VII.
While thus were constituted the mixed methods whose principal applications we have just indicated, the pure geometers were not inactive. Poinsot, the creator of the theory of couples, developed, by a method purely geometric, 'that, said he, where one never for a moment loses from view the object of the research,' the theory of the rotation of a solid body that the researches of d'Alembert, Euler and Lagrange seemed to have exhausted: Chasles made a precious contribution to kinematic by his beautiful theorems on the displacement of a solid body, which have since been extended by other elegant methods to the case where the motion has divers degrees of freedom. He made known those beautiful propositions on attraction in general, which figure without disadvantage beside those of Green and Gauss.
Chasles and Steiner met in the study of the attraction of ellipsoids and showed thus once more that geometry has its designated place in the highest questions of the integral calculus.
Steiner did not disdain at the same time to occupy himself with the elementary parts of geometry. His researches on the contacts of circles and conics, on isoperimetric problems, on parallel surfaces, on the center of gravity of curvature, excited the admiration of all by their simplicity and their depth.
Chasles introduced his principle of correspondence between two variable objects which has given birth to so many applications; but here analysis re-took its place to study the principle in its essence, make it precise and generalize it.
It was the same concerning the famous theory of characteristics and the numerous researches of de Jonquieres, Chasles, Cremona and still others, which gave the foundations of a new branch of the science, Enumerative Geometry.
During many years, the celebrated postulate of Chasles was admitted without any objection: a crowd of geometers believed they had established it in a manner irrefutable.
But, as Zeuthen then said, it is very difficult to recognize whether, in demonstrations of this sort, there does not exist always some weak point that their author has not perceived; and, in fact, Halphen, after fruitless efforts, crowned finally all these researches by clearly indicating in what cases the postulate of Chasles may be admitted and in what cases it must be rejected.
VIII.
Such are the principal works which restored geometric synthesis to honor and assured to it, in the course of the last century, the place belonging to it in mathematical research.
