xxiv TRANSLATOR'S INTRODUCTION. 



then informed me how much farther alre 

 had been attained on this way, and later 

 expressed himself about the great acquisitio 

 which is offered to the few expert judges ii, 

 the Appendix to your book." 



The ' ' latest edition ' ' mentioned appeared 

 in 1851, and the passage referred to is: "This 

 proof [of the parallel-axiom] has been sought 

 in manifold ways by acute mathematicians, 

 but yet until now not found with complete 

 sufficiency. So long as it fails, the theorem, 

 as all founded on it, remains a hypothesis, 

 whose validity for our life indeed is suffici- 

 ently proven by experience, whose general, 

 necessary exactness, however, .could be 

 doubted without absurdity." 



Alas! that this feeble utterance should have 

 seemed sufficient for more than thirty years 

 to the associate of Gauss and Schweikart, the 

 latter certainly one of the independent discov- 

 erers of the non-Euclidean geometry. But 

 then, since neither of these sufficiently real- 

 ized the transcendent importance of the mat- 

 ter to publish any of their thoughts on the 

 subject, a more adequate conception of the 

 issues at stake could scarcely be expected of 

 the scholar and colleague. How different with 

 Bolyai Janos and Lobachevski, who claimed 



