A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

PROCLUS 533
The commentary opens with two Prologues. The first is
on mathematics in general and its relation to, and use in,
philosophy, from which Proclus passes to the classification of
mathematics. Prologue II deals with geometry generally and
its subject-matter according to Plato, Aristotle and others.
After this section comes the famous summary (pp. 64-8)
ending with a eulogium of Euclid, with particular reference
to the admirable discretion shown in the selection of the propositions
which should constitute the Elements of geometry,
the ordering of the whole subject-matter, the exactness and
the conclusiveness of the demonstrations, and the power with
which every question is handled. Generalities follow, such as
the discussion of the nature of elements, the distinction between
theorems and problems according to different authorities, and
finally a division of Book I into three main sections, (1) the
construction and properties of triangles and their parts and
the comparison between triangles in respect of their angles
and sides, (2) the properties of parallels and parallelograms
and their construction from certain data, and (3) the bringing
of triangles and parallelograms into relation as regards area.
Coming to the Book itself, Proclus deals historically and
critically with all the definitions, postulates and axioms in
order. The notes on the postulates and axioms are preceded
by a general discussion of the principles of geometry, hypotheses,
postulates and axioms, and their relation to one
another ; here as usual Proclus quotes the opinions of all the
important authorities. Again, when he comes to Prop. 1, he
discusses once more the difference between theorems and
problems, then sets out and explains the
formal divisions of
a proposition, the enunciation (rrporacrLs), the setting-out
(eKOeais), the definition or specification (8iopianos), the construction
(Karaa-Kevrj), the 2
jroof
(av/jL7repaor/ia), and finally a number of
(dTToSeigts), the conclusion
other technical terms,
e.g. things said to be given, in the various senses of this term,
the lemma, the case, the porism in its two senses, the objection
(evo-Tacris), the reduction of a problem, reductio ad absurdum,
analysis and synthesis.
In his comments on the separate propositions Proclus
generally proceeds in this way : first he gives explanations
regarding Euclid's proofs, secondly he gives a few different
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.