Part 3

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."

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...."

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." Only a few small engines were made under the patent.

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. 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."

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). This is the linkage that was attributed much later to John Scott Russell (1808-1882), the prominent naval architect. 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.

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. 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.

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. The fact that the same linkage is shown in an engraving of 1769 (fig. 18) further confuses the issue.

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, 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.

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

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. 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."

There is a persistent rumor that Professor Chebyshev sought to demonstrate the impossibility of constructing any linkage, regardless of the number of links, that would generate a straight line; but I have found only a dubious statement in the Grande Encyclopédie 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...." 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.

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

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

On a Friday evening in January 1874 Albemarle Street in London was filled with carriages, each maneuvering to unload its charge of gentlemen and their ladies at the door of the venerable hall of the Royal Institution. Amidst a "mighty rustling of silks," the elegant crowd made its way to the auditorium for one of the famous weekly lectures. The speaker on this occasion was James Joseph Sylvester, a small intense man with an enormous head, sometime professor of mathematics at the University of Virginia, in America, and more recently at the Royal Military Academy in Woolwich. He spoke from the same rostrum that had been occupied by Davy, Faraday, Tyndall, Maxwell, and many other notable scientists. Professor Sylvester's subject was "Recent Discoveries in Mechanical Conversion of Motion."

Remarking upon the popular appeal of most of the lectures, a contemporary observer noted that while many listeners might prefer to hear Professor Tyndall expound on the acoustic opacity of the atmosphere, "those of a higher and drier turn of mind experience ineffable delight when Professor Sylvester holds forth on the conversion of circular into parallel motion."

Sylvester's aim was to bring the Peaucellier linkage to the notice of the English-speaking world, as it had been brought to his attention by Chebyshev--during a recent visit of the Russian to England--and to give his listeners some insight into the vastness of the field that he saw opened by the discovery of the French soldier.

"The perfect parallel motion of Peaucellier looks so simple," he observed, "and moves so easily that people who see it at work almost universally express astonishment that it waited so long to be discovered." But that was not his reaction at all. The more one reflects upon the problem, Sylvester continued, he "wonders the more that it was ever found out, and can see no reason why it should have been discovered for a hundred years to come. Viewed a priori there was nothing to lead up to it. It bears not the remotest analogy (except in the fact of a double centring) to Watt's parallel motion or any of its progeny."

It must be pointed out, parenthetically at least, that James Watt had not only had to solve the problem as best he could, but that he had no inkling, so far as experience was concerned, that a solvable problem existed.

Sylvester interrupted his panegyric long enough to enumerate some of the practical results of the Peaucellier linkage. He said that Mr. Penrose, the eminent architect and surveyor to St. Paul's Cathedral, had "put up a house-pump worked by a negative Peaucellier cell, to the great wonderment of the plumber employed, who could hardly believe his senses when he saw the sling attached to the piston-rod moving in a true vertical line, instead of wobbling as usual from side to side." Sylvester could see no reason "why the perfect parallel motion should not be employed with equal advantage in the construction of ordinary water-closets." The linkage was to be employed by "a gentleman of fortune" in a marine engine for his yacht, and there was talk of using it to guide a piston rod "in certain machinery connected with some new apparatus for the ventilation and filtration of the air of the Houses of Parliament." In due course, Mr. Prim, "engineer to the Houses," was pleased to show his adaptation of the Peaucellier linkage to his new blowing engines, which proved to be exceptionally quiet in their operation (fig. 25). A bit on the ludicrous side, also, was Sylvester's 78-bar linkage that traced a straight line along the line connecting the two fixed centers of the linkage.

Before dismissing with a smile the quaint ideas of our Victorian forbears, however, it is well to ask, 88 years later, whether some rather elaborate work reported recently on the synthesis of straight-line mechanisms is more to the point, when the principal objective appears to be the moving of an indicator on a "pleasing, expanded" (i.e., squashed flat) radio dial.

But Professor Sylvester was more interested, really, in the mathematical possibilities of the Peaucellier linkage, as no doubt our modern investigators are. Through a compounding of Peaucellier mechanisms, he had already devised square-root and cube-root extractors, an angle trisector, and a quadratic-binomial root extractor, and he could see no limits to the computing abilities of linkages as yet undiscovered.

Sylvester recalled fondly, in a footnote to his lecture, his experience with a little mechanical model of the Peaucellier linkage at an earlier dinner meeting of the Philosophical Club of the Royal Society. The Peaucellier model had been greeted by the members with lively expressions of admiration "when it was brought in with the dessert, to be seen by them after dinner, as is the laudable custom among members of that eminent body in making known to each other the latest scientific novelties." And Sylvester would never forget the reaction of his brilliant friend Sir William Thomson (later Lord Kelvin) upon being handed the same model in the Athenaeum Club. After Sir William had operated it for a time, Sylvester reached for the model, but he was rebuffed by the exclamation "No! I have not had nearly enough of it--it is the most beautiful thing I have ever seen in my life."

The aftermath of Professor Sylvester's performance at the Royal Institution was considerable excitement amongst a limited company of interested mathematicians. Many alternatives to the Peaucellier straight-line linkage were suggested by several writers of papers for learned journals.

In the summer of 1876, after Sylvester had departed from England to take up his post as professor of mathematics in the new Johns Hopkins University in Baltimore, Alfred Bray Kempe, a young barrister who pursued mathematics as a hobby, delivered at London's South Kensington Museum a lecture with the provocative title "How to Draw a Straight Line."

In order to justify the Peaucellier linkage, Kempe belabored the point that a perfect circle could be generated by means of a pivoted bar and a pencil, while the generation of a straight line was most difficult if not impossible until Captain Peaucellier came along. A straight line could be drawn along a straight edge; but how was one to determine whether the straight edge was straight? He did not weaken his argument by suggesting the obvious possibility of using a piece of string. Kempe had collaborated with Sylvester in pursuing the latter's first thoughts on the subject, and one result, that to my mind exemplifies the general direction of their thinking, was the Sylvester-Kempe "parallel motion" (fig. 26).

Enthusiastic as Kempe was, however, he injected an apologetic note in his lecture. "That these results are valuable cannot I think be doubted," he said, "though it may well be that their great beauty has led some to attribute to them an importance which they do not really possess...." He went on to say that 50 years earlier, before the great improvements in the production of true plane surfaces, the straight-line mechanisms would have been more important than in 1876, but he added that "linkages have not at present, I think, been sufficiently put before the mechanician to enable us to say what value should really be set upon them."