420 POPULAR SCIENCE MONTHLY. 



demonstrates that each of these singularities can not be considered as 

 equivalent to a certain group of ordinary singularities and his re- 

 searches close for a time this difficult and important question. 



Analysis and geometry, Steiner, Cayley, Salmon, Cremona, meet in 

 the study of surfaces of the third order, and, in conformity with 

 the anticipations of Steiner, this theory becomes as simple and as 

 easy as that of surfaces of the second order. 



The algebraic ruled surfaces, so important for applications, are 

 studied by Chasles, by Cayley, of whom we find the influence and the 

 mark in all mathematical researches, by Cremona, Salmon, La Gour- 

 nerie; so they will be later by Pluecker in a work to which we must 

 return. 



The study of the general surface of the fourth order would seem 

 to be still too difficult; but that of the particular surfaces of this order 

 with multiple points or multiple lines is commenced, by Pluecker for the 

 surface of waves, by Steiner, Kummer, Cayley, Moutard, Laguerre, 

 Cremona and many other investigators. 



As for the theory of algebraic skew curves, grown rich in its 

 elementary parts, it receives finally, by the labors of Halphen and of 

 Noether, whom it is impossible for us here to separate, the most 

 notable extensions. 



A new theory with a great future is born by the labors of 

 Chasles, of Clebsch and of Cremona; it concerns the study of all the 

 algebraic curves which can be traced on a determined surface. 



Homography and correlation, those two methods of transforma- 

 tion which have been the distant origin of all the preceding researches, 

 receive from them in their turn an unexpected extension; they are 

 not the only methods which make a single element correspond to a single 

 element, as might have shown a particular transformation briefly in- 

 dicated by Poncelet in the ' Traite des proprietes projectives.' 



Pluecker defines the transformation by reciprocal radii vectores 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 teach 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. 



