viii TRANSLATOR'S INTRODUCTION. 



Sir Henry Savile, in his Praelectiones on 

 Euclid, Oxford, 1621, p. 140, says: "In pul- 

 cherrimo Geometriae corpora duo sunt naevi, 

 duae labes ..." etc., and these two blemishes 

 are the theory of parallels and the doctrine of 

 proportion; the very points in the Elements 

 which now arouse our wondering admiration. 

 But down to our very nineteenth century an 

 ever renewing stream of mathematicians tried 

 to wash away the first of these supposed stains 

 from the most beauteous body of Geometry. 



The year 1799 finds two extraordinary young 

 men striving thus 



4 ' To gild refined gold, to paint the lily, 

 To cast a perfume o'er the violet." 



At the end of that year Gauss from Braun- 

 schweig writes to Bolyai Farkas in Klausen- 

 burg (Kolozsvar) as follows: [Abhandlungen 

 der Koeniglichen Gesellschaft der Wissen- 

 schaften zu Goettingen, Bd. 22, 1877.] 



" 1 very much regret, that I did not make use 

 of our former proximity, to find out more 

 about your investigations in regard to the first 

 grounds of geometry; I should certainly thereby 

 have spared myself much vain labor, and would 

 have become more restful than any one, such 



