Title: Kinematics of Mechanisms from the Time of Watt
Author: Ferguson, Eugene S., 1916-2004
Language: English
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CONTRIBUTIONS FROM

THE MUSEUM OF HISTORY AND TECHNOLOGY

PAPER 27



KINEMATICS OF MECHANISMS FROM THE TIME OF WATT

_Eugene S. Ferguson_


JAMES WATT, KINEMATIC SYNTHESIST      187

TO DRAW A STRAIGHT LINE               199

SCHOLARS AND MACHINES                 209

MECHANICIANS AND MECHANISMS           216

MECHANISMS IN AMERICA, 1875-1955      223

ADDITIONAL REFERENCES                 229



KINEMATICS OF MECHANISMS FROM THE TIME OF WATT


_In an inventive tour de force that seldom, if ever, has been equalled
for its brilliance and far-reaching consequences, James Watt radically
altered the steam engine not only by adding a separate condenser but by
creating a whole new family of linkages. His approach was largely
empirical, as we use the word today._

_This study suggests that, despite the glamor of today's sophisticated
methods of calculation, a highly developed intuitive sense, reinforced
by a knowledge of the past, is still indispensable to the design of
successful mechanisms._

THE AUTHOR: _Eugene S. Ferguson, formerly curator of mechanical and
civil engineering in the United States National Museum, Smithsonian
Institution, is now professor of mechanical engineering at Iowa State
University of Science and Technology._

In engineering schools today, a student is introduced to the kinematics
of mechanisms by means of a course of kinematic analysis, which is
concerned with principles underlying the motions occurring in
mechanisms. These principles are demonstrated by a study of mechanisms
already in existence, such as the linkage of a retractable landing gear,
computing mechanisms, mechanisms used in an automobile, and the like. A
systematic, if not rigorous, approach to the design of gears and cams
also is usually presented in such a course. Until recently, however, no
serious attempt was made to apply the principles developed in kinematic
analysis to the more complex problem of kinematic synthesis of linkages.
By kinematic synthesis is meant the designing of a linkage to produce a
given series of motions for a particular purpose.

That a rational--numerical or geometrical--approach to kinematic
synthesis is possible is a relatively recent idea, not yet fully
accepted; but it is this idea that is responsible for the intense
scholarly interest in the kinematics of mechanisms that has occurred in
this country within the last 10 years.

This scholarly activity has resulted in the rediscovery of many earlier
works on the subject, and nearly all the scholars now working in this
field have acknowledged in one way or another their debt to those who
arrived on the scene at an earlier time than they. There have been
occasional reviews of the sequence and nature of developments, but the
emphasis naturally has been upon the recent past. It seems to me that
there is something to be gained in looking beyond our own generation, or
even beyond the time of Franz Reuleaux (1829-1905), who is generally
credited with originating many of our modern concepts of mechanism
analysis and design, and to inquire into the ideas that made possible
Reuleaux's contributions.

     _Take to Kinematics. It will repay you. It is more fecund than
     geometry; it adds a fourth dimension to space._

     --Chebyshev to Sylvester, 1873

While no pretense of completeness is made, I have tried in this paper to
trace the high points in the development of kinematic analysis and
synthesis, both in academic circles and in the workshop, noting where
possible the influence of one upon the other. If I have devoted more
space to particular people and episodes than is warranted by their
contributions to the modern treatment of the subject, it is because I
have found that the history of kinematics of mechanisms, like the
history of any other branch of engineering, is more interesting and more
plausible if it is recognized that its evolutionary development is the
result of human activity. This history was wrought by people like us, no
less intelligent and no less subject than we are to environment, to a
subjective way of looking at things, and to a heritage of ideas and
beliefs.

I have selected the period from the time of Watt because modern
mechanisms originated with him, and I have emphasized the first century
of the period because by 1885 many of the ideas of modern kinematics of
mechanisms were well developed. Linkages are discussed, to the virtual
exclusion of gears and cams, because much of the scholarly work in
kinematic synthesis is presently directed toward the design of linkages
and because linkages provide a convenient thread for a narrative that
would have become unnecessarily complex if detailed treatment of gears
and cams had been included. I have brought the narrative down to the
present by tracing kinematics as taught in American engineering
schools, closing with brief mention of the scholarly activity in
kinematics in this country since 1950. An annotated list of additional
references is appended as an encouragement to further work in the
history of the subject.


James Watt, Kinematic Synthesist

James Watt (1736-1819), improver of the steam engine, was a highly
gifted designer of mechanisms, although his background included no
formal study of mechanisms. Indeed, the study of mechanisms, without
immediate regard to the machines in which they were used, was not
introduced until after Watt's important work had been completed, while
the actual design of mechanisms had been going on for several centuries
before the time of Watt.

Mechanisms that employed screws, cams, and gears were certainly in use
by the beginning of the Christian era. While I am not aware of
unequivocal evidence of the existence of four-bar linkages before the
16th century, their widespread application by that time indicates that
they probably originated much earlier. A tantalizing 13th-century sketch
of an up-and-down sawmill (fig. 1) suggests, but does not prove, that
the four-bar linkage was then in use. Leonardo da Vinci (1452-1519)
delineated, if he did not build, a crank and slider mechanism, also for
a sawmill (fig. 2). In the 16th century may be found the conversion of
rotary to reciprocating motion (strictly speaking, an oscillation
through a small arc of a large circle) and vice versa by use of linkages
of rigid members (figs. 3 and 4), although the conversion of rotary to
reciprocating motion was at that time more frequently accomplished by
cams and intermittent gearing. Nevertheless, the idea of linkages was a
firmly established part of the repertory of the machine builder before
1600. In fact one might have wondered in 1588, when Agostino Ramelli
published his book on machines,[1] whether linkages had not indeed
reached their ultimate stage of development. To illustrate my point, I
have selected the plate of Ramelli that most appeals to me (fig. 5),
although the book exhibits more than 200 other machines of comparable
complexity and ingenuity.

[Footnote 1: Agostino Ramelli, _Le Diverse et Artificiose Machine_,
Paris, 1588.]

[Illustration: Figure 1.--Up-and-down sawmill of the 13th century. The
guide mechanism at lower left, attached to the saw blade, appears to be
a 4-bar linkage. After Robert Willis, trans. and ed., _Facsimile of the
Sketch-Book of Wilars de Honecort_ (London, 1859, pl. 43).]

[Illustration: Figure 2.--Slider-crank mechanism of Leonardo da Vinci
(1452-1519), redrawn from his manuscript notebooks. A frame saw is
depicted at the lower end of the guides. From Theodor Beck, _Beiträge
zur Geschichte des Maschinenbaues_ (Berlin, 1899, p. 323).]

[Illustration: Figure 3.--Blowing engine by Vanuccio Biringuccio, about
1540, showing conversion of motion of the waterwheel shaft from rotation
to oscillation. From Theodor Beck, _Beiträge zur Geschichte des
Maschinenbaues_ (Berlin, 1899, p. 120).]

[Illustration: Figure 4.--Grain mill, 1588, showing conversion of motion
of the operating bars from oscillation to rotation. Note the
fly-weights, predecessors of the flywheel. From Agostino Ramelli, _Le
Diverse et Artificiose Machine_ (Paris, 1588, pl. opposite p. 199).]

[Illustration: Figure 5.--Machine for raising water. Such a machine was
built in Spain during the 16th century and was operated for some 80
years. From Agostino Ramelli, _Le Diverse et Artificiose Machine_
(Paris, 1588, p. 199).]

There was a vast difference, both in conception and execution, between
the linkages of Ramelli and those of James Watt some 200 years later.
Watt was responsible for initiating profound changes in mechanical
technology, but it should be recognized that the mechanic arts had,
through centuries of slow development, reached the stage where his
genius could flourish. The knowledge and ability to provide the
materials and tools necessary for Watt's researches were at hand, and
through the optimism and patient encouragement of his partner, Matthew
Boulton, they were placed at his disposal.

Watt's genius was nowhere more evident than in his synthesis of
linkages. An essential ingredient in the success of Watt's linkages,
however, was his partner's appreciation of the entirely new order of
refinement that they called for. Matthew Boulton, who had been a
successful manufacturer of buttons and metal novelties long before his
partnership with Watt was formed, had recognized at once the need for
care in the building of Watt's steam engine. On February 7, 1769, he had
written Watt:[2] "I presumed that your engine would require money,
very accurate workmanship and extensive correspondence to make it turn
out to the best advantage and that the best means of keeping up the
reputation and doing the invention justice would be to keep the
executive part of it out of the hands of the multitude of empirical
engineers, who from ignorance, want of experience and want of necessary
convenience, would be very liable to produce bad and inaccurate
workmanship; all of which deficiencies would affect the reputation of
the invention." Boulton expected to build the engines in his shop "with
as great a difference of accuracy as there is between the blacksmith and
the mathematical instrument maker." The Soho Works of Boulton and Watt,
in Birmingham, England, solved for Watt the problem of producing "in
great" (that is, in sizes large enough to be useful in steam engines)
the mechanisms that he devised.[3]

[Footnote 2: Henry W. Dickinson, _James Watt, Craftsman & Engineer_,
Cambridge, Cambridge University Press, 1936, pp. 52-53.]

[Footnote 3: James P. Muirhead, _The Origin and Progress of the
Mechanical Inventions of James Watt_, London, 1854, vol. 1, pp. 56, 64.
This work, in three volumes, contains letters, other documents, and
plates of patent specification drawings.]

The contributions of Boulton and Watt to practical mechanics "in great"
cannot be overestimated. There were in the 18th century instrument
makers and makers of timekeepers who had produced astonishingly
accurate work, but such work comprised relatively small items, all being
within the scope of a bench lathe, hand tools, and superb handwork. The
rapid advancement of machine tools, which greatly expanded the scope of
the machine-building art, began during the Boulton and Watt partnership
(1775-1800).

In April 1775 the skirmish at Concord between American colonists and
British redcoats marked the beginning of a war that was to determine for
the future the course of political events in the Western Hemisphere.

Another event of April 1775 occurring in Birmingham now appears to have
been one that marked the beginning of a new era of technological
advance. It was near the end of this month that Boulton, at the Soho
Works, wrote to his partner and commented upon receiving the cast iron
steam engine cylinder that had been finished in John Wilkinson's boring
mill:

     ... it seems tolerably true, but is an inch thick and weighs about
     10 cwt. Its diameter is about as much above 18 inches as the tin
     one was under, and therefore it is become necessary to add a brass
     hoop to the piston, which is made almost two inches broad.[4]

[Footnote 4: _Ibid._, vol. 2, p. 84.]

This cylinder indeed marked the turning point in the discouragingly long
development of the Watt steam engine, which for 10 years had occupied
nearly all of Watt's thoughts and all the time he could spare from the
requirements of earning a living. Although there were many trials ahead
for the firm of Boulton and Watt in further developing and perfecting
the steam engine, the crucial problem of leakage of steam past the
piston in the cylinder had now been solved by Wilkinson's new boring
mill, which was the first large machine tool capable of boring a
cylinder both round and straight.

The boring mill is pertinent to the development of linkages "in great,"
being the first of a new class of machine tools that over the next 50 or
60 years came to include nearly all of the basic types of heavy
chip-removing tools that are in use today. The development of tools was
accelerated by the inherent accuracy required of the linkages that were
originated by Watt. Once it had been demonstrated that a large and
complex machine, such as the steam engine, could be built accurately
enough so that its operation would be relatively free of trouble, many
outstanding minds became engaged in the development of machines and
tools. It is interesting, however, to see how Watt and others grappled
with the solutions of problems that resulted from the advance of the
steam engine.

During the 1770's the demand for continuous, dependable power applied to
a rotating shaft was becoming insistent, and much of Boulton's and
Watt's effort was directed toward meeting this demand. Mills of all
kinds used water or horses to turn "wheel-work," but, while these
sources of power were adequate for small operations, the quantity of
water available was often limited, and the use of enormous horse-whims
was frequently impracticable.

The only type of steam engine then in existence was the Newcomen beam
engine, which had been introduced in 1712 by Thomas Newcomen, also an
Englishman. This type of engine was widely used, mostly for pumping
water out of mines but occasionally for pumping water into a reservoir
to supply a waterwheel. It was arranged with a vertical steam cylinder
located beneath one end of a large pivoted working beam and a vertical
plunger-type pump beneath the other end. Heavy, flat chains were secured
to a sector at each end of the working beam and to the engine and pump
piston rods in such a way that the rods were always tangent to a circle
whose center was at the beam pivot. The weight of the reciprocating pump
parts pulled the pump end of the beam down; the atmosphere, acting on
the open top of the piston in the steam cylinder, caused the engine end
of the beam to be pulled down when the steam beneath the piston was
condensed. The chains would of course transmit force from piston to beam
only in tension.

It is now obvious that a connecting rod, a crank, and a sufficiently
heavy flywheel might have been used in a conventional Newcomen engine in
order to supply power to a rotating shaft, but contemporary evidence
makes it clear that this solution was by no means obvious to Watt nor to
his contemporaries.

At the time of his first engine patent, in 1769, Watt had devised a
"steam wheel," or rotary engine, that used liquid mercury in the lower
part of a toroidal chamber to provide a boundary for steam spaces
successively formed by flap gates within the chamber. The practical
difficulties of construction finally ruled out this solution to the
problem of a rotating power source, but not until after Boulton and
Watt had spent considerable effort and money on it.[5]

[Footnote 5: Henry W. Dickinson and Rhys Jenkins, _James Watt and the
Steam Engine_, Oxford, Clarendon Press, 1927, pp. 146-148, pls. 14, 31.
This work presents a full and knowledgeable discussion, based on primary
material, of the development of Watt's many contributions to mechanical
technology. It is ably summarized in Dickinson, _op. cit._ (footnote
2).]

In 1777 a speaker before the Royal Society in London observed that in
order to obtain rotary output from a reciprocating steam engine, a crank
"naturally occurs in theory," but that in fact the crank is impractical
because of the irregular rate of going of the engine and its variable
length of stroke. He said that on the first variation of length of
stroke the machine would be "either broken to pieces, or turned
back."[6] John Smeaton, in the front rank of English steam engineers of
his time, was asked in 1781 by His Majesty's Victualling-Office for his
opinion as to whether a steam-powered grain mill ought to be driven by a
crank or by a waterwheel supplied by a pump. Smeaton's conclusion was
that the crank was quite unsuited to a machine in which regularity of
operation was a factor. "I apprehend," he wrote, "that no motion
communicated from the reciprocating beam of a fire engine can ever act
perfectly equal and steady in producing a circular motion, like the
regular efflux of water in turning a waterwheel." He recommended,
incidentally, that a Boulton and Watt steam engine be used to pump water
to supply the waterwheel.[7] Smeaton had thought of a flywheel, but he
reasoned that a flywheel large enough to smooth out the halting, jerky
operation of the steam engines that he had observed would be more of an
encumbrance than a pump, reservoir, and waterwheel.[8]

[Footnote 6: John Farey, _A Treatise on the Steam Engine_, London, 1827,
pp. 408-409.]

[Footnote 7: _Reports of the Late John Smeaton, F.R.S._, London, 1812,
vol. 2, pp. 378-380.]

[Footnote 8: Farey, _op. cit._ (footnote 6), p. 409.]

The simplicity of the eventual solution of the problem was not clear to
Watt at this time. He was not, as tradition has it, blocked merely by
the existence of a patent for a simple crank and thus forced to invent
some other device as a substitute.

Matthew Wasbrough, of Bristol, the engineer commonly credited with the
crank patent, made no mention of a crank in his patent specification,
but rather intended to make use of "racks with teeth," or "one or more
pullies, wheels, segments of wheels, to which are fastened rotchets and
clicks or palls...." He did, however, propose to "add a fly or flys, in
order to render the motion more regular and uniform." Unfortunately for
us, he submitted no drawings with his patent specification.[9]

[Footnote 9: British Patent 1213, March 10, 1779.]

James Pickard, of Birmingham, like Boulton, a buttonmaker, in 1780
patented a counterweighted crank device (fig. 6) that was expected to
remove the objection to a crank, which operated with changing leverage
and thus irregular power. In figure 6, the counterweighted wheel,
revolving twice for each revolution of the crank (A), would allow the
counterweight to descend while the crank passed the dead-center position
and would be raised while the crank had maximum leverage. No mention of
a flywheel was made in this patent.[10]

[Footnote 10: British Patent 1263, August 23, 1780.]

[Illustration: Figure 6.--One of the steam engine "Crank Patents" that
hindered James Watt's progress. This patent, granted to James Pickard in
1780, claimed only the arrangement of counterweights, not the crank. The
crank pin to which the connecting rod was attached is at _Aa_. From
British Patent 1263, August 23, 1780.]

Wasbrough, finding that his "rotchets and clicks" did not serve,
actually used, in 1780, a crank with a flywheel. Watt was aware of this,
but he remained unconvinced of the superiority of the crank over other
devices and did not immediately appreciate the regulating ability of a
flywheel.[11] In April 1781 Watt wrote to Boulton, who was then out of
town: "I know from experiment that the other contrivance, which you saw
me try, performs at least as well, and has in fact many advantages over
the crank."[12] The "other contrivance" probably was his swash wheel
which he built and which appeared on his next important patent
specification (fig. 7a). Also in this patent were four other devices,
one of which was easily recognizable as a crank, and two of which were
eccentrics (fig. 7a, b). The fourth device was the well-known
sun-and-planet gearing (fig. 7e).[13] In spite of the similarity of the
simple crank to the several variations devised by Watt, this patent drew
no fire from Wasbrough or Pickard, perhaps because no reasonable person
would contend that the crank itself was a patentable feature, or perhaps
because the similarity was not at that time so obvious. However, Watt
steered clear of directly discernible application of cranks because he
preferred to avoid a suit that might overthrow his or other patents. For
example, if the Wasbrough and Pickard patents had been voided, they
would have become public property; and Watt feared that they might
"get into the hands of men more ingenious," who would give Boulton and
Watt more competition than Wasbrough and Pickard.[14]

[Footnote 11: Dickinson and Jenkins, _op. cit._ (footnote 5), pp. 150,
154.]

[Footnote 12: _Ibid._, p. 154.]

[Footnote 13: William Murdock, at this time a Boulton and Watt erector,
may have suggested this arrangement. _Ibid._, p. 56.]

[Footnote 14: Muirhead, _op. cit._ (footnote 3), vol. 3, note on p. 39.]

[Illustration: Figure 7.--James Watt's five alternative devices for the
conversion of reciprocating motion to rotary motion in a steam engine.
(British Patent 1306, October 25, 1781). From James P. Muirhead, _The
Origin and Progress of the Mechanical Inventions of James Watt_ (London,
1854, vol. 3, pls. 3-5, 7).]

[Illustration: (a) "Inclined wheel." The vertical shaft at _D_ is
rotated by action of wheels _H_ and _J_ on cam, or swash plate, _ABC_.
Boulton and Watt tried this device but discarded it.]

[Illustration: (b) Counterweighted crank wheel.]

[Illustration: (c) "Eccentric wheel" with external yoke hung from
working beam. The wheel pivots at _C_.]

[Illustration: (d) "Eccentric wheel" with internal driving wheel hung
from working beam. Wheel _B_ is pivoted at center of shaft _A_.]

[Illustration: (e) Sun-and-planet gearing. This is the idea actually
employed in Boulton and Watt engines. As the optional link _JK_ held the
gearwheel centers always equidistant, the annular guide _G_ was not
used.]

The sun-and-planet arrangement, with gears of equal size, was adopted by
Watt for nearly all the rotative engines that he built during the term
of the "crank patents." This arrangement had the advantage of turning
the flywheel through two revolutions during a single cycle of operation
of the piston, thus requiring a flywheel only one-fourth the size of the
flywheel needed if a simple crank were used. The optional link (JK of
fig. 7e) was used in the engines as built.

From the first, the rotative engines were made double-acting--that is,
work was done by steam alternately in each end of the cylinder. The
double-acting engine, unlike the single-acting pumping engine, required
a piston rod that would push as well as pull. It was in the solution of
this problem that Watt's originality and sure judgment were most clearly
demonstrated.

A rack and sector arrangement (fig. 8) was used on some engines. The
first one, according to Watt, "has broke out several teeth of the rack,
but works steady."[15] A little later he told a correspondent that his
double-acting engine "acts so powerfully that it has broken all its
tackling repeatedly. We have now tamed it, however."[16]

[Footnote 15: James Watt, March 31, 1783, quoted in Dickinson and
Jenkins, _op. cit._ (footnote 5), p. 140.]

[Footnote 16: Watt to De Luc, April 26, 1783, quoted in Muirhead, _op.
cit._ (footnote 3), vol. 2, p. 174.]

[Illustration: Figure 8.--Watt engine of 1782 (British Patent 1321,
March 12, 1782) showing the rack and sector used to guide the upper end
of the piston rod and to transmit force from piston to working beam.
This engine, with a 30-inch cylinder and an 8-foot stroke, was arranged
for pumping. Pump rod _SS_ is hung from sector of the working beam. From
James P. Muirhead, _The Origin and Progress of the Mechanical Inventions
of James Watt_ (London, 1854, vol. 3, pl. 15).]

It was about a year later that the straight-line linkage[17] was thought
out. "I have started a new hare," Watt wrote to his partner. "I have
got a glimpse of a method of causing the piston-rod to move up and down
perpendicularly, by only fixing it to a piece of iron upon the beam,
without chains, or perpendicular guides, or untowardly frictions,
arch-heads, or other pieces of clumsiness.... I have only tried it in a
slight model yet, so cannot build upon it, though I think it a very
probable thing to succeed, and one of the most ingenious simple pieces
of mechanism I have contrived...."[18]

[Footnote 17: Watt's was a four-bar linkage. All four-bar straight-line
linkages that have no sliding pairs trace only an approximately straight
line. The exact straight-line linkage in a single plane was not known
until 1864 (see p. 204). In 1853 Pierre-Frédéric Sarrus (1798-1861), a
French professor of mathematics at Strasbourg, devised an accordion-like
spatial linkage that traced a true straight line. Described but not
illustrated (Académie des Sciences, Paris, _Comptes rendus_, 1853, vol.
36, pp. 1036-1038, 1125), the mechanism was forgotten and twice
reinvented; finally, the original invention was rediscovered by an
English writer in 1905. For chronology, see Florian Cajori, _A History
of Mathematics_, ed. 2, New York, 1919, p. 301.]

[Footnote 18: Muirhead, _op. cit._ (footnote 3), vol. 2, pp. 191-192.]

Watt's marvelously simple straight-line linkage was incorporated into a
large beam engine almost immediately, and the usually pessimistic and
reserved inventor was close to a state of elation when he told Boulton
that the "new central perpendicular motion answers beyond expectation,
and does not make the shadow of a noise."[19] This linkage, which was
included in an extensive patent of 1784, and two alternative devices are
illustrated here (fig. 9). One of the alternatives is a guided crosshead
(fig. 9, top right).

[Footnote 19: _Ibid._, p. 202.]

[Illustration: Figure 9.--Watt's mechanisms for guiding the upper end of
the piston rod of a double-acting engine (British Patent 1432, April 28,
1784). _Top left_, straight-line linkage; _top right_, crosshead and
guide arrangement; _lower left_, piston rod _A_ is guided by sectors _D_
and _E_, suspended by flexible cords. From James P. Muirhead, _The
Origin and Progress of the Mechanical Inventions of James Watt_ (London,
1854, vol. 3, pls. 21, 22).]

Brilliant as was the conception of this linkage, it was followed up by a
synthesis that is very little short of incredible. In order to make the
linkage attached to the beam of his engines more compact, Watt had
plumbed his experience for ideas; his experience had yielded up the work
done much earlier on a drafting machine that made use of a
pantograph.[20] Watt combined his straight-line linkage with a
pantograph, one link becoming a member of the pantograph.

[Footnote 20: "It has only one fault," he had told a friend on December
24, 1773, after describing the drafting machine to him, "which is, that
it will not do, because it describes conic sections instead of straight
lines." _Ibid._, p. 71.]

The length of each oscillating link of the straight-line linkage was
thus reduced to one-fourth instead of one-half the beam length, and the
entire mechanism could be constructed so that it would not extend
beyond the end of the working beam. This arrangement soon came to be
known as Watt's "parallel motion" (fig. 10).[21] Years later Watt told
his son: "Though I am not over anxious after fame, yet I am more proud
of the parallel motion than of any other mechanical invention I have
ever made."[22]

[Footnote 21: Throughout the 19th century the term "parallel motion" was
used indiscriminately to refer to any straight-line linkage. I have not
discovered the origin of the term. Watt did not use it in his patent
specification, and I have not found it in his writings or elsewhere
before 1808 (see footnote 22). _The Cyclopaedia_ (Abraham Rees, ed.,
London, 1819, vol. 26) defined parallel motion as "a term used among
practical mechanics to denote the rectilinear motion of a piston-rod,
&c. in the direction of its length; and contrivances, by which such
alternate rectilinear motions are converted into continuous rotatory
ones, or _vice versa_...." Robert Willis in his _Principles of
Mechanism_ (London, 1841, p. 399) described parallel motion as "a term
somewhat awkwardly applied to a combination of jointed rods, the purpose
of which is to cause a point to describe a straight line...." A. B.
Kempe in _How to Draw a Straight Line_ (London, 1877, p. 49) wrote: "I
have been more than once asked to get rid of the objectionable term
'parallel motion.' I do not know how it came to be employed, and it
certainly does not express what is intended. The expression, however,
has now become crystallised, and I for one cannot undertake to find a
solvent."]

[Footnote 22: Muirhead, _op. cit._ (footnote 3), vol. 3, note on p. 89.]

[Illustration: Figure 10.--Watt's "parallel motion." Engine's working
beam is pivoted at _A_. Pivot _F_ is attached to the engine frame. From
Dyonysius Lardner, _The Steam Engine_ (Philadelphia, 1852), pl. 5
(American ed. 5 from London ed. 5).]

The Watt four-bar linkage was employed 75 years after its inception by
the American Charles B. Richards when, in 1861, he designed his first
high-speed engine indicator (fig. 11). Introduced into England the
following year, the Richards Indicator was an immediate success, and
many thousands were sold over the next 20 or 30 years.[23]

[Footnote 23: Charles T. Porter, _Engineering Reminiscences_, New York,
1908, pp. 58-59, 90.]

[Illustration: Figure 11.--Richards high-speed engine indicator of 1861,
showing application of the Watt straight-line linkage. (_USNM 307515_;
_Smithsonian photo 46570_).]

In considering the order of synthetic ability required to design the
straight-line linkage and to combine it with a pantograph, it should be
kept in mind that this was the first one of a long line of such
mechanisms.[24] Once the idea was abroad, it was only to be expected
that many variations and alternative solutions should appear. One
wonders, however, what direction the subsequent work would have taken
if Watt had not so clearly pointed the way.

[Footnote 24: At least one earlier straight-line linkage, an arrangement
later ascribed to Richard Roberts, had been depicted before Watt's
patent (Pierre Patte, _Mémoirs sur les objets les plus importants de
l'architecture_, Paris, 1769, p. 229 and pl. 11). However, this linkage
(reproduced here in figure 18) had no detectable influence on Watt or on
subsequent practice.]

In 1827 John Farey, in his exhaustive study of the steam engine, wrote
perhaps the best contemporary view of Watt's work. Farey as a young man
had several times talked with the aging Watt, and he had reflected upon
the nature of the intellect that had caused Watt to be recognized as a
genius, even within his own lifetime. In attempting to explain Watt's
genius, Farey set down some observations that are pertinent not only to
kinematic synthesis but to the currently fashionable term "creativity."

In Farey's opinion Watt's inventive faculty was far superior to that of
any of his contemporaries; but his many and various ideas would have
been of little use if he had not possessed a very high order of
judgment, that "faculty of distinguishing between ideas; decomposing
compound ideas into more simple elements; arranging them into classes,
and comparing them together...."

Farey was of the opinion that while a mind like Watt's could produce
brilliant new ideas, still the "common stock of ideas which are current
amongst communities and professions, will generally prove to be of a
better quality than the average of those new ideas, which can be
produced by any individual from the operation of his own mind, without
assistance from others." Farey concluded with the observation that "the
most useful additions to that common stock, usually proceed from the
individuals who are well acquainted with the whole series."[25]

[Footnote 25: Farey, _op. cit._ (footnote 6), pp. 651, 652.]


To Draw a Straight Line

During most of the century after James Watt had produced his parallel
motion, the problem of devising a linkage, one point of which would
describe a straight line, was one that tickled the fancies of
mathematicians, of ingenious mechanics, and of gentlemanly dabblers in
ideas. The quest for a straight-line mechanism more accurate than that
of Watt far outlasted the pressing practical need for such a device.
Large metal planing machines were well known by 1830, and by midcentury
crossheads and crosshead guides were used on both sides of the Atlantic
in engines with and without working beams.

By 1819 John Farey had observed quite accurately that, in England at
least, many other schemes had been tried and found wanting and that "no
methods have been found so good as the original engine; and we
accordingly find, that all the most established and experienced
manufacturers make engines which are not altered in any great feature
from Mr. Watt's original engine...."[26]

[Footnote 26: In Rees, _op. cit._ (footnote 21), vol. 34 ("Steam
Engine"). John Farey was the writer of this article (see Farey, _op.
cit._, p. vi).]

Two mechanisms for producing a straight line were introduced before the
Boulton and Watt monopoly ended in 1800. Perhaps the first was by Edmund
Cartwright (1743-1823), who is said to have had the original idea for a
power loom. This geared device (fig. 12), was characterized
patronizingly by a contemporary American editor as possessing "as much
merit as can possibly be attributed to a gentleman engaged in the
pursuit of mechanical studies for his own amusement."[27] Only a few
small engines were made under the patent.[28]

[Footnote 27: _Emporium of Arts and Sciences_, December 1813, new ser.,
vol. 2, no. 1, p. 81.]

[Footnote 28: Farey, _op. cit._ (footnote 6), p. 666.]

[Illustration: Figure 12.--Cartwright's geared straight-line mechanism
of about 1800. From Abraham Rees, _The Cyclopaedia_ (London, 1819,
"Steam Engine," pl. 5).]

The properties of a hypocycloid were recognized by James White, an
English engineer, in his geared design which employed a pivot located on
the pitch circle of a spur gear revolving inside an internal gear. The
diameter of the pitch circle of the spur gear was one-half that of the
internal gear, with the result that the pivot, to which the piston rod
was connected, traced out a diameter of the large pitch circle (fig.
13). White in 1801 received from Napoleon Bonaparte a medal for this
invention when it was exhibited at an industrial exposition in
Paris.[29] Some steam engines employing White's mechanism were built,
but without conspicuous commercial success. White himself rather agreed
that while his invention was "allowed to possess curious properties, and
to be a _pretty_ thing, opinions do not all concur in declaring it,
essentially and generally, a _good_ thing."[30]

[Footnote 29: H. W. Dickinson, "James White and His 'New Century of
Inventions,'" _Transactions of the Newcomen Society_, 1949-1951, vol.
27, pp. 175-179.]

[Footnote 30: James White, _A New Century of Inventions_, Manchester,
1822, pp. 30-31, 338. A hypocycloidal engine used in Stourbridge,
England, is in the Henry Ford Museum.]

[Illustration: Figure 13.--James White's hypocycloidal straight-line
mechanism, about 1800. The fly-weights (at the ends of the diagonal arm)
functioned as a flywheel. From James White, _A New Century of
Inventions_ (Manchester, 1822, pl. 7).]

The first of the non-Watt four-bar linkages appeared shortly after 1800.
The origin of the grasshopper beam motion is somewhat obscure, although
it came to be associated with the name of Oliver Evans, the American
pioneer in the employment of high-pressure steam. A similar idea,
employing an isosceles linkage, was patented in 1803 by William
Freemantle, an English watchmaker (fig. 14).[31] This is the linkage
that was attributed much later to John Scott Russell (1808-1882), the
prominent naval architect.[32] An inconclusive hint that Evans had
devised his straight-line linkage by 1805 appeared in a plate
illustrating his _Abortion of the Young Steam Engineer's Guide_
(Philadelphia, 1805), and it was certainly used on his Columbian engine
(fig. 15), which was built before 1813. The Freemantle linkage, in
modified form, appeared in Rees's _Cyclopaedia_ of 1819 (fig. 16), but
it is doubtful whether even this would have been readily recognized as
identical with the Evans linkage, because the connecting rod was at the
opposite end of the working beam from the piston rod, in accordance with
established usage, while in the Evans linkage the crank and connecting
rod were at the same end of the beam. It is possible that Evans got his
idea from an earlier English periodical, but concrete evidence is
lacking.

[Footnote 31: British Patent 2741, November 17, 1803.]

[Footnote 32: William J. M. Rankine, _Manual of Machinery and Millwork_,
ed. 6, London, 1887, p. 275.]

[Illustration: Figure 14.--Freemantle straight-line linkage, later
called the Scott Russell linkage. From British Patent 2741, November 17,
1803.]

[Illustration: Figure 15.--Oliver Evans' "Columbian" engine, 1813,
showing the Evans, or "grasshopper," straight-line linkage. From
_Emporium of Arts and Sciences_ (new ser., vol. 2, no. 3, April 1814,
pl. opposite p. 380).]

[Illustration: Figure 16.--Modified Freemantle linkage, 1819, which is
kinematically the same as the Evans linkage. Pivots _D_ and _E_ are
attached to engine frame. From Abraham Rees, _The Cyclopaedia_ (London,
1819, "Parallel Motions," pl. 3).]

If the idea did in fact originate with Evans, it is strange that he did
not mention it in his patent claims, or in the descriptions that he
published of his engines.[33] The practical advantage of the Evans
linkage, utilizing as it could a much lighter working beam than the Watt
or Freemantle engines, would not escape Oliver Evans, and he was not a
man of excessive modesty where his own inventions were concerned.

[Footnote 33: Greville and Dorothy Bathe, _Oliver Evans_, Philadelphia,
1935, pp. 88, 196, and _passim_.]

Another four-bar straight-line linkage that became well known was
attributed to Richard Roberts of Manchester (1789-1864), who around 1820
had built one of the first metal planing machines, which machines helped
make the quest for straight-line linkages largely academic. I have not
discovered what occasioned the introduction of the Roberts linkage, but
it dated from before 1841. Although Roberts patented many complex
textile machines, an inspection of all of his patent drawings has failed
to provide proof that he was the inventor of the Roberts linkage.[34]
The fact that the same linkage is shown in an engraving of 1769 (fig.
18) further confuses the issue.[35]

[Footnote 34: Robert Willis (_op. cit._ [Footnote 21] p. 411) credited
Richard Roberts with the linkage. Roberts' 15 British patent drawings
exhibit complex applications of cams, levers, guided rods, cords, and so
forth, but no straight-line mechanism. In his patent no. 6258 of April
13, 1832, for a steam engine and locomotive carriage, Roberts used
Watt's "parallel motion" on a beam driven by a vertical cylinder.]

[Footnote 35: This engraving appeared as plate 11 in Pierre Patte's 1769
work (_op. cit._ footnote 24). Patte stated that the machine depicted in
his plate 11 was invented by M. de Voglie and was actually used in
1756.]

[Illustration: Figure 17.--Straight-line linkage (before 1841)
attributed to Richard Roberts by Robert Willis. From A. B. Kempe, _How
to Draw a Straight Line_ (London, 1877, p. 10).]

[Illustration: Figure 18.--Machine for sawing off pilings under water,
about 1760, designed by De Voglie. The Roberts linkage operates the bar
(_Q_ in detailed sketch) at the rear of the machine below the operators.
The significance of the linkage apparently was not generally recognized.
A similar machine depicted in Diderot's _Encyclopédie_, published
several years later, did not employ the straight-line linkage. From
Pierre Patte, _Memoirs sur les objets plus importants de l'architecture_
(Paris, 1769, pl. 11).]

The appearance in 1864 of Peaucellier's exact straight-line linkage went
nearly unnoticed. A decade later, when news of its invention crossed
the Channel to England, this linkage excited a flurry of interest, and
variations of it occupied mathematical minds for several years. For at
least 10 years before and 20 years after the final solution of the
problem, Professor Chebyshev,[36] a noted mathematician of the
University of St. Petersburg, was interested in the matter. Judging by
his published works and his reputation abroad, Chebyshev's interest
amounted to an obsession.

[Footnote 36: This is the Library of Congress spelling]

Pafnutï[)i] L'vovich Chebyshev was born in 1821, near Moscow, and
entered the University of Moscow in 1837. In 1853, after visiting France
and England and observing carefully the progress of applied mechanics in
those countries, he read his first paper on approximate straight-line
linkages, and over the next 30 years he attacked the problem with new
vigor at least a dozen times. He found that the two principal
straight-line linkages then in use were Watt's and Evans'. Chebyshev
noted the departure of these linkages from a straight line and
calculated the deviation as of the fifth degree, or about 0.0008 inch
per inch of beam length. He proposed a modification of the Watt linkage
to refine its accuracy but found that he would have to more than double
the length of the working beam. Chebyshev concluded ruefully that his
modification would "present great practical difficulties."[37]

[Footnote 37: _Oeuvres de P. L. Tchebychef_, 2 vols., St. Petersburg,
1899-1907, vol. 1, p. 538; vol. 2, pp. 57, 85.]

At length an idea occurred to Chebyshev that would enable him to
approach if not quite attain a true straight line. If one mechanism was
good, he reasoned, two would be better, _et cetera, ad infinitum_. The
idea was simply to combine, or compound, four-link approximate linkages,
arranging them in such a way that the errors would be successively
reduced. Contemplating first a combination of the Watt and Evans
linkages (fig. 19), Chebyshev recognized that if point D of the Watt
linkage followed nearly a straight line, point A of the Evans linkage
would depart even less from a straight line. He calculated the deviation
in this case as of the 11th degree. He then replaced Watt's linkage by
one that is usually called the Chebyshev straight-line mechanism (fig.
20), with the result that precision was increased to the 13th
degree.[38] The steam engine that he displayed at the Vienna Exhibition
in 1873 employed this linkage--the Chebyshev mechanism compounded with
the Evans, or approximate isosceles, linkage. An English visitor to the
exhibition commented that "the motion is of little or no practical use,
for we can scarcely imagine circumstances under which it would be more
advantageous to use such a complicated system of levers, with so many
joints to be lubricated and so many pins to wear, than a solid guide of
some kind; but at the same time the arrangement is very ingenious and in
this respect reflects great credit on its designer."[39]

[Footnote 38: _Ibid._, vol. 2, pp. 93, 94.]

[Footnote 39: _Engineering_, October 3, 1873, vol. 16, p. 284.]

[Illustration: Figure 19.--Pafnutï[)i] L'vovich Chebyshev (1821-1894),
Russian mathematician active in analysis and synthesis of straight-line
mechanisms. From _Ouvres de P. L. Tchebychef_ (St. Petersburg, 1907,
vol. 2, frontispiece).]

[Illustration: Figure 20.--Chebyshev's combination (about 1867) of
Watt's and Evans' linkages to reduce errors inherent in each. Points
_C_, _C'_, and _C"_ are fixed; _A_ is the tracing point. From _Oeuvres
de P. L. Tchebychef_ (St. Petersburg, 1907, vol. 2, p. 93).]

[Illustration: Figure 21.--_Top_: Chebyshev straight-line linkage, 1867;
from A. B. Kempe, _How to Draw a Straight Line_ (London, 1877, p. 11).
_Bottom_: Chebyshev-Evans combination, 1867; from _Oeuvres de P. L.
Tchebychef_ (St. Petersburg, 1907, vol. 2, p. 94). Points _C_, _C'_, and
_C"_ are fixed. _A_ is the tracing point.]

There is a persistent rumor that Professor Chebyshev sought to
demonstrate the impossibility of constructing any linkage, regardless of
the number of links, that would generate a straight line; but I have
found only a dubious statement in the _Grande Encyclopédie_[40] of the
late 19th century and a report of a conversation with the Russian by an
Englishman, James Sylvester, to the effect that Chebyshev had "succeeded
in proving the nonexistence of a five-bar link-work capable of producing
a perfect parallel motion...."[41] Regardless of what tradition may have
to say about what Chebyshev said, it is of course well known that
Captain Peaucellier was the man who finally synthesized the exact
straight-line mechanism that bears his name.

[Footnote 40: _La Grande Encyclopédie_, Paris, 1886 ("Peaucellier").]

[Footnote 41: James Sylvester, "Recent Discoveries in Mechanical
Conversion of Motion," _Notices of the Proceedings of the Royal
Institution of Great Britain_, 1873-1875, vol. 7, p. 181. The fixed link
was not counted by Sylvester; in modern parlance this would be a
six-link mechanism.]

[Illustration: Figure 22.--Peaucellier exact straight-line linkage,
1873. From A. B. Kempe, _How to Draw a Straight Line_ (London, 1877, p.
12).]

[Illustration: Figure 23.--Model of the Peaucellier "Compas Composé,"
deposited in Conservatoire National des Arts et Métiers, Paris, 1875.
Photo courtesy of the Conservatoire.] [Illustration: Figure 24.--James
Joseph Sylvester (1814-1897), mathematician and lecturer on
straight-line linkages. From _Proceedings of the Royal Society of
London_ (1898, vol. 63, opposite p. 161).]

Charles-Nicolas Peaucellier, a graduate of the Ecole Polytechnique and a
captain in the French corps of engineers, was 32 years old in 1864 when
he wrote a short letter to the editor of _Nouvelles Annales de
mathématiques_ (ser. 2, vol. 3, pp. 414-415) in Paris. He called
attention to what he termed "compound compasses," a class of linkages
that included Watt's parallel motion, the pantograph, and the polar
planimeter. He proposed to design linkages to describe a straight line,
a circle of any radius no matter how large, and conic sections, and he
indicated in his letter that he had arrived at a solution.

This letter stirred no pens in reply, and during the next 10 years the
problem merely led to the filling of a few academic pages by Peaucellier
and Amédée Mannheim (1831-1906), also a graduate of Ecole Polytechnique,
a professor of mathematics, and the designer of the Mannheim slide rule.
Finally, in 1873, Captain Peaucellier gave his solution to the readers
of the _Nouvelles Annales_. His reasoning, which has a distinct flavor
of discovery by hindsight, was that since a linkage generates a curve
that can be expressed algebraically, it must follow that any algebraic
curve can be generated by a suitable linkage--it was only necessary to
find the suitable linkage. He then gave a neat geometric proof,
suggested by Mannheim, for his straight-line "compound compass."[42]

[Footnote 42: Charles-Nicholas Peaucellier, "Note sur une question de
geométrie de compas," _Nouvelles Annales de mathématiques_, 1873, ser.
2, vol. 12, pp. 71-78. A sketch of Mannheim's work is in Florian Cajori,
_A History of the Logarithmic Slide Rule_, New York, about 1910,
reprinted in _String Figures and Other Monographs_, New York, Chelsea
Publishing Company, 1960.]

On a Friday evening in January 1874 Albemarle Street in London was
filled with carriages, each maneuvering to unload its charge of
gentlemen and their ladies at the door of the venerable hall of the
Royal Institution. Amidst a "mighty rustling of silks," the elegant
crowd made its way to the auditorium for one of the famous weekly
lectures. The speaker on this occasion was James Joseph Sylvester, a
small intense man with an enormous head, sometime professor of
mathematics at the University of Virginia, in America, and more recently
at the Royal Military Academy in Woolwich. He spoke from the same
rostrum that had been occupied by Davy, Faraday, Tyndall, Maxwell, and
many other notable scientists. Professor Sylvester's subject was "Recent
Discoveries in Mechanical Conversion of Motion."[43]

[Footnote 43: Sylvester, _op. cit._ (footnote 41), pp. 179-198. It
appears from a comment in this lecture that Sylvester was responsible
for the word "linkage." According to Sylvester, a linkage consists of an
even number of links, a "link-work" of an odd number. Since the fixed
member was not considered as a link by Sylvester, this distinction
became utterly confusing when Reuleaux's work was published in 1876.
Although "link" was used by Watt in a patent specification, it is not
probable that he ever used the term "link-work"--at any rate, my search
for his use of it has been fruitless. "Link work" is used by Willis
(_op. cit._ footnote 21), but the term most likely did not originate
with him. I have not found the word "linkage" used earlier than
Sylvester.]

Remarking upon the popular appeal of most of the lectures, a
contemporary observer noted that while many listeners might prefer to
hear Professor Tyndall expound on the acoustic opacity of the
atmosphere, "those of a higher and drier turn of mind experience
ineffable delight when Professor Sylvester holds forth on the conversion
of circular into parallel motion."[44]

[Footnote 44: Bernard H. Becker, _Scientific London_, London, 1874, pp.
45, 50, 51.]

Sylvester's aim was to bring the Peaucellier linkage to the notice of
the English-speaking world, as it had been brought to his attention by
Chebyshev--during a recent visit of the Russian to England--and to give
his listeners some insight into the vastness of the field that he saw
opened by the discovery of the French soldier.[45]

[Footnote 45: Sylvester, _op. cit._ (footnote 41), p. 183; _Nature_,
November 13, 1873, vol. 9, p. 33.]

"The perfect parallel motion of Peaucellier looks so simple," he
observed, "and moves so easily that people who see it at work almost
universally express astonishment that it waited so long to be
discovered." But that was not his reaction at all. The more one reflects
upon the problem, Sylvester continued, he "wonders the more that it was
ever found out, and can see no reason why it should have been
discovered for a hundred years to come. Viewed _a priori_ there was
nothing to lead up to it. It bears not the remotest analogy (except in
the fact of a double centring) to Watt's parallel motion or any of its
progeny."[46]

[Footnote 46: Sylvester, _op. cit._ (footnote 41), p. 181.]

It must be pointed out, parenthetically at least, that James Watt had
not only had to solve the problem as best he could, but that he had no
inkling, so far as experience was concerned, that a solvable problem
existed.

Sylvester interrupted his panegyric long enough to enumerate some of the
practical results of the Peaucellier linkage. He said that Mr. Penrose,
the eminent architect and surveyor to St. Paul's Cathedral, had "put up
a house-pump worked by a negative Peaucellier cell, to the great
wonderment of the plumber employed, who could hardly believe his senses
when he saw the sling attached to the piston-rod moving in a true
vertical line, instead of wobbling as usual from side to side."
Sylvester could see no reason "why the perfect parallel motion should
not be employed with equal advantage in the construction of ordinary
water-closets." The linkage was to be employed by "a gentleman of
fortune" in a marine engine for his yacht, and there was talk of using
it to guide a piston rod "in certain machinery connected with some new
apparatus for the ventilation and filtration of the air of the Houses of
Parliament." In due course, Mr. Prim, "engineer to the Houses," was
pleased to show his adaptation of the Peaucellier linkage to his new
blowing engines, which proved to be exceptionally quiet in their
operation (fig. 25).[47] A bit on the ludicrous side, also, was
Sylvester's 78-bar linkage that traced a straight line along the line
connecting the two fixed centers of the linkage.[48]

[Footnote 47: _Ibid._, pp. 182, 183, 188, 193.]

[Footnote 48: Kempe, _op. cit._ (footnote 21), p. 17.]

[Illustration: Figure 25.--Mr. Prim's blowing engine used for
ventilating the House of Commons, 1877. The crosshead of the
reciprocating air pump is guided by a Peaucellier linkage shown at the
center. The slate-lined air cylinders had rubber-flap inlet and exhaust
valves and a piston whose periphery was formed by two rows of brush
bristles. Prim's machine was driven by a steam engine. Photograph by
Science Museum, London.]

Before dismissing with a smile the quaint ideas of our Victorian
forbears, however, it is well to ask, 88 years later, whether some
rather elaborate work reported recently on the synthesis of
straight-line mechanisms is more to the point, when the principal
objective appears to be the moving of an indicator on a "pleasing,
expanded" (i.e., squashed flat) radio dial.[49]

[Footnote 49: _Machine Design_, December 1954, vol. 26, p. 210.]

But Professor Sylvester was more interested, really, in the mathematical
possibilities of the Peaucellier linkage, as no doubt our modern
investigators are. Through a compounding of Peaucellier mechanisms, he
had already devised square-root and cube-root extractors, an angle
trisector, and a quadratic-binomial root extractor, and he could see no
limits to the computing abilities of linkages as yet undiscovered.[50]

[Footnote 50: Sylvester, _op. cit._ (footnote 41), p. 191.]

Sylvester recalled fondly, in a footnote to his lecture, his experience
with a little mechanical model of the Peaucellier linkage at an earlier
dinner meeting of the Philosophical Club of the Royal Society. The
Peaucellier model had been greeted by the members with lively
expressions of admiration "when it was brought in with the dessert, to
be seen by them after dinner, as is the laudable custom among members of
that eminent body in making known to each other the latest scientific
novelties." And Sylvester would never forget the reaction of his
brilliant friend Sir William Thomson (later Lord Kelvin) upon being
handed the same model in the Athenaeum Club. After Sir William had
operated it for a time, Sylvester reached for the model, but he was
rebuffed by the exclamation "No! I have not had nearly enough of it--it
is the most beautiful thing I have ever seen in my life."[51]

[Footnote 51: _Ibid._, p. 183.]

The aftermath of Professor Sylvester's performance at the Royal
Institution was considerable excitement amongst a limited company of
interested mathematicians. Many alternatives to the Peaucellier
straight-line linkage were suggested by several writers of papers for
learned journals.[52]

[Footnote 52: For a summary of developments and references, see Kempe,
_op. cit._ (footnote 21), pp. 49-51. Two of Hart's six-link exact
straight-line linkages referred to by Kempe are illustrated in Henry M.
Cundy and A. P. Rollett, _Mathematical Models_, Oxford, Oxford
University Press, 1952, pp. 204-205. Peaucellier's linkage was of eight
links.]

In the summer of 1876, after Sylvester had departed from England to take
up his post as professor of mathematics in the new Johns Hopkins
University in Baltimore, Alfred Bray Kempe, a young barrister who
pursued mathematics as a hobby, delivered at London's South Kensington
Museum a lecture with the provocative title "How to Draw a Straight
Line."[53]

[Footnote 53: Kempe, _op. cit._ (footnote 21), p. 26.]

In order to justify the Peaucellier linkage, Kempe belabored the point
that a perfect circle could be generated by means of a pivoted bar and a
pencil, while the generation of a straight line was most difficult if
not impossible until Captain Peaucellier came along. A straight line
could be drawn along a straight edge; but how was one to determine
whether the straight edge was straight? He did not weaken his argument
by suggesting the obvious possibility of using a piece of string. Kempe
had collaborated with Sylvester in pursuing the latter's first thoughts
on the subject, and one result, that to my mind exemplifies the general
direction of their thinking, was the Sylvester-Kempe "parallel motion"
(fig. 26).

[Illustration: Figure 26.--Sylvester-Kempe translating linkage, 1877.
The upper and lower plates remain parallel and equidistant. From A. B.
Kempe, _How to Draw a Straight Line_ (London, 1877, p. 37).]

[Illustration: Figure 27.--Gaspard Monge (1746-1818), professor of
mathematics at the Ecole Polytechnique from 1794 and founder of the
academic discipline of machine kinematics, From _Livre du Centenaire,
1794-1894, Ecole Polytechnique_ (Paris, 1895, vol. 1, frontispiece).]

Enthusiastic as Kempe was, however, he injected an apologetic note in
his lecture. "That these results are valuable cannot I think be
doubted," he said, "though it may well be that their great beauty has
led some to attribute to them an importance which they do not really
possess...." He went on to say that 50 years earlier, before the great
improvements in the production of true plane surfaces, the straight-line
mechanisms would have been more important than in 1876, but he added
that "linkages have not at present, I think, been sufficiently put
before the mechanician to enable us to say what value should really be
set upon them."[54]

[Footnote 54: _Ibid._, pp. 6-7. I have not pursued the matter of cognate
linkages (the Watt and Evans linkages are cognates) because the
Roberts-Chebyshev theorem escaped my earlier search, as it had
apparently escaped most others until 1958. See R. S. Hartenberg and J.
Denavit, "The Fecund Four-Bar," _Transactions of the Fifth Conference on
Mechanisms_, Cleveland, Penton Publishing Company, 1958, pp. 194-206,
reprinted in _Machine Design_, April 16, 1959, vol. 31, pp. 149-152. See
also A. E. R. de Jonge, "The Correlation of Hinged Four-Bar
Straight-Line Motion Devices by Means of the Roberts Theorem and a New
Proof of the Latter," _Annals of the New York Academy of Sciences_,
March 18, 1960, vol. 84, art. 3, pp. 75-145 (published separately).]

It was during this same summer of 1876, at the Loan Exhibition of
Scientific Apparatus in the South Kensington Museum, that the work of
Franz Reuleaux, which was to have an important and lasting influence on
kinematics everywhere, was first introduced to English engineers. Some
300 beautifully constructed teaching aids, known as the Berlin kinematic
models, were loaned to the exhibition by the Royal Industrial School in
Berlin, of which Reuleaux was the director. These models were used by
Prof. Alexander B. W. Kennedy of University College, London, to help
explain Reuleaux's new and revolutionary theory of machines.[55]

[Footnote 55: Alexander B. W. Kennedy, "The Berlin Kinematic Models,"
_Engineering_, September 15, 1876, vol. 22, pp. 239-240.]


Scholars and Machines

When, in 1829, André-Marie Ampère (1775-1836) was called upon to prepare
a course in theoretical and experimental physics for the Collège de
France, he first set about determining the limits of the field of
physics. This exercise suggested to his wide-ranging intellect not only
the definition of physics but the classification of all human knowledge.
He prepared his scheme of classification, tried it out on his physics
students, found it incomplete, returned to his study, and produced
finally a two-volume work wherein the province of kinematics was first
marked out for all to see and consider.[56] Only a few lines could be
devoted to so specialized a branch as kinematics, but Ampère managed to
capture the central idea of the subject.

[Footnote 56: André-Marie Ampère, _Essai sur la philosophie des
sciences, une exposition analytique d'une classification naturelle de
toutes les connaissances humaines_, 2 vols., Paris, 1838 (for origin of
the project, see vol. 1, pp. v, xv).]

Cinématique (from the Greek word for movement) was, according to Ampère,
the science "in which movements are considered in themselves
[independent of the forces which produce them], as we observe them in
solid bodies all about us, and especially in the assemblages called
machines."[57] Kinematics, as the study soon came to be known in
English,[58] was one of the two branches of elementary mechanics, the
other being statics.

[Footnote 57: _Ibid._, vol. 1, pp. 51-52.]

[Footnote 58: Willis (_op. cit._ footnote 21) adopted the word
"kinematics," and this Anglicization subsequently became the standard
term for this branch of mechanics.]

In his definition of kinematics, Ampère stated what the faculty of
mathematics at the Ecole Polytechnique, in Paris, had been groping
toward since the school's opening some 40 years earlier. The study of
mechanisms as an intellectual discipline most certainly had its origin
on the left bank of the Seine, in this school spawned, as suggested by
one French historian,[59] by the great _Encyclopédie_ of Diderot and
d'Alembert.

[Footnote 59: G. Pinet, _Histoire de l'Ecole Polytechnique_, Paris,
1887, pp. viii-ix. In their forthcoming book on kinematic synthesis, R.
S. Hartenberg and J. Denavit will trace the germinal ideas of Jacob
Leupold and Leonhard Euler of the 18th century.]

Because the Ecole Polytechnique had such a far-reaching influence upon
the point of view from which mechanisms were contemplated by scholars
for nearly a century after the time of Watt, and by compilers of
dictionaries of mechanical movements for an even longer time, it is
well to look for a moment at the early work that was done there. If one
is interested in origins, it might be profitable for him to investigate
the military school in the ancient town of Mézières, about 150 miles
northeast of Paris. It was here that Lazare Carnot, one of the principal
founders of the Ecole Polytechnique, in 1783 published his essay on
machines,[60] which was concerned, among other things, with showing the
impossibility of "perpetual motion"; and it was from Mézières that
Gaspard Monge and Jean Hachette[61] came to Paris to work out the system
of mechanism classification that has come to be associated with the
names of Lanz and Bétancourt.

[Footnote 60: Lazare N. M. Carnot, _Essai sur les machines en général_,
Mézières, 1783 (later published as _Principes fondamentaux de
l'equilibre et du mouvement_, Paris, 1803).]

[Footnote 61: Biographical notices of Monge and Hachette appear in
_Encyclopaedia Britannica_, ed. 11. See also _L'Ecole Polytechnique,
Livre du Centenaire_, Paris, 1895, vol. 1, p. 11ff.]

Gaspard Monge (1746-1818), who while a draftsman at Mézières originated
the methods of descriptive geometry, came to the Ecole Polytechnique as
professor of mathematics upon its founding in 1794, the second year of
the French Republic. According to Jean Nicolas Pierre Hachette
(1769-1834), who was junior to Monge in the department of descriptive
geometry, Monge planned to give a two-months' course devoted to the
elements of machines. Having barely gotten his department under way,
however, Monge became involved in Napoleon's ambitious scientific
mission to Egypt and, taking leave of his family and his students,
embarked for the distant shores.

"Being left in charge," wrote Hachette, "I prepared the course of which
Monge had given only the first idea, and I pursued the study of machines
in order to analyze and classify them, and to relate geometrical and
mechanical principles to their construction." Changes of curriculum
delayed introduction of the course until 1806, and not until 1811 was
his textbook ready, but the outline of his ideas was presented to his
classes in chart form (fig. 28). This chart was the first of the widely
popular synoptical tables of mechanical movements.[62]

[Footnote 62: Jean N. P. Hachette, _Traité élémentaire des machines_,
Paris, 1811, p. v.]

[Illustration: Figure 28.--Hachette's synoptic chart of elementary
mechanisms, 1808. This was the first of many charts of mechanical
movements that enjoyed wide popularity for over 100 years.

From Jean N. P. Hachette, _Traité Élémentaire des Machines_ (Paris,
1811, pl. 1).]

Hachette classified all mechanisms by considering the conversion of one
motion into another. His elementary motions were continuous circular,
alternating circular, continuous rectilinear, and alternating
rectilinear. Combining one motion with another--for example, a treadle
and crank converted alternating circular to continuous circular
motion--he devised a system that supplied a frame of reference for the
study of mechanisms. In the U.S. Military Academy at West Point,
Hachette's treatise, in the original French, was used as a textbook in
1824, and perhaps earlier.[63]

[Footnote 63: This work was among the books sent back by Sylvanus Thayer
when he visited France in 1816 to observe the education of the French
army cadets. Thayer's visit resulted in his adopting the philosophy of
the Ecole Polytechnique in his reorganization of the U.S. Military
Academy and, incidentally, in his inclusion of Hachette's course in the
Academy's curriculum (U.S. Congress, _American State Papers_,
Washington, 1832-1861, Class v, Military Affairs, vol. 2, p. 661: Sidney
Forman, _West Point_, New York, 1950, pp. 36-60). There is a collection
of miscellaneous papers (indexed under Sylvanus Thayer and William
McRee, U.S. National Archives, RG 77, Office, Chief of Engineers, Boxes
1 and 6) pertaining to the U.S. Military Academy of this period, but I
found no mention of kinematics in this collection.]

Lanz and Bétancourt, scholars from Spain at the Ecole Polytechnique,
plugged some of the gaps in Hachette's system by adding continuous and
alternating curvilinear motion, which doubled the number of combinations
to be treated, but the advance of their work over that of Hachette was
one of degree rather than of kind.[64]

[Footnote 64: Phillipe Louis Lanz and Augustin de Bétancourt, _Essai sur
la composition des machines_, Paris, 1808. Hachette's chart and an
outline of his elementary course on machines is bound with the Princeton
University Library copy of the Lanz and Bétancourt work. This copy
probably represents the first textbook of kinematics. Bétancourt was
born in 1760 in Teneriffe, attended the military school in Madrid, and
became inspector-general of Spanish roads and canals. He was in England
before 1789, learning how to build Watt engines, and he introduced the
engines to Paris in 1790 (see Farey, _op. cit._, p. 655). He entered
Russian service in 1808 and died in St. Petersburg in 1826 J. C.
Poggendorff, _Biographisches-literarisches Handwörterbuch für Mathematik
..._, Leipzig, 1863, vol. 1.]

[Illustration: Figure 29.--Robert Willis (1800-1875), Jacksonian
Professor, Cambridge University, and author of _Principles of
Mechanism_, one of the landmark books in the development of kinematics
of mechanisms. Photo courtesy Gonville and Caius College, Cambridge
University.]

Giuseppe Antonio Borgnis, an Italian "engineer and member of many
academies" and professor of mechanics at the University of Pavia in
Italy, in his monumental, nine-volume _Traité complet de méchanique
appliquée aux arts_, caused a bifurcation of the structure built upon
Hachette's foundation of classification when he introduced six orders of
machine elements and subdivided these into classes and species. His six
orders were _récepteurs_ (receivers of motion from the prime mover),
_communicateurs_, _modificateurs_ (modifiers of velocity), _supports_
(e.g., bearings), _regulateurs_ (e.g., governors), and _operateurs_,
which produced the final effect.[65]

[Footnote 65: Giuseppe Antonio Borgnis, _Théorie de la mécanique
usuelle_ in _Traité complet de mécanique appliquée aux arts_, Paris,
1818, vol. 1, pp. xiv-xvi.]

The brilliant Gaspard-Gustave de Coriolis (1792-1843)--remembered mainly
for a paper of a dozen pages explaining the nature of the acceleration
that bears his name[66]--was another graduate of the Ecole Polytechnique
who wrote on the subject of machines. His book,[67] published in 1829,
was provoked by his recognition that the designer of machines needed
more knowledge than his undergraduate work at the Ecole Polytechnique
was likely to give him. Although he embraced a part of Borgnis'
approach, adopting _récepteurs_, _communicateurs_, and _operateurs_,
Coriolis indicated by the title of his book that he was more concerned
with forces than with relative displacements. However, the attractively
simple three-element scheme of Coriolis became well fixed in French
thinking.[68]

[Footnote 66: Gaspard-Gustave de Coriolis, "Memoire sur les equations du
mouvement relatif des systèmes de corps," _Journal de l'Ecole
Polytechnique_, 1835, vol. 15, pp. 142-154.]

[Footnote 67: Gaspard-Gustave de Coriolis, _De Calcul de l'effet des
machines_, Paris, 1829. In this book Coriolis proposed the now generally
accepted equation, work = force × distance (pp. iii, 2).]

[Footnote 68: The renowned Jean Victor Poncelet lent weight to this
scheme. (See Franz Reuleaux, _Theoretische Kinematik: Grundzüge einer
Theorie des Maschinenwesens_, Braunschweig, 1875, translated by
Alexander B. W. Kennedy as _The Kinematics of Machinery: Outlines of a
Theory of Machines_, London, 1876, pp. 11, 487. I have used the Kennedy
translation in the Reuleaux references throughout the present work.)]

Michel Chasles (1793-1880), another graduate of the Ecole Polytechnique,
contributed some incisive ideas in his papers on instant centers[69]
published during the 1830's, but their tremendous importance in
kinematic analysis was not recognized until much later.

[Footnote 69: The instant center was probably first recognized by Jean
Bernoulli (1667-1748) in his "De Centro Spontaneo Rotationis" (_Johannis
Bernoulli ... Opera Omnia ..._, Lausanne, 1742, vol. 4, p. 265ff.).]

[Illustration: Figure 30.--Franz Reuleaux (1829-1905). His _Theoretische
Kinematik_, published in 1875, provided the basis for modern kinematic
analysis. Photo courtesy Deutsches Museum, Munich.]

Acting upon Ampère's clear exposition of the province of kinematics and
excluding, as Ampère had done, the consideration of forces, an
Englishman, Robert Willis, made the next giant stride forward in the
analysis of mechanisms. Willis was 37 years old in 1837 when he was
appointed professor of natural and experimental philosophy at Cambridge.
In the same year Professor Willis--a man of prodigious energy and
industry and an authority on archeology and architectural history as
well as mechanisms--read his important paper "On the Teeth of Wheels"
before the Institution of Civil Engineers[70] and commenced at Cambridge
his lectures on kinematics of mechanisms that culminated in his 1841
book _Principles of Mechanism_.[71]

[Footnote 70: Robert Willis, "On the Teeth of Wheels," _Transactions of
the Institution of Civil Engineers of London_, 1838, vol. 2, pp.
89-112.]

[Footnote 71: Willis, _op. cit._ (footnote 21). Through the kindness of
its owner (Mr. Warren G. Ogden of North Andover, Massachusetts), I have
had access to Willis' own copy of his 1841 edition of _Principles of
Mechanism_. The book is interleaved, and it contains notes made by
Willis from time to time until at least 1870, when the second edition
was issued. Corrections, emendations, notations of some of his sources
(for example, the De Voglie linkage mentioned in footnote 35 above),
notes to himself to "examine the general case" and "examine the modern
forms" of straight-line devices are interspersed with references to
authors that had borrowed from his work without acknowledgment. Of one
author Willis writes an indignant "He ignores my work."]

It seemed clear to Willis that the problem of devising a mechanism for a
given purpose ought to be attacked systematically, perhaps
mathematically, in order to determine "all the forms and arrangements
that are applicable to the desired purpose," from which the designer
might select the simplest or most suitable combination. "At present," he
wrote, "questions of this kind can only be solved by that species of
intuition which long familiarity with a subject usually confers upon
experienced persons, but which they are totally unable to communicate to
others."

In analyzing the process by which a machine was designed, Willis
observed: "When the mind of a mechanician is occupied with the
contrivance of a machine, he must wait until, in the midst of his
meditations, some happy combination presents itself to his mind which
may answer his purpose." He ventured the opinion that at this stage of
the design process "the motions of the machine are the principal subject
of contemplation, rather than the forces applied to it, or the work it
has to do." Therefore he was prepared to adopt without reservation
Ampère's view of kinematics, and, if possible, to make the science
useful to engineers by stating principles that could be applied without
having to fit the problem at hand into the framework of the systems of
classification and description that had gone before. He appraised the
"celebrated system" of Lanz and Bétancourt as "a merely popular
arrangement, notwithstanding the apparently scientific simplicity of the
scheme." He rejected this scheme because "no attempt is made to subject
the motions to calculation, or to reduce these laws to general formulas,
for which indeed the system is totally unfitted."

Borgnis had done a better job, Willis thought, in actually describing
machinery, with his "orders" based upon the functions of machine
elements or mechanisms within the machine, but again there was no means
suggested by which the kinematics of mechanisms could be systematically
investigated.

Although Willis commenced his treatise with yet another "synoptical
table of the elementary combinations of pure mechanism," his view
shifted quickly from description to analysis. He was consistent in his
pursuit of analytical methods for "pure mechanism," eschewing any
excursions into the realm of forces and absolute velocities. He grasped
the important concept of relative displacements of machine elements, and
based his treatment upon "the proportions and relations between the
velocities and directions of the pieces, and not upon their actual and
separate motions."[72]

[Footnote 72: _Ibid._, pp. iv, x-xii, xxi, 15.]

That he did not succeed in developing the "formulas" that would enable
the student to determine "all the forms and arrangements that are
applicable to the desired purpose"--that he did not present a rational
approach to synthesis--is not to be wondered at. Well over a century
later we still are nibbling at the fringes of the problem. Willis did,
nonetheless, give the thoughtful reader a glimpse of the most powerful
tool for kinematic synthesis that has yet been devised; namely,
kinematic analysis, in which the argument is confined to the relative
displacements of points on links of a mechanism, and through which the
designer may grasp the nature of the means at his disposal for the
solution of any particular problem.

As remarked by Reuleaux a generation later, there was much in Professor
Willis's book that was wrong, but it was an original, thoughtful work
that departed in spirit if not always in method from its predecessors.
_Principles of Mechanism_ was a prominent landmark along the road to a
rational discipline of machine-kinematics.

A phenomenal engineer of the 19th century was the Scottish professor of
civil engineering at the University of Glasgow, William John MacQuorn
Rankine. Although he was at the University for only 17 years--he died at
the age of 52, in 1872--he turned out during that time four thick
manuals on such diverse subjects as civil engineering, ship-building,
thermodynamics, and machinery and mill-work, in addition to literally
hundreds of papers, articles, and notes for scientific journals and the
technical press. Endowed with apparently boundless energy, he found time
from his studies to command a battalion of rifle volunteers and to
compose and sing comic and patriotic songs. His manuals, often used as
textbooks, were widely circulated and went through many editions.
Rankine's work had a profound effect upon the practice of engineering by
setting out principles in a form that could be grasped by people who
were dismayed by the treatment usually found in the learned journals.

When Rankine's book titled _A Manual of Machinery and Millwork_ was
published in 1869 it was accurately characterized by a reviewer as
"dealing with the _principles_ of machinery and millworks, and as such
it is entirely distinct from [other works on the same subject] which
treat more of the practical applications of such principles than of the
principles themselves."[73]

[Footnote 73: _Engineering_, London, August 13, 1869, vol. 8, p. 111.]

Rankine borrowed what appeared useful from Willis' _Principles of
Mechanism_ and from other sources. His treatment of kinematics was not
as closely reasoned as the later treatises of Reuleaux and Kennedy,
which will be considered below. Rankine did, however, for the first time
show the utility of instant centers in velocity analysis, although he
made use only of the instant centers involving the fixed link of a
linkage. Like others before him, he considered the fixed link of a
mechanism as something quite different from the movable links, and he
did not perceive the possibilities opened up by determining the instant
center of two movable links.

Many other books dealing with mechanisms were published during the
middle third of the century, but none of them had a discernible
influence upon the advance of kinematical ideas.[74] The center of
inquiry had by the 1860's shifted from France to Germany. Only by
scattered individuals in England, Italy, and France was there any
impatience with the well-established, general understanding of the
machine-building art.

[Footnote 74: Several such books are referred to by Reuleaux, _op. cit._
(footnote 68), pp. 12-16.]

In Germany, on the other hand, there was a surge of industrial activity
that attracted some very able men to the problems of how machines ought
to be built. Among the first of these was Ferdinand Redtenbacher
(1809-1863), professor of mechanical engineering in the polytechnic
school in Karlsruhe, not far from Heidelberg. Redtenbacher, although he
despaired of the possibility of finding a "true system on which to base
the study of mechanisms," was nevertheless a factor in the development
of such a system. He had young Franz Reuleaux in his classes for two
years, from 1850. During that time the older man's commanding presence,
his ability as a lecturer, and his infectious impatience with the
existing order influenced Reuleaux to follow the scholar's trail that
led him to eminence as an authority of the first rank.[75]

[Footnote 75: See Carl Weihe, "Franz Reuleaux und die Grundlagen seiner
Kinematik," Deutsches Museum, Munich, _Abhandlung und Berichte_, 1942,
p. 2; Friedrich Klemm, _Technik: Eine Geschichte ihrer Probleme_,
Freiburg and Munich, Verlag Karl Alber, 1954, translated by Dorothea W.
Singer as _A History of Western Technology_, New York, Charles
Scribner's Sons, 1959, p. 317.]

Before he was 25 years old Franz Reuleaux published, in collaboration
with a classmate, a textbook whose translated title would be
_Constructive Lessons for the Machine Shop_.[76] His several years in
the workshop, before and after coming under Redtenbacher's influence,
gave his works a practical flavor, simple and direct. According to one
observer, Reuleaux's book exhibited "a recognition of the claims of
practice such as Englishmen do not generally associate with the writings
of a German scientific professor."[77]

[Footnote 76: See Weihe, _op. cit._ (footnote 75), p. 3; Hans Zopke,
"Professor Franz Reuleaux," _Cassier's Magazine_, December 1896, vol.
11, pp. 133-139; _Transactions of the American Society of Mechanical
Engineers_, 1904-1905, vol. 26, pp. 813-817.]

[Footnote 77: _Engineering_, London, September 8, 1876, vol. 22, p.
197.]

Reuleaux's original ideas on kinematics, which are responsible for the
way in which we look at mechanisms today, were sufficiently formed in
1864 for him to lecture upon them.[78] Starting in 1871, he published
his findings serially in the publication of the Verein zur Beförderung
des Gewerbefleisses in Preussen (Society for the Advancement of Industry
in Prussia), of which he was editor. In 1875 these articles were brought
together in the book that established his fame--_Theoretische
Kinematik...._[79]

[Footnote 78: A. E. Richard de Jonge, "What is Wrong with Kinematics and
Mechanisms?" _Mechanical Engineering_, April 1942, vol. 64, pp. 273-278
(comments on this paper are in _Mechanical Engineering_, October 1942,
vol. 64, pp. 744-751); Zopke, _op. cit._ (footnote 76), p. 135.]

[Footnote 79: Reuleaux, _op. cit._ (footnote 68). This was not the last
of Reuleaux's books. His trilogy on kinematics and machine design is
discussed by De Jonge, _op. cit._ (footnote 78).]

In the introduction of this book, Reuleaux wrote:

     In the development of every exact science, its substance having
     grown sufficiently to make generalization possible, there is a time
     when a series of changes bring it into clearness. This time has
     most certainly arrived for the science of kinematics. The number of
     mechanisms has grown almost out of measure, and the number of ways
     in which they are applied no less. It has become absolutely
     impossible still to hold the thread which can lead in any way
     through this labyrinth by the existing methods.[80]

[Footnote 80: Reuleaux, _op. cit._ (footnote 68), p. 23.]

Reuleaux's confidence that it would be his own work that would bring
order out of confusion was well founded. His book had already been
translated into Italian and was being translated into French when, only
a year after its publication, it was presented by Prof. Alexander B. W.
Kennedy in English translation.[81]

[Footnote 81: _Ibid._, p. iii.]

The book was enthusiastically reviewed by the weekly London journal
_Engineering_,[82] and it was given lengthy notice by the rival journal,
_The Engineer_. The editor of _The Engineer_ thought that the
mechanician would find in it many new ideas, that he would be "taught to
detect hitherto hidden resemblances, and that he must part--reluctantly,
perhaps--with many of his old notions." "But," added the editor with
considerable justice, "that he [the mechanician] would suddenly
recognize in Professor Reuleaux's 'kinematic notation,' 'analysis,' and
'synthesis,' the long-felt want of his professional existence we do not
for a moment believe."[83] Indeed, the fresh and sharp ideas of Reuleaux
were somewhat clouded by a long (600-page) presentation; and his
kinematic notation, which required another attempt at classification,
did not simplify the presentation of radically new ideas.[84]

[Footnote 82: _Engineering_, _loc. cit._ (footnote 77).]

[Footnote 83: _The Engineer_, London, March 30 and April 13, 1877, vol.
43, pp. 211-212, 247-248.]

[Footnote 84: It is perhaps significant that the first paper of the
First Conference on Mechanisms at Purdue University was Allen S. Hall's
"Mechanisms and Their Classification," which appeared in _Machine
Design_, December 1953, vol. 25, pp. 174-180. The place of
classification in kinematic synthesis is suggested in Ferdinand
Freudenstein's "Trends in Kinematics of Mechanisms," _Applied Mechanics
Reviews_, September 1959, vol. 12, pp. 587-590.]

[Illustration: Figure 31.--Alexander Blackie William Kennedy
(1847-1928), translator of Reuleaux' _Theoretische Kinematik_ and
discoverer of Kennedy's "Law of Three Centers." From _Minutes of the
Proceedings of the Institution of Civil Engineers_ (1907, vol. 167,
frontispiece).]

Nevertheless, no earlier author had seen the problem of kinematic
analysis so clearly or had introduced so much that was fresh, new, and
of lasting value.

Reuleaux was first to state the concept of the pair; by his concept of
the expansion of pairs he was able to show similarities in mechanisms
that had no apparent relation. He was first to recognize that the fixed
link of a mechanism was kinematically the same as the movable links.
This led him to the important notion of inversion of linkages, fixing
successively the various links and thus changing the function of the
mechanism. He devoted 40 pages to showing, with obvious delight, the
kinematic identity of one design after another of rotary steam engines,
demolishing for all time the fond hopes of ingenious but ill-informed
inventors who think that improvements and advances in mechanism design
consist in contortion and complexity.

The chapter on synthesis was likewise fresh, but it consisted of a
discussion, not a system; and Reuleaux stressed the idea that I have
mentioned above in connection with Willis' book, that synthesis will be
successful in proportion to the designer's understanding and
appreciation of analysis. Reuleaux tried to put the designer on the
right track by showing him clearly "the essential simplicity of the
means with which we have to work" and by demonstrating to him "that the
many things which have to be done can be done with but few means, and
that the principles underlying them all lie clearly before us."[85]

[Footnote 85: Reuleaux, _op. cit._ (footnote 68), p. 582.]

It remained for Sir Alexander Blackie William Kennedy (1847-1928) and
Robert Henry Smith (1852-1916) to add to Reuleaux's work the elements
that would give kinematic analysis essentially its modern shape.

Kennedy, the translator of Reuleaux's book, became professor of
engineering at the University College in London in 1874, and eventually
served as president both of the Institution of Mechanical Engineers and
of the Institution of Civil Engineers. Smith, who had taught in the
Imperial University of Japan, was professor of engineering at Mason
College, now a part of Birmingham University, in England.

While Reuleaux had used instant centers almost exclusively for the
construction of centrodes (paths of successive positions of an instant
center), Professor Kennedy recognized that instant centers might be used
in velocity analysis. His book, _Mechanics of Machinery_, was published
in 1886 ("partly through pressure of work and partly through ill-health,
this book appears only now"). In it he developed the law of three
centers, now known as Kennedy's theorem. He noted that his law of three
centers "was first given, I believe, by Aronhold, although its previous
publication was unknown to me until some years after I had given it in
my lectures."[86] In fact, the law had been published by Siegfried
Heinrich Aronhold (1819-1884) in his "Outline of Kinematic Geometry,"
which appeared in 1872 alongside Reuleaux's series in the journal that
Reuleaux edited. Apparently Reuleaux did not perceive its particular
significance at that time.[87]

[Footnote 86: Alexander B. W. Kennedy, _The Mechanics of Machinery_, ed.
3, London, 1898, pp. vii, x.]

[Footnote 87: Siegfried Heinrich Aronhold, "Outline of Kinematic
Geometry," _Verein zur Beförderung des Gewerbefleisses in Preussen_,
1872, vol. 51, pp. 129-155. Kennedy's theorem is on pp. 137-138.]

[Illustration: Figure 32.--Robert Henry Smith (1852-1916), originator of
velocity and acceleration polygons for kinematic analysis. Photo
courtesy the Librarian, Birmingham Reference Library, England.]

Kennedy, after locating instant centers, determined velocities by
calculation and accelerations by graphical differentiation of
velocities, and he noted in his preface that he had been unable, for a
variety of reasons, to make use in his book of Smith's recent work.
Professor Kennedy at least was aware of Smith's surprisingly advanced
ideas, which seem to have been generally ignored by Americans and
Englishmen alike.

Professor Smith, in a paper before the Royal Society of Edinburgh in
1885, stated clearly the ideas and methods for construction of velocity
and acceleration diagrams of linkages.[88] For the first time, velocity
and acceleration "images" of links (fig. 33) were presented. It is
unfortunate that Smith's ideas were permitted to languish for so long a
time.

[Footnote 88: Robert H. Smith, "A New Graphic Analysis of the Kinematics
of Mechanisms," _Transactions of the Royal Society of Edinburgh_,
1882-1885, vol. 32, pp. 507-517, and pl. 82. Smith used this paper as
the basis for a chapter in his _Graphics or the Art of Calculating by
Drawing Lines_, London, 1889, pp. 144-162. In a footnote of his paper,
Smith credited Fleeming Jenkin (1833-1885) with suggesting the term
"image." After discarding as "practically useless" Kennedy's graphical
differentiation, Smith complained that he had "failed to find any
practical use" for Reuleaux's "method of centroids, more properly called
axoids." Such statements were not calculated to encourage Kennedy and
Reuleaux to advertise Smith's fame; however, I found no indication that
either one took offense at the criticism. Smith's velocity and
acceleration diagrams were included (apparently embalmed, so far as
American engineers were concerned) in _Encyclopaedia Britannica_, ed.
11, 1910, vol. 17, pp. 1008-1009.]

[Illustration: Figure 33.--Smith's velocity image (the two figures at
top), and his velocity, mechanism, and acceleration diagrams, 1885. The
image of link BACD is shown as figure _bacd_. The lines _pa_, _pb_,
_pc_, and _pd_ are velocity vectors. This novel, original, and powerful
analytical method was not generally adopted in English or American
schools until nearly 50 years after its inception. From _Transactions of
the Royal Society of Edinburgh_ (1882-1885, vol. 32, pl. 82).]

By 1885 nearly all the tools for modern kinematic analysis had been
forged. Before discussing subsequent developments in analysis and
synthesis, however, it will be profitable to inquire what the
mechanician--designer and builder of machines--was doing while all of
this intellectual effort was being expended.


Mechanicians and Mechanisms

While the inductive process of recognizing and stating true principles
of the kinematics of mechanisms was proceeding through three generations
of French, English, and finally German scholars, the actual design of
mechanisms went ahead with scant regard for what the scholars were doing
and saying.

After the demonstration by Boulton and Watt that large mechanisms could
be wrought with sufficient precision to be useful, the English tool
builders Maudslay, Roberts, Clement, Nasmyth, and Whitworth developed
machine tools of increasing size and truth. The design of other
machinery kept pace with--sometimes just behind, sometimes just ahead
of--the capacity and capability of machine tools. In general, there was
an increasing sophistication of mechanisms that could only be accounted
for by an increase of information with which the individual designer
could start.

Reuleaux pointed out in 1875 that the "almost feverish progress made in
the regions of technical work" was "not a consequence of any increased
capacity for intellectual action in the race, but only the perfecting
and extending of the tools with which the intellect works." These tools,
he said, "have increased in number just like those in the modern
mechanical workshop--the men who work them remain the same." Reuleaux
went on to say that the theory and practice of machine-kinematics had
"carried on a separate existence side by side." The reason for this
failure to apply theory to practice, and vice versa, must be sought in
the defects of the theory, he thought, because "the mechanisms
themselves have been quietly developed in practical machine-design, by
invention and improvement, regardless of whether or not they were
accorded any direct and proper theoretical recognition." He pointed out
that the theories had thus far "furnished no new mechanisms."[89]

[Footnote 89: Reuleaux, _op. cit._ (footnote 68), p. 8.]

It is reasonable, therefore, to ask what was responsible for the
appearance of new mechanisms, and then to see what sort of mechanisms
had their origins in this period.

It is immediately evident to a designer that the progress in mechanisms
came about through the spread of knowledge of what had already been
done; but designers of the last century had neither the leisure nor
means to be constantly visiting other workshops, near and far, to
observe and study the latest developments. In the 1800's, as now, word
must in the main be spread by the printed page.

Hachette's chart (fig. 28) had set the pattern for display of mechanical
contrivances in practical journals and in the large number of mechanical
dictionaries that were compiled to meet an apparent demand for such
information. It is a little surprising, however, to find how persistent
were some of Hachette's ideas that could only have come from the
uppermost superficial layer of his cranium. See, for example, his
"anchored ferryboat" (fig. 34). This device, employed by Hachette to
show conversion of continuous rectilinear motion into alternating
circular motion, appeared in one publication after another throughout
the 19th century. As late as 1903 the ferryboat was still anchored in
Hiscox's _Mechanical Movements_, although the tide had changed (fig.
35).[90]

[Footnote 90: Gardner D. Hiscox, ed., _Mechanical Movements_, ed. 10,
New York, 1903, p. 151. The ferryboat did not appear in the 1917
edition.]

[Illustration: Figure 34.--Hachette's ferryboat of 1808, a "machine" for
converting continuous rectilinear motion into alternating circular
motion. From Phillipe Louis Lanz and Augustin de Bétancourt, _Essai sur
la composition des machines_ (Paris, 1808, pl. 2).]

[Illustration: Figure 35.--Ferryboat from Gardner D. Hiscox, ed.,
_Mechanical Movements_ (ed. 10, New York, 1903, p. 151).]

During the upsurge of the Lyceum--or working-man's institute--movement
in the 1820's, Jacob Bigelow, Rumford professor of applied science at
Harvard University, gave his popular lectures on the "Elements of
Technology" before capacity audiences in Boston. In preparing his
lecture on the elements of machinery, Bigelow used as his authorities
Hachette, Lanz and Bétancourt, and Olinthus Gregory's mechanical
dictionary, an English work in which Hachette's classification scheme
was copied and his chart reproduced.[91]

[Footnote 91: Jacob Bigelow, _Elements of Technology_, ed. 2, Boston,
1831, pp. 231-256; Olinthus Gregory, _A Treatise of Mechanics_, 3 vols.,
ed. 3, London, 1815.]

A translation of the work of Lanz and Bétancourt[92] under the title
_Analytical Essay on the Construction of Machines_, was published about
1820 at London by Rudolph Ackermann (for whom the Ackermann steering
linkage was named), and their synoptic chart was reprinted again in 1822
in Durham.[93] In the United States, _Appleton's Dictionary of
Machines_[94] (1851) adopted the same system and used the same figures.
Apparently the wood engraver traced directly onto his block the figures
from one of the reprints of Lanz and Bétancourt's chart because the
figures are in every case exact mirror images of the originals.

[Footnote 92: Rudolph Ackermann, _Analytical Essay on the Construction
of Machines_, London, about 1820, a translation of Lanz and Bétancourt,
_op. cit._ (footnote 64).]

[Footnote 93: Thomas Fenwick, _Essays on Practical Mechanics_, ed. 3,
Durham, England, 1822.]

[Footnote 94: _Appleton's Dictionary of Machines, Mechanics,
Engine-Work, and Engineering_, 2 vols., New York, 1851 ("Motion").]

In the _Dictionary of Engineering_[95] (London, 1873), the figures were
redrawn and dozens of mechanisms were added to the repertory of
mechanical motions; the result was a fair catalog of sound ideas. The
ferryboat still tugged at its anchor cable, however.[96] _Knight's
American Mechanical Dictionary_,[97] a classic of detailed pictorial
information compiled by a U.S. patent examiner, contained well over
10,000 finely detailed figures of various kinds of mechanical
contrivances. Knight did not have a separate section on mechanisms, but
there was little need for one of the Hachette variety, because his whole
dictionary was a huge and fascinating compendium of ideas to be filed
away in the synthetic mind. One reason for the popularity and usefulness
of the various pictorial works was the peculiar ability of a wood or
steel engraving to convey precise mechanical information, an advantage
not possessed by modern halftone processes.

[Footnote 95: E. F. and N. Spon, _Dictionary of Engineering_, London
1873, pp. 2421-2452.]

[Footnote 96: _Ibid._, p. 2447.]

[Footnote 97: Edward H. Knight, _Knight's American Mechanical
Dictionary_, 3 vols., New York 1874-1876.]

[Illustration: Figure 36.--Typical mechanisms from E. F. and N. Spon,
_Dictionary of Engineering_ (London, 1873, pp. 2426, 2478).]

Many patent journals and other mechanical periodicals concerned with
mechanics were available in English from the beginning of the 19th
century, but few of them found their way into the hands of American
mechanicians until after 1820. Oliver Evans (1755-1819) had much to say
about "the difficulties inventive mechanics labored under for want of
published records of what had preceded them, and for works of reference
to help the beginner."[98] In 1817 the _North American Review_ also
remarked upon the scarcity of engineering books in America.[99]

[Footnote 98: George Escol Sellers in _American Machinist_, July 12,
1884, vol. 7, p. 3.]

[Footnote 99: _North-American Review and Miscellaneous Journal_, 1819,
new ser., vol. 8, pp. 13-15, 25.]

The _Scientific American_, which appeared in 1845 as a patent journal
edited by the patent promoter Rufus Porter, carried almost from its
beginning a column or so entitled "Mechanical Movements," in which one
or two mechanisms--borrowed from an English work that had borrowed from
a French work--were illustrated and explained. The _American Artisan_
began a similar series in 1864, and in 1868 it published a compilation
of the series as _Five Hundred and Seven Mechanical Movements_,
"embracing all those which are most important in dynamics, hydraulics,
hydrostatics, pneumatics, steam engines ... and miscellaneous
machinery."[100] This collection went through many editions; it was last
revived in 1943 under the title _A Manual of Mechanical Movements_.
This 1943 edition included photographs of kinematic models.[101]

[Footnote 100: Henry T. Brown, ed., _Five Hundred and Seven Mechanical
Movements_, New York, 1868.]

[Footnote 101: Will M. Clark, _A Manual of Mechanical Movements_, Garden
City, New York, 1943.]

Many readers are already well acquainted with the three volumes of
_Ingenious Mechanisms for Designers and Inventors_,[102] a work that
resulted from a contest, announced by _Machinery_ (vol. 33, p. 405) in
1927, in which seven prizes were offered for the seven best articles on
unpublished ingenious mechanisms.

[Footnote 102: _Ingenious Mechanisms for Designers and Inventors_ (vols.
1 and 2 edited by F. D. Jones, vol. 3 edited by H. L. Horton), New York,
Industrial Press, 1930-1951.]

There was an interesting class of United States patents called
"Mechanical Movements" that comprised scores of patents issued
throughout the middle decades of the 19th century. A sampling of these
patents shows that while some were for devices used in particular
machines--such as a ratchet device for a numbering machine, a locking
index for gunmaking machinery, and a few gear trains--the great majority
were for converting reciprocating motion to rotary motion. Even a
cursory examination of these patents reveals an appalling absence of
sound mechanical sense, and many of them appear to be attempts at
"perpetual motion," in spite of an occasional disclaimer of such intent.

Typical of many of these patented devices was a linkage for
"multiplying" the motion of a flywheel, proposed in 1841 by Charles
Johnson of Amity, Illinois (fig. 37). "It is not pretended that there is
any actual gain of power," wrote Mr. Johnson; and probably he meant it.
The avowed purpose of his linkage was to increase the speed of a
flywheel and thus decrease its size.[103]

[Footnote 103: U.S. Patent 2295, October 11, 1841.]

[Illustration: Figure 37.--Johnson's "converting motion," 1841. The
linkage causes the flywheel to make two revolutions for each
double-stroke of the engine piston rod B. From U.S. Patent 2295, October
11, 1841.]

An Englishman who a few years earlier had invented a "new Motion" had
claimed that his device would supersede the "ordinary crank in steam
engines," the beam, parallel motion, and "external flywheel," reduce
friction, neutralize "all extra contending power," and leave nothing for
the piston to do "but the work intended to be done."

A correspondent of the _Repertory of Patent Inventions_ made short work
of this device: "There is hardly one assertion that can be supported by
proof," he wrote, "and most of them are palpable misstatements." The
writer attacked "the 'beetle impetus wheel,' which he [the inventor]
thinks us all so beetle-headed, as not to perceive to be a flywheel,"
and concluded with the statement: "In short the whole production evinces
gross ignorance either of machinery, if the patentee really believed
what he asserted, or of mankind, if he did not."[104]

[Footnote 104: _Repertory of Patent Inventions_, ser. 3, October 1828,
vol. 7, pp. 196-200, and December 1828, vol. 7, pp. 357-361.]

Although many of the mechanisms for which patents were taken out were
designed by persons who would make no use of the principles involved
even if such principles could at that time have been clearly stated, it
is a regrettable fact that worthless mechanisms often got as much space
as sound ones in patent journals, and objections such as the one above
were infrequent. The slanted information thus conveyed to the young
mechanician, who was just accumulating his first kinematic repertory,
was at times sadly misleading.

From even this sketchy outline of the literature on the subject, it
should be fairly evident that there has been available to the
mechanician an enormous quantity of information about mechanical
linkages and other devices. Whatever one may think of the quality of the
literature, it has undoubtedly had influence not only in supplying
designers with information but in forming a tradition of how one ought
to supply the background that will enable the mind to assemble and
synthesize the necessary mechanism for a given purpose.[105]

[Footnote 105: Some additional catalogs of "mechanical movements" are
listed in the selected references at the end of this paper.]

Some of the mechanisms that have been given names--such as the Watt
straight-line linkage and the Geneva stop--have appeared in textbook
after textbook. Their only excuse for being seems to be that the authors
must include them or risk censure by colleagues. Such mechanisms are
more interesting to a reader, certainly, when he has some idea of what
the name has to do with the mechanism, and who originated it. One such
mechanism is the drag link.

After I had learned of the drag link (as most American engineering
students do), I wondered for awhile, and eventually despaired of making
any sense out of the term. What, I wanted to know, was being dragged?
Recently, in Nicholson's _Operative Mechanic and British Machinist_
(1826), I ran across the sketch reproduced here as figure 38. This
figure, explained Mr. Nicholson (in vol. 1, p. 32) "represents the
coupling link used by Messrs. Boulton and Watt in their portable steam
engines. A, a strong iron pin, projecting from one of the arms of the
fly-wheel B; D, a crank connected with the shaft C; and E, a link to
couple the pin A and the crank D together, so the motion may be
communicated to the shaft C." So the drag link was actually a link of a
coupling. Nothing could be more logical. A drag link mechanism now makes
sense to me.

[Illustration: Figure 38.--Drag link coupling used on Boulton and Watt
portable engines. The link E drags one shaft when the other turns. From
John Nicholson, _The Operative Mechanic, and British Machinist_
(Philadelphia, 1826, vol. I, pl. 5).]

Directly related to the drag link coupling were the patents of John
Oldham (1779-1840), an Irish engineer who is remembered mainly for the
coupling that bears his name (fig. 39). His three patents, which were
for various forms of steamboat feathering paddle wheels, involved
linkages kinematically similar to the drag link coupling, although it is
quite unlikely that Oldham recognized the similarity. However, for his
well-known coupling, which employs an inversion of the elliptical
trammel mechanism, I have found no evidence of a patent. Probably it was
part of the machinery that he designed for the Bank of Ireland's
printing house, of which Oldham was manager for many years. "Mr. Oldham
and his beautiful system" were brought to the Bank of England in 1836,
where Oldham remained until his death in 1840.[106]

[Footnote 106: Oldham's paddle-wheel patents were British Patents 4169
(October 10, 1817), 4429 (January 15, 1820), and 5445 (February 1,
1827). Robert Willis (_op. cit._ footnote 21, p. 167) noticed the
existence of the coupling. Drawings or descriptions of the banknote
machinery apparently have not been published though they probably still
exist in the banks' archives. The quotation is from Frederick G. Hall,
_The Bank of Ireland 1783-1946_, Dublin, 1949. John Francis in his
_History of the Bank of England_ (London, 1848, vol. 2, p. 232) wrote:
"The new machinery for printing the notes, which was introduced by Mr.
Oldham ... is well worthy of a visit, but would be uninteresting to
delineate."]

[Illustration: Figure 39.--_Top_, Original Oldham coupling built before
1840, using a cross (instead of a center disk), as sketched by Robert
Willis in personal copy of his _Principles of Mechanism_ (London, 1841,
p. 167). _Bottom_, Oldham coupling as illustrated in Alexander B. W.
Kennedy, _Kinematics of Machinery_, a translation of Franz Reuleaux'
_Theoretische Kinematik_ (London, 1876, pp. 315-316).]

The Geneva stop mechanism (fig. 40) was properly described by Willis as
a device to permit less than a full revolution of the star wheel and
thus to prevent overwinding of a watch spring. It was called Geneva stop
because it was used in Geneva watches. The Geneva wheel mechanism, which
permits full rotation of the star wheel and which is frequently used
for intermittent drives, was improperly called a Geneva stop in a
recent textbook probably because the logical origin of the term had been
lost.

[Illustration: Figure 40.--Geneva stop mechanism first used in Geneva
watches to prevent overwinding. The starwheel B had one convex surface
(_g-f_, dotted) so the wheel could be turned less than a full
revolution. After Robert Willis, _Principles of Mechanism_ (London,
1841, p. 266).]

The name for the Scotch yoke seems to be of fairly recent origin, the
linkage being called by a Scotsman in 1869 a "crank and slot-headed
sliding rod" (fig. 41). I suppose that it is now known as a Scotch yoke
because, in America at least, a "Scotch" was a slotted bar that was
slipped under a collar on a string of well-drilling tools to support
them while a section was being added (fig. 42).

[Illustration: Figure 41.--Scotch yoke, described as a "crank and
slot-headed sliding rod." From W. J. M. Rankine, _A Manual of Machinery
and Millwork_ (ed. 6, London, 1887, p. 169).]

[Illustration: Figure 42.--A "Scotch" supporting the top member of a
string of well-drilling tools while a section is being added, 1876. From
Edward H. Knight, _Knight's American Mechanical Dictionary_ (New York,
1876, p. 2057).]

It was surprising to me to find that the Ackermann steering linkage,
used today on most automobiles, was patented in 1818 when Detroit was
still a frontier town.[107] Furthermore, the man who took out the patent
described himself as Rudolph Ackermann, publisher and printseller. I
thought I had the necessary clue to the linkage's origin when I noticed
that the first English translation of the Lanz and Bétancourt treatise
was published by Ackermann, but the connection finally proved to be more
logical, if less direct. Ackermann (1764-1834), son of a Bavarian coach
builder, had spent a number of years designing coaches for English
gentlemen in London, where he made his home. One of his more notable
commissions was for the design of Admiral Nelson's funeral car in 1805.
The Ackermann steering linkage was not actually Ackermann's invention,
although he took out the British patent in his name and promoted the
introduction of the running gear of which the linkage was a part (fig.
43). The actual inventor was Ackermann's friend George Lankensperger of
Munich, coachmaker to the King of Bavaria. The advantage of being able
to turn a carriage around in a limited area without danger of
oversetting was immediately obvious, and while there was considerable
opposition by English coachmakers to an innovation for which a premium
had to be paid, the invention soon "made its way from its own intrinsic
merit," as Ackermann predicted it would.[108]

[Footnote 107: British Patent 4212, January 27, 1818.]

[Footnote 108: Rudolph Ackermann, _Observations on Ackermann's Patent
Moveable Axles_, London, 1819. It was interesting to me to note an
abstract of W. A. Wolfe's paper "Analytical Design of an Ackermann
Steering Linkage" in _Mechanical Engineering_, September 1958, vol. 80,
p. 92.]

[Illustration: Figure 43.--Ackermann steering linkage of 1818, currently
used in automobiles. This linkage was invented by George Lankensperger,
coachmaker to the King of Bavaria. From _Dinglers Polytechnisches
Journal_ (1820, vol. 1, pl. 7).]

The Whitworth quick-return mechanism (fig. 44) was first applied to a
slotter, or vertical shaper, in 1849, and was exhibited in 1851 at the
Great Exhibition in London.[109] Willis' comments on the mechanism are
reproduced in figure 44. I hope that Sir Joseph Whitworth (1803-1887)
will be remembered for sounder mechanical contrivances than this.

[Footnote 109: The quick-return mechanism (British Patent 12907,
December 19, 1849) was perhaps first publicly described in Charles
Tomlinson, ed., _Cyclopaedia of Useful Arts and Manufactures_, London,
1854, vol. 1, p. cxliv.]

[Illustration: Figure 44.--Quick-return mechanism. _Top_, Early
representation of the quick-return mechanism patented by Whitworth in
1849, from William Johnson, ed., _The Imperial Cyclopaedia of machinery_
(Glasgow, about 1855, pl. 88). _Middle_, Sketch by Robert Willis from
his copy of _Principles of Mechanism_ (London, 1841, p. 264), which
"shews Whitworth dissected into a simpler form"; it is as obscure as
most subsequent attempts have been to explain this mechanism without a
schematic diagram. _Bottom_, Linkage that is kinematically equivalent to
Whitworth's, from Robert Willis, _Principles of Mechanism_ (London,
1841, p. 264).]


Mechanisms in America, 1875-1955

Engineering colleges in the United States were occupied until the late
1940's with extending, refining, and sharpening the tools of analysis
that had been suggested by Willis, Rankine, Reuleaux, Kennedy, and
Smith. The actual practice of kinematic synthesis went on apace, but
designers often declined such help as the analytical methods might give
them and there was little exchange of ideas between scholars and
practitioners.

The capability and precision of machine tools were greatly enhanced
during this period, although, with the exception of the centerless
grinder, no significant new types of tools appeared. The machines that
were made with machine tools increased in complexity and, with the
introduction of ideas that made mass production of complex mechanical
products economically feasible, there was an accelerating increase in
quantity. The adoption of standards for all sorts of component parts
also had an important bearing upon the ability of a designer
economically to produce mechanisms that operated very nearly as he hoped
they would.

The study of kinematics has been considered for nearly 80 years as a
necessary part of the mechanical engineer's training, as the dozens of
textbooks that have been published over the years make amply clear.
Until recently, however, one would look in vain for original work in
America in the analysis or rational synthesis of mechanisms.

One of the very earliest American textbooks of kinematics was the 1883
work of Charles W. MacCord (1836-1915), who had been appointed professor
of mechanical drawing at Stevens Institute of Technology in Hoboken
after serving John Ericsson, designer of the _Monitor_, as chief
draftsman during the Civil War.[110] Based upon the findings of Willis
and Rankine, MacCord's _Kinematics_ came too early to be influenced by
Kennedy's improvements upon Reuleaux's work.

[Footnote 110: A biographical notice and a bibliography of MacCord
appears in _Morton Memorial: A History of the Stevens Institute of
Technology_, Hoboken, 1905, pp. 219-222.]

When the faculty at Washington University in St. Louis introduced in
1885 a curriculum in "dynamic engineering," reflecting a
dissatisfaction with the traditional branches of engineering, kinematics
was a senior subject and was taught from Rankine's _Machinery and
Millwork_.[111]

[Footnote 111: _Transactions of the American Society of Mechanical
Engineers_, 1885-1886, vol. 7, p. 757.]

At Massachusetts Institute of Technology, Peter Schwamb, professor of
machine design, put together in 1885 a set of printed notes on the
kinematics of mechanisms, based on Reuleaux's and Rankine's works. Out
of these notes grew one of the most durable of American textbooks, first
published in 1904.[112] In the first edition of this work, acceleration
was mentioned only once in passing (on p. 4). Velocities in linkages
were determined by orthogonal components transferred from link to link.
Instant centers were used only to determine velocities of various points
on the same link. Angular velocity ratios were frequently noted. In the
third edition, published in 1921, linear and angular accelerations were
defined, but no acceleration analyses were made. Velocity analyses were
altered without essential change. The fourth edition (1930) was
essentially unchanged from the previous one. Treatment of velocity
analysis was improved in the fifth edition (1938) and acceleration
analysis was added. A sixth edition, further revised by Prof. V. L.
Doughtie of the University of Texas, appeared in 1947.

[Footnote 112: Peter Schwamb and Allyne L. Merrill, _Elements of
Mechanism_, New York, 1904. In addition to the work of Reuleaux and
Rankine, the authors acknowledged their use of the publications of
Charles MacCord, Stillman W. Robinson, Thomas W. Goodeve, and William C.
Unwin. For complete titles see the list of selected references.]

Before 1900, several other books on mechanisms had been published, and
all followed one or another of the patterns of their predecessors.
Professors Woods and Stahl, at the Universities of Illinois and Purdue,
respectively, who published their _Elementary Mechanism_ in 1885, said
in their preface what has been said by many other American authors and
what should have been said by many more. "We make little claim to
originality of the subject-matter," wrote Woods and Stahl, "free use
having been made of all available matter on the subject.... Our claim to
consideration is based almost entirely on the manner in which the
subject has been presented." Not content with this disclaimer, they
continued: "There is, in fact, very little room for such originality,
the ground having been almost completely covered by previous
writers."[113]

[Footnote 113: Arthur T. Woods and Albert W. Stahl, _Elementary
Mechanism_, New York, 1885.]

The similarity and aridity of kinematics textbooks in this country from
around 1910 are most striking. The generation of textbook writers
following MacCord, Woods and Stahl, Barr of Cornell, Robinson of Ohio
State, and Schwamb and Merrill managed to squeeze out any remaining
juice in the subject, and the dessication and sterilization of textbooks
was nearly complete when my generation used them in the 1930's.
Kinematics was then, in more than one school, very nearly as it was
characterized by an observer in 1942--"on an intellectual par with
mechanical drafting."[114] I can recall my own naïve belief that a
textbook contained all that was known of the subject; and I was not
disabused of my belief by my own textbook or by my teacher. I think I
detect in several recent books a fresh, less final, and less tidy
treatment of the kinematics of mechanisms, but I would yet recommend
that anyone who thinks of writing a textbook take time to review,
carefully and at first hand, not only the desk copies of books that he
has accumulated but a score or more of earlier works, covering the last
century at least. Such a study should result in a better appreciation of
what constitutes a contribution to knowledge and what constitutes merely
the ringing of another change.

[Footnote 114: _Mechanical Engineering_, October 1942, vol. 64, p. 745.]

The author of the contentious article that appeared in _Mechanical
Engineering_ in 1942 under the title "What is Wrong with Kinematics and
Mechanisms?" made several pronouncements that were questioned by various
readers, but his remarks on the meagerness of the college courses of
kinematics and the "curious fact" that the textbooks "are all strangely
similar in their incompleteness" went unchallenged and were, in fact,
quite timely.[115]

[Footnote 115: De Jonge, _op. cit._ (footnote 78).]

It appears that in the early 1940's the general classroom treatment of
accelerations was at a level well below the existing knowledge of the
subject, for in a series of articles by two teachers at Purdue attention
was called to the serious consequences of errors in acceleration
analysis occasioned by omitting the Coriolis component.[116] These
authors were reversing a trend that had been given impetus by an article
written in 1920 by one of their predecessors, Henry N. Bonis. The
earlier article, appearing in a practical-and-proud-of-it technical
magazine, demonstrated how the acceleration of a point on a flywheel
governor might be determined "without the use of the fictitious
acceleration of Coriolis." The author's analysis was right enough, and
he closed his article with the unimpeachable statement that "it is
better psychologically for the student and practically for the engineer
to understand the fundamentals thoroughly than to use a complex formula
that may be misapplied." However, many readers undoubtedly read only the
lead paragraph, sagely nodded their heads when they reached the word
"fictitious," which confirmed their half-formed conviction that anything
as abstruse as the Coriolis component could have no bearing upon a
practical problem, and turned the page to the "practical kinks"
section.[117]

[Footnote 116: A. S. Hall and E. S. Ault, "How Acceleration Analysis Can
Be Improved," _Machine Design_, February 1943, vol. 15, pp. 100-102,
162, 164; and March 1943, vol. 15, pp. 90-92, 168, 170. See also A. S.
Hall, "Teaching Coriolis' Law," _Journal of Engineering Education_, June
1948, vol. 38, pp. 757-765.]

[Footnote 117: Henry N. Bonis, "The Law of Coriolis," _American
Machinist_, November 18, 1920, vol. 53, pp. 928-930. See also
"Acceleration Determinations," _American Machinist_, November 25 and
December 2, 1920, vol. 53, pp. 977-981 and 1027-1029.]

Less than 20 years ago one might have read in _Mechanical Engineering_
that "Practical machinery does not originate in mathematical formulas
nor in beautiful vector diagrams." While this remark was in a letter
evoked by an article, and was not a reflection of editorial policy, it
was nevertheless representative of an element in the American tradition
of engineering. The unconscious arrogance that is displayed in this
statement of the "practical" designer's creed is giving way to
recognition of the value of scholarly work. Lest the scholar develop
arrogance of another sort, however, it is well to hear the author of
the statement out. "A drafting machine is a useful tool," he wrote. "It
is not a substitute for a draftsman."[118]

[Footnote 118: _Mechanical Engineering_, October 1942, vol. 64, p. 746.]

The scholarly interest in a subject is fairly represented by the papers
that are published in the transactions of professional societies and,
more recently, by original papers that appear in specialized magazines.
From 1900 to 1930 there were few papers on mechanisms, and most of those
that did appear were concerned with descriptions of new "mechanical
motions." In the 1930's the number of papers reported in _Engineering
Index_ increased sharply, but only because the editors had begun to
include foreign-language listings.

There has been in Germany a thread of continuity in the kinematics of
mechanisms since the time of Reuleaux. While most of the work has had to
do with analysis, the teasing question of synthesis that Reuleaux raised
in his work has never been ignored. The developments in Germany and
elsewhere have been ably reviewed by others,[119] and it is only to be
noted here that two of the German papers, published in 1939 in
_Maschinenbau_, appear to have been the sparks for the conflagration
that still is increasing in extent and intensity. According to summaries
in _Engineering Index_, R. Kraus, writing on the synthesis of the
double-crank mechanism, drew fire from the Russian Z. S. Bloch, who, in
1940, discussed critically Kraus's articles and proceeded to give the
outline of the "correct analysis of the problem" and a general numerical
solution for the synthesis of "any four-bar linkage."[120] Russian work
in mechanisms, dating back to Chebyshev and following the "Chebyshev
theory of synthesis" in which algebraic methods are used to determine
paths of minimum deviation from a given curve, has also been reviewed
elsewhere,[121] and I can add nothing of value.

[Footnote 119: Grodzinski, Bottema, De Jonge, and Hartenberg and
Denavit. For complete titles see list of selected references.]

[Footnote 120: My source, as noted, is _Engineering Index_. Kraus's
articles are reported in 1939 and Bloch's in 1940, both under the
section heading "Mechanisms."]

[Footnote 121: A. E. Richard de Jonge, "Are the Russians Ahead in
Mechanism Analysis?" _Machine Design_, September 1951, vol. 23, pp. 127,
200-208; O. Bottema, "Recent Work on Kinematics," _Applied Mechanics
Reviews_, April 1953, vol. 6, pp. 169-170.]

When, after World War II, some of the possibilities of kinematic
synthesis were recognized in the United States, a few perceptive
teachers fanned the tinder into an open flame.

The first publication of note in this country on the synthesis of
linkages was a practical one, but in conception and undertaking it was a
bold enterprise. In a book by John A. Hrones and G. L. Nelson,
_Analysis of the Four Bar Linkage_ (1951), the four-bar crank-and-rocker
mechanism was exhaustively analyzed mechanically and the results were
presented graphically. This work was faintly praised by a Dutch scholar,
O. Bottema, who observed that the "complicated analytical theory of the
three-bar [sic] curve has undoubtedly kept the engineer from using it"
and who went on to say that "we fully understand the publication of an
atlas by Hrones and Nelson containing thousands of trajectories which
must be very useful in many design problems."[122] Nevertheless, the
authors furnished designers with a tool that could be readily, almost
instantly, understood (fig. 45), and the atlas has enjoyed wide
circulation.[123] The idea of a geometrical approach to synthesis has
been exploited by others in more recent publications,[124] and it is
likely that many more variations on this theme will appear.

[Footnote 122: Bottema, _op. cit._ (footnote 121).]

[Footnote 123: In 1851 Robert Willis had designed a coupler-point
path-generating machine (fig. 46) that could have been used to produce a
work similar to that of Hrones and Nelson.]

[Footnote 124: R. S. Hartenberg and J. Denavit, "Systematic Mechanism
Design," _Machine Design_, September 1954, vol. 26, pp. 167-175, and
October 1954, vol. 26, pp. 257-265; A. S. Hall, A. R. Holowenko, and H.
G. Laughlin, "Four-Bar Lever Crank Mechanism," _Design News_, September
15, 1957, vol. 12, pp. 130-139, October 1, 1957, vol. 12, pp. 145-154,
and October 15, 1957, vol. 12, pp. 132-141. For a nomographic approach,
with particular application to computers, see Antonin Svoboda,
_Computing Mechanisms and Linkages_, New York, 1948.]

[Illustration: Figure 45.--Paths of 11 points on the coupler
(horizontal) link are plotted through one cycle. Dashes indicate equal
time intervals. From John A. Hrones and G. L. Nelson, _Analysis of the
Four Bar Linkage_ (New York, 1951, p. 635).]

[Illustration: Figure 46.--Coupler-point path-generating machine for
four-bar linkage. This device, built by Professor Willis as a teaching
aid for demonstrating straight-line linkages, could have been adapted to
produce a plate like the one shown in figure 45. From Robert Willis, _A
System of Apparatus for the Use of Lecturers and Experimenters_ ...
(London 1851, pl. 3).]

Pursuit of solutions to the "complicated analytical theory" of linkages
was stimulated by publication of Ferdinand Freudenstein's "Analytical
Approach to the Design of Four-Link Mechanisms" in 1954,[125] and an
increasing interest in the problem is indicated by the extensive
literature that has appeared in the last five years.

[Footnote 125: _Transactions of the American Society of Mechanical
Engineers_, 1954, vol. 76, pp. 483-492. See also _Transactions of the
American Society of Mechanical Engineers_, 1955, vol. 77, pp. 853-861,
and 1956, vol. 78, pp. 779-787.]

The proper role of rational methods in the synthesis of mechanisms is
not yet clear. "While we may talk about kinematic synthesis," wrote two
of today's leaders in the field, "we are really talking about a hope for
the future rather than a great reality of the present."[126] When the
mental equipment and the enthusiasm of scholars who are devoting their
time to the problems of kinematic synthesis are considered, however, it
is difficult to see how important new ideas can fail to be produced.

[Footnote 126: R. S. Hartenberg and J. Denavit, "Kinematic Synthesis,"
_Machine Design_, September 6, 1956, vol. 28, pp. 101-105.]

An annual Conference on Mechanisms, sponsored by Purdue University and
_Machine Design_, was inaugurated in 1953 and has met with a lively
response. Among other manifestations of current interest in mechanisms,
the contributions of Americans to international conferences on
mechanisms reflects the growing recognition of the value of scholarly
investigation of the kind that can scarcely hope to yield immediately
tangible results.

While we look to the future, one may ask how a lengthy view of the past
can be justified. It seems to me that there is inherent in the almost
feverish activity of the present the danger of becoming so preoccupied
with operational theory that the goals may become clouded and the
synthesis (let us put it less elegantly: the design) of mechanisms may
never quite come into focus. If one knows nothing of the past, I wonder
how he can with any confidence decide in what direction he must turn in
order to face the future.


Acknowledgment

I am grateful to Professors Richard S. Hartenberg and Allen S. Hall,
Jr., for reading the manuscript, making helpful comments, and suggesting
material that I had not found. The errors, however, are mine.


Additional References

The following list of additional reference material on kinematics may be
of help to readers who desire to do independent research. The material
is listed according to the section headings in the text of the present
article.


TO DRAW A STRAIGHT LINE

KEMPE, A. B. _How to Draw a Straight Line._ London, 1877.

Contains a useful bibliography. Reprinted in _Squaring the Circle and
Other Monographs_, New York, Chelsea Publishing Company, 1953.

Much attention has been given to straight-line mechanisms since the time
of Kempe; at least a half dozen articles have appeared in the United
States since 1950, but I did not investigate the literature published
after 1877.


SCHOLARS AND MACHINES

BECK, THEODOR. _Beiträge zur Geschichte des Maschinenbaues._ Berlin,
1899.

Reviews of early works, such as those by Leonardo da Vinci, Biringuccio,
Besson, Zonca, etc.

BORGNIS, GIUSEPPE ANTONIO. _Traité complet de mécanique appliquée aux
arts._ Paris, 1818-1821, 9 vols.

Contains several hundred finely detailed plates of machines.

LABOULAYE, CHARLES. _Traité de cinématique ou théorie des mécanismes._
Paris, 1861 (ed. 2).

This work was quoted frequently by Laboulaye's contemporaries.

ROYAL SOCIETY OF LONDON. _Catalogue of Scientific Papers, 1800-1900,
Author Index._ London, 1867-1902, and Cambridge, 1914-1925.

----. _Catalogue of Scientific Papers, 1800-1900, Subject Index._
London, 1909, vol. 2.

This subject index was started in 1908, and by 1914 three volumes (the
third in two parts) had been published; however, this subject index was
never completed. Volume 2, titled _Mechanics_, has some 200 entries
under "Linkages." It is interesting to note that both of the Royal
Society's monumental catalogs grew out of a suggestion made by Joseph
Henry at a British Association meeting in Glasgow in 1855.

WEISBACH, JULIUS. _The Mechanics of the Machinery of Transmission_, vol.
3, pt. 1, sec. 2 of _Mechanics of Engineering and Machinery_, translated
by J. F. Klein. New York, 1890 (ed. 2).


MECHANISMS AND MECHANICIANS

BARBER, THOMAS W. _Engineer's Sketch-Book._ London, 1890 (ed. 2).

HERKIMER, HERBERT. _Engineer's Illustrated Thesaurus._ New York, 1952.

PERIODICALS. _Artizan_, from 1843; _Practical Mechanic and Engineer's
Magazine_, from 1841; _Repertory of Arts and Manufactures_, from 1794;
_Newton's London Journal of Arts and Science_, from 1820. (The preceding
periodicals have many plates of patent specification drawings.) _The
Engineer_, November 10, 1933, vol. 156, p. 463, and _Engineering_,
November 10, 1933, vol. 136, p. 525. (Recent English views questioning
the utility of kinematics.)

TATE, THOMAS. _Elements of Mechanism._ London, 1851.

Contains figures from Lanz and Bétancourt (1808).

WYLSON, JAMES. _Mechanical Inventor's Guide._ London, 1859.

Contains figures from Henry Adcock, _Adcock's Engineers' Pocket-Book,
1858_.


MECHANISMS IN AMERICA, 1875-1955

ALBERT, CALVIN D., AND ROGERS, F. D. _Kinematics of Machinery._ New
York, 1931.

Contains a bibliography that includes works not mentioned in the present
paper.

BARR, JOHN H. _Kinematics of Machinery._ New York, 1899.

An early textbook. The author taught at Cornell University.

BEGGS, JOSEPH S. _Mechanism._ New York, 1955.

Contains an extensive and useful bibliography.

BOTTEMA, O. "Recent Work on Kinematics," _Applied Mechanics Reviews_,
April 1953, vol. 6, pp. 169-170.


CONFERENCE ON MECHANISMS.

This conference was sponsored by Purdue University and _Machine Design_.
Transactions of the first two conferences appeared as special sections
in _Machine Design_, December 1953, vol. 25, pp. 173-220, December 1954,
vol. 26, pp. 187-236, and in collected reprints. Papers of the third and
fourth conferences (May 1956 and October 1957) appeared in _Machine
Design_ over several months following each conference and in collected
reprints. Papers of the fifth conference (October 1958) were collected
and preprinted for conference participants; subsequently, all papers
appeared in _Machine Design_. Collected reprints and preprints are
available (May 1960) from Penton Publishing Company, Cleveland, Ohio.

DE JONGE, A. E. RICHARD. "Kinematic Synthesis of Mechanisms,"
_Mechanical Engineering_, July 1940, vol. 62, pp. 537-542.

----. "A Brief Account of Modern Kinematics," _Transactions of the
American Society of Mechanical Engineers_, 1943, vol. 65, pp. 663-683.

GOODEVE, THOMAS M. _The Elements of Mechanism._ London, 1903.

An early textbook.

GRODZINSKI, PAUL, AND MCEWEN, EWEN. "Link Mechanisms in Modern
Kinematics," _Journal and Proceedings of the Institution of Mechanical
Engineers_, 1954, vol. 168, pp. 877-896.

This article evoked interesting discussion. It is unfortunate that
Grodzinski's periodical, _Mechanism, An International Bibliography_,
which was published in London in 1956-1957 and which terminated shortly
after his death, has not been revived. Grodzinski's incisive views and
informative essays are valuable and interesting.

HARTENBERG, R. S. "Complex Numbers and Four-Bar Linkages," _Machine
Design_, March 20, 1958, vol. 30, pp. 156-163.

This is an excellent primer. The author explains complex numbers in his
usual lucid fashion.

HARTENBERG, R. S., AND DENAVIT, J. "Kinematic Synthesis," _Machine
Design_, September 6, 1956, vol. 28, pp. 101-105.

MACCORD, CHARLES. _Kinematics._ New York, 1883.

An early textbook.

ROBINSON, STILLMAN W. _Principles of Mechanism._ New York, 1896.

An early textbook. The author taught at Ohio State University.

UNWIN, WILLIAM C. _The Elements of Machine Design._ New York, 1882 (ed.
4).

An early textbook. The author taught at Royal Indian Engineering
College, in England.


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