A history of Greek mathematics - Wilbourhall.org

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