Hachette classified all mechanisms by considering the conversion of one motion into another. His elementary motions were continuous circular, alternating circular, continuous rectilinear, and alternating rectilinear. Combining one motion with another--for example, a treadle and crank converted alternating circular to continuous circular motion--he devised a system that supplied a frame of reference for the study of mechanisms. In the U.S. Military Academy at West Point, Hachette's treatise, in the original French, was used as a textbook in 1824, and perhaps earlier.[63] [Footnote 63: This work was among the books sent back by Sylvanus Thayer when he visited France in 1816 to observe the education of the French army cadets. Thayer's visit resulted in his adopting the philosophy of the Ecole Polytechnique in his reorganization of the U.S. Military Academy and, incidentally, in his inclusion of Hachette's course in the Academy's curriculum (U.S. Congress, _American State Papers_, Washington, 1832-1861, Class v, Military Affairs, vol. 2, p. 661: Sidney Forman, _West Point_, New York, 1950, pp. 36-60). There is a collection of miscellaneous papers (indexed under Sylvanus Thayer and William McRee, U.S. National Archives, RG 77, Office, Chief of Engineers, Boxes 1 and 6) pertaining to the U.S. Military Academy of this period, but I found no mention of kinematics in this collection.] Lanz and Bétancourt, scholars from Spain at the Ecole Polytechnique, plugged some of the gaps in Hachette's system by adding continuous and alternating curvilinear motion, which doubled the number of combinations to be treated, but the advance of their work over that of Hachette was one of degree rather than of kind.[64] [Footnote 64: Phillipe Louis Lanz and Augustin de Bétancourt, _Essai sur la composition des machines_, Paris, 1808. Hachette's chart and an outline of his elementary course on machines is bound with the Princeton University Library copy of the Lanz and Bétancourt work. This copy probably represents the first textbook of kinematics. Bétancourt was born in 1760 in Teneriffe, attended the military school in Madrid, and became inspector-general of Spanish roads and canals. He was in England before 1789, learning how to build Watt engines, and he introduced the engines to Paris in 1790 (see Farey, _op. cit._, p. 655). He entered Russian service in 1808 and died in St. Petersburg in 1826 J. C. Poggendorff, _Biographisches-literarisches Handwörterbuch für Mathematik ..._, Leipzig, 1863, vol. 1.] [Illustration: Figure 29.--Robert Willis (1800-1875), Jacksonian Professor, Cambridge University, and author of _Principles of Mechanism_, one of the landmark books in the development of kinematics of mechanisms. Photo courtesy Gonville and Caius College, Cambridge University.]