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

246 TRIGONOMETRY
TRIGONOMETRY:
XVII
HIPPARCHUS, MENELAUS,
PTOLEMY
We have seen that Sphaeric, the geometry of the sphere,
was very early studied, because it was required so soon as
astronomy became mathematical ; with the Pythagoreans the
word Sphaeric, applied to one of the subjects of the quadrivium,
actually meant astronomy. The subject was so far advanced
before Euclid's time that there was in existence a regular
textbook containing the principal propositions about great
and small circles on the sphere, from which both Autolycus
and Euclid quoted the propositions as generally known.
These propositions, with others of purely astronomical interest,
were collected afterwards in a work entitled Sphaerica,
in three Books, by Theodosius.
Suidas has a notice, s. v. QeoSocnos, which evidently confuses
the author of the Sphaerica with another Theodosius,
'
a Sceptic philosopher, since it calls him Theodosius, a philosopher
', and attributes to him, besides the mathematical works,
'
Sceptic chapters ' and a commentary on the chapters of
Theudas. Now the commentator on Theudas must have
belonged, at the earliest, to the second half of the second
century A.D., whereas our Theodosius was earlier than Menelaus
{fi.
about A. D. 100), who quotes him by name. The next
notice by Suidas is of yet another Theodosius, a poet, who
came from Tripolis. Hence it was at one time supposed that
our Theodosius was of Tripolis. But Vitruvius x mentions a
Theodosius who invented a sundial ? for any climate ; '
and
Strabo, in speaking of certain Bithynians distinguished in
their particular sciences, refers to ' Hipparchus, Theodosius
and his sons, mathematicians 2 . We conclude that our Theo-
dosius was of
Bithynia and not later in date than Vitruvius
(say 20 B.C.); but the order in which Strabo gives the
names makes it not unlikely that he was contemporary with
Hipparchus, while the character of his Sphaerica suggests a
date even earlier
rather than later.
Works by Theodosius.
Two other works of Theodosius besides the Sphaerica,
namely On habitations and On Days and Nights, seem to
have been included in the 'Little Astronomy' [jxiKpos d&Tpovofjiovfj.ei'os,
sc. tottos). These two treatises need not detain us
long. They are extant in Greek (in the great MS. Vaticanus
Graecus 204 and others), but the Greek text has not apparently
yet been published. In the first, On habitations, in 12
propositions, Theodosius explains the different phenomena due
to the daily rotation of the earth, and the particular portions
of the whole system which are visible to inhabitants of the
different zones.
In the second, On Days and Nights, containing
13 and 19 propositions in the two Books respectively,
Theodosius considers the arc of the ecliptic described by the
sun each day, with a view to determining the conditions to be
satisfied in order that the solstice may occur in the meridian
at a given place, and in order that the day and the night may
really be equal at the equinoxes; he shows also that the
variations in the day and night must recur exactly after
a certain time, if the length of the solar year is commensurable
with that of the day, while on the contrary assumption
they will not recur so exactly.
In addition to the works bearing on astronomy, Theodosius
is said l to have written a commentary, now lost, on the e