The heroic age of geometry

Abstract

The remarks which I shall have the honor to make today are addressed primarily, if not exclusively, to the older generation. The prophet Joel describes the golden age to come by saying Your old men shall dream dreams, and your young men shall see visions. That distinction is eternally valid. Youth has visions of the future. Age dreams of the past. Our younger members are facing forward, looking towards the splendid mathematical discoveries that will be made during the coming decades, among which will be included those which they themselves will have made. It stirs the blood and fortifies the courage to feel that one is called upon to contribute in this way to the advancement of science, and that a grateful posterity will recognize what one has done. If the present passion for materialism continues, the number of contributions to mathematical science in the next hundred years may not greatly exceed half a million. Go on, oh younger generation, happy in the thought that the mathematicians of a century hence will, by an unfailing instinct, pick out your own particular contributions from all the rest! For you who, like myself, belong to the generation that is passing, who have reached the age to dream dreams, I propose a humbler task. I propose that we look backward. The most notable epoch in all the long history of geometry, the heroic age, was almost exactly a hundred years ago. I t was the moment when geometric supremacy was passing from France to Germany, a fact which the French geometers were slow to recognize, for Chasles, in his Apercu historique des methodes en geometrie, regretted that he could not