THE MODERN GEOMETRY. 



suits the two last named discovering the general law of reciprocity or du- 

 ality. 



The foundation proper of the modern geometry was laid by Poncelet in 

 his Traite des proprietesprojectiveades figures (Paris, 1828): "Aggrandir lea 

 resources de la simple Geometrie, en generaliser lea conceptions et le langage or- 

 dinairement assez restreints, les rapprocher deceuxde la Geometrie analytique, 

 et surtout offrir des moyens generaux, propres d demontrer et dfaire decouvrir, 

 d'une maniere facile, cette clause de proprietes dont jowissent les figures quand 

 on Us considere d'une maniere purement dbstraite et independamment d'au- 

 cune grandeur dbsolue et determinee, tel est Vobjet qu'on s'est specialement 

 propose dans cet outrage" 



The new ideas found in Germany especially fruitful soil. Mobius, 

 PlucTcer, Steiner, Grassman, and many others, proceeding in part from 

 entirely different points of view, opened out an abundance of new direc- 

 tions which have not yet been thoroughly explored, and which, in union 

 with other investigations, have caused a thorough change in our concep- 

 tions of space relations, whose latest phases are indicated by the names of 

 Riemann, Helmholtz and Lie-Tdein, 



In this development period, also, still existed the two parties in analyti- 

 cal and synthetic, or pure geometry. Plucks held the analytical relations 

 as the most general, and which were with advantage to be illustrated and 

 interpreted geometrically ; while Steiner recognized in the space figure 

 itself the true object and most efficient aid of investigation. Both direc- 

 tions the modern analytic and synthetic lead naturally to the same results. 

 With reference to the methods, however, they diverge the nearer the ideas 

 and transformations of geometry approach the generality and ease of the 

 algebraic method, thus rendering possible an abandonment of this last. 

 Thus, while analytical geometry, through the theory of determinants of 

 Hesse, came into ever closer connection with analysis a direction in which 

 English and Italian investigators as Salmon, Cayley, Cremona brilliantly 

 assisted, the Erlangen Professor von Staudt cut loose from algebraic formu- 

 lae and metrical relations, and gave us the geometry of position (Numberg, 

 1847, Beitr. z. Geom. d. Lage). 



After von Staudt, the strict geometry of position remained a long time 

 disregarded, while the synthetic geometry of Steiner has enjoyed, without 

 intermission till the present day, a special preference on the part of mathe- 

 maticians. One reason may indeed be that mathematicians take little in- 

 terest in an independence of geometry to which analysis can lay no claim ; 

 but another, still more potent, is the extremely condensed, almost schematic 

 presentation of von Staudt, which has not exactly an encouraging effect 

 upon every one. 



Culmann gave the impulse to a change in this respect. In his graphical 

 statics he rests directly upon the work of von Staudt, and, with something 

 more than boldness, assumes a knowledge of the geometry of position 

 among all practical men. Such a course was not indispensable for the 

 foundation of his method, and impeded the spread of the graphical stat- 

 ics ; but by it the geometry of position gained. This last had next, of 

 necessity, to be introduced into the Zurich Polytechnic, and thus arose the 



