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

POSIDONIUS 221
222 SUCCESSORS OF THE GREAT GEOMETERS
statement of Eratosthenes (based apparently upon hearsay)
from them cannot be proved without the admission of something
else as well which has not been included in the said
that at Syene, which is under the summer tropic, and
throughout a circle round it of 300 stades in diameter, the
principles, and he intended by means of these criticisms to
upright gnomon throws no shadow at noon. It follows from
destroy the whole of geometry. We 1 can understand, therefore,
that the tract of Posidonius was a serious work.
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
A definition of the centre of gravity by one Posidonius a
diameter of the sun is 3,000,000 stades. The assumption ' that
Stoic ' is quoted in Heron's Mechanics, but, as the writer goes
the sun's circle is 10,000 times as large as a great circle of the
on to say that Archimedes introduced a further distinction, we
earth was presumably taken from Archimedes, who had proved
may fairly assume that the Posidonius in question is not
in the Sand-reckoner that the diameter of the sun's orbit is
Posidonius of Rhodes, but another, perhaps Posidonius of
less than 10,000 times that of the earth; Posidonius in fact
Alexandria, a pupil of Zeno of Cittium in the third century
B.C.
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
We now come to Geminus, a very important authority on
of the mean diameter of the earth, the estimates of these
many questions belonging to the history of mathematics, as is
astronomers give for the diameter of the sun the figures 6|,
shown by the numerous quotations from him in Proclus's
12§, and 5^ respectively; Posidonius's estimate gives 39J, the
Commentary on Euclid, Book I. His date and birthplace are
true figure being 108-9.
uncertain, and the discussions on the subject now form a whole
In elementary geometry Posidonius is credited by Proclus
literature for which reference must be made to Manitius's
with certain definitions. He defined figure ' ' as confining
edition of the so-called Gemini elementa astronomiae (Teubner,
'
limit' (wepa? crvyK\dov) 1
and 'parallels' as 'those lines which,
1898) and the article 'Geminus' in Pauly-Wissowa's Realbeing
in one plane, neither converge nor diverge, but have all
Encyclopadie. The doubts begin with his name. Petau, who
the perpendiculars equal which are drawn from the points of
included the treatise mentioned in his Uranologion (Paris,
one line to the other'. 2 (Both these definitions are included
1630), took it to be the Latin Geminus. Manitius, the latest
in the Definitions of Heron.) He also distinguished seven
editor, satisfied himself that it was Geminus, a Greek name,
species of quadrilaterals, and had views on the distinction
judging from the fact that it consistently appears with the
between theorem and problem. Another indication of his
properispomenon accent in Greek {Teiuvos), while it is also
interest in the fundamentals of elementary geometry is the
found in inscriptions with the spelling Pe/xe^o?; Manitius
fact 3 that he wrote a separate work in refutation of the
suggests the derivation from ye/z, as 'EpyTvos from epy, and
Epicurean Zeno of Sidon, who had objected to the very beginnings
of the Elements on the ground that they contained un-
Greek names 'IktIvos, Kparivos. Now, however, we are told
'A\e£tvos from aAe£ ; he compares also the unmistakably
proved assumptions. Thus, said Zeno, even Eucl.1. 1 requires it
(by Tittel) that the name is, after all, the Latin Ge'mmus,
to be admitted that two straight lines cannot have a common
that '
re/jLiuo? came to be so written through false analogy
segment ; '
and, as regards the proof ' ' of this fact deduced
with 'AXegivos, &c, and that Te^elvos, if the reading is
from the bisection of a circle by its diameter, he would object
correct, is also wrongly formed on the model of Avrccvelvo^,
that it has to be assumed that two arcs of circles cannot have
Aypnnrdva. The occurrence of a Latin name in a centre
a common part. Zeno argued generally that, even if we
of Greek culture need not surprise us, since Romans settled in
admit the fundamental principles of geometry, the deductions
such centres in large numbers during the last century B.C.
Geminus, however, in spite of his name, was thoroughly Greek.
1
2
Proclus on Eucl. I, p. 143. 8.
lb., p. 176. 6-10.
3 lb., pp. 199. 14-200. 3.
1
Proclus on Eucl. I, pp. 214. 18-215. 13, p. 216. 10-19, p. 217. 10-23.