4i 8 POPULAR SCIENCE MONTHLY. 



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 ' Geometrie superieure ' 

 (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 transformation 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 con- 

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

 of polars, and finally on the construction of curves and surfaces 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, above all Pluecker, 

 perfected the geometry of Descartes and constituted an analytic system 

 in a manner adequate to the discoveries of the geometers. 



It is to Bobillier and to Pluecker that we owe the method called 

 abridged notation. Bobillier consecrated to it some pages truly new 

 in the last volumes of the Annates of Gergonne. 



Pluecker commenced to develop it in his first work, soon followed 

 by a series of works where are established in a fully conscious manner 

 the foundations of the modern analytic geometry. It is to him that 

 we owe tangential coordinates, trilinear coordinates, employed with 

 homogeneous equations, and finally the employment of canonical forms 

 whose validity was recognized by the method, so deceptive sometimes, 

 but so fruitful, called the enumeration of constants. 



All these happy acquisitions infused new blood into Descartes' 

 analysis and put it in condition to give their full signification to 

 the conceptions of which the geometry called synthetic had been unable 

 to make itself completely mistress. 



Pluecker, to whom it is without doubt just to adjoin Bobillier. 

 carried off by a premature death, should be regarded as the veritable 



