TRANSLATOR'S INTRODUCTION. xiii 



him up mornings? Now at last, true man of 

 science, he acknowledges it indemonstrable by 

 spreading it in all its ugly length among his 

 postulates. 



Since Schiaparelli has restored the astron- 

 omical system of Eudoxus, and Hultsch has 

 published the writings of Autolycus, we see 

 that Euclid knew surface-spherics, was famil- 

 iar with triangles whose angle-sum is more 

 than a straight angle. Did he ever think to 

 carry out for himself the beautiful system of 

 geometry which comes from the contradiction 

 of his indemonstrable postulate; which exists 

 if there be straights produced indefinitely from 

 less than two right angles yet nowhere meet- 

 ing; which is real if the triangle's angle-sum 

 is less than a straight angle? 



Of how naturally the three systems of geom- 

 etry flow from just exactly the attempt we 

 suppose Euclid to have made, the attempt to 

 demonstrate his postulate fifth, we have a most 

 romantic example in the work of the Italian 

 priest, Saccheri, who died the twenty-fifth of 

 October, 1733. He studied Euclid in the edi- 

 tion of Clavius, where the fifth postulate is 

 given as Axiom 13. Saccheri says it should 

 not be called an axiom, but ought to be dem- 

 onstrated. He tries this seemingly simple 



