Chapter 3

[Footnote 42: Charles-Nicholas Peaucellier, "Note sur une question de geometrie de compas," _Nouvelles Annales de mathematiques_, 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.]

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).

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, Andre-Marie Ampere (1775-1836) was called upon to prepare a course in theoretical and experimental physics for the College 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 Ampere managed to capture the central idea of the subject.

[Footnote 56: Andre-Marie Ampere, _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).]

Cinematique (from the Greek word for movement) was, according to Ampere, 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, Ampere 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 _Encyclopedie_ 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 Mezieres, 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 Mezieres 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 Betancourt.

[Footnote 60: Lazare N. M. Carnot, _Essai sur les machines en general_, Mezieres, 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 Mezieres 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, _Traite elementaire des machines_, Paris, 1811, p. v.]

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 Betancourt, 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 Betancourt, _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 Betancourt work. This copy probably represents the first textbook of kinematics. Betancourt 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 Handwoerterbuch fuer Mathematik ..._, Leipzig, 1863, vol. 1.]

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 _Traite complet de mechanique appliquee 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 _recepteurs_ (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, _Theorie de la mecanique usuelle_ in _Traite complet de mecanique appliquee 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 _recepteurs_, _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 systemes 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 x distance (pp. iii, 2).]

[Footnote 68: The renowned Jean Victor Poncelet lent weight to this scheme. (See Franz Reuleaux, _Theoretische Kinematik: Grundzuege 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.).]

Acting upon Ampere's clear exposition of the province of kinematics and excluding, as Ampere 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.]