A history of Greek mathematics - Wilbourhall.org

POSIDONIUS 221
statement of Eratosthenes (based apparently upon hearsay)
that at Syene, which is under the summer tropic, and
throughout a circle round it of 300 stades in diameter, the
It follows from
upright gnomon throws no shadow at noon.
this that the diameter of the sun occupies a portion of the
sun's circle 3,000,000 stades in length ;
in other words, the
diameter of the sun is 3,000,000 stades. The assumption that
the sun's circle is
10,000 times as large as a great circle of the
earth was presumably taken from Archimedes, who had proved
in the Sand-reckoner that the diameter of the sun's orbit is
less than 10,000 times that of the earth; Posidonius in fact
took the maximum value to be the true value ; but his estimate
of the sun's size is far nearer the truth than the estimates
of Aristarchus, Hipparchus, and Ptolemy.
Expressed in terms
of the mean diameter of the earth, the estimates of these
astronomers give for the diameter of the sun the figures 6|,
12§, and 5^ respectively; Posidonius's estimate gives 39J, the
true figure being 108-9.
In elementary geometry Posidonius is credited by Proclus
with certain definitions. He defined figure ' ' as confining
'
limit' (wepa? crvyK\dov) 1
and 'parallels' as 'those lines which,
being in one plane, neither converge nor diverge, but have all
the perpendiculars equal which are drawn from the points of
one line to the other'. 2 (Both these definitions are included
in the Definitions of Heron.) He also distinguished seven
species of quadrilaterals, and had views on the distinction
between theorem and problem. Another indication of his
interest in the fundamentals of elementary geometry is the
fact 3 that he wrote a separate work in refutation of the
Epicurean Zeno of Sidon, who had objected to the very beginnings
of the Elements on the ground that they contained unproved
assumptions. Thus, said Zeno, even Eucl.1. 1 requires it
to be admitted that two straight lines cannot have a common
'
segment ; '
and, as regards the proof ' ' of this fact deduced
from the bisection of a circle by its diameter, he would object
that it has to be assumed that two arcs of circles cannot have
a common part. Zeno argued generally that, even if we
admit the fundamental principles of geometry, the deductions
1
Proclus on Eucl. I, p. 143. 8.
3 lb., pp. 199. 14-200. 3.
2 lb., p. 176. 6-10.