A history of Greek mathematics - Wilbourhall.org

534 COMMENTATORS AND BYZANTINES
cases, mainly for the sake of practice, and thirdly he addresses
himself to refuting objections which cavillers had taken or
might take to particular propositions or arguments. He does
not seem to have had any notion of correcting or improving
Euclid; only in one place does he propose anything of his
own to get over a difficulty which he finds in Euclid ; this is
where he tries to prove the parallel-postulate, after giving
Ptolemy's attempt to prove it and pointing out objections to
Ptolemy's proof.
The book is evidently almost entirely a compilation, though
a compilation ' in the better sense of the term '. The onus
probandi is on any one who shall assert that anything in it is
Proclus's own ; very few things can with certainty be said to
be so. Instances are (1) remarks on certain things which he
quotes from Pappus, since Pappus was the last of the commentators
whose works he seems to have used, (2) a defence
of Geminus against Carpus, who criticized Geminus's view of
the difference between theorems and problems, and perhaps
(3) criticisms of certain attempts by Apollonius to improve on
Euclid's proofs and constructions ;
but the only substantial
example is (4) the attempted proof of the parallel-postulate,
based on an axiom ' ' to the effect that, ' if from one point two
straight lines forming an angle be produced ad infinitum, the
distance between them when so produced ad infinitum exceeds
any finite magnitude (i. e. length) ', an assumption which
purports to be the equivalent of a statement in Aristotle. 1
Philoponus says that Proclus as well as Ptolemy wrote a whole
book on the parallel-postulate. 2
It is still not quite certain whether Proclus continued his
commentaries beyond Book I. He certainly intended to do so,
for, speaking of the trisection of an angle by means of certain
curves, he says, we may perhaps more appropriately examine
'
these things on the third Book, where the writer of the
Elements bisects a given circumference ', and again, after
saying that of all parallelograms which have the same perimeter
the square is the greatest and the rhomboid least of
'
all', he adds, ' But this we will prove in another place, for it
is more appropriate to the discussion of the hypotheses of the
1
De caelo, i. 5, 271 b 28-30.
2
Philoponus on Anal. Post. i. 10, p. 214 a 9-12, Brandis.