have by degrees greatly increased in number. The two sides of the question, the theoretical and the applied, have thus carried on a separate existence side by side; but their union has not yet been accomplished. The cause of this must be sought in the systems alone, for the applications, the mechanisms themselves, have been quietly developed in practical machine-design, by invention and improvement, regardless of whether or not they were accorded any direct and proper theoretical recognition. Indeed the theories used hitherto have contributed to this development only in regard to the form of execution in certain cases, as e.g. in the methods of drawing wheel-teeth, &c.; they have however furnished no new mechanisms. This circumstance is very remarkable; it explains the conservative tenacity with which practical mechanists, where they have adopted any theory whatever, have always fallen back upon the old and apparently natural ideas of Monge in spite of the promises held out by the more recent theories; this the technical journals everywhere sufficiently show.
I believe I have proved the insufficiency of the theoretical Kinematics hitherto taught, and the necessity for some reform. The question now comes, wherein exactly does the error of the existing methods lie?
Monge's classification, however natural it appears, does not in the first place correspond to the real nature of the matter. Did it really do so—did it resemble, for instance, the classifications of Linnæus and Cuvier in organic nature—it would, like them, be able to make its footing firm; and the obstinate conservatism above referred to may perhaps be explained by the existence of a dim feeling that some analogous relation must also exist here. But this is not the case. Even if it were supposed that the function of the science did not extend beyond arranging the mechanisms in classes, still their division cannot be made according to the changes of motion, for this would necessitate endless repetitions. Almost all mechanisms might be looked for, and might also be placed, in at least two such classes, and most in from four to six or even ten to fifteen, for they can be used, and in practice are actually employed, for just so many different kinds of change. Willis, who strictly investigated this very point, and had a strong perception of the necessity for logical order, showed a distrust of the elasticity of the basis of his own
