vi TRANSLATOR'S INTRODUCTION. 



Bolyai speaks of it as Euclid's Axiom XI. 

 Todhunter has it as twelfth of the Axioms. 



Clavius (1574) gives it as Axiom 13. 



The Harpur Euclid separates it by forty- 

 eight pages from the other axioms. 



It is not used in the first twenty-eight pro- 

 positions of Euclid. Moreover, when at length 

 used, it appears as the inverse of a proposition 

 already demonstrated, the seventeenth, and is 

 only needed to prove the inverse of another 

 proposition already demonstrated, the twenty- 

 seventh. 



Now the great Lambert expressly says that 

 Proklus demanded a proof of this assumption 

 because when inverted it is demonstrable. 



All this suggested, at Europe's renaissance, 

 not a doubt of the necessary external reality 

 and exact applicability of the assumption, but 

 the possibility of deducing it from the other 

 assumptions and the twenty-eight propositions 

 already proved by Euclid without it. 



Euclid demonstrated things more axiomatic 

 by far. He proves what every dog knows, 

 that any two sides of a triangle are together 

 greater than the third. 



Yet after he has finished his demonstration, 

 that straight lines making with a transversal 

 equal alternate angles are parallel, in order to 



