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How to Draw a Straight Lineby Daina Taimina

Cornell university's Reuleaux kinematic model collection includesmany linkages; the most popular of these among mathematicians isthe Peaucellier-Lipkin linkage S35. This article is a short introduction(not complete) to the history of the problem of how to change circularmotion into straight-line motion and vice versa. Somemathematicians formulated this problem as: "How can you draw astraight line?" The peaucellier-Lipkin linkage was the first precisesolution to this problem.

When using a compass to draw a circle, we are not starting with amodel of a circle; instead we are using a fundamental property ofcircles that the points on a circle are at a fixed distance from itscenter, which is Euclid's definition of a circle. Is there a tool (servingthe role of a compass) that will draw a straight line? If, in this case,we want to use Euclid's definition: "A straight line is a line which liesevenly with the points on itself" it will not be of much help. One cansay, "We can use a straightedge for constructing a straight line!"Well, how do you know that your straightedge is straight? How canyou check that something is straight? What does "straight" mean?Think about it!

As we can see in some 13th-century drawings of a sawmill (at right),mechanisms for changing circular motion to straight-line motionwere in use in the 13th-century and probably originated much earlier.In 1588 Agostino Ramelli published his book on machines wherelinkages were widely used. But, of course, there is a vast differencebetween the linkages of Ramelli and those of James Watt(1736-1819), a pioneer of the improved steam engine and a highlygifted designer of mechanisms. Watt's partner, machine builder,Matthew Boulton, built engines in his shop "...with as great adifference of accuracy as there is between the blacksmith and themathematical instrument maker." [Fergusson 1962]

It took Watt several years to design a straight-line linkage that wouldchange straight-line motion into circular motion. He wrote to Boulton:

"I have got a glimpse of a method of causing thepiston-rod to move up and down perpendicularly, byonly fixing it to a piece of iron upon the beam, withoutchains, or perpendicular guides, or untowardly frictions,archheads, or other pieces of clumsiness…. I haveonly tried it in a slight model yet, so cannot build uponit, though I think it a very probable thing to succeed,and one of the most ingenious simple pieces ofmechanisms I have contrived…". [Fergusson 1962]

Peaucellier-Lipkin linkage

13th Century hydraulic sawmill

Years later Watt told his son: "Though I am not over anxious afterfame, yet I am more proud of the parallel motion than of any othermechanical invention I have ever made." [Fergusson 1962]

"Parallel motion" is a name Watt used for his linkage (see modelS24), which was included in an extensive patent of 1784. Watt'slinkage was a good solution to the practical problem. But thissolution did not satisfy mathematicians who knew that it only tracedan approximate straight line. An exact straight-line linkage in theplane was not known until 1864. In 1853 Pierre-Frederic Sarrus(1798-1861), a French professor of mathematics at Strassbourg,devised an accordion-like spatial linkage that traced exact straightline but it still was not a solution of the planar problem.

There were several attempts to solve this problem before Peucellier.Other linkages in this Reuleaux model collection are connected withsome of the names of 19th century mathematicians who tried tosolve the problem of how to draw a precise straight line. Reuleauxthought that these mechanisms were so important that he designed39 straight line mechanisms for his collection, including those ofWatt, Roberts, Evans, Chebyshev, Peuaucellier-Lipkin, Cartwrightand some of his own design. See all models in the S-series.

The appearance in 1864 of Peaucellier's exact straight-line linkagewent nearly unnoticed. Charles Nicolas Peaucellier (1832-1913) wasa captain in the French army. He announced his "inversor" linkage in1864 - in the form of a question and without explaining the solution -in a letter to the Nouvelles Annales. Eventually Peaucellier becamea general and (as claimed by J.J. Sylvester) was in command of thefortress of Toul.

For at least 10 years before and 20 years after Peaucellier's finalsolution of the problem, Professor P.L. Chebyshev, a notedmathematician at the University of St. Petersburg was interested inthe matter. Judging by his published works and his reputationabroad, his interest amounted to an obsession. In 1853, after visitingFrance and England and observing carefully the progress of appliedmechanics in those countries, he wrote his first paper onapproximate straight-line linkages, and over the next 30 years heattacked the problem with new vigor at least a dozen times.Chebyshev noted the departure of Watt's and Evans linkages from astraight line and calculated the deviation as of the fifth degree, orabout 0.0008 inch per inch of beam length. He proposed amodification of Watt's linkage to refine the accuracy but concludedthat it would "present great practical difficulties."Then he got an ideathat if one mechanism would be good, two would be better. So hecombined two linkages and got as a result, what is usually calledChebyshev's linkage, in which precision was increased to 13thdegree. The steam engine he displayed at the Vienna Exhibition of1873 employed this linkage.

In 1871 Lipmann I. Lipkin (1851-1875) independently discovered thesame straight-line linkage as Peaucellier and demonstrated aworking model at the World Exhibition in Vienna 1873. After thatPeaucellier published details of his discovery with a proof of hissolution acknowledging Lipkin's independent discovery. Sylvesterclaims the French government awarded Peaucellier the "PrixMontyon" (1875) for his invention, whereas Lipkin received a

Pafnuty Lvovich Chebyshev (top) and JamesJoseph Sylvester

"substantial reward from the Russian government."[Kempe 1877]There is not much we know about Lipkin. Some sources mentionedthat he was born in Lithuania and was Chebyshev's student but diedbefore completing his doctoral dissertation.

In January 1874 James Joseph Sylvester (1814-1897) delivered alecture "Recent Discoveries in Mechanical Conversion of Motion."Sylvester's aim was to bring the Peaucellier-Lipkin linkage to thenotice of the English-speaking world.Sylvester learned about thisproblem from Chebyshev - during a recent visit of the Russian toEngland.

"The perfect parallel motion of Peaucellier looks sosimple, " he observed, "and moves so easily thatpeople who see it at work almost universally expressastonishment that it waited so long to bediscovered." [Fergusson 1962]

Later Mr. Prim, "engineer to the Houses" (the Houses of Parliamentin London) was pleased to show his adaptation of Peaucellierlinkage in his new "blowing engines" for the ventilation and filtrationof the Houses. Those engines proved to be exceptionally quiet intheir operation. [Kempe 1877]

Sylvester recalled his experience with a little mechanical model ofthe Peaucellier linkage at a dinner meeting of the Philosophical Clubof the Royal Society. The Peaucellier model had been greeted bythe members with lively expressions of admiration

"when it was brought in with the dessert, to be seen bythem after dinner, as is the laudable custom amongmembers of that eminent body in making known toeach other the latest scientific novelties." [Fergusson1962]

And Sylvester would never forget the reaction of his brilliant friendSir William Thomson (later Lord Kelvin) upon being handed thesame model in the Athenaeum Club. After Sir William had operatedit for a time, Sylvester reached for the model, but he was rebuffed bythe exclamation:

"No! I have not had nearly enough of it - it is the mostbeautiful thing I have ever seen in my life." [Fergusson1962]

In summer of 1876 Alfred Bray Kempe, a barrister who pursuedmathematics as a hobby, delivered at London's South KensingtonMuseum a lecture with the provocative title "How to Draw a StraightLine" which in the next year was published in a small book. In thisbook you can find pictures of the linkages we have mentioned here.Kempe essentially knew that linkages (rigid bars constrained to aplane and joined at their ends by rivets) are capable of drawing anyalgebraic curve. Other authors provided more complete proofsduring the period 1877-1902. More about the many connectionsbetween linkages and such problems of modern mathematics asalgebraic completeness, rigidity, NP completeness can be read in

Sir William Thomson (top) and Alfred BrayKempe

Warren D. Smith paper "Plane mechanisms and the 'downhillprinciple'".

Peaucellier-Lipkin linkage is also used in computer science to provetheorems about workspace topology in robotics [3]. Some historyabout linkages and discussion of the Peaucellier-Lipkin linkage is in:http://www.ams.org/new-in-math/cover/linkages1.html.

References

1. Kempe, A. B. How to Draw a Straight Line, London: Macmillanand Co. 1877. online

2. Fergusson, Eugene S. Kinematics of Mechanisms from theTime of Watt, United States National Museum Bulletin 228,Smithsonian Institute, Washington D.C., 1962, pp. 185-230.online

3. Hopkroft, J., Joseph, D., Whitesides, S. "Movement problemsfor 2-Dimensional Linkages", SIAM J. Comput. Vol. 13, No.3,August 1984.