DEVELOPMENT OF GEOMETRIC METHODS 547 



of geometry of three dimensions, analogous to those of which the 

 school of Monge made such frequent use. 



In this first part of his work, von Staudt neglected entirely im- 

 aginary elements. It is only in the Beitrage, his second work, that 

 he succeeds, by a very original extension of the method of Chasles, 

 in defining geometrically an isolated imaginary element and dis- 

 tinguishing it from its conjugate. 



This extension, although rigorous, is difficult and very abstract. 

 It may be defined in substance as follows: Two conjugate imaginary 

 points may always be considered as the double points of an involu- 

 tion on a real straight; and just as one passes from an imaginary to 

 its conjugate by changing i into i, so one may distinguish the two 

 imaginary points by making correspond to each of them one of the 

 two different senses which may be attributed to the straight. In this 

 there is something a little artificial; the development of the theory 

 erected on such foundations is necessarily complicated. By methods 

 purely projective, von Staudt establishes a calculus of cross-ratios of 

 the most general imaginary elements. Like all geometry, the pro- 

 jective geometry employs the notion of order and order engenders 

 number; we are not astonished therefore that von Staudt has been 

 able to constitute his calculus; but we must admire the ingenuity 

 displayed in attaining it. In spite of the efforts of distinguished 

 geometers who have essayed to simplify its exposition, we fear that 

 this part of the geometry of von Staudt, like the geometry otherwise 

 so interesting of the profound thinker Grassmann, cannot prevail 

 against the analytical methods which have won to-day favor almost 

 universal. Life is short; geometers know and also practice the 

 principle of least action. Despite these fears, which should discour- 

 age no one, it seems to us that under the first form given it by von 

 Staudt, projective geometry must become the necessary companion 

 of descriptive geometry, that it is called to renovate this geometry 

 in its spirit, its procedures, and its applications. 



This has already been comprehended in many countries, and 

 notably in Italy, where the great geometer Cremona did not disdain 

 to write for the schools an elementary treatise on projective geometry. 



IX 



In the preceding articles, we have essayed to follow and bring out 

 clearly the most remote consequences of the methods of Monge and 

 Poncelet. In creating tangential coordinates and homogeneous coor- 

 dinates, Pluecker seemed to have exhausted all that the method of 

 projections and that of reciprocal polars give to analysis. 



It remained for him, toward the end of his life, to return to his 

 first researches to give them an extension enlarging to an unexpected 

 degree the domain of geometry. 



