DEVELOPMENT OF GEOMETRIC METHODS 541 



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 line 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 

 always admire the beautiful theories he has left us on homofocal 

 surfaces 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- 

 tions, and consequently would not have been able to define the cross- 

 ratio of four elements when these ceased to be real in whole or in 

 part. If Chasles had been able to establish the notion of the cross- 

 ratio of imaginary elements, a formula he gives in the Geometric 

 suprieure (p. 118 of the new edition) would have immediately 

 furnished him that beautiful definition of angle as logarithm of a 

 cross-ratio which enabled Laguerre, our regretted confrere, to give 

 the complete solution, sought so long, of the problem of the trans- 

 formation of relations which contain at the same time angles and 

 segments in homography and correlation. 



Like Chasles, Steiner, the great and profound geometer, followed 

 the way of pure geometry; but he has neglected to give us a complete 

 exposition of the methods upon which he depended. However, they 

 may be characterized by saying that they rest upon the introduction 

 of those elementary geometric forms which Desargues had already 

 considered, on the development he was able to give to Bobillier's 

 theory of polars, and finally on the construction of curves and sur- 

 faces of higher degrees by the aid of sheaves or nets of curves of 

 lower orders. In default of recent researches, analysis would suffice 

 to show that the field thus embraced has just the extent of that into 

 which the analysis of Descartes introduces us without effort. 



IV 



While Chasles, Steiner, and, later, as we shall see, von Staudt, were 

 intent on constituting a rival doctrine to analysis and set in some 

 sort altar against altar, Gergonne, Bobillier, Sturm, and above all 

 Pluecker, perfected the geometry of Descartes and constituted an 



