DEVELOPMENT OF GEOMETRIC METHODS. 423 



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 the}' 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 ' Beitrage zur Geometrie der Lage,' 

 published in 1856, that is to say, four years after the ' Geometrie 

 superieure.' 



Chasles, as we have seen, had devoted himself to constituting a 

 body of doctrine independent of Descartes' analysis and had not com- 

 pletely 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 re- 

 quires frequent analytic transformations, gives to the ' Geometrie 

 superieure ' 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 considera- 

 tions 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 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 — /, so one may distinguish the two 



