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." [2]
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…". [2]
"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." [2]
"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."[1] 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." [2]
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. [1]
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." [2]
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." [2]
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":
http://www.cise.ufl.edu/~mpf/pu-planelink.pdf
Peuacellier-Lipkin
linkage is also used in computer science to prove theorems about workspace
topology in robotics [3].
Some
history about 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: 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.