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

DIONYSODORUS 219
generates the tore, and c the distance of its centre from the
axis of revolution,
(volume of tore) \ttc 2 .d = ^7rd 2 :^cd i
2
that is, (volume of tore) = \tt . cd
2 ,
which is of course the product of the area of the generating
circle and the length of the path of its centre of gravity. The
form in which the result is stated, namely that the tore is to
the cylinder with height d and radius c as the generating
circle of the tore is to half the parallelogram cd, indicates
quite clearly that Dionysodorus proved his result by the same
procedure as that employed by Archimedes in the Method and
in the book On Conoids and Sjjheroids ; and indeed the proof
on Archimedean lines is not difficult.
Before passing to the mathematicians who are identified
with the discovery and development of trigonometry, it will
be convenient, I think, to dispose of two more mathematicians
belonging to the last century B.C., although this involves
a slight departure from chronological order ; I mean Posidonius
and Geminus.
Posidonius, a Stoic, the teacher of Cicero, is known as
Posidonius of Apamea (where he was born) or of Rhodes
(where he taught) ; his date may be taken as approximately
135-51 B.C. In pure mathematics he is mainly quoted as the
author of certain definitions, or for views on technical terms,
e.g. theorem and problem ' ' ' ', and subjects belonging to elementary
geometry. More important were his contributions
to mathematical geography and astronomy. He gave his
great work on geography the title On the Ocean, using the
word which had always had such a fascination for the Greeks
;
its contents are known to us through the copious quotations
from it in Strabo ; it dealt with physical as well as mathematical
geography, the zones, the tides and their connexion
with the moon, ethnography and all sorts of observations made
during extensive travels. His astronomical book bore the
title Meteorologica or nepl uereoopcov, and, while Geminus
wrote a commentary on or exposition of this
work, we may
assign to it a number of views quoted from Posidonius in
220 SUCCESSORS OF THE GREAT GEOMETERS
Cleomedes's work De motu circulari corporum caelestium.
Posidonius also wrote a separate tract on the size of the sun.
The two things which are sufficiently important to deserve
mention here are (1) Posidonius's measurement of the circumference
of the earth, (2) his hypothesis as to the distance and
size of the sun. *
(1) He estimated the circumference of the earth in this
way. He assumed (according to Cleomedes l ) that, whereas
the star Canopus, invisible in Greece, was just seen to graze the
horizon at Rhodes, rising and setting again immediately, the
'
meridian altitude of the same star at Alexandria was a fourth
part of a sign, that is, one forty-eighth part of the zodiac
circle' (= 7-|°) ; and he observed that the distance between
the two places (supposed to lie on the same meridian) ' was
considered to be 5,000 stades'. The circumference of the
earth was thus made out to be 240,000 stades.
Unfortunately
the estimate of the difference of latitude, 7^°, was very far
from correct, the true difference being 5^° only ; moreover
the estimate of 5,000 stades for the distance was incorrect*
being only the maximum estimate put upon it by mariners,
while some put it
of the shadows of gnomons, found it
at 4,000 and Eratosthenes, by observations
to be 3,750 stades only.
Strabo, on the other hand, says that Posidonius favoured the
latest of the measurements which gave the smallest dimensions
to the earth, namely about 180,000 stades'. 2 This is
evidently 48 times 3,750, so that Posidonius combined Eratosthenes's
figure of 3,750 stades with the incorrect estimate
of 7\° for the difference of latitude, although Eratosthenes
presumably obtained the figure of 3,750 stades from his own
estimate (250,000 or 252,000) of the circumference of the earth
combined with an estimate of the difference of latitude which
was about 5-1° and therefore near the truth.
5
(2) Cleomedes 3 tells us that Posidonius supposed the circle
in which the sun apparently moves round the earth to be
10,000 times the size of a circular section of the earth through
its centre, and that with this assumption he combined the
1
Cleomedes, Be motu circulari, i. 10, pp. 92-4.
2
Strabo, ii. c. 95.
3
Cleomedes, op. cit. ii. 1, pp. 144-6, p. 98. 1-5.