## Some history

2. Some history

The idea of a chain of linked rods that can move with respect to each other, like the links on the robot arm of the International Space Station, suggests issues of rigidity and flexibility. Thus, it is not surprising that the mathematical investigation of linkages has from time to time overlapped with the mathematics of rigidity. However, people interested in rigidity are interested in rigidity of structures other than path linkages, for example, polyhedra. We will also be interested in linkages where the rods lie in a plane but the collection of rods can be more general than a chain. We will also allow circuits of rods.

One of the earliest people to raise the question of flexibility of structures was the awesome Leonhard Euler (1707-1783). Euler, whose collected works and letters have yet to be completely published (!), raised the question of the rigidity of closed polyhedra surfaces in 3-dimensional space. A conjecture which he made was not shown to be wrong until Robert Connelly (Cornell University) published a counter-example in 1978. (A model that Connelly used to help demonstrate what he did is on display at the Smithsonian's National Museum of American History in Washington, DC.) Another famous mathematician who pioneered work in rigidity was A.L. Cauchy (1789-1857). He proved an important theorem about the rigidity of convex polyhedra though Ernst Steinitz, many years later, had to correct a subtle error in his proof. Cauchy's seminal work, though interfacing with modern work on linkages, did not directly affect an understanding of mathematical issues involving linkages in his day.

A major figure in investigating linkages was A. B. Kempe. Kempe, who was neither an academic nor a mathematician by profession, but, rather, a lawyer, is most famous for producing a proof of the 4-color problem: the vertices of a planar graph can be colored with 4 or fewer colors so that vertices joined by an edge get different colors. In 1877 hepublished a small book: How to Draw a Straight Line; A Lecture on Linkages. The book is beautifully illustrated with illustrations of models prepared by Kempe's brother, H. R. Kempe, and grew out of a collaboration with J. J. Sylvester (1814-1897). Kempe in surveying work done prior to his own, mentions that of James Watt (1784) in conjunction with Watt's work on the steam engine. He also mentions the work of Sylvester, Richard Roberts (Manchester), P. L. Chebyshev (1821-1894), Harry Hart (1848-1920), M. V. Liguine, William Kingdon Clifford (1845-1879), Jules Antoine Lissajous (1822-1880), Samuel Roberts(1827-1913), and Arthur Cayley (1821-1895). (For some additional historical information and details of what was being done during this early period see the paper of Warren Smith in the references.) Chebyshev's work was especially broad in scope, and during a visit to London, he apparently inspired Sylvester, who had not been active mathematically for some time prior to his visit, to take up the subject of linkages.