x TRANSLATOR'S INTRODUCTION. 



try, and that it is the system regnant in the 

 external space of our physical experience. 



The first is false; the second can never be 

 proven. 



Before another quarter of a century, Bolyai 

 Janos, then unborn, had created another pos- 

 sible universe; and, strangely enough, though 

 nothing renders it impossible that the space of 

 our physical experience may, this very year, 

 be satisfactorily shown to belong to Bolyai 

 Janos, yet the same is not true for Euclid. 



To decide our space is Bolyai's, one need 

 only show a single rectilineal triangle whose 

 angle-sum measures less than a straight angle. 

 And this could be shown to exist by imperfect 

 measurements, such as human measurements 

 must always be. For example, if our instru- 

 ments for angular measurement could be 

 brought to measure an angle to within one 

 millionth of a second, then if the lack were as 

 great as two millionths of a second, we could 

 make certain its existence. 



But to prove Euclid's system, we must show 

 that a triangle's angle-sum is exactly a straight 

 angle, which nothing human can ever do. 



However this is anticipating, for in 1799 it 

 seems that the mind of the elder Bolyai, Bolyai 

 Farkas, was in precisely the same state as 



