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diff --git a/27106-8.txt b/27106-8.txt new file mode 100644 index 0000000..cfc579b --- /dev/null +++ b/27106-8.txt @@ -0,0 +1,2919 @@ +The Project Gutenberg EBook of Kinematics of Mechanisms from the Time of +Watt, by Eugene S. Ferguson + +This eBook is for the use of anyone anywhere at no cost and with +almost no restrictions whatsoever. You may copy it, give it away or +re-use it under the terms of the Project Gutenberg License included +with this eBook or online at www.gutenberg.org + + +Title: Kinematics of Mechanisms from the Time of Watt + +Author: Eugene S. Ferguson + +Release Date: October 31, 2008 [EBook #27106] + +Language: English + +Character set encoding: ISO-8859-1 + +*** START OF THIS PROJECT GUTENBERG EBOOK KINEMATICS OF MECHANISMS *** + + + + +Produced by Chris Curnow, Viv, Joseph Cooper and the Online +Distributed Proofreading Team at http://www.pgdp.net + + + + + +[Transcriber's notes: + +1) Peaucillier is a printers error and has been changed to Peaucellier. + +2) The 4 characters at the end of the word 'Pafnutï[)i]' denote a letter +'i' with a breve accent.] + + +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. + + +GOVERNMENT PRINTING OFFICE: 1962 + +For sale by the Superintendent of Documents, U.S. Government Printing +Office Washington 25, D. C.--Price 40 cents + + + + + +End of the Project Gutenberg EBook of Kinematics of Mechanisms from the Time +of Watt, by Eugene S. 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