358 
EUCLID 
it appeared, is and will doubtless remain the greatest mathe 
matical text-book of all time. Scarcely any other book 
except the Bible can have circulated more widely the world 
over, or been more edited and studied. Even in Greek times 
the most accomplished mathematicians occupied themselves 
with it; Heron, Pappus, Porphyry, Proclus and Simplicius 
wrote commentaries; Theon of Alexandria re-edited it, alter 
ing the language here and there, mostly with a view to 
greater clearness and consistency, and interpolating inter 
mediate steps, alternative proofs, separate ‘cases’, porisrns 
(corollaries) and lemmas (the most important addition being 
the second part of VI. 33 relating to sectors). Even the great 
Apollonius was moved by Euclid's work to discuss the first 
principles of geometry; his treatise on the subject was in 
fact a criticism of Euclid, and none too successful at that; 
some alternative definitions given by him have point, but his 
alternative solutions of some of the easy problems in Book I 
do not constitute any improvement, and his attempt to prove 
the axioms (if one may judge by the case quoted by Proclus, 
that of Axiom 1) was thoroughly misconceived. 
Apart from systematic commentaries on the whole work or 
substantial parts of it, there were already in ancient times 
discussions and controversies on special subjects dealt with by 
Euclid, and particularly his theory of parallels. The fifth 
Postulate was a great stumbling-block. We know from 
Aristotle that up to his time the theory of parallels had not 
been put on a scientific basis 1 : there was apparently some 
petitio principii lurking in it. It seems therefore clear that 
Euclid was the first to apply the bold remedy of laying down 
the indispensable principle of the theory in the form of an 
indemonstrable Postulate. But geometers were not satisfied 
with this solution. Posidonius and Geminus tried to get 
over the difficulty by substituting an equidistance theory of 
parallels. Ptolemy actually tried to prove Euclid’s postulate, 
as also did Proclus, and (according to Simplicius) one Diodorus, 
as well as ‘ Aganis ’; the attempt of Ptolemy is given by 
Proclus along with his own, while that of ‘ Aganis ’ is repro 
duced from Simplicius by the Arabian commentator an- 
Nairizi. 
1 Anal. Prior, ii. 16. 65 a 4.