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

PROCLUS 535
second Book \ But at the time when the commentary on
Book I was written he was evidently uncertain whether he
would be able to continue it, for at the end he says, ' For my
part, if I should be able to discuss the other Books in the
same way, I should give thanks to the gods ; but, if other
cares should draw me away, I beg those who are attracted by
this subject to complete the exposition of
well,
the other Books as
following the same method and addressing themselves
throughout to the deeper and more sharply defined questions
involved '} Wachsmuth, finding a Vatican manuscript containing
a collection of scholia on Books I, II, V, VI, X, headed Eis ra
EvKXeiSou (TToi)(€ia TTpoXanfiavojieva €K toou UpoKXov (nropdSrju
Kal kolt €7TLTo/xriu, and seeing that the scholia on Book I were
extracts from the extant commentary of Proclus, concluded
that those on the other Books were also from Proclus; but
the 7r/oo- in TTpoXa\x$avbii£va rather suggests that only the
scholia to Book I are from Proclus. Heiberg found and
published in 1903 a scholium to X. 9, in which Proclus is
expressly quoted as the authority, but he does not regard
this circumstance as conclusive. On the other hand, Heiberg
has noted two facts which go against the view that Proclus
wrote on the later Books: (1) the scholiast's copy of
Proclus was not much better than our manuscripts ; in
particular, it had the same lacunae in the notes to I. 36,
37, and I. 41-3; this makes it improbable that the scholiast
had further commentaries of Proclus which have vanished
for us ; (2) there is no trace in the scholia of the notes
which Proclus promised in the passages already referred to.
All, therefore, that we can say is that, while the Wachsmuth
scholia may be extracts from Proclus, it is on the whole
improbable.
Hypotyposis of Astronomical Hypotheses.
Another extant work of Proclus which should be referred
to is his Hypotyposis of Astronomical Hypotheses, a sort of
readable and easy introduction to the astronomical system
of Hipparchus and Ptolemy. It has been well edited by
Manitius (Teubner, 1909). Three things may be noted as
1
Proclus on Eucl. I, p. 432. 9-15.
536 COMMENTATORS AND BYZANTINES
regards this work. It contains 1 a description of the method
of measuring the sun's apparent diameter by means of
Heron's water-clock, which, by comparison with the corresponding
description in Theon's commentary to the Syntaxis
of Ptolemy, is seen to have a common source with it. That
source is Pappus, and, inasmuch as Proclus has a figure (reproduced
by Manitius in his text from one set of manuscripts)
corresponding to the description, while the text of Theon has
no figure, it is clear that Proclus drew directly on Pappus,
who doubtless gave, in
his account of the procedure, a figure
taken from Heron's own work on water-clocks. A simple
proof of the equivalence of the epicycle and eccentric hypotheses
is quoted by Proclus from one Hilarius of Antioch. 2
An interesting passage is that in chap. 4 (p. 130, 18) where
Sosigenes the Peripatetic is said to have recorded in his work
1
on reacting spheres ' that an annular eclipse of the sun is
sometimes observed at times of perigee ; this is, so far as
I know, the only allusion in ancient times to annular eclipses,
and Proclus himself questions the correctness of Sosigenes's
statement.
Commentary on
the Republic.
The commentary of Proclus on the Republic contains some
passages of great interest to the historian of mathematics.
The most important is that 3 in which Proclus indicates that
Props. 9, 10 of Euclid, Book II, are Pythagorean propositions
invented for the purpose of proving geometrically the
fundamental property of the series of ' side-' and ' diameter-'
numbers, giving successive approximations to the value of
\/2 (see vol. i, p. 93). The explanation 4 of the passage in
Plato about the Geometrical Number is defective and disappointing,
but it contains an interesting reference to one
Paterius, of date presumably intermediate between Nestorius
and Proclus. Paterius is said to have made a calculation, in
units and submultiples, of the lengths of different segments of
1
2
Proclus, Hypotyposis, c. 4, pp. 120-22.
lb., c. 3, pp. 76, 17 sq.
3
Prodi Diadochi in Platonis Rempublicam Commentarii, ed. Kroil,
vol. ii, p. 27.
4 lb., vol. ii, pp. 36-42.