A history of Greek mathematics - Wilbourhall.org

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.