How to Draw a Straight Line
By Daina Taimina
Reuleaux kinematic model collection at Cornell University has many linkages; the most popular of these among mathematicians is Peaucellier-Lipkin linkage. This article is a short introduction (not complete) to the history of the problem of how to change circular motion into straight-line motion and vice versa. Some mathematicians formulated this problem as - how to draw a straight line?
When using a compass to draw a circle, we are not starting with a model of a circle; instead we are using a fundamental property of circles that the points on a circle are at a fixed distance from a center. Or we can say we use Euclid's definition of a circle. Is there a tool(serving the 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 lies evenly with the points on itself" it will not be of much help. One can say - but we can use a straightedge for constructing a straight line! Well, how do you know that your straightedge is straight? How can you check that something is straight? What does "straight" mean? Think about it!
As we can see in some 13th century drawings of sawmill four bar linkage was in use and probably was originated much earlier. In 1588 Agostino Ramelli published his book on machines where linkages were widely used. (This book is in Cornell Kroch Library Rare Books and Manuscripts collection.) But, of course, there is a vast difference between the linkages of Ramelli and those of James Watt (1736-1819), pioneer of improved steam engine and highly gifted designer of mechanisms. Watt had a good partner, machine builder, Mathew Boulton who build engines in his shop
"with as great a difference of accuracy as there is between the blacksmith and the mathematical instrument maker." 
It took Watt several years to design the straight-line linkage that would change straight-line motion to circular one. He wrote to Boulton:
" 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, archheads, 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 mechanisms I have contrived…". 
"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." 
"Parallel motion" is a name Watt used for his linkage, which was included in an extensive patent of 1784. Watt's linkage was a good solution to the practical problem. But this solution did not satisfied mathematicians who knew that all four bar straight-line linkages that have no sliding pairs trace only an approximate straight line. An exact straight-line linkage in a plane 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 straight line but it still was not a solution of the planar problem.
There were several attempts to solve this problem before Peucellier. Other linkages in Reuleaux model collection are connected with some of the names of 19th century mathematicians who tried to solve a problem how to draw a precise straight line. Reuleaux thought that these mechanisms were so important that he designed 39 straight line mechanisms for his collection, including those of Watt, Roberts, Evans, Chebyshev, Peuaucellier-Lipkin, Cartwright and some of his own design.
The appearance in 1864 of Peaucellier's exact straight-line linkage went nearly unnoticed. Charles Nicolas Peaucellier (1832-1913) was a French "Capitaine de genie a Nice". He announced his "inversor" linkage in 1864 - in the form of a question and without explaining the solution - in a letter to the Nouvelles Annales. Eventually Peaucellier became a general and (as claimed by J.J. Sylvester) was in command of the fortress of Toul.
For at least 10 years before and 20 years after Peaucellier's final solution of the problem, Professor P.L. Chebyshev, a noted mathematician at the University of St. Petersburg was interested in the matter. Judging by his published works and his reputation abroad, his interest amounted to an obsession. In 1853, after visiting France and England and observing carefully the progress of applied mechanics in those countries, he wrote 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. Chebyshev noted the departure of Watt's and Evans 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 Watt's linkage to refine the accuracy but concluded that it would "present great practical difficulties."Then he got an idea that if one mechanism would be good, two would be better. So he combined two linkages and got as a result, what is usually called Chebyshev's linkage, in which precision was increased to 13th degree. The steam engine he displayed at the Vienna Exhibition of 1873 employed this linkage.
In 1871 Lipmann I. Lipkin (1851-1875) independently discovered the same straight-line linkage as Peaucellier and demonstrated a working model at the World Exhibition in Vienna 1873. After that Peaucellier published details of his discovery with a proof of his solution acknowledging Lipkin's independent discovery. Sylvester claims the French government awarded Peaucellier the "Prix Montyon" (1875) for his invention, whereas Lipkin received a "substantial reward from the Russian government." There is not much we know about Lipkin. Some sources mentioned that he was born in Lithuania and was Chebyshev's student but died before completing his doctoral dissertation.
In January 1874 James Joseph Sylvester (1814-1897) delivered a lecture "Recent Discoveries in Mechanical Conversion of Motion." Sylvester's aim was to bring the Peaucellier-Lipkin linkage to the notice of the English-speaking world.Sylvester learned about this problem from Chebyshev - during a recent visit of the Russian to England.
"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." 
Later Mr. Prim, "engineer to the Houses" (the Houses of Parliament in London) was pleased to show his adaptation of Peaucellier linkage in his new "blowing engines" for the ventilation and filtration of the Houses. Those engines proved to be exceptionally quiet in their operation. 
Sylvester recalled his experience with a little mechanical model of the Peaucellier linkage at a 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." 
In summer of 1876 Alfred Bray Kempe, a 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" which in the next year was published in a small book. In this book you can find pictures of the linkages we have mentioned here. Kempe essentially knew that linkages ( rigid bars constrained to a plane and joined at their ends by rivets) are capable of drawingany algebraic curve. Other authors provided more complete proofs during the period 1877-1902. More about the many connections between linkages and such problems of modern mathematics as algebraic completeness, rigidity, NP completeness can be read in Warren D. Smith paper "Plane mechanisms and the 'downhill principle":
Peuacellier-Lipkin linkage is also used in computer science to prove theorems about workspace topology in robotics .
Some history about linkages and discussion of the Peaucellier-Lipkin linkage is in:
1. Kempe, A. B. How to Draw a Straight Line, London: Macmillan and Co. 1877
2. Fergusson, Eugene S. Kinematics of Mechanisms from the Time of Watt, United States National Museum Bulletin 228, Smithsonian Institute, Washington D.C., 1962, pp. 185-230.
3. Hopkroft, J., Joseph, D., Whitesides, S. Movement problems for 2-Dimensional Linkages", SIAM J. Comput. Vol. 13, No.3, August 1984.