Page:Popular Science Monthly Volume 66.djvu/427

Rh Numerous and illustrious workers took part in this great geometric movement, but we must recognize that its chiefs and leaders were Chasles and Steiner. So brilliant were their marvelous discoveries that they threw into the shade, at least momentarily, the publications of other modest geometers, less preoccupied perhaps in finding brilliant applications, fitted to evoke love for geometry, than to establish this science itself on an absolutely solid foundation.

Their works have received perhaps a recompense more tardy, but their influence grows each day; it will without doubt increase still more. To pass them over in silence would be without doubt to neglect one of the principal factors which will enter into future researches.

We allude at this moment above all to von Staudt. His geometric works were published in two books of grand interest: the 'Geometrie der Lage,' issued in 1847, and the 'Beiträge zur Geometrie der Lage,' published in 1856, that is to say, four years after the 'Géométrie supérieure.'

Chasles, as we have seen, had devoted himself to constituting a body of doctrine independent of Descartes' analysis and had not completely succeeded. We have already indicated one of the criticisms that can be made upon this system: the imaginary elements are there defined only by their symmetric functions, which necessarily excludes them from a multitude of researches. On the other hand, the constant employment of cross ratio, of transversals and of involution, which requires frequent analytic transformations, gives to the 'Géométrie supérieure' a character almost exclusively metric which removes it notably from the methods of Poncelet. Returning to these methods, von Staudt devoted himself to constituting a geometry freed from all metric relation and resting exclusively on relations of situation.

This is the spirit in which was conceived his first work, the 'Geometrie der Lage' of 1847. The author there takes as point of departure the harmonic properties of the complete quadrilateral and those of homologic triangles, demonstrated uniquely by considerations 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 imaginary elements. It is only in the Beiträge, his second work, that he succeeds, by a very original extension of the method of Chasles, in defining geometrically an isolated imaginary element and distinguishing 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 involution 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