DEVELOPMENT OF GEOMETRIC METHODS. 419 



initiator of those methods of modern analysis where the employment of 

 homogeneous coordinates permits treating simultaneously and, so to 

 say, without the reader perceiving it, together with one figure all those 

 deducible from it by homography and correlation. 



V. 



Parting from this moment, a period opens brilliant for geometric 

 researches of every nature. 



The analysts interpret all their results and are occupied in trans- 

 lating them by constructions. 



The geometers are intent on discovering in every question some 

 general principle, usually undemonstrable without the aid of analysis, 

 in order to make flow from it without effort a crowd of particular 

 consequences, solidly bound to one another and to the principle whence 

 they are derived. Otto Hesse, brilliant disciple of Jacobi, develops 

 in an admirable manner that method of homogeneous coordinates to 

 which Pluecker perhaps had not attached its full value. Boole dis- 

 covers in the polars of Bobillier the first notion of a covariant; 

 the theory of forms is created by the labors of Cayley, Sylvester, 

 Hermite, Brioschi. Later Aronhold, Clebsch and Gordan and other 

 geometers still living gave to it its final notation, established the 

 fundamental theorem relative to the limitation of the number of 

 covariant forms and so gave it all its amplitude. 



The theory of surfaces of the second order, built up principally 

 by the school of Monge, was enriched by a multitude of elegant 

 properties, established principally by 0. Hesse, who found later in 

 Paul Serret a worthy emulator and continuer. 



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 de- 

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

 to the researches of Pluecker relative to the singularities called 

 ordinary 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, Halphen 



