A history of Greek mathematics - Wilbourhall.org

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