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O appreciate the progress geometry has made during the century just ended, it is of advantage to cast a rapid glance over the state of mathematical science at the beginning of the nineteenth century.

We know that, in the last period of his life, Lagrange, fatigued by the researches in analysis and mechanics, which assured him, however, an immortal glory, neglected mathematics for chemistry, which, according to him, was easy as algebra, for physics, for philosophic speculations.

This mood of Lagrange we almost always find at certain moments of the life of the greatest savants. The new ideas which came to them in the fecund period of youth and which they introduced into the common domain have given them all they could have expected; they have fulfilled their task and feel the need of turning their mental activity towards wholly new subjects. This need, as we recognize, manifested itself with particular force at the epoch of Lagrange. At this moment, in fact, the program of researches opened to geometers by the discovery of the infinitesimal calculus appeared very nearly finished up. Some differential equations more or less complicated to integrate, some chapters to add to the integral calculus, and one seemed about to touch the very outmost bounds of science.

Laplace had achieved the explanation of the system of the world and laid the foundations of molecular physics. New ways opened before the experimental sciences and prepared the astonishing development they received in the course of the century just ended. Ampère, Poisson, Fourier and Cauchy himself, the creator of the theory of imaginaries, were occupied above all in studying the application of the analytic methods to mechanics, and seemed to believe that outside this new domain, which they hastened to cover, the outlines of theory and science were finally fixed.

Modern geometry, a glory we must claim for it, came, after the end of the eighteenth century, to contribute in large measure to the renewing of all mathematical science, by offering to research a way