540 GEOMETRY 



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

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

 Descartes's analysis were established the principles which serve as 

 foundation for the TraiU des proprietes projectiles. 



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 Geometrie superieure in the 

 Faculty of Science of Paris, endeavored to constitute a geometric 

 doctrine entirely independent and autonomous. He has expounded 

 it in two works of high importance, the Traite de geometric supe- 

 rieure, which dates from 1852, and the Traite des sections conigues, 

 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 

 advance 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 

 imaginaries, which supplies the place of the principle of continuity 

 and furnishes demonstrations as general as those of analytic geo- 

 metry; (3) the simultaneous demonstration of propositions which are 

 correlative, that is to say, which correspond in virtue of the principle 

 of duality. 



Chasles studies indeed in his work homography and correlation; 

 but he avoids systematically in his exposition the employment of 

 transformations of figures, which, he thinks, cannot take the place of 

 direct demonstrations since they mask the origin and the true nature 

 of the properties obtained by their means. 



There is truth in this judgment, but the advance itself of the science 

 permits us to declare it too severe. If it happens often that, em- 

 ployed without discernment, transformations multiply uselessly the 

 number of theorems, it must be recognized that they often aid us to 

 better understand the nature of the propositions even to which they 

 have been applied. Is it not the employment of Poncelet's projection 

 which has led to the so fruitful distinction between projective proper- 

 ties and metric properties, which has taught us also the high import- 

 ance of that cross-ratio whose essential property is found already 

 in Pappus, and of which the fundamental r61e has begun to appear 

 after fifteen centuries only in the researches of modern geometry? 



The introduction of the principle of signs was not so new as Chasles 

 supposed at the time he wrote his TraiU de Geometric superieure. 



Moebius, in his Barycentrische Calcul, had already given issue to 

 a desideratum of Carnot, and employed the signs in a way the largest 



