545 



rational or of deficiency zero; this led him, toward the end of his 

 too short life, to envisage what may be called also rational surfaces, 

 those which can be simply represented by a plane. This was a vast 

 field for research, opened already for the elementary cases by Chasles, 

 and in which Clebsch was followed by Cremona and many other 

 savants. It was on this occasion that Cremona, generalizing his re- 

 searches on plane geometry, made known not indeed the totality of 

 birational transformations of space, but certain of the most interest- 

 ing among these transformations. 



The extension of the notion of deficiency to algebraic surfaces is 

 already commenced; already also works of high value have shown 

 that the theory of integrals, simple or multiple, of algebraic differ- 

 entials will find, in the study of surfaces as in that of curves, an ample 

 field of important applications; but it is not proper for the reporter 

 on geometry to dilate on this subject . 



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, 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 desig- 

 nated 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 centre 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 anatysis retook 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. 



