TRANSLATOR'S INTRODUCTION. ix 



as I, can be, so long as on such a subject there 

 yet remains so much to be wished for. 



In my own work thereon I myself have ad- 

 vanced far (though my other wholly hetero- 

 geneous employments leave me little time 

 therefor) but the way, which I have hit upon, 

 leads not so much to the goal, which one 

 wishes, as much more to making doubtful the 

 truth of geometry. 



Indeed I have come upon much, which with 

 most no doubt would pass for a proof, but 

 which in my eyes proves as good as nothing. 



For example, if one could prove, that a rec- 

 tilineal triangle is possible, whose content may 

 be greater, than any given surface, then I am 

 in condition, to prove with perfect rigor all 

 geometry. 



Most would indeed l,et that pass as an axiom; 

 I not; it might well be possible, that, how far 

 apart soever one took the three vertices of the 

 triangle in space, yet the content was always 

 under a given limit. 



I have more such theorems, but in none do I 

 find anything satisfying." 



From this letter we clearly see that in 1799 

 Gauss was still trying to prove that Euclid's 

 is the only non-contradictory system of geome- 



