4 i 6 POPULAR SCIENCE MONTHLY. 



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 impor- 

 tance to bring out a fundamental difference between the tendencies of 

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

 unknown. 



III. 



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

 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' 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 con- 

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

 to recognize that his own objections, applicable without doubt to cer- 

 tain transcendent figures, were without force in the applications made 

 by the author of the ' Traite des proprietes projectives.' 



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

 in the clearest manner that the* geometric system of Poncelet rested 

 on an analytic foundation, and besides we know, by the untoward 

 publication of the manuscripts of Saratoff, that by the aid of Descartes' 

 analysis were established the principles which serve as foundation for 

 the ' Traite des proprietes projectives.' 



Younger than Poncelet, who besides abandoned geometry for 

 mechanics where his works had a preponderant influence, Chasles, for 

 whom was created in 1847 a chair of Geometric superieure in the Faculty 

 of Science of Paris, endeavored to constitute a geometric doctrine en- 

 tirely independent and autonomous. He has expounded it in two 

 works of high importance, the ' Traite de geometrie superieure,' which 

 dates from 1852, and the ' Traite des sections coniques,' unhappily 

 unfinished and of which the first part alone appeared in 1865. 



In the preface of the first of these works he indicates very clearly 

 the three fundamental points which permit the new doctrine to share 

 the advantages of analysis and which to him appear to mark an ad- 

 vance in the cultivation of the science. These are: (1) The intro- 

 duction of the principle of signs, which simplifies at once the enuncia- 

 tions and the demonstrations, and gives to Carnot's analysis of trans- 

 versals all the scope of which it is susceptible; (2) the introduction of 



