[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 equalledfor its brilliance and far-reaching consequences, James Watt radicallyaltered the steam engine not only by adding a separate condenser but bycreating a whole new family of linkages. His approach was largelyempirical, as we use the word today. _

_This study suggests that, despite the glamor of today's sophisticatedmethods of calculation, a highly developed intuitive sense, reinforcedby a knowledge of the past, is still indispensable to the design ofsuccessful mechanisms. _

THE AUTHOR: _Eugene S. Ferguson, formerly curator of mechanical andcivil engineering in the United States National Museum, SmithsonianInstitution, is now professor of mechanical engineering at Iowa StateUniversity of Science and Technology. _

In engineering schools today, a student is introduced to the kinematicsof mechanisms by means of a course of kinematic analysis, which isconcerned with principles underlying the motions occurring inmechanisms. These principles are demonstrated by a study of mechanismsalready in existence, such as the linkage of a retractable landing gear, computing mechanisms, mechanisms used in an automobile, and the like. Asystematic, if not rigorous, approach to the design of gears and camsalso is usually presented in such a course. Until recently, however, noserious attempt was made to apply the principles developed in kinematicanalysis to the more complex problem of kinematic synthesis of linkages. By kinematic synthesis is meant the designing of a linkage to produce agiven series of motions for a particular purpose. 

That a rational--numerical or geometrical--approach to kinematicsynthesis is possible is a relatively recent idea, not yet fullyaccepted; but it is this idea that is responsible for the intensescholarly interest in the kinematics of mechanisms that has occurred inthis country within the last 10 years. 

This scholarly activity has resulted in the rediscovery of many earlierworks on the subject, and nearly all the scholars now working in thisfield have acknowledged in one way or another their debt to those whoarrived on the scene at an earlier time than they. There have beenoccasional reviews of the sequence and nature of developments, but theemphasis naturally has been upon the recent past. It seems to me thatthere is something to be gained in looking beyond our own generation, oreven beyond the time of Franz Reuleaux (1829-1905), who is generallycredited with originating many of our modern concepts of mechanismanalysis and design, and to inquire into the ideas that made possibleReuleaux'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 totrace the high points in the development of kinematic analysis andsynthesis, both in academic circles and in the workshop, noting wherepossible the influence of one upon the other. If I have devoted morespace to particular people and episodes than is warranted by theircontributions to the modern treatment of the subject, it is because Ihave found that the history of kinematics of mechanisms, like thehistory of any other branch of engineering, is more interesting and moreplausible if it is recognized that its evolutionary development is theresult of human activity. This history was wrought by people like us, noless intelligent and no less subject than we are to environment, to asubjective way of looking at things, and to a heritage of ideas andbeliefs. 

I have selected the period from the time of Watt because modernmechanisms originated with him, and I have emphasized the first centuryof the period because by 1885 many of the ideas of modern kinematics ofmechanisms were well developed. Linkages are discussed, to the virtualexclusion of gears and cams, because much of the scholarly work inkinematic synthesis is presently directed toward the design of linkagesand because linkages provide a convenient thread for a narrative thatwould have become unnecessarily complex if detailed treatment of gearsand cams had been included. I have brought the narrative down to thepresent by tracing kinematics as taught in American engineeringschools, closing with brief mention of the scholarly activity inkinematics in this country since 1950. An annotated list of additionalreferences is appended as an encouragement to further work in thehistory of the subject. 

James Watt, Kinematic Synthesist

James Watt (1736-1819), improver of the steam engine, was a highlygifted designer of mechanisms, although his background included noformal study of mechanisms. Indeed, the study of mechanisms, withoutimmediate regard to the machines in which they were used, was notintroduced until after Watt's important work had been completed, whilethe actual design of mechanisms had been going on for several centuriesbefore the time of Watt. 

Mechanisms that employed screws, cams, and gears were certainly in useby the beginning of the Christian era. While I am not aware ofunequivocal evidence of the existence of four-bar linkages before the16th century, their widespread application by that time indicates thatthey probably originated much earlier. A tantalizing 13th-century sketchof an up-and-down sawmill (fig. 1) suggests, but does not prove, thatthe four-bar linkage was then in use. Leonardo da Vinci (1452-1519)delineated, if he did not build, a crank and slider mechanism, also fora sawmill (fig. 2). In the 16th century may be found the conversion ofrotary to reciprocating motion (strictly speaking, an oscillationthrough a small arc of a large circle) and vice versa by use of linkagesof rigid members (figs. 3 and 4), although the conversion of rotary toreciprocating motion was at that time more frequently accomplished bycams and intermittent gearing. Nevertheless, the idea of linkages was afirmly established part of the repertory of the machine builder before1600. In fact one might have wondered in 1588, when Agostino Ramellipublished his book on machines, [1] whether linkages had not indeedreached their ultimate stage of development. To illustrate my point, Ihave selected the plate of Ramelli that most appeals to me (fig. 5), although the book exhibits more than 200 other machines of comparablecomplexity and ingenuity. 

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

[Illustration: Figure 1. --Up-and-down sawmill of the 13th century. Theguide mechanism at lower left, attached to the saw blade, appears to bea 4-bar linkage. After Robert Willis, trans. And ed. , _Facsimile of theSketch-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 isdepicted at the lower end of the guides. From Theodor Beck, _Beiträgezur Geschichte des Maschinenbaues_ (Berlin, 1899, p. 323). ]

[Illustration: Figure 3. --Blowing engine by Vanuccio Biringuccio, about1540, showing conversion of motion of the waterwheel shaft from rotationto oscillation. From Theodor Beck, _Beiträge zur Geschichte desMaschinenbaues_ (Berlin, 1899, p. 120). ]

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

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

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

Watt's genius was nowhere more evident than in his synthesis oflinkages. An essential ingredient in the success of Watt's linkages, however, was his partner's appreciation of the entirely new order ofrefinement that they called for. Matthew Boulton, who had been asuccessful manufacturer of buttons and metal novelties long before hispartnership with Watt was formed, had recognized at once the need forcare in the building of Watt's steam engine. On February 7, 1769, he hadwritten Watt:[2] "I presumed that your engine would require money, very accurate workmanship and extensive correspondence to make it turnout to the best advantage and that the best means of keeping up thereputation and doing the invention justice would be to keep theexecutive part of it out of the hands of the multitude of empiricalengineers, who from ignorance, want of experience and want of necessaryconvenience, would be very liable to produce bad and inaccurateworkmanship; all of which deficiencies would affect the reputation ofthe invention. " Boulton expected to build the engines in his shop "withas great a difference of accuracy as there is between the blacksmith andthe mathematical instrument maker. " The Soho Works of Boulton and Watt, in Birmingham, England, solved for Watt the problem of producing "ingreat" (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 theMechanical Inventions of James Watt_, London, 1854, vol. 1, pp. 56, 64. This work, in three volumes, contains letters, other documents, andplates of patent specification drawings. ]

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

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

Another event of April 1775 occurring in Birmingham now appears to havebeen one that marked the beginning of a new era of technologicaladvance. It was near the end of this month that Boulton, at the SohoWorks, wrote to his partner and commented upon receiving the cast ironsteam engine cylinder that had been finished in John Wilkinson's boringmill:

 . . . 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 longdevelopment of the Watt steam engine, which for 10 years had occupiednearly all of Watt's thoughts and all the time he could spare from therequirements of earning a living. Although there were many trials aheadfor the firm of Boulton and Watt in further developing and perfectingthe steam engine, the crucial problem of leakage of steam past thepiston in the cylinder had now been solved by Wilkinson's new boringmill, which was the first large machine tool capable of boring acylinder 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 or60 years came to include nearly all of the basic types of heavychip-removing tools that are in use today. The development of tools wasaccelerated by the inherent accuracy required of the linkages that wereoriginated by Watt. Once it had been demonstrated that a large andcomplex machine, such as the steam engine, could be built accuratelyenough so that its operation would be relatively free of trouble, manyoutstanding minds became engaged in the development of machines andtools. It is interesting, however, to see how Watt and others grappledwith the solutions of problems that resulted from the advance of thesteam engine. 

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

The only type of steam engine then in existence was the Newcomen beamengine, which had been introduced in 1712 by Thomas Newcomen, also anEnglishman. This type of engine was widely used, mostly for pumpingwater out of mines but occasionally for pumping water into a reservoirto supply a waterwheel. It was arranged with a vertical steam cylinderlocated beneath one end of a large pivoted working beam and a verticalplunger-type pump beneath the other end. Heavy, flat chains were securedto a sector at each end of the working beam and to the engine and pumppiston rods in such a way that the rods were always tangent to a circlewhose center was at the beam pivot. The weight of the reciprocating pumpparts pulled the pump end of the beam down; the atmosphere, acting onthe open top of the piston in the steam cylinder, caused the engine endof the beam to be pulled down when the steam beneath the piston wascondensed. The chains would of course transmit force from piston to beamonly in tension. 

It is now obvious that a connecting rod, a crank, and a sufficientlyheavy flywheel might have been used in a conventional Newcomen engine inorder to supply power to a rotating shaft, but contemporary evidencemakes it clear that this solution was by no means obvious to Watt nor tohis 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 lowerpart of a toroidal chamber to provide a boundary for steam spacessuccessively formed by flap gates within the chamber. The practicaldifficulties of construction finally ruled out this solution to theproblem of a rotating power source, but not until after Boulton andWatt had spent considerable effort and money on it. [5]

[Footnote 5: Henry W. Dickinson and Rhys Jenkins, _James Watt and theSteam Engine_, Oxford, Clarendon Press, 1927, pp. 146-148, pls. 14, 31. This work presents a full and knowledgeable discussion, based on primarymaterial, of the development of Watt's many contributions to mechanicaltechnology. It is ably summarized in Dickinson, _op. Cit. _ (footnote2). ]

In 1777 a speaker before the Royal Society in London observed that inorder to obtain rotary output from a reciprocating steam engine, a crank"naturally occurs in theory, " but that in fact the crank is impracticalbecause of the irregular rate of going of the engine and its variablelength of stroke. He said that on the first variation of length ofstroke the machine would be "either broken to pieces, or turnedback. "[6] John Smeaton, in the front rank of English steam engineers ofhis time, was asked in 1781 by His Majesty's Victualling-Office for hisopinion as to whether a steam-powered grain mill ought to be driven by acrank or by a waterwheel supplied by a pump. Smeaton's conclusion wasthat the crank was quite unsuited to a machine in which regularity ofoperation was a factor. "I apprehend, " he wrote, "that no motioncommunicated from the reciprocating beam of a fire engine can ever actperfectly equal and steady in producing a circular motion, like theregular efflux of water in turning a waterwheel. " He recommended, incidentally, that a Boulton and Watt steam engine be used to pump waterto supply the waterwheel. [7] Smeaton had thought of a flywheel, but hereasoned that a flywheel large enough to smooth out the halting, jerkyoperation of the steam engines that he had observed would be more of anencumbrance 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 toWatt at this time. He was not, as tradition has it, blocked merely bythe existence of a patent for a simple crank and thus forced to inventsome other device as a substitute. 

Matthew Wasbrough, of Bristol, the engineer commonly credited with thecrank patent, made no mention of a crank in his patent specification, but rather intended to make use of "racks with teeth, " or "one or morepullies, wheels, segments of wheels, to which are fastened rotchets andclicks or palls. . . . " He did, however, propose to "add a fly or flys, inorder to render the motion more regular and uniform. " Unfortunately forus, 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 1780patented a counterweighted crank device (fig. 6) that was expected toremove the objection to a crank, which operated with changing leverageand thus irregular power. In figure 6, the counterweighted wheel, revolving twice for each revolution of the crank (A), would allow thecounterweight to descend while the crank passed the dead-center positionand would be raised while the crank had maximum leverage. No mention ofa 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" thathindered James Watt's progress. This patent, granted to James Pickard in1780, claimed only the arrangement of counterweights, not the crank. Thecrank pin to which the connecting rod was attached is at _Aa_. FromBritish 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 otherdevices and did not immediately appreciate the regulating ability of aflywheel. [11] In April 1781 Watt wrote to Boulton, who was then out oftown: "I know from experiment that the other contrivance, which you sawme try, performs at least as well, and has in fact many advantages overthe crank. "[12] The "other contrivance" probably was his swash wheelwhich he built and which appeared on his next important patentspecification (fig. 7a). Also in this patent were four other devices, one of which was easily recognizable as a crank, and two of which wereeccentrics (fig. 7a, b). The fourth device was the well-knownsun-and-planet gearing (fig. 7e). [13] In spite of the similarity of thesimple crank to the several variations devised by Watt, this patent drewno fire from Wasbrough or Pickard, perhaps because no reasonable personwould contend that the crank itself was a patentable feature, or perhapsbecause the similarity was not at that time so obvious. However, Wattsteered clear of directly discernible application of cranks because hepreferred to avoid a suit that might overthrow his or other patents. Forexample, if the Wasbrough and Pickard patents had been voided, theywould have become public property; and Watt feared that they might"get into the hands of men more ingenious, " who would give Boulton andWatt 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 theconversion of reciprocating motion to rotary motion in a steam engine. (British Patent 1306, October 25, 1781). From James P. Muirhead, _TheOrigin 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_ isrotated 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 fromworking beam. The wheel pivots at _C_. ]

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

[Illustration: (e) Sun-and-planet gearing. This is the idea actuallyemployed in Boulton and Watt engines. As the optional link _JK_ held thegearwheel centers always equidistant, the annular guide _G_ was notused. ]

The sun-and-planet arrangement, with gears of equal size, was adopted byWatt for nearly all the rotative engines that he built during the termof the "crank patents. " This arrangement had the advantage of turningthe flywheel through two revolutions during a single cycle of operationof the piston, thus requiring a flywheel only one-fourth the size of theflywheel needed if a simple crank were used. The optional link (JK offig. 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. Thedouble-acting engine, unlike the single-acting pumping engine, requireda piston rod that would push as well as pull. It was in the solution ofthis problem that Watt's originality and sure judgment were most clearlydemonstrated. 

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

[Footnote 15: James Watt, March 31, 1783, quoted in Dickinson andJenkins, _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 endof 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 arrangedfor pumping. Pump rod _SS_ is hung from sector of the working beam. FromJames P. Muirhead, _The Origin and Progress of the Mechanical Inventionsof James Watt_ (London, 1854, vol. 3, pl. 15). ]

It was about a year later that the straight-line linkage[17] was thoughtout. "I have started a new hare, " Watt wrote to his partner. "I havegot a glimpse of a method of causing the piston-rod to move up and downperpendicularly, 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 aslight model yet, so cannot build upon it, though I think it a veryprobable thing to succeed, and one of the most ingenious simple piecesof mechanism I have contrived. . . . "[18]

[Footnote 17: Watt's was a four-bar linkage. All four-bar straight-linelinkages that have no sliding pairs trace only an approximately straightline. The exact straight-line linkage in a single plane was not knownuntil 1864 (see p. 204). In 1853 Pierre-Frédéric Sarrus (1798-1861), aFrench professor of mathematics at Strasbourg, devised an accordion-likespatial linkage that traced a true straight line. Described but notillustrated (Académie des Sciences, Paris, _Comptes rendus_, 1853, vol. 36, pp. 1036-1038, 1125), the mechanism was forgotten and twicereinvented; finally, the original invention was rediscovered by anEnglish writer in 1905. For chronology, see Florian Cajori, _A Historyof 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 alarge beam engine almost immediately, and the usually pessimistic andreserved inventor was close to a state of elation when he told Boultonthat the "new central perpendicular motion answers beyond expectation, and does not make the shadow of a noise. "[19] This linkage, which wasincluded in an extensive patent of 1784, and two alternative devices areillustrated 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 ofthe piston rod of a double-acting engine (British Patent 1432, April 28, 1784). _Top left_, straight-line linkage; _top right_, crosshead andguide arrangement; _lower left_, piston rod _A_ is guided by sectors _D_and _E_, suspended by flexible cords. From James P. Muirhead, _TheOrigin 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 asynthesis that is very little short of incredible. In order to make thelinkage attached to the beam of his engines more compact, Watt hadplumbed his experience for ideas; his experience had yielded up the workdone much earlier on a drafting machine that made use of apantograph. [20] Watt combined his straight-line linkage with apantograph, one link becoming a member of the pantograph. 

[Footnote 20: "It has only one fault, " he had told a friend on December24, 1773, after describing the drafting machine to him, "which is, thatit will not do, because it describes conic sections instead of straightlines. " _Ibid. _, p. 71. ]

The length of each oscillating link of the straight-line linkage wasthus reduced to one-fourth instead of one-half the beam length, and theentire mechanism could be constructed so that it would not extendbeyond the end of the working beam. This arrangement soon came to beknown as Watt's "parallel motion" (fig. 10). [21] Years later Watt toldhis son: "Though I am not over anxious after fame, yet I am more proudof the parallel motion than of any other mechanical invention I haveever made. "[22]

[Footnote 21: Throughout the 19th century the term "parallel motion" wasused indiscriminately to refer to any straight-line linkage. I have notdiscovered the origin of the term. Watt did not use it in his patentspecification, and I have not found it in his writings or elsewherebefore 1808 (see footnote 22). _The Cyclopaedia_ (Abraham Rees, ed. , London, 1819, vol. 26) defined parallel motion as "a term used amongpractical mechanics to denote the rectilinear motion of a piston-rod, &c. In the direction of its length; and contrivances, by which suchalternate rectilinear motions are converted into continuous rotatoryones, or _vice versa_. . . . " Robert Willis in his _Principles ofMechanism_ (London, 1841, p. 399) described parallel motion as "a termsomewhat awkwardly applied to a combination of jointed rods, the purposeof 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: "Ihave 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 itcertainly does not express what is intended. The expression, however, has now become crystallised, and I for one cannot undertake to find asolvent. "]

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

[Illustration: Figure 10. --Watt's "parallel motion. " Engine's workingbeam is pivoted at _A_. Pivot _F_ is attached to the engine frame. FromDyonysius 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 bythe American Charles B. Richards when, in 1861, he designed his firsthigh-speed engine indicator (fig. 11). Introduced into England thefollowing year, the Richards Indicator was an immediate success, andmany 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 thestraight-line linkage and to combine it with a pantograph, it should bekept in mind that this was the first one of a long line of suchmechanisms. [24] Once the idea was abroad, it was only to be expectedthat many variations and alternative solutions should appear. Onewonders, however, what direction the subsequent work would have takenif Watt had not so clearly pointed the way. 

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

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

In Farey's opinion Watt's inventive faculty was far superior to that ofany of his contemporaries; but his many and various ideas would havebeen of little use if he had not possessed a very high order ofjudgment, that "faculty of distinguishing between ideas; decomposingcompound 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 producebrilliant new ideas, still the "common stock of ideas which are currentamongst communities and professions, will generally prove to be of abetter quality than the average of those new ideas, which can beproduced by any individual from the operation of his own mind, withoutassistance from others. " Farey concluded with the observation that "themost useful additions to that common stock, usually proceed from theindividuals 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 parallelmotion, the problem of devising a linkage, one point of which woulddescribe a straight line, was one that tickled the fancies ofmathematicians, of ingenious mechanics, and of gentlemanly dabblers inideas. The quest for a straight-line mechanism more accurate than thatof Watt far outlasted the pressing practical need for such a device. Large metal planing machines were well known by 1830, and by midcenturycrossheads and crosshead guides were used on both sides of the Atlanticin engines with and without working beams. 

By 1819 John Farey had observed quite accurately that, in England atleast, many other schemes had been tried and found wanting and that "nomethods have been found so good as the original engine; and weaccordingly find, that all the most established and experiencedmanufacturers make engines which are not altered in any great featurefrom Mr. Watt's original engine. . . . "[26]

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

Two mechanisms for producing a straight line were introduced before theBoulton and Watt monopoly ended in 1800. Perhaps the first was by EdmundCartwright (1743-1823), who is said to have had the original idea for apower loom. This geared device (fig. 12), was characterizedpatronizingly by a contemporary American editor as possessing "as muchmerit as can possibly be attributed to a gentleman engaged in thepursuit of mechanical studies for his own amusement. "[27] Only a fewsmall 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 mechanismof about 1800. From Abraham Rees, _The Cyclopaedia_ (London, 1819, "Steam Engine, " pl. 5). ]

The properties of a hypocycloid were recognized by James White, anEnglish engineer, in his geared design which employed a pivot located onthe pitch circle of a spur gear revolving inside an internal gear. Thediameter of the pitch circle of the spur gear was one-half that of theinternal gear, with the result that the pivot, to which the piston rodwas connected, traced out a diameter of the large pitch circle (fig. 13). White in 1801 received from Napoleon Bonaparte a medal for thisinvention when it was exhibited at an industrial exposition inParis. [29] Some steam engines employing White's mechanism were built, but without conspicuous commercial success. White himself rather agreedthat while his invention was "allowed to possess curious properties, andto 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 ofInventions, '" _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-linemechanism, about 1800. The fly-weights (at the ends of the diagonal arm)functioned as a flywheel. From James White, _A New Century ofInventions_ (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, althoughit came to be associated with the name of Oliver Evans, the Americanpioneer in the employment of high-pressure steam. A similar idea, employing an isosceles linkage, was patented in 1803 by WilliamFreemantle, an English watchmaker (fig. 14). [31] This is the linkagethat was attributed much later to John Scott Russell (1808-1882), theprominent naval architect. [32] An inconclusive hint that Evans haddevised his straight-line linkage by 1805 appeared in a plateillustrating 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, inmodified form, appeared in Rees's _Cyclopaedia_ of 1819 (fig. 16), butit is doubtful whether even this would have been readily recognized asidentical with the Evans linkage, because the connecting rod was at theopposite end of the working beam from the piston rod, in accordance withestablished usage, while in the Evans linkage the crank and connectingrod were at the same end of the beam. It is possible that Evans got hisidea from an earlier English periodical, but concrete evidence islacking. 

[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, latercalled 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 iskinematically the same as the Evans linkage. Pivots _D_ and _E_ areattached 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 didnot mention it in his patent claims, or in the descriptions that hepublished of his engines. [33] The practical advantage of the Evanslinkage, utilizing as it could a much lighter working beam than the Wattor Freemantle engines, would not escape Oliver Evans, and he was not aman 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 wasattributed to Richard Roberts of Manchester (1789-1864), who around 1820had built one of the first metal planing machines, which machines helpedmake the quest for straight-line linkages largely academic. I have notdiscovered what occasioned the introduction of the Roberts linkage, butit dated from before 1841. Although Roberts patented many complextextile machines, an inspection of all of his patent drawings has failedto 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) creditedRichard Roberts with the linkage. Roberts' 15 British patent drawingsexhibit complex applications of cams, levers, guided rods, cords, and soforth, but no straight-line mechanism. In his patent no. 6258 of April13, 1832, for a steam engine and locomotive carriage, Roberts usedWatt's "parallel motion" on a beam driven by a vertical cylinder. ]

[Footnote 35: This engraving appeared as plate 11 in Pierre Patte's 1769work (_op. Cit. _ footnote 24). Patte stated that the machine depicted inhis plate 11 was invented by M. De Voglie and was actually used in1756. ]

[Illustration: Figure 17. --Straight-line linkage (before 1841)attributed to Richard Roberts by Robert Willis. From A. B. Kempe, _Howto 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_, publishedseveral years later, did not employ the straight-line linkage. FromPierre 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 wentnearly unnoticed. A decade later, when news of its invention crossedthe Channel to England, this linkage excited a flurry of interest, andvariations of it occupied mathematical minds for several years. For atleast 10 years before and 20 years after the final solution of theproblem, Professor Chebyshev, [36] a noted mathematician of theUniversity of St. Petersburg, was interested in the matter. Judging byhis published works and his reputation abroad, Chebyshev's interestamounted to an obsession. 

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

Pafnutï[)i] L'vovich Chebyshev was born in 1821, near Moscow, andentered the University of Moscow in 1837. In 1853, after visiting Franceand England and observing carefully the progress of applied mechanics inthose countries, he read his first paper on approximate straight-linelinkages, and over the next 30 years he attacked the problem with newvigor at least a dozen times. He found that the two principalstraight-line linkages then in use were Watt's and Evans'. Chebyshevnoted the departure of these linkages from a straight line andcalculated the deviation as of the fifth degree, or about 0. 0008 inchper inch of beam length. He proposed a modification of the Watt linkageto refine its accuracy but found that he would have to more than doublethe length of the working beam. Chebyshev concluded ruefully that hismodification 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 toapproach if not quite attain a true straight line. If one mechanism wasgood, he reasoned, two would be better, _et cetera, ad infinitum_. Theidea was simply to combine, or compound, four-link approximate linkages, arranging them in such a way that the errors would be successivelyreduced. Contemplating first a combination of the Watt and Evanslinkages (fig. 19), Chebyshev recognized that if point D of the Wattlinkage followed nearly a straight line, point A of the Evans linkagewould depart even less from a straight line. He calculated the deviationin this case as of the 11th degree. He then replaced Watt's linkage byone that is usually called the Chebyshev straight-line mechanism (fig. 20), with the result that precision was increased to the 13thdegree. [38] The steam engine that he displayed at the Vienna Exhibitionin 1873 employed this linkage--the Chebyshev mechanism compounded withthe Evans, or approximate isosceles, linkage. An English visitor to theexhibition commented that "the motion is of little or no practical use, for we can scarcely imagine circumstances under which it would be moreadvantageous to use such a complicated system of levers, with so manyjoints to be lubricated and so many pins to wear, than a solid guide ofsome kind; but at the same time the arrangement is very ingenious and inthis 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-linemechanisms. From _Ouvres de P. L. Tchebychef_ (St. Petersburg, 1907, vol. 2, frontispiece). ]

[Illustration: Figure 20. --Chebyshev's combination (about 1867) ofWatt's and Evans' linkages to reduce errors inherent in each. Points_C_, _C'_, and _C"_ are fixed; _A_ is the tracing point. From _Oeuvresde 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 todemonstrate the impossibility of constructing any linkage, regardless ofthe number of links, that would generate a straight line; but I havefound only a dubious statement in the _Grande Encyclopédie_[40] of thelate 19th century and a report of a conversation with the Russian by anEnglishman, James Sylvester, to the effect that Chebyshev had "succeededin proving the nonexistence of a five-bar link-work capable of producinga perfect parallel motion. . . . "[41] Regardless of what tradition may haveto say about what Chebyshev said, it is of course well known thatCaptain Peaucellier was the man who finally synthesized the exactstraight-line mechanism that bears his name. 

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

[Footnote 41: James Sylvester, "Recent Discoveries in MechanicalConversion of Motion, " _Notices of the Proceedings of the RoyalInstitution of Great Britain_, 1873-1875, vol. 7, p. 181. The fixed linkwas not counted by Sylvester; in modern parlance this would be asix-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. --JamesJoseph Sylvester (1814-1897), mathematician and lecturer onstraight-line linkages. From _Proceedings of the Royal Society ofLondon_ (1898, vol. 63, opposite p. 161). ]

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

This letter stirred no pens in reply, and during the next 10 years theproblem merely led to the filling of a few academic pages by Peaucellierand 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 readersof the _Nouvelles Annales_. His reasoning, which has a distinct flavorof discovery by hindsight, was that since a linkage generates a curvethat can be expressed algebraically, it must follow that any algebraiccurve can be generated by a suitable linkage--it was only necessary tofind 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 degeométrie 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, ChelseaPublishing Company, 1960. ]

On a Friday evening in January 1874 Albemarle Street in London wasfilled with carriages, each maneuvering to unload its charge ofgentlemen and their ladies at the door of the venerable hall of theRoyal Institution. Amidst a "mighty rustling of silks, " the elegantcrowd made its way to the auditorium for one of the famous weeklylectures. The speaker on this occasion was James Joseph Sylvester, asmall intense man with an enormous head, sometime professor ofmathematics at the University of Virginia, in America, and more recentlyat the Royal Military Academy in Woolwich. He spoke from the samerostrum that had been occupied by Davy, Faraday, Tyndall, Maxwell, andmany other notable scientists. Professor Sylvester's subject was "RecentDiscoveries in Mechanical Conversion of Motion. "[43]

[Footnote 43: Sylvester, _op. Cit. _ (footnote 41), pp. 179-198. Itappears from a comment in this lecture that Sylvester was responsiblefor the word "linkage. " According to Sylvester, a linkage consists of aneven number of links, a "link-work" of an odd number. Since the fixedmember was not considered as a link by Sylvester, this distinctionbecame utterly confusing when Reuleaux's work was published in 1876. Although "link" was used by Watt in a patent specification, it is notprobable that he ever used the term "link-work"--at any rate, my searchfor his use of it has been fruitless. "Link work" is used by Willis(_op. Cit. _ footnote 21), but the term most likely did not originatewith him. I have not found the word "linkage" used earlier thanSylvester. ]

Remarking upon the popular appeal of most of the lectures, acontemporary observer noted that while many listeners might prefer tohear Professor Tyndall expound on the acoustic opacity of theatmosphere, "those of a higher and drier turn of mind experienceineffable delight when Professor Sylvester holds forth on the conversionof 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 ofthe English-speaking world, as it had been brought to his attention byChebyshev--during a recent visit of the Russian to England--and to givehis listeners some insight into the vastness of the field that he sawopened 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, " heobserved, "and moves so easily that people who see it at work almostuniversally express astonishment that it waited so long to bediscovered. " But that was not his reaction at all. The more one reflectsupon the problem, Sylvester continued, he "wonders the more that it wasever found out, and can see no reason why it should have beendiscovered for a hundred years to come. Viewed _a priori_ there wasnothing to lead up to it. It bears not the remotest analogy (except inthe fact of a double centring) to Watt's parallel motion or any of itsprogeny. "[46]

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

It must be pointed out, parenthetically at least, that James Watt hadnot only had to solve the problem as best he could, but that he had noinkling, so far as experience was concerned, that a solvable problemexisted. 

Sylvester interrupted his panegyric long enough to enumerate some of thepractical results of the Peaucellier linkage. He said that Mr. Penrose, the eminent architect and surveyor to St. Paul's Cathedral, had "put upa house-pump worked by a negative Peaucellier cell, to the greatwonderment of the plumber employed, who could hardly believe his senseswhen he saw the sling attached to the piston-rod moving in a truevertical line, instead of wobbling as usual from side to side. "Sylvester could see no reason "why the perfect parallel motion shouldnot be employed with equal advantage in the construction of ordinarywater-closets. " The linkage was to be employed by "a gentleman offortune" in a marine engine for his yacht, and there was talk of usingit to guide a piston rod "in certain machinery connected with some newapparatus for the ventilation and filtration of the air of the Houses ofParliament. " In due course, Mr. Prim, "engineer to the Houses, " waspleased to show his adaptation of the Peaucellier linkage to his newblowing engines, which proved to be exceptionally quiet in theiroperation (fig. 25). [47] A bit on the ludicrous side, also, wasSylvester's 78-bar linkage that traced a straight line along the lineconnecting 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 forventilating the House of Commons, 1877. The crosshead of thereciprocating air pump is guided by a Peaucellier linkage shown at thecenter. The slate-lined air cylinders had rubber-flap inlet and exhaustvalves and a piston whose periphery was formed by two rows of brushbristles. Prim's machine was driven by a steam engine. Photograph byScience Museum, London. ]

Before dismissing with a smile the quaint ideas of our Victorianforbears, however, it is well to ask, 88 years later, whether somerather elaborate work reported recently on the synthesis ofstraight-line mechanisms is more to the point, when the principalobjective 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 mathematicalpossibilities of the Peaucellier linkage, as no doubt our moderninvestigators are. Through a compounding of Peaucellier mechanisms, hehad already devised square-root and cube-root extractors, an angletrisector, and a quadratic-binomial root extractor, and he could see nolimits 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 experiencewith a little mechanical model of the Peaucellier linkage at an earlierdinner meeting of the Philosophical Club of the Royal Society. ThePeaucellier model had been greeted by the members with livelyexpressions of admiration "when it was brought in with the dessert, tobe seen by them after dinner, as is the laudable custom among members ofthat eminent body in making known to each other the latest scientificnovelties. " And Sylvester would never forget the reaction of hisbrilliant friend Sir William Thomson (later Lord Kelvin) upon beinghanded the same model in the Athenaeum Club. After Sir William hadoperated it for a time, Sylvester reached for the model, but he wasrebuffed by the exclamation "No! I have not had nearly enough of it--itis 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 RoyalInstitution was considerable excitement amongst a limited company ofinterested mathematicians. Many alternatives to the Peaucellierstraight-line linkage were suggested by several writers of papers forlearned 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 exactstraight-line linkages referred to by Kempe are illustrated in Henry M. Cundy and A. P. Rollett, _Mathematical Models_, Oxford, OxfordUniversity Press, 1952, pp. 204-205. Peaucellier's linkage was of eightlinks. ]

In the summer of 1876, after Sylvester had departed from England to takeup his post as professor of mathematics in the new Johns HopkinsUniversity in Baltimore, Alfred Bray Kempe, a young barrister whopursued mathematics as a hobby, delivered at London's South KensingtonMuseum a lecture with the provocative title "How to Draw a StraightLine. "[53]

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

In order to justify the Peaucellier linkage, Kempe belabored the pointthat a perfect circle could be generated by means of a pivoted bar and apencil, while the generation of a straight line was most difficult ifnot impossible until Captain Peaucellier came along. A straight linecould be drawn along a straight edge; but how was one to determinewhether the straight edge was straight? He did not weaken his argumentby suggesting the obvious possibility of using a piece of string. Kempehad collaborated with Sylvester in pursuing the latter's first thoughtson the subject, and one result, that to my mind exemplifies the generaldirection 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 ofmathematics at the Ecole Polytechnique from 1794 and founder of theacademic 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 inhis lecture. "That these results are valuable cannot I think bedoubted, " he said, "though it may well be that their great beauty hasled some to attribute to them an importance which they do not reallypossess. . . . " He went on to say that 50 years earlier, before the greatimprovements in the production of true plane surfaces, the straight-linemechanisms would have been more important than in 1876, but he addedthat "linkages have not at present, I think, been sufficiently putbefore the mechanician to enable us to say what value should really beset upon them. "[54]

[Footnote 54: _Ibid. _, pp. 6-7. I have not pursued the matter of cognatelinkages (the Watt and Evans linkages are cognates) because theRoberts-Chebyshev theorem escaped my earlier search, as it hadapparently escaped most others until 1958. See R. S. Hartenberg and J. Denavit, "The Fecund Four-Bar, " _Transactions of the Fifth Conference onMechanisms_, Cleveland, Penton Publishing Company, 1958, pp. 194-206, reprinted in _Machine Design_, April 16, 1959, vol. 31, pp. 149-152. Seealso A. E. R. De Jonge, "The Correlation of Hinged Four-BarStraight-Line Motion Devices by Means of the Roberts Theorem and a NewProof 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 ofScientific Apparatus in the South Kensington Museum, that the work ofFranz Reuleaux, which was to have an important and lasting influence onkinematics everywhere, was first introduced to English engineers. Some300 beautifully constructed teaching aids, known as the Berlin kinematicmodels, were loaned to the exhibition by the Royal Industrial School inBerlin, of which Reuleaux was the director. These models were used byProf. Alexander B. W. Kennedy of University College, London, to helpexplain 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 preparea course in theoretical and experimental physics for the Collège deFrance, he first set about determining the limits of the field ofphysics. This exercise suggested to his wide-ranging intellect not onlythe definition of physics but the classification of all human knowledge. He prepared his scheme of classification, tried it out on his physicsstudents, found it incomplete, returned to his study, and producedfinally a two-volume work wherein the province of kinematics was firstmarked out for all to see and consider. [56] Only a few lines could bedevoted to so specialized a branch as kinematics, but Ampère managed tocapture the central idea of the subject. 

[Footnote 56: André-Marie Ampère, _Essai sur la philosophie dessciences, une exposition analytique d'une classification naturelle detoutes les connaissances humaines_, 2 vols. , Paris, 1838 (for origin ofthe 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 insolid bodies all about us, and especially in the assemblages calledmachines. "[57] Kinematics, as the study soon came to be known inEnglish, [58] was one of the two branches of elementary mechanics, theother 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 standardterm for this branch of mechanics. ]

In his definition of kinematics, Ampère stated what the faculty ofmathematics at the Ecole Polytechnique, in Paris, had been gropingtoward since the school's opening some 40 years earlier. The study ofmechanisms as an intellectual discipline most certainly had its originon the left bank of the Seine, in this school spawned, as suggested byone French historian, [59] by the great _Encyclopédie_ of Diderot andd'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 JacobLeupold and Leonhard Euler of the 18th century. ]

Because the Ecole Polytechnique had such a far-reaching influence uponthe point of view from which mechanisms were contemplated by scholarsfor nearly a century after the time of Watt, and by compilers ofdictionaries of mechanical movements for an even longer time, it iswell to look for a moment at the early work that was done there. If oneis interested in origins, it might be profitable for him to investigatethe military school in the ancient town of Mézières, about 150 milesnortheast of Paris. It was here that Lazare Carnot, one of the principalfounders of the Ecole Polytechnique, in 1783 published his essay onmachines, [60] which was concerned, among other things, with showing theimpossibility of "perpetual motion"; and it was from Mézières thatGaspard Monge and Jean Hachette[61] came to Paris to work out the systemof mechanism classification that has come to be associated with thenames of Lanz and Bétancourt. 

[Footnote 60: Lazare N. M. Carnot, _Essai sur les machines en général_, Mézières, 1783 (later published as _Principes fondamentaux del'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 originatedthe methods of descriptive geometry, came to the Ecole Polytechnique asprofessor of mathematics upon its founding in 1794, the second year ofthe French Republic. According to Jean Nicolas Pierre Hachette(1769-1834), who was junior to Monge in the department of descriptivegeometry, Monge planned to give a two-months' course devoted to theelements of machines. Having barely gotten his department under way, however, Monge became involved in Napoleon's ambitious scientificmission 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 whichMonge had given only the first idea, and I pursued the study of machinesin order to analyze and classify them, and to relate geometrical andmechanical principles to their construction. " Changes of curriculumdelayed introduction of the course until 1806, and not until 1811 washis textbook ready, but the outline of his ideas was presented to hisclasses in chart form (fig. 28). This chart was the first of the widelypopular 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 elementarymechanisms, 1808. This was the first of many charts of mechanicalmovements 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 onemotion into another. His elementary motions were continuous circular, alternating circular, continuous rectilinear, and alternatingrectilinear. Combining one motion with another--for example, a treadleand crank converted alternating circular to continuous circularmotion--he devised a system that supplied a frame of reference for thestudy of mechanisms. In the U. S. Military Academy at West Point, Hachette's treatise, in the original French, was used as a textbook in1824, and perhaps earlier. [63]

[Footnote 63: This work was among the books sent back by Sylvanus Thayerwhen he visited France in 1816 to observe the education of the Frencharmy cadets. Thayer's visit resulted in his adopting the philosophy ofthe Ecole Polytechnique in his reorganization of the U. S. MilitaryAcademy and, incidentally, in his inclusion of Hachette's course in theAcademy's curriculum (U. S. Congress, _American State Papers_, Washington, 1832-1861, Class v, Military Affairs, vol. 2, p. 661: SidneyForman, _West Point_, New York, 1950, pp. 36-60). There is a collectionof miscellaneous papers (indexed under Sylvanus Thayer and WilliamMcRee, U. S. National Archives, RG 77, Office, Chief of Engineers, Boxes1 and 6) pertaining to the U. S. Military Academy of this period, but Ifound 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 andalternating curvilinear motion, which doubled the number of combinationsto be treated, but the advance of their work over that of Hachette wasone of degree rather than of kind. [64]

[Footnote 64: Phillipe Louis Lanz and Augustin de Bétancourt, _Essai surla composition des machines_, Paris, 1808. Hachette's chart and anoutline of his elementary course on machines is bound with the PrincetonUniversity Library copy of the Lanz and Bétancourt work. This copyprobably represents the first textbook of kinematics. Bétancourt wasborn in 1760 in Teneriffe, attended the military school in Madrid, andbecame inspector-general of Spanish roads and canals. He was in Englandbefore 1789, learning how to build Watt engines, and he introduced theengines to Paris in 1790 (see Farey, _op. Cit. _, p. 655). He enteredRussian 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), JacksonianProfessor, Cambridge University, and author of _Principles ofMechanism_, one of the landmark books in the development of kinematicsof mechanisms. Photo courtesy Gonville and Caius College, CambridgeUniversity. ]

Giuseppe Antonio Borgnis, an Italian "engineer and member of manyacademies" and professor of mechanics at the University of Pavia inItaly, in his monumental, nine-volume _Traité complet de méchaniqueappliquée aux arts_, caused a bifurcation of the structure built uponHachette's foundation of classification when he introduced six orders ofmachine elements and subdivided these into classes and species. His sixorders 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 mécaniqueusuelle_ in _Traité complet de mécanique appliquée aux arts_, Paris, 1818, vol. 1, pp. Xiv-xvi. ]

The brilliant Gaspard-Gustave de Coriolis (1792-1843)--remembered mainlyfor a paper of a dozen pages explaining the nature of the accelerationthat bears his name[66]--was another graduate of the Ecole Polytechniquewho wrote on the subject of machines. His book, [67] published in 1829, was provoked by his recognition that the designer of machines neededmore knowledge than his undergraduate work at the Ecole Polytechniquewas 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 concernedwith forces than with relative displacements. However, the attractivelysimple three-element scheme of Coriolis became well fixed in Frenchthinking. [68]

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

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

[Footnote 68: The renowned Jean Victor Poncelet lent weight to thisscheme. (See Franz Reuleaux, _Theoretische Kinematik: Grundzüge einerTheorie des Maschinenwesens_, Braunschweig, 1875, translated byAlexander B. W. Kennedy as _The Kinematics of Machinery: Outlines of aTheory of Machines_, London, 1876, pp. 11, 487. I have used the Kennedytranslation 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 inkinematic analysis was not recognized until much later. 

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

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

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

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

[Footnote 71: Willis, _op. Cit. _ (footnote 21). Through the kindness ofits owner (Mr. Warren G. Ogden of North Andover, Massachusetts), I havehad access to Willis' own copy of his 1841 edition of _Principles ofMechanism_. The book is interleaved, and it contains notes made byWillis from time to time until at least 1870, when the second editionwas 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 modernforms" of straight-line devices are interspersed with references toauthors that had borrowed from his work without acknowledgment. Of oneauthor Willis writes an indignant "He ignores my work. "]

It seemed clear to Willis that the problem of devising a mechanism for agiven purpose ought to be attacked systematically, perhapsmathematically, in order to determine "all the forms and arrangementsthat are applicable to the desired purpose, " from which the designermight select the simplest or most suitable combination. "At present, " hewrote, "questions of this kind can only be solved by that species ofintuition which long familiarity with a subject usually confers uponexperienced persons, but which they are totally unable to communicate toothers. "

In analyzing the process by which a machine was designed, Willisobserved: "When the mind of a mechanician is occupied with thecontrivance of a machine, he must wait until, in the midst of hismeditations, some happy combination presents itself to his mind whichmay answer his purpose. " He ventured the opinion that at this stage ofthe design process "the motions of the machine are the principal subjectof contemplation, rather than the forces applied to it, or the work ithas to do. " Therefore he was prepared to adopt without reservationAmpère's view of kinematics, and, if possible, to make the scienceuseful to engineers by stating principles that could be applied withouthaving to fit the problem at hand into the framework of the systems ofclassification and description that had gone before. He appraised the"celebrated system" of Lanz and Bétancourt as "a merely populararrangement, notwithstanding the apparently scientific simplicity of thescheme. " He rejected this scheme because "no attempt is made to subjectthe 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 describingmachinery, with his "orders" based upon the functions of machineelements or mechanisms within the machine, but again there was no meanssuggested by which the kinematics of mechanisms could be systematicallyinvestigated. 

Although Willis commenced his treatise with yet another "synopticaltable of the elementary combinations of pure mechanism, " his viewshifted quickly from description to analysis. He was consistent in hispursuit of analytical methods for "pure mechanism, " eschewing anyexcursions into the realm of forces and absolute velocities. He graspedthe important concept of relative displacements of machine elements, andbased his treatment upon "the proportions and relations between thevelocities and directions of the pieces, and not upon their actual andseparate motions. "[72]

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

That he did not succeed in developing the "formulas" that would enablethe student to determine "all the forms and arrangements that areapplicable to the desired purpose"--that he did not present a rationalapproach to synthesis--is not to be wondered at. Well over a centurylater we still are nibbling at the fringes of the problem. Willis did, nonetheless, give the thoughtful reader a glimpse of the most powerfultool for kinematic synthesis that has yet been devised; namely, kinematic analysis, in which the argument is confined to the relativedisplacements of points on links of a mechanism, and through which thedesigner may grasp the nature of the means at his disposal for thesolution of any particular problem. 

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

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

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

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

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

Many other books dealing with mechanisms were published during themiddle third of the century, but none of them had a discernibleinfluence upon the advance of kinematical ideas. [74] The center ofinquiry had by the 1860's shifted from France to Germany. Only byscattered individuals in England, Italy, and France was there anyimpatience with the well-established, general understanding of themachine-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 activitythat attracted some very able men to the problems of how machines oughtto be built. Among the first of these was Ferdinand Redtenbacher(1809-1863), professor of mechanical engineering in the polytechnicschool in Karlsruhe, not far from Heidelberg. Redtenbacher, although hedespaired of the possibility of finding a "true system on which to basethe study of mechanisms, " was nevertheless a factor in the developmentof such a system. He had young Franz Reuleaux in his classes for twoyears, from 1850. During that time the older man's commanding presence, his ability as a lecturer, and his infectious impatience with theexisting order influenced Reuleaux to follow the scholar's trail thatled him to eminence as an authority of the first rank. [75]

[Footnote 75: See Carl Weihe, "Franz Reuleaux und die Grundlagen seinerKinematik, " 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, CharlesScribner's Sons, 1959, p. 317. ]

Before he was 25 years old Franz Reuleaux published, in collaborationwith a classmate, a textbook whose translated title would be_Constructive Lessons for the Machine Shop_. [76] His several years inthe workshop, before and after coming under Redtenbacher's influence, gave his works a practical flavor, simple and direct. According to oneobserver, Reuleaux's book exhibited "a recognition of the claims ofpractice such as Englishmen do not generally associate with the writingsof 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 MechanicalEngineers_, 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 theway in which we look at mechanisms today, were sufficiently formed in1864 for him to lecture upon them. [78] Starting in 1871, he publishedhis findings serially in the publication of the Verein zur Beförderungdes Gewerbefleisses in Preussen (Society for the Advancement of Industryin Prussia), of which he was editor. In 1875 these articles were broughttogether in the book that established his fame--_TheoretischeKinematik. . . . _[79]

[Footnote 78: A. E. Richard de Jonge, "What is Wrong with Kinematics andMechanisms?" _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 lastof Reuleaux's books. His trilogy on kinematics and machine design isdiscussed 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 bringorder out of confusion was well founded. His book had already beentranslated into Italian and was being translated into French when, onlya 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 themechanician would find in it many new ideas, that he would be "taught todetect hitherto hidden resemblances, and that he must part--reluctantly, perhaps--with many of his old notions. " "But, " added the editor withconsiderable justice, "that he [the mechanician] would suddenlyrecognize in Professor Reuleaux's 'kinematic notation, ' 'analysis, ' and'synthesis, ' the long-felt want of his professional existence we do notfor a moment believe. "[83] Indeed, the fresh and sharp ideas of Reuleauxwere somewhat clouded by a long (600-page) presentation; and hiskinematic 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 theFirst Conference on Mechanisms at Purdue University was Allen S. Hall's"Mechanisms and Their Classification, " which appeared in _MachineDesign_, December 1953, vol. 25, pp. 174-180. The place ofclassification in kinematic synthesis is suggested in FerdinandFreudenstein's "Trends in Kinematics of Mechanisms, " _Applied MechanicsReviews_, September 1959, vol. 12, pp. 587-590. ]

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

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

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

The chapter on synthesis was likewise fresh, but it consisted of adiscussion, not a system; and Reuleaux stressed the idea that I havementioned above in connection with Willis' book, that synthesis will besuccessful in proportion to the designer's understanding andappreciation of analysis. Reuleaux tried to put the designer on theright track by showing him clearly "the essential simplicity of themeans with which we have to work" and by demonstrating to him "that themany things which have to be done can be done with but few means, andthat 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) andRobert Henry Smith (1852-1916) to add to Reuleaux's work the elementsthat would give kinematic analysis essentially its modern shape. 

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

While Reuleaux had used instant centers almost exclusively for theconstruction of centrodes (paths of successive positions of an instantcenter), Professor Kennedy recognized that instant centers might be usedin velocity analysis. His book, _Mechanics of Machinery_, was publishedin 1886 ("partly through pressure of work and partly through ill-health, this book appears only now"). In it he developed the law of threecenters, now known as Kennedy's theorem. He noted that his law of threecenters "was first given, I believe, by Aronhold, although its previouspublication was unknown to me until some years after I had given it inmy lectures. "[86] In fact, the law had been published by SiegfriedHeinrich Aronhold (1819-1884) in his "Outline of Kinematic Geometry, "which appeared in 1872 alongside Reuleaux's series in the journal thatReuleaux edited. Apparently Reuleaux did not perceive its particularsignificance 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 KinematicGeometry, " _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 ofvelocity and acceleration polygons for kinematic analysis. Photocourtesy the Librarian, Birmingham Reference Library, England. ]

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

Professor Smith, in a paper before the Royal Society of Edinburgh in1885, stated clearly the ideas and methods for construction of velocityand acceleration diagrams of linkages. [88] For the first time, velocityand acceleration "images" of links (fig. 33) were presented. It isunfortunate that Smith's ideas were permitted to languish for so long atime. 

[Footnote 88: Robert H. Smith, "A New Graphic Analysis of the Kinematicsof Mechanisms, " _Transactions of the Royal Society of Edinburgh_, 1882-1885, vol. 32, pp. 507-517, and pl. 82. Smith used this paper asthe basis for a chapter in his _Graphics or the Art of Calculating byDrawing 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 graphicaldifferentiation, Smith complained that he had "failed to find anypractical use" for Reuleaux's "method of centroids, more properly calledaxoids. " Such statements were not calculated to encourage Kennedy andReuleaux to advertise Smith's fame; however, I found no indication thateither one took offense at the criticism. Smith's velocity andacceleration diagrams were included (apparently embalmed, so far asAmerican engineers were concerned) in _Encyclopaedia Britannica_, ed. 11, 1910, vol. 17, pp. 1008-1009. ]

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

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

Mechanicians and Mechanisms

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

After the demonstration by Boulton and Watt that large mechanisms couldbe wrought with sufficient precision to be useful, the English toolbuilders Maudslay, Roberts, Clement, Nasmyth, and Whitworth developedmachine tools of increasing size and truth. The design of othermachinery kept pace with--sometimes just behind, sometimes just aheadof--the capacity and capability of machine tools. In general, there wasan increasing sophistication of mechanisms that could only be accountedfor by an increase of information with which the individual designercould start. 

Reuleaux pointed out in 1875 that the "almost feverish progress made inthe regions of technical work" was "not a consequence of any increasedcapacity for intellectual action in the race, but only the perfectingand extending of the tools with which the intellect works. " These tools, he said, "have increased in number just like those in the modernmechanical workshop--the men who work them remain the same. " Reuleauxwent on to say that the theory and practice of machine-kinematics had"carried on a separate existence side by side. " The reason for thisfailure to apply theory to practice, and vice versa, must be sought inthe defects of the theory, he thought, because "the mechanismsthemselves have been quietly developed in practical machine-design, byinvention and improvement, regardless of whether or not they wereaccorded any direct and proper theoretical recognition. " He pointed outthat 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 theappearance of new mechanisms, and then to see what sort of mechanismshad their origins in this period. 

It is immediately evident to a designer that the progress in mechanismscame about through the spread of knowledge of what had already beendone; but designers of the last century had neither the leisure normeans to be constantly visiting other workshops, near and far, toobserve and study the latest developments. In the 1800's, as now, wordmust in the main be spread by the printed page. 

Hachette's chart (fig. 28) had set the pattern for display of mechanicalcontrivances in practical journals and in the large number of mechanicaldictionaries that were compiled to meet an apparent demand for suchinformation. It is a little surprising, however, to find how persistentwere some of Hachette's ideas that could only have come from theuppermost superficial layer of his cranium. See, for example, his"anchored ferryboat" (fig. 34). This device, employed by Hachette toshow conversion of continuous rectilinear motion into alternatingcircular motion, appeared in one publication after another throughoutthe 19th century. As late as 1903 the ferryboat was still anchored inHiscox'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 1917edition. ]

[Illustration: Figure 34. --Hachette's ferryboat of 1808, a "machine" forconverting continuous rectilinear motion into alternating circularmotion. From Phillipe Louis Lanz and Augustin de Bétancourt, _Essai surla 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--movementin the 1820's, Jacob Bigelow, Rumford professor of applied science atHarvard University, gave his popular lectures on the "Elements ofTechnology" before capacity audiences in Boston. In preparing hislecture on the elements of machinery, Bigelow used as his authoritiesHachette, Lanz and Bétancourt, and Olinthus Gregory's mechanicaldictionary, an English work in which Hachette's classification schemewas 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 about1820 at London by Rudolph Ackermann (for whom the Ackermann steeringlinkage was named), and their synoptic chart was reprinted again in 1822in Durham. [93] In the United States, _Appleton's Dictionary ofMachines_[94] (1851) adopted the same system and used the same figures. Apparently the wood engraver traced directly onto his block the figuresfrom one of the reprints of Lanz and Bétancourt's chart because thefigures are in every case exact mirror images of the originals. 

[Footnote 92: Rudolph Ackermann, _Analytical Essay on the Constructionof 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 wereredrawn and dozens of mechanisms were added to the repertory ofmechanical motions; the result was a fair catalog of sound ideas. Theferryboat still tugged at its anchor cable, however. [96] _Knight'sAmerican Mechanical Dictionary_, [97] a classic of detailed pictorialinformation compiled by a U. S. Patent examiner, contained well over10, 000 finely detailed figures of various kinds of mechanicalcontrivances. Knight did not have a separate section on mechanisms, butthere was little need for one of the Hachette variety, because his wholedictionary was a huge and fascinating compendium of ideas to be filedaway in the synthetic mind. One reason for the popularity and usefulnessof the various pictorial works was the peculiar ability of a wood orsteel engraving to convey precise mechanical information, an advantagenot possessed by modern halftone processes. 

[Footnote 95: E. F. And N. Spon, _Dictionary of Engineering_, London1873, pp. 2421-2452. ]

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

[Footnote 97: Edward H. Knight, _Knight's American MechanicalDictionary_, 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 withmechanics were available in English from the beginning of the 19thcentury, but few of them found their way into the hands of Americanmechanicians until after 1820. Oliver Evans (1755-1819) had much to sayabout "the difficulties inventive mechanics labored under for want ofpublished records of what had preceded them, and for works of referenceto help the beginner. "[98] In 1817 the _North American Review_ alsoremarked 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 journaledited by the patent promoter Rufus Porter, carried almost from itsbeginning a column or so entitled "Mechanical Movements, " in which oneor two mechanisms--borrowed from an English work that had borrowed froma French work--were illustrated and explained. The _American Artisan_began a similar series in 1864, and in 1868 it published a compilationof the series as _Five Hundred and Seven Mechanical Movements_, "embracing all those which are most important in dynamics, hydraulics, hydrostatics, pneumatics, steam engines . . . And miscellaneousmachinery. "[100] This collection went through many editions; it was lastrevived 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 MechanicalMovements_, New York, 1868. ]

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

Many readers are already well acquainted with the three volumes of_Ingenious Mechanisms for Designers and Inventors_, [102] a work thatresulted from a contest, announced by _Machinery_ (vol. 33, p. 405) in1927, in which seven prizes were offered for the seven best articles onunpublished 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 issuedthroughout the middle decades of the 19th century. A sampling of thesepatents shows that while some were for devices used in particularmachines--such as a ratchet device for a numbering machine, a lockingindex for gunmaking machinery, and a few gear trains--the great majoritywere for converting reciprocating motion to rotary motion. Even acursory examination of these patents reveals an appalling absence ofsound 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 CharlesJohnson of Amity, Illinois (fig. 37). "It is not pretended that there isany actual gain of power, " wrote Mr. Johnson; and probably he meant it. The avowed purpose of his linkage was to increase the speed of aflywheel and thus decrease its size. [103]

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

[Illustration: Figure 37. --Johnson's "converting motion, " 1841. Thelinkage causes the flywheel to make two revolutions for eachdouble-stroke of the engine piston rod B. From U. S. Patent 2295, October11, 1841. ]

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

A correspondent of the _Repertory of Patent Inventions_ made short workof this device: "There is hardly one assertion that can be supported byproof, " he wrote, "and most of them are palpable misstatements. " Thewriter 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 evincesgross ignorance either of machinery, if the patentee really believedwhat 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 weredesigned by persons who would make no use of the principles involvedeven if such principles could at that time have been clearly stated, itis a regrettable fact that worthless mechanisms often got as much spaceas sound ones in patent journals, and objections such as the one abovewere infrequent. The slanted information thus conveyed to the youngmechanician, who was just accumulating his first kinematic repertory, was at times sadly misleading. 

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

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

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

After I had learned of the drag link (as most American engineeringstudents do), I wondered for awhile, and eventually despaired of makingany 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. Thisfigure, explained Mr. Nicholson (in vol. 1, p. 32) "represents thecoupling link used by Messrs. Boulton and Watt in their portable steamengines. A, a strong iron pin, projecting from one of the arms of thefly-wheel B; D, a crank connected with the shaft C; and E, a link tocouple the pin A and the crank D together, so the motion may becommunicated to the shaft C. " So the drag link was actually a link of acoupling. Nothing could be more logical. A drag link mechanism now makessense to me. 

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

Directly related to the drag link coupling were the patents of JohnOldham (1779-1840), an Irish engineer who is remembered mainly for thecoupling that bears his name (fig. 39). His three patents, which werefor various forms of steamboat feathering paddle wheels, involvedlinkages kinematically similar to the drag link coupling, although it isquite unlikely that Oldham recognized the similarity. However, for hiswell-known coupling, which employs an inversion of the ellipticaltrammel mechanism, I have found no evidence of a patent. Probably it waspart of the machinery that he designed for the Bank of Ireland'sprinting house, of which Oldham was manager for many years. "Mr. Oldhamand 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 theexistence of the coupling. Drawings or descriptions of the banknotemachinery apparently have not been published though they probably stillexist 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 todelineate. "]

[Illustration: Figure 39. --_Top_, Original Oldham coupling built before1840, using a cross (instead of a center disk), as sketched by RobertWillis 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 asa device to permit less than a full revolution of the star wheel andthus to prevent overwinding of a watch spring. It was called Geneva stopbecause it was used in Geneva watches. The Geneva wheel mechanism, whichpermits full rotation of the star wheel and which is frequently usedfor intermittent drives, was improperly called a Geneva stop in arecent textbook probably because the logical origin of the term had beenlost. 

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

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

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

[Illustration: Figure 42. --A "Scotch" supporting the top member of astring of well-drilling tools while a section is being added, 1876. FromEdward 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 wasstill a frontier town. [107] Furthermore, the man who took out the patentdescribed himself as Rudolph Ackermann, publisher and printseller. Ithought I had the necessary clue to the linkage's origin when I noticedthat the first English translation of the Lanz and Bétancourt treatisewas published by Ackermann, but the connection finally proved to be morelogical, if less direct. Ackermann (1764-1834), son of a Bavarian coachbuilder, had spent a number of years designing coaches for Englishgentlemen in London, where he made his home. One of his more notablecommissions 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 theintroduction of the running gear of which the linkage was a part (fig. 43). The actual inventor was Ackermann's friend George Lankensperger ofMunich, coachmaker to the King of Bavaria. The advantage of being ableto turn a carriage around in a limited area without danger ofoversetting was immediately obvious, and while there was considerableopposition by English coachmakers to an innovation for which a premiumhad to be paid, the invention soon "made its way from its own intrinsicmerit, " as Ackermann predicted it would. [108]

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

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

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

The Whitworth quick-return mechanism (fig. 44) was first applied to aslotter, or vertical shaper, in 1849, and was exhibited in 1851 at theGreat Exhibition in London. [109] Willis' comments on the mechanism arereproduced 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 CharlesTomlinson, ed. , _Cyclopaedia of Useful Arts and Manufactures_, London, 1854, vol. 1, p. Cxliv. ]

[Illustration: Figure 44. --Quick-return mechanism. _Top_, Earlyrepresentation of the quick-return mechanism patented by Whitworth in1849, from William Johnson, ed. , _The Imperial Cyclopaedia of machinery_(Glasgow, about 1855, pl. 88). _Middle_, Sketch by Robert Willis fromhis copy of _Principles of Mechanism_ (London, 1841, p. 264), which"shews Whitworth dissected into a simpler form"; it is as obscure asmost subsequent attempts have been to explain this mechanism without aschematic diagram. _Bottom_, Linkage that is kinematically equivalent toWhitworth'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 late1940's with extending, refining, and sharpening the tools of analysisthat had been suggested by Willis, Rankine, Reuleaux, Kennedy, andSmith. The actual practice of kinematic synthesis went on apace, butdesigners often declined such help as the analytical methods might givethem and there was little exchange of ideas between scholars andpractitioners. 

The capability and precision of machine tools were greatly enhancedduring this period, although, with the exception of the centerlessgrinder, no significant new types of tools appeared. The machines thatwere made with machine tools increased in complexity and, with theintroduction of ideas that made mass production of complex mechanicalproducts economically feasible, there was an accelerating increase inquantity. The adoption of standards for all sorts of component partsalso had an important bearing upon the ability of a designereconomically to produce mechanisms that operated very nearly as he hopedthey would. 

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

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

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

When the faculty at Washington University in St. Louis introduced in1885 a curriculum in "dynamic engineering, " reflecting adissatisfaction with the traditional branches of engineering, kinematicswas a senior subject and was taught from Rankine's _Machinery andMillwork_. [111]

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

At Massachusetts Institute of Technology, Peter Schwamb, professor ofmachine design, put together in 1885 a set of printed notes on thekinematics of mechanisms, based on Reuleaux's and Rankine's works. Outof these notes grew one of the most durable of American textbooks, firstpublished in 1904. [112] In the first edition of this work, accelerationwas mentioned only once in passing (on p. 4). Velocities in linkageswere determined by orthogonal components transferred from link to link. Instant centers were used only to determine velocities of various pointson the same link. Angular velocity ratios were frequently noted. In thethird edition, published in 1921, linear and angular accelerations weredefined, but no acceleration analyses were made. Velocity analyses werealtered without essential change. The fourth edition (1930) wasessentially unchanged from the previous one. Treatment of velocityanalysis was improved in the fifth edition (1938) and accelerationanalysis 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 ofMechanism_, New York, 1904. In addition to the work of Reuleaux andRankine, the authors acknowledged their use of the publications ofCharles 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, andall 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, saidin their preface what has been said by many other American authors andwhat should have been said by many more. "We make little claim tooriginality of the subject-matter, " wrote Woods and Stahl, "free usehaving been made of all available matter on the subject. . . . Our claim toconsideration is based almost entirely on the manner in which thesubject has been presented. " Not content with this disclaimer, theycontinued: "There is, in fact, very little room for such originality, the ground having been almost completely covered by previouswriters. "[113]

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

The similarity and aridity of kinematics textbooks in this country fromaround 1910 are most striking. The generation of textbook writersfollowing MacCord, Woods and Stahl, Barr of Cornell, Robinson of OhioState, and Schwamb and Merrill managed to squeeze out any remainingjuice in the subject, and the dessication and sterilization of textbookswas nearly complete when my generation used them in the 1930's. Kinematics was then, in more than one school, very nearly as it wascharacterized by an observer in 1942--"on an intellectual par withmechanical drafting. "[114] I can recall my own naïve belief that atextbook contained all that was known of the subject; and I was notdisabused of my belief by my own textbook or by my teacher. I think Idetect in several recent books a fresh, less final, and less tidytreatment of the kinematics of mechanisms, but I would yet recommendthat anyone who thinks of writing a textbook take time to review, carefully and at first hand, not only the desk copies of books that hehas accumulated but a score or more of earlier works, covering the lastcentury at least. Such a study should result in a better appreciation ofwhat constitutes a contribution to knowledge and what constitutes merelythe ringing of another change. 

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

The author of the contentious article that appeared in _MechanicalEngineering_ in 1942 under the title "What is Wrong with Kinematics andMechanisms?" made several pronouncements that were questioned by variousreaders, but his remarks on the meagerness of the college courses ofkinematics and the "curious fact" that the textbooks "are all strangelysimilar 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 ofaccelerations was at a level well below the existing knowledge of thesubject, for in a series of articles by two teachers at Purdue attentionwas called to the serious consequences of errors in accelerationanalysis occasioned by omitting the Coriolis component. [116] Theseauthors were reversing a trend that had been given impetus by an articlewritten in 1920 by one of their predecessors, Henry N. Bonis. Theearlier article, appearing in a practical-and-proud-of-it technicalmagazine, demonstrated how the acceleration of a point on a flywheelgovernor might be determined "without the use of the fictitiousacceleration of Coriolis. " The author's analysis was right enough, andhe closed his article with the unimpeachable statement that "it isbetter psychologically for the student and practically for the engineerto understand the fundamentals thoroughly than to use a complex formulathat may be misapplied. " However, many readers undoubtedly read only thelead paragraph, sagely nodded their heads when they reached the word"fictitious, " which confirmed their half-formed conviction that anythingas abstruse as the Coriolis component could have no bearing upon apractical problem, and turned the page to the "practical kinks"section. [117]

[Footnote 116: A. S. Hall and E. S. Ault, "How Acceleration Analysis CanBe 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_, June1948, vol. 38, pp. 757-765. ]

[Footnote 117: Henry N. Bonis, "The Law of Coriolis, " _AmericanMachinist_, November 18, 1920, vol. 53, pp. 928-930. See also"Acceleration Determinations, " _American Machinist_, November 25 andDecember 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 formulasnor in beautiful vector diagrams. " While this remark was in a letterevoked by an article, and was not a reflection of editorial policy, itwas nevertheless representative of an element in the American traditionof engineering. The unconscious arrogance that is displayed in thisstatement of the "practical" designer's creed is giving way torecognition of the value of scholarly work. Lest the scholar developarrogance of another sort, however, it is well to hear the author ofthe statement out. "A drafting machine is a useful tool, " he wrote. "Itis 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 papersthat 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 thosethat did appear were concerned with descriptions of new "mechanicalmotions. " In the 1930's the number of papers reported in _EngineeringIndex_ increased sharply, but only because the editors had begun toinclude foreign-language listings. 

There has been in Germany a thread of continuity in the kinematics ofmechanisms since the time of Reuleaux. While most of the work has had todo with analysis, the teasing question of synthesis that Reuleaux raisedin his work has never been ignored. The developments in Germany andelsewhere have been ably reviewed by others, [119] and it is only to benoted here that two of the German papers, published in 1939 in_Maschinenbau_, appear to have been the sparks for the conflagrationthat still is increasing in extent and intensity. According to summariesin _Engineering Index_, R. Kraus, writing on the synthesis of thedouble-crank mechanism, drew fire from the Russian Z. S. Bloch, who, in1940, discussed critically Kraus's articles and proceeded to give theoutline of the "correct analysis of the problem" and a general numericalsolution for the synthesis of "any four-bar linkage. "[120] Russian workin mechanisms, dating back to Chebyshev and following the "Chebyshevtheory of synthesis" in which algebraic methods are used to determinepaths of minimum deviation from a given curve, has also been reviewedelsewhere, [121] and I can add nothing of value. 

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

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

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

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

The first publication of note in this country on the synthesis oflinkages was a practical one, but in conception and undertaking it was abold enterprise. In a book by John A. Hrones and G. L. Nelson, _Analysis of the Four Bar Linkage_ (1951), the four-bar crank-and-rockermechanism was exhaustively analyzed mechanically and the results werepresented graphically. This work was faintly praised by a Dutch scholar, O. Bottema, who observed that the "complicated analytical theory of thethree-bar [sic] curve has undoubtedly kept the engineer from using it"and who went on to say that "we fully understand the publication of anatlas by Hrones and Nelson containing thousands of trajectories whichmust be very useful in many design problems. "[122] Nevertheless, theauthors furnished designers with a tool that could be readily, almostinstantly, understood (fig. 45), and the atlas has enjoyed widecirculation. [123] The idea of a geometrical approach to synthesis hasbeen exploited by others in more recent publications, [124] and it islikely 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-pointpath-generating machine (fig. 46) that could have been used to produce awork similar to that of Hrones and Nelson. ]

[Footnote 124: R. S. Hartenberg and J. Denavit, "Systematic MechanismDesign, " _Machine Design_, September 1954, vol. 26, pp. 167-175, andOctober 1954, vol. 26, pp. 257-265; A. S. Hall, A. R. Holowenko, and H. G. Laughlin, "Four-Bar Lever Crank Mechanism, " _Design News_, September15, 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 equaltime intervals. From John A. Hrones and G. L. Nelson, _Analysis of theFour Bar Linkage_ (New York, 1951, p. 635). ]

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

Pursuit of solutions to the "complicated analytical theory" of linkageswas stimulated by publication of Ferdinand Freudenstein's "AnalyticalApproach to the Design of Four-Link Mechanisms" in 1954, [125] and anincreasing interest in the problem is indicated by the extensiveliterature that has appeared in the last five years. 

[Footnote 125: _Transactions of the American Society of MechanicalEngineers_, 1954, vol. 76, pp. 483-492. See also _Transactions of theAmerican 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 isnot yet clear. "While we may talk about kinematic synthesis, " wrote twoof today's leaders in the field, "we are really talking about a hope forthe future rather than a great reality of the present. "[126] When themental equipment and the enthusiasm of scholars who are devoting theirtime to the problems of kinematic synthesis are considered, however, itis 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 livelyresponse. Among other manifestations of current interest in mechanisms, the contributions of Americans to international conferences onmechanisms reflects the growing recognition of the value of scholarlyinvestigation of the kind that can scarcely hope to yield immediatelytangible results. 

While we look to the future, one may ask how a lengthy view of the pastcan be justified. It seems to me that there is inherent in the almostfeverish activity of the present the danger of becoming so preoccupiedwith operational theory that the goals may become clouded and thesynthesis (let us put it less elegantly: the design) of mechanisms maynever quite come into focus. If one knows nothing of the past, I wonderhow he can with any confidence decide in what direction he must turn inorder 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 suggestingmaterial that I had not found. The errors, however, are mine. 

Additional References

The following list of additional reference material on kinematics may beof help to readers who desire to do independent research. The materialis listed according to the section headings in the text of the presentarticle. 

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 andOther Monographs_, New York, Chelsea Publishing Company, 1953. 

Much attention has been given to straight-line mechanisms since the timeof Kempe; at least a half dozen articles have appeared in the UnitedStates since 1950, but I did not investigate the literature publishedafter 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 auxarts. _ 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 (thethird in two parts) had been published; however, this subject index wasnever completed. Volume 2, titled _Mechanics_, has some 200 entriesunder "Linkages. " It is interesting to note that both of the RoyalSociety's monumental catalogs grew out of a suggestion made by JosephHenry 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_, translatedby 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'sMagazine_, from 1841; _Repertory of Arts and Manufactures_, from 1794;_Newton's London Journal of Arts and Science_, from 1820. (The precedingperiodicals have many plates of patent specification drawings. ) _TheEngineer_, November 10, 1933, vol. 156, p. 463, and _Engineering_, November 10, 1933, vol. 136, p. 525. (Recent English views questioningthe 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. _ NewYork, 1931. 

Contains a bibliography that includes works not mentioned in the presentpaper. 

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 sectionsin _Machine Design_, December 1953, vol. 25, pp. 173-220, December 1954, vol. 26, pp. 187-236, and in collected reprints. Papers of the third andfourth conferences (May 1956 and October 1957) appeared in _MachineDesign_ over several months following each conference and in collectedreprints. Papers of the fifth conference (October 1958) were collectedand preprinted for conference participants; subsequently, all papersappeared in _Machine Design_. Collected reprints and preprints areavailable (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 theAmerican 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 ModernKinematics, " _Journal and Proceedings of the Institution of MechanicalEngineers_, 1954, vol. 168, pp. 877-896. 

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

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

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

HARTENBERG, R. S. , AND DENAVIT, J. "Kinematic Synthesis, " _MachineDesign_, 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 EngineeringCollege, in England. 

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