Page:Popular Science Monthly Volume 66.djvu/438

434 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 important 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 'Aperçu 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, without 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.