422 POPULAR SCIENCE MONTHLY. 



VII. 



While thus were constituted the mixed methods whose principal 

 applications we have just indicated, the pure geometers were not in- 

 active. Poinsot, the creator of the theory of couples, developed, by a 

 method purely geometric, ' that, said he, where one never for a mo- 

 ment 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 con- 

 tribution 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 gen- 

 eral, 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 conies, 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 ad- 

 mitted 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. 



