DEVELOPMENT OF GEOMETRIC METHODS. 539 



In 1832 Steiner brought out at Berlin his great work: Systemat- 

 ische Entwickelung der Abhaengigkeit der geometrischen Gestalten von 

 einander, and, the following year, Die geometrischen Konstruktionen 

 ausgefuehrt mittels der geraden Linie und eines festen Kreises, where 

 was confirmed by the most elegant examples a proposition of Pon- 

 celet's relative to the employment of a single circle for the geometric 

 constructions. 



Finally, in 1830, Chasles sent to the Academy of Brussels, which 

 happily inspired had offered a prize for a study of the principles of 

 modern geometry, his celebrated Apergu historique sur I'origine et 

 le developpement des methodes en geometric, followed by Memoire 

 sur deux principes generaux de la science : la dualite et I'homographie, 

 which was published only in 1837. 



Time would fail us to give a worthy appreciation of these beautiful 

 works and to apportion the share of each. Moreover, to what would 

 such a study conduct us, but to a new verification of the general laws 

 of the development of science ? When the times are ripe, when the 

 fundamental principles have been recognized and enunciated, nothing 

 stops the march of ideas ; the same discoveries, or discoveries almost 

 equivalent, appear at nearly the same instant, and in places the most 

 diverse. Without undertaking a discussion of this sort, which, besides, 

 might appear useless or become irritating, it is, however, of import- 

 ance to bring out a fundamental difference between the tendencies 

 of the great geometers, who, about 1830, gave to geometry a scope 

 before unknown. 



Ill 



Some, like Chasles and Steiner, who consecrated their entire lives 

 to research in pure geometry, opposed what they called synthesis to 

 analysis, and, adopting in the ensemble if not in detail the tendencies 

 of Poncelet, proposed to constitute an independent doctrine, rival of 

 Descartes's analysis. 



Poncelet could not content himself with the insufficient resources 

 furnished by the method of projections; to attain imaginaries he 

 created that famous principle of continuity which gave birth to such 

 long discussions between him and Cauchy. 



Suitably enunciated, this principle is excellent and can render 

 great service. Poncelet was wrong in refusing to present it as a simple 

 consequence of analysis; and Cauchy, on the other hand, was not 

 willing to recognize that his own objections, applicable without 

 doubt to certain transcendent figures, were without force in the 

 applications made by the author of the Traite des proprietes pro- 

 jectives. 



Whatever be the opinion of such a discussion, it showed at least 

 in the clearest manner that the geometric system of Poncelet rested 



