inches divided each into ten parts; and this for
  gauging, etc. On the other a foot divided into 100 parts.

Later further changes were introduced in Coggeshall’s rule.[36]

It is worthy of note that Coggeshall’s slide rule book, The Art of
Practical Measuring, was reviewed in the Acta eruditorum, anno 1691, p.
473; hence Leupold’s description[37] of the rectilinear slide rule in his
Theatrum arithmetico-geometricum, Leipzig, 1727, Cap. XIII, p. 71, is not
the earliest reference to the rectilinear rule found in German
publications. The above date is earlier even than Biler’s reference to a
circular slide rule in his Descriptio instrumenti mathematici universalis
of 1696.

Two noted slide rules for gauging were described by Tho. Everard,
Philomath, in his Stereometry made easie, London, 1684. He designates his
lines by the capital letters A, B, C, D, E. On the first instrument, A on
the rule, and B and C on the slide, have each two radiuses of numbers, D
has only one, while E has three. The second rule is described in an
Appendix; it is one foot long, with two slides enabling the rule to be
extended to 3 feet.

Everard’s instruments were made in London by Isaac Carver who, soon after,
himself wrote a sixteen-page Description and Use of a New Sliding Rule,
projected from the Tables in the Gauger’s Magazine, London, 1687, which was
“printed for William Hunt” and bound in one volume with a book by Hunt,
called The Gauger’s Magazine, London, 1687. This appears to be the same
William Hunt who later brought out descriptions of his own of slide rules.
The instrument described by Carver “consists of three pieces, two whereof
are moveable to be drawn out till the whole be 36 inches long.” It has
several non-logarithmic graduations, together with logarithmic lines marked
A, B, C, D, of which A, B, C are “double lines,” and D a “single line” used
for squares and square roots. It is designed for the determination of the
vacuity of a “spheroidal cask lying,” a “spheroidal cask standing,” and a
“parabolical cask lying.”

Another seventeenth century writer on the slide rule is John Atkinson, whom
we have mentioned earlier. He says:[38] “The Lines of Numbers, Sines and
Tangents, are set double, that is, one on each side, as the middle piece
slides: which middle piece is so contrived, to slip to and fro easily, to
slide out, and to be put in any side uppermost, in order to bring those
Lines together (or against one another) most proper for solving the
Question, wrought by Sliding-Gunter.”

The data presented in this article show that, while the earliest slide
rules were of the circular type, the later slide rules of the seventeenth
century were of the rectilinear type.[39]

                                                         January 12, 1915.


[1]F. Cajori, History of the Logarithmic Slide Rule and Allied Instruments,
   New York, 1909, pp. 7-14, also Addenda i-vi.

[2]F. Cajori, “On the Invention of the Slide Rule,” in Colorado College
   Publication, Engineering Series Vol. 1, 1910. An abstract of this is
   given in Nature (London), Vol. 82, 1909, p. 267.

[3]F. Cajori, History etc., p. 14.

[4]Art. “Slide Rule” in the Penny Cyclopaedia and in the English
   Cyclopaedia [Arts and Sciences].

[5]Anthony Wood, Athenae oxonienses (Ed. P. Bliss), London, Vol. III, 1817,
   p. 423.

[6]The full title of the book which Wingate published on this subject in
   Paris is as follows:

   L’Vsage | de la | Reigle de | Proportion | en l’Arithmetique & |
   Geometrie. | Par Edmond Vvingate, | Gentil-homme Anglois. |

   Εἂν ἧς φιλεµαθὴς, ἕση ἥση πολυµαθὴς.

   In tenui, sed nõ tenuis vsusve, laborne. |

   A Paris, | Chez Melchior Mondiere, | demeurant en l’Isle du Palais, | à
   la | ruë de Harlay aux deux Viperes. | M. DC. XXIV. | Auec Priuilege du
   Roy. |

   Back of the title page is the announcement:

   Notez que la Reigle de Proportion en toutes façons se vend à Paris chez
   Melchior Tauernier, Graueur & Imprimeur du Roy pour les Tailles douces,
   demeurant en l’Isle du Palais sur le Quay qui regarde la Megisserie à
   l’Espic d’or.

[7]The title-page of the edition of 1658 is as follows:

   The Use of the Rule of Proportion in Arithmetick & Geometrie. First
   published at Paris in the French tongue, and dedicated to Monsieur, the
   then king’s onely Brother (now Duke of Orleance). By Edm. Wingate, an
   English Gent. And now translated into English by the Author. Whereinto
   is now also inserted the Construction of the same Rule, & a farther use
   thereof . . . 2nd edition inlarged and amended. London, 1658.

[8]Memories of the Life of that Learned Antiquary, Elias Ashmole, Esq.;
   Drawn up by himself by way of Diary. With Appendix of original Letters.
   Publish’d by Charles Burman, Esq., London, 1717, p. 23.

[9]Mathematical Tables, 1811, p. 36, and art. “Gunter’s Line” in his Phil.
   and Math. Dictionary, London, 1815.

[10]To the English Gentrie, and all others studious of the Mathematicks,
   which shall bee readers hereof. The just Apologie of Wil: Ovghtred,
   against the slaunderous insimulations of Richard Delamain, in a Pamphlet
   called Grammelogia, or the Mathematicall Ring, or Mirifica logarithmorum
   projectio circularis. We shall refer to this document as Epistle. It was
   published without date in 32 unnumbered pages of fine print, and was
   bound in with Oughtred’s Circles of Proportion, in the editions of 1633
   and 1639. In the 1633 edition it is inserted at the end of the volume
   just after the Addition vnto the Vse of the Instrument etc., and in that
   of 1639 immediately after the preface. It was omitted from the Oxford
   edition of 1660. The Epistle was also published separately. There is a
   separate copy in the British Museum, London. Aubrey, in his Brief Lives,
   edited by A. Clark, Vol. II, Oxford, 1898, p. 113, says quaintly, “He
   writt a stitch’t pamphlet about 163(?4) against . . . Delamaine.”

[11]Thomas Browne is mentioned by Stone in his Mathematical Instruments,
   London 1723, p. 16. See also Cajori, History of the Slide Rule, New
   York, 1909, p. 15.

[12]The Description and Use of a Joynt-Rule: . . . also the use of Mr.
   White’s Rule for measuring of Board and Timber, round and square; With
   the manner of Vsing the Serpentine-line of Numbers, Sines, Tangents, and
   Versed Sines. By J. Brown, Philom., London, 1661.

[13]A Collection of Centers and Useful Proportions on the Line of Numbers,
   by John Brown, 1662(?), 16 pages; Description and Use of the Triangular
   Quadrant, by John Brown, London, 1671; Wingate’s Rule of Proportion in
   Arithmetick and Geometry: or Gunter’s Line. Newly rectified by Mr. Brown
   and Mr. Atkinson, Teachers of the Mathematicks, London, 1683; The
   Description and Use of the Carpenter’s-Rule: Together with the Use of
   the Line of Numbers commonly call’d Gunter’s-Line, by John Brown,
   London, 1704.

[14]William Leybourn, op. cit., pp. 129, 130, 132, 133.

[15]James Atkinson’s edition of Andrew Wakely’s The Mariners Compass
   Rectified, London, 1694 [Wakely’s preface dated 1664, Atkinson’s
   preface, 1693]. Atkinson adds An Appendix containing Use of Instruments
   most useful in Navigation. Our quotation is from this Appendix, p. 199.

[16]R. Delamain, The Making, Description, and Use of a small portable
   Instrument . . . called a Horizontall Quadrant, etc., London, 1631.

[17]Oughtred’s description of his circular slide rule of 1632 and his
   rectilinear slide rule of 1633, as well as a drawing of the circular
   slide rule, are reproduced in Cajori’s History of the Slide Rule,
   Addenda, pp. ii-vi.

[18]The full title of the Grammelogia I is as follows:

   Gram̄elogia | or, | The Mathematicall Ring. | Shewing (any
   reasonable Capacity that hath | not Arithmeticke) how to resolve and
   worke | all ordinary operations of Arithmeticke. | And those which are
   most difficult with greatest | facilitie: The extraction of Roots, the
   valuation of | Leases, &c. The measuring of Plaines | and Solids. | With
   the resolution of Plaine and Sphericall | Triangles. | And that onely by
   an Ocular Inspection, | and a Circular Motion. | Naturae secreta tempus
   aperit. | London printed by John Haviland, 1630.

[19]Grammelogia III is the same as Grammelogia I, except for the addition
   of an appendix, entitled:

   De la Mains | Appendix | Vpon his | Mathematicall | Ring. Attribuit
   nullo (praescripto tempore) vitae | vsuram nobis ingeniique Deus. |
   London, |

   . . . The next line or two of this title-page which probably contained
   the date of publication, were cut off by the binder in trimming the
   edges of this and several other pamphlets for binding into one volume.

[20]Grammelogia IV has two title pages. The first is Mirifica Logarithmoru’
   Projectio Circularis. There follows a diagram of a circular slide rule,
   with the inscription within the innermost ring: Nil Finis, Motvs,
   Circvlvs vllvs Habet. The second title page is as follows:

   Grammelogia | Or, the Mathematicall Ring. | Extracted from the
   Logarythmes, and projected Circular: Now published in the | inlargement
   thereof unto any magnitude fit for use: shewing any reason- | able
   capacity that hath not Arithmeticke how to resolve and worke, | all
   ordinary operations of Arithmeticke: | And those that are most difficult
   with greatest facilitie, the extracti- | on of Rootes, the valuation of
   Leases, &c. the measuring of Plaines and Solids, | with the resolution
   of Plaine and Sphericall Triangles applied to the | Practicall parts of
   Geometrie, Horologographie, Geographie | Fortification, Navigation,
   Astronomie, &c. | And that onely by an ocular inspection, and a Circular
   motion, Invented and first published, by R. Delamain, Teacher, and
   Student of the Mathematicks. | Naturae secreta tempus aperit. |

   There is no date. There follows the diagram of a second circular slide
   rule, with the inscription within the innermost ring: Typus proiectionis
   Annuli adaucti vt in Conslusione Lybri praelo commissi, Anno 1630
   promisi. There are numerous drawings in the Grammelogia, all of which,
   excepting the drawings of slide rules on the engraved title-pages of
   Grammelogia IV and V, were printed upon separate pieces of paper and
   then inserted by hand into the vacant spaces on the printed pages
   reserved for them. Some drawings are missing, so that the Bodleian
   Grammelogia IV differs in this respect slightly from the copy in the
   British Museum and from the British Museum copy of Grammelogia V.

[21]Epistle, p. (8).

[22]Aubrey, op. cit., Vol. II., p. 111.

[23]Rigaud, Correspondence of Scientific Men during the 17th Century, Vol.
   I, Oxford, 1841, p. 11.

[24]Dictionary of National Biography, Art. “Delamain, Richard.” See also
   Rev. Charles J. Robinson, Taylors’ School, from A.D. 1562 to 1874, Vol.
   I, 1882, p. 151; Journal of the House of Commons, Vol. IV., p. 197b;
   Sixth Report of the Royal Commission on Historical Manuscripts, Part I,
   Report and Appendix, London, 1877. In this Appendix, p. 82, we read the

   Oct. 22 [1645] Petition of Sarah Delamain, relict of Richard Delamain.
   Petitioner’s husband was servant to the King, and one of His Majesty’s
   engineers for the fortification of the kingdom, and his tutor in
   mathematical arts; but upon the breaking out of the war he deserted the
   Court, and was called by the State to several employments, in fortifying
   the towns of Northampton, Newport, and Abingdon; and was also abroad
   with the armies as Quartermaster-General of the Foot, and therein died.
   Petitioner is left a disconsolate widow with ten children, the four
   least of whom are now afflicted with sickness, and petitioner has
   nothing left to support them. There are several considerable sums of
   money due to the petitioner, as well from the King as the State. Prays
   that she may have some relief amongst other widows. See L. J., VII. 6.

[25]Anthony Wood, Athenae Oxonienses (Edition Bliss) Vol. IV., London,
   1820, p. 34.

[26]The New Artificial Gauging Line or Rod: together with rules concerning
   the use thereof: Invented and written by WILLIAM OUGHTRED, etc., London,
   1633. The copy we have seen is in the Bodleian Library, Oxford. The book
   is small sized and has 40 pages.

[27]Oughtred, op. cit., p. 11.

[28]S. J. Rigaud, Correspondence of Scientific Men of the 17th Century,
   Oxford, Vol. I, 1841, p. 17.

[29]Rigaud, loc. cit., p. 22.

[30]Rigaud, loc. cit., pp. 30, 31.

[31]Oughtred, An Addition vnto the Vse of the Instrument called the Circles
   of Proportion, London, 1633, p. 63.

[32]F. Cajori, History of the Slide Rule, New York, 1909, pp. 16-22,
   Addenda, pp. vi-ix.

[33]W. Leybourn, op. cit., 1673, Preface, and pp. 128-29.

[34]Cajori op. cit., Addenda, p. ix.

[35]William Leybourn, op. cit., 1673, p. 35.

[36]See Cajori, op. cit., pp. 20, 28, Addenda, p. ix.

[37]See F. Cajori, “A Note on the History of the Slide Rule,” Bibliotheca
   mathematica, 3 F., Vol. 10, pp. 161-163.

[38]John Atkinson, op. cit., 1694, p. 204.

[39]Probably the oldest slide rule now in existence is owned by St. John’s
   College, Oxford, and is in the form of a brass disc, 1 ft. 6 in. in
   diameter. It was exhibited along with other instruments in May, 1919.
   According to the Catalogue of a Loan Exhibition of Early Scientific
   Instruments in Oxford, opened May 16, 1919, the instrument is inscribed
   with the name of the maker (“Elias Allen fecit”) and with the name of
   the donor, Georgius Barkham. It is dated 1635, which is only three years
   after the first publication of Oughtred’s description of his circular
   slide rule. It is stated in the Catalogue: “Unfortunately all the
   movable parts but the base-plate and a couple of thumb-screws are
   missing. The face of the instrument is engraved with Oughtred’s
   Horizontal Instrument. The back is engraved with eleven Circles of
   Proportion as described in Arthur Haughton’s book, a copy of which was
   presented to St. John’s College by George Barkham, to explain the use of
   the instrument.” As Arthur Haughton’s Oxford edition of Oughtred’s
   Circles of Proportion did not appear until 1660, it would seem that the
   instrument was probably not presented to the College before 1660. As far
   as is known, the next oldest slide rule is of the year 1654, kept in the
   South Kensington Museum, London, and is described in Nature of March 5,
   1914. It is a rectilinear rule, “of boxwood, well made, and bound
   together with brass at the two ends. It is of the square type, a little
   more than 2 ft. in length, and bears the logarithmic lines first
   described by Edmund Gunter. Of these, the num, sin and tan lines are
   arranged in pairs, identical and contiguous, one line in each pair being
   on the fixed part, and the other on the slide.” The instrument is
   inscribed, “Made by Robert Bissaker for T. W., 1654.” Nowhere else have
   we seen reference to Robert Bissaker. His slide rule seems to antedate
   the “Whites rule” mentioned above. [This foot-note was added on October
   15, 1919.]


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    1. Triassic Ichthyosauria, with Special Reference to the
        American Forms, by John C. Merriam. September, 1908          $3.00

    2. The Fauna of Rancho La Brea, by John C. Merriam.

      Part I, Occurrence                                               .30

      Part II, Canidae                                                 .80

  Volume II. The Silva of California, by Willis Linn Jepson. 1909
                                                        $5.00; cloth, 7.50

  Volume III. Business Cycles, by Wesley C. Mitchell. 1913
                                                        $5.00; cloth, 7.50

  Volume IV. 1. The Battle of the Seven Arts, by Louis J. Paetow.
    July, 1914                                                         .75

  University life, issued quarterly. George M. Calhoun, Editor.
  Price, $1.00 per year. Current, Volume XXII.

                       Transcriber’s Notes

--Inserted subchapter headings from the Table of Contents into the

--Corrected two page numbers in the Table of Contents

--Corrected a Greek quotation in footnote 6 by reference to the
  original sources.

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