In France, meanwhile, progress which has been of great importance to Kinematics has taken place in the region of Geometry. The geometrical method of treating the motion of rigid bodies developed by Euler in the last century was again taken up, and soon further extended, by Chasles and (especially) by Poinsot. The works of the latter,—Théorie de la rotation des corps, and Théorie des cônes circulaires roulants, gave a great impulse also to the employment of geometrical methods of representation in the study of mechanisms. The propositions of Euler, which had hitherto possessed no more than a purely theoretical and abstract interest, were now formed by the French kinematists into fundamental doctrines. They breathed fresh life into what had become a somewhat dull study. Under their influence appeared Girault's Elémens de géom. appl. à la transformation du mouvement 1858,—Belanger's Cinématique 1864,—and Haton's Traité des mécanismes 1864; the two first specially rich in geometrical, i.e. in theoretical parts, the third bearing rather on the application of theory to the mechanisms themselves. All three books, however, valuable and important though they be, fall into the old difficulties as to classification as soon as they enter upon the applications of the science. They all differ from Monge, for the inadequacy of the "ancien système" became too evident to escape detection in the light of the new ideas,—nevertheless they all remained, for better or worse, partly involved in it. They differ too among themselves, and each seems to hesitate in his own particular way between Monge and Willis. Girault and Belanger take their principal divisions from the changes of motion,—but in entirely different ways,—using the various methods of transmitting the motion as subdivisions, Belanger with the addition of Willis' relative velocities. Haton recognises the want of the old system, and points out, for example, that to arrange toothed wheels according to it would require in all some 21 different classes;—he himself takes his principal divisions from the methods of transmission; these he divides into nine classes, of which Rollers, Guide-bars, Eccentrics, Toothed-wheels, Connecting Rods, and Cords form the first six, the three last bearing in common the fatal designation appareils accessoires. One whole third of the subject therefore has been cut off,—placed as it were like a note below the text.
Nor has this new geometrical development, into the further
