DEVELOPMENT OF GEOMETRIC METHODS 557 



A single objection can be made to studies of this sort, and was 

 already formulated by Poisson: the absence of all real foundation, of 

 all substratum permitting the presentation, under aspects visible and 

 in some sort palpable, of the results obtained. 



The extension of the methods of descriptive geometry, and above 

 all the employment of Pluecker's conceptions on the generation of 

 space, will contribute to take away from this objection much of its 

 force. 



I would have liked to speak to you also of the method of equi- 

 pollences, of which we find the germ in the posthumous works of 

 Gauss, of Hamilton's quaternions, of Grassmann's methods, and in 

 general of systems of complex units, of the analysis situs, so inti- 

 mately connected with the theory of functions, of the geometry 

 called kinematic, of the theory of abaci, of geometrography, of the 

 applications of geometry to natural philosophy or to the arts. But 

 1 fear, if I branched out beyond measure, some analyst, as has hap- 

 pened before, would accuse geometry of wishing to monopolize 

 everything. 



My admiration for analysis, grown so fruitful and so powerful in 

 our time, would not permit me to conceive such a thought. But if 

 some reproach of this sort could be formulated to-day, it is not to 

 geometry, it is to analysis it would be proper, I believe, to address it. 

 The circle in which the mathematical studies appeared to be inclosed 

 at the beginning of the nineteenth century has been broken on all 

 sides. 



The old problems present themselves to us under a new form, new 

 problems offer themselves, whose study occupies legions of workers. 



The number of those who cultivate pure geometry has become 

 prodigiously restricted. Therein is a danger against which it is im- 

 portant to provide. We must not forget that, if analysis has acquired 

 means of investigation which it lacked heretofore, it owes them in 

 great part to the conceptions introduced by the geometers. Geometry 

 must not remain in some sort entombed in its triumph. It is in its 

 school we have learned; our successors must learn never to be blindly 

 proud of methods too general, to envisage the questions in themselves 

 and to find, in the conditions particular to each problem, perhaps 

 a direct way towards a solution, perhaps the means of applying in 

 an appropriate manner the general procedures which every science 

 should gather. 



As Chasles said at the beginning of the Apercu, historique, "The 

 doctrines of pure geometry offer often, and in a multitude of ques- 

 tions, that simple and natural way which, penetrating to the very 

 source of the truths, lays bare the mysterious chain which binds them 

 to each other and makes us know them individually in the way most 

 luminous and most complete." 



