vi TRANSLATOR S INTRODUCTION. 



Bolyai speaks of it as Euclid s Axiom XI. 

 Todhunter has it as twelfth of the Axioms. 



Clavitis (1374) gives it as Axiom 13. 



The Harpur Euclid separates it by forty- 

 eight pages from the other axioms. 



It is not used in tin- 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 accessary external reality 

 and exact applicability &amp;lt;&amp;gt;f the assumption, but 

 the possibility of deducing it from the other 

 aumptions and the twenty-eight propositions 

 already proved bv Kuclid without it. 



Kuclid demonstrated things more axiomatic 

 by far. He pnve&amp;lt; 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 &amp;gt;traight lines making with a transversal 

 eoual alternate angles are parallel, in order to 



