544 GEOMETRY 



single element, as might have shown a particular transformation 

 briefly indicated* by Poncelet in the Traite des proprietes projectives. 



Pluecker defines the transformation by reciprocal radii vector es or 

 inversion, of which Sir W. Thomson and Liouville hasten to show all 

 the importance, as well for mathematical physics as for geometry. 



A contemporary of Moebius and Pluecker, Magnus believed he had 

 found the most general transformation which makes a point corre- 

 spond to a point, but the researches of Cremona show us that the 

 transformation of Magnus is only the first term of a series of bira- 

 tional transformations which the great Italian geometer teaches us to 

 determine methodically, at least for the figures of plane geometry. 



The Cremona transformations long retained a great interest, 

 though later researches have shown us that they reduce always to 

 a series of successive applications of the transformation of Magnus. 



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 con- 

 ceptions of modern geometry; but the moment has come to indicate 

 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 Riemann 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 J 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 equa- 

 tions. 



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 Riemann, 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 

 departure. 



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 practiced, 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 



