TRANSLATOR'S INTRODUCTION. xi 



that of his friend Gauss. Both were intensely 

 trying to prove what now we know is inde- 

 monstrable. And perhaps Bolyai got nearer 

 than Gauss to the unattainable. In his * 4 Kurzer 

 Grundriss eines Versuchs," etc., p. 46, we read: 

 "Koennten jede 3 Punkte, die nicht in einer 

 Geraden sind, in eine Sphaere fallen, so waere 

 das Eucl. Ax. XI. bewiesen." Frischauf calls 

 this "das anschaulichste Axiom." But in his 

 Autobiography written in Magyar, of which 

 my Life of Bolyai contains the first transla- 

 tion ever made, Bolyai Farkas says: "Yet I 

 could not become satisfied with my different 

 treatments of the question of parallels, which 

 was ascribable to the long discontinuance of 

 my studies, or more probably it was due to 

 myself that I drove this problem to the point 

 which robbed my rest, deprived me of tran- 

 quillity." 



It is wellnigh certain that Euclid tried his 

 own calm, immortal genius, and the genius of 

 his race for perfection, against this self-same 

 question. If so, the benign intellectual pride 

 of the founder of the mathematical school of 

 the greatest of universities, Alexandria, would 

 not let the question cloak itself in the obscuri- 

 ties of the infinitely great or the infinitely 

 small. He would say to himself: "Can I prove 



