72 Proceedings of the Royal Irish Academy. 



Paris, 1814, in Greek, Latin, and Frencli, mainly based upon a very 

 ancient ATS. which had remained unknown until examined by 

 Peyrard. 



"Tliis MS. (190), may be called the Vatican, as it formerly be- 

 longed to that Library, and was sent from Eome to Paris by the 

 Comte de Peluse. 



This MS. bears all the characters of the end of the ninth century, 

 while all the other MSS. belong to much more recent times. _ It is 

 remarkable that the Vatican MS. places the base of the doctrine of 

 parallel lines among the Postulates and not among the Axioms ; and 

 the Arabic text translated by Campanus seems to have been taken 

 from some old source similar to the Vatican MS. 



We may, therefore, as I think, reasonably conclude^ that Euclid 

 shared the opinion of the moderate geometers, who think that the 

 foundation of the doctrine of parallel lines is not self-evident, but 

 takes the form of a Postulate, equivalent to the converse of the seven- 

 teenth of the first. 



