DEVELOPMENT OF GEOMETRIC METHODS 543 



The properties of the polars of algebraic curves are developed by 

 Pluecker and above all by Steiner. The study, already old, of curves 

 of the third order is rejuvenated and enriched by a crowd of new 

 elements. Steiner, the first, studies by pure geometry the double 

 tangents of curves of the fourth order, and Hesse, after him, applies 

 the methods of algebra to this beautiful question, as well as to that 

 of points of inflection of curves of the third order. 



The notion of class introduced by Gergonne, the study of a para- 

 dox in part elucidated by Poncelet and relative to the respective 

 degrees of two curves reciprocal polars one of the other, give birth 

 to the researches of Pluecker relative to the singularities called ordi- 

 nary of algebraic plane curves. The celebrated formulas to which 

 Pluecker is thus conducted are later extended by Cayley and by 

 other geometers to algebraic skew curves, by Cayley again and by 

 Salmon to algebraic surfaces. 



The singularities of higher order are in their turn taken up by 

 the geometers; contrary to an opinion then very widespread, Hal- 

 phen demonstrates that each of these singularities cannot be con- 

 sidered as equivalent to a certain group of ordinary singularities, and 

 his researches 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 ele- 

 mentary 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 transformation 

 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 



