546 GEOMETRY 



During many years, the celebrated postulate of Chasles was ad- 

 mitted without any objection: a crowd of geometers believed they 

 had established it in a manner irrefutable. 



But, as Zeuthen then said, it is very difficult to recognize whether, 

 in demonstrations of this sort, there does not exist always some weak 

 point that their author has not perceived; and, in fact, Halphen, 

 after fruitless efforts, crowned finally all these researches by clearly 

 indicating in what cases the postulate of Chasles may be admitted 

 and in what cases it must be rejected. 



VIII 



Such are the principal works which restored geometric synthesis 

 to honor and assured to it, in the course of the last century, the place 

 belonging to it in mathematical research. 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 geo- 

 meters, 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 per- 

 haps a recompense more tardy, but their influence grows each day; 

 it will assuredly 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 great interest: the Geometric der Lage, issued in 1847, and the 

 Beitrage zur Geometric der Lage, published in 1856, that is to say, 

 four years after the Geometric superieure. Chasles, as we have seen, 

 had devoted himself to constituting a body of doctrine independent 

 of Descartes's 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 sym- 

 metric functions, which necessarily exclude 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 Geometrie superieure a char- 

 acter 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 Geo- 

 metrie 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 



