Page:Popular Science Monthly Volume 66.djvu/422

418, 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 'Géometrie supérieure' (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 confrère, 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 considered, 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.

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 Annales 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