DEVELOPMENT OF GEOMETRIC METHODS. 4*7 



imaginaries, which supplies the place of the principle of continuity 

 and furnishes demonstrations as general as-those of analytic geometry; 

 (3) the simultaneous demonstration of propositions which are cor- 

 relative, 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, can not 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, 

 employed 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 pro- 

 jection which has led to the so fruitful distinction between projective 

 properties and metric properties, which has taught us also the high 

 importance of that cross ratio whose essential property is found already 

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

 after fifteen centuries only in the researches of modern geometry ? 



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

 Chasles supposed at the time he wrote his ' Traite de Geometrie 

 superieure.' 



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

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

 and most precise, defining for the first time the sign of a segment 

 and even that of an area. 



Later he succeeded in extending the use of signs to lengths not 

 laid off on the same straight and to angles not formed about the same 

 point. 



Besides Grassmann, whose mind has so much analogy to that of 

 Moebius, had necessarily employed the principle of signs in the defini- 

 tions which serve as basis for his methods, so original, of studying the 

 properties of space. 



The second characteristic which Chasles assigns to his system of 

 geometry is the employment of imaginaries. Here, his method was 

 really new and he illustrates it by examples of high interest. One will 

 alwavs admire the beautiful theories he has left us on homofocal sur- 

 faces of the second degree, where all the known properties and others 

 new, as varied as elegant, flow from the general principle that they 

 are inscribed in the same developable circumscribed to the circle at 

 infinity. 



But Chasles introduced imaginaries only by their symmetric func- 



