434 POPULAR SCIENCE MONTHLY. 



of the theory of abaci, of geometrography, of the applications of 

 geometry to natural philosophy or to the arts. But I fear, if I 

 branched out beyond measure, some analyst, as has happened 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 enclosed 

 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 

 questions, that way simple and natural 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.' 



Cultivate therefore geometry, which has its own advantages, with- 

 out wishing, on all points, to make it equal to its rival. 



For the rest, if we were tempted to neglect it, it would soon find in 

 the applications of mathematics, as it did once before, means to rise 

 up again and develop itself anew. It is like the giant Anteus who 

 recovered his strength in touching the earth. 



