542 GEOMETRY 



analytic system in a manner adequate to the discoveries of the 

 geometers. It is to Bobillier and to Pluecker that we owe the method 

 called abridged notation. Bobillier consecrated to it some pages truly 

 new in the last volumes of the Annales of Gergonne. 



Pluecker commenced to develop it in his first work, soon followed 

 by a series of works where are established in a fully conscious manner 

 the foundations of the modern analytic geometry. It is to him that 

 we owe tangential coordinates, trilinear coordinates, employed with 

 homogeneous equations, and finally the employment of canonical 

 forms whose validity was recognized by the method, so deceptive 

 sometimes, but so fruitful, called the enumeration of constants. 



All these happy acquisitions infused new blood into Descartes 's 

 analysis and put it in condition to give their full signification to the 

 conceptions of which the geometry called synthetic had been unable 

 to make itself completely mistress. 



Pluecker, to whom it is without doubt just to adjoin Bobillier, 

 carried off by a premature death, should be regarded as the veritable 

 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. 



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 ana- 

 lysis, in order to make flow from it without effort a crowd of particu- 

 lar 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 discovers 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, estab- 

 lished 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 O. Hesse, who found later in 

 Paul Serret a worthy emulator and continuer. 



