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

HIPPARCHUS 255
that the lengths of the mean synodic, the sidereal, the
anomalistic and the draconitic month obtained by Hipparchus
agree exactly with Babylonian cuneiform tables of date not
later than Hipparchus, and it is clear that Hipparchus was
in full possession of all the results established by Babylonian
astronomy.
Improved estimates of sizes and distances of sun
and moon.
4. Hipparchus improved on Aristarchus's calculations of the
sizes and distances of the sun and moon, determining the
apparent diameters more exactly and noting the changes in
them ; he made the mean distance of the sun 1,245 D, the mean
distance of the moon 33|D, the diameters of the sun and
moon 1 2 J D and J D respectively, where D is the mean
diameter of
the earth.
Epicycles and eccentrics.
5. Hipparchus, in investigating the motions of the sun, moon
and planets, proceeded on the alternative hypotheses of epicycles
and eccentrics ; he did not invent these hypotheses,
which were already fully understood and discussed by
Apollonius. While the motions of the sun and moon could
with difficulty be accounted for by the simple epicycle and
eccentric hypotheses, Hipparchus found that for the planets it
was necessary to combine the two, i.e. to superadd epicycles to
motion in eccentric circles.
Catalogue of stars.
6. He compiled a catalogue of fixed stars including 850 or
more such stars; apparently he was the first to state their
positions in terms of coordinates in relation to the ecliptic
(latitude and longitude), and his table distinguished the
apparent sizes of the stars. His work was continued by
Ptolemy, who produced a catalogue of 1,022 stars which,
owing to an error in his solar tables affecting all his longitudes,
has by many erroneously been supposed to be a mere
reproduction of Hipparchus's catalogue. That Ptolemy took
many observations himself seems certain. 1
1
See two papers by Dr. J. L. E. Dreyer in] the Monthly Notices of the
Royal Astronomical Society, 1917, pp. 528-39, and 1918.. pp. 343-9.
256 TRIGONOMETRY
Improved Instruments.
7. He made great improvements in the instruments used for
observations. Among those which he used were an improved
dioptra, a ' meridian-instrument ' designed for observations in
the meridian only, and a universal instrument {acrTpoXdPov
opyavov) for more general use. He also made a globe on
which he showed the positions of the fixed stars as determined
by him ; it appears that he showed a larger number of stars
on his globe than in his catalogue.
Geography.
In geography Hipparchus wrote a criticism of Eratosthenes,
in great part unfair. He checked Eratosthenes's data by
means of a sort of triangulation ; he insisted on the necessity
of applying astronomy to geography, of fixing the position of
places by latitude and longitude, and of determining longitudes
by observations of lunar eclipses.
Outside the domain of astronomy and geography, Hipparchus
wrote a book On things borne doivn by their weight from
which Simplicius (on Aristotle's De caelo, p. 264 sq.) quotes
two propositions. It is possible, however, that even in this
work Hipparchus may have applied his doctrine to the case of
the heavenly bodies.
In pure mathematics he is said to have considered a problem
in permutations and combinations, the problem of finding the
number of different possible combinations of 10 axioms or
assumptions, which he made to be 103,049 (v. I. 101,049)
or 310,952 according as the axioms were affirmed or denied 1 :
When
it seems impossible to make anything of these figures.
'
the Fihrist attributes to him works On the art of algebra,
known by the title of the Rules ' and On the division of numbers
we have no confirmation ', : Suter suspects some ' confusion,
in view of the fact that the article immediately following in
the Fihrist is on Diophantus, who also ' wrote on the art of
algebra
1
Plutarch, Quaest. Conviv, viii. 9. 3, 732 f, De Stoicorum repugn. 29.
1047 d.