Chapter 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

To Draw a Straight Line

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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