424 POPULAR SCIENCE MONTHLY. 



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, can not prevail 

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

 universal. 



Life is short; geometers know and also practise the principle of 

 least action. Despite these fears, which should discourage no one, 

 it seems to us that under the first form given it by von Staudt, pro- 

 jective 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 could 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. 



Preceded by innumerable researches on systems of straight lines, 

 due to Poinsot, Moebius, Chasles, Dupin, Malus, Hamilton, Kummer, 

 Transon, above all to Cayley, who first introduced the notion of the 

 coordinates of the straight, researches originating perhaps in statics 

 and kinematics, perhaps in geometrical optics, Pluecker's geometry of 

 the straight line will always be regarded as the part of his work where 

 are met the newest and most interesting ideas. 



That Pluecker first set up a methodic study of the straight line, 

 that already is important, but that is nothing beside what he dis- 

 covered. It is sometimes said that the principle of duality shows that 



