DEVELOPMENT OF GEOMETRIC METHODS. 421 



VI. 



All the works we have enumerated, others to which we shall return 

 later, find their origin and, in some sort, their first motive in the 

 conceptions of modern geometry; but the moment has come to indi- 

 cate rapidly another source of great advances for geometric studies. 

 Legendre's theory of elliptic functions, too much neglected by the 

 French geometers, is developed and extended by Abel and Jacobi. 

 With these great geometers, soon followed by Eiemann and Weier- 

 strass, the theory of Abelian functions which, later, algebra would try 

 to follow solely with its own resources, brought to the geometry of 

 curves and surfaces a contribution whose importance will continue to 

 grow. 



Already, Jacobi had employed the analysis of elliptic functions 

 in the demonstration of Poncelet's celebrated theorems on inscribed 

 and circumscribed polygons, inaugurating thus a chapter since en- 

 riched by a multitude of elegant results; he had obtained also, by 

 methods pertaining to geometry, the integration of Abelian equations. 



But it was Clebsch who first showed in a long series of works all 

 the importance of the notion of deficiency (Geschlecht, genre) of a 

 curve, due to Abel and Eiemann, in developing a crowd of results 

 and elegant solutions that the employment of Abelian integrals would 

 seem, so simple was it, to connect with their veritable point of de- 

 parture. 



The study of points of inflection of curves of the third order, that 

 of double tangents of curves of the fourth order and, in general, the 

 theory of osculation on which the ancients and the moderns had so 

 often practised, were connected with the beautiful problem of the 

 division of elliptic functions and Abelian functions. 



In one of his memoirs, Clebsch had studied the curves which are 

 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 mutiple, of algebraic differentials 

 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. 



