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.]