x TRANSLATOR S INTROIH VTION. 



try. and that it is tin- system regnant in the 

 external space of our physical experience. 



The first is false; the second ran never he 

 proven. 



I&amp;gt;efore another quarter of a century, Bolyai 

 Janus, then unhorn, 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 -atisfactorilv shown to belong to Bolyai 

 Janus, yet the same is not true for Euclid. 



To decide our space is Holyai s, one need 

 only show a single rectilineal triangle whose 

 angle-sum measures less than a straight angle. 

 And this could he shown to exist bv imperfect 

 meaMiivmriit&amp;gt;, such as human measurements 

 must alwa\s he. For example, if uur instru- 

 mrnts fur angular meastiivment could he 

 brought to measure an angle to within one 

 millionth of a -rcond, then if the lack were as 

 givat a&amp;gt; two millionths of a second, we could 

 make certain its rxisten- 



But to prove Kuclid s system, we must show 

 that a triangU- s angle-sum is c.vcn t/v a straight 

 angle, which nothing human can ever do. 



However this is anticipating, for in 1 7 ( ) ( ) it 

 seemfl that tin- mind of the elder Holvai, Bolvai 

 Karkas. was in precisely tile same state as 



