Page:Popular Science Monthly Volume 66.djvu/428

424 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 projective 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, 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.

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 coordinates, 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 discovered. It is sometimes said that the principle of duality shows that