A history of Greek mathematics - Wilbourhall.org

258 TRIGONOMETRY
in relation to one another ; they could not Calculate the actual
times. As Hipparchus proved corresponding propositions by
means of numbers, we can only conclude that he used propositions
in spherical trigonometry, calculating arcs from others
which are given, by means of tables. (3) In the only work
of his which survives, the Commentary on the Phaenomena
of Eudoxus and Aratus (an early work anterior to the
discovery of the precession of the equinoxes), Hipparchus
states that (presumably in the latitude of Rhodes) a star which
lies 27^° north of the equator describes above the horizon an
arc containing 3 minutes less than 15/24ths of the whole
circle 1 ;
then, after some more inferences, he says, ' For each
of the aforesaid facts is proved by means of lines (8ia rS>v
ypafificov) in the general treatises on these matters compiled
by me '. In other places 2 of the Commentary he alludes to
a work On simultaneous risings (ra irepl tcov o-vvavaroXcov),
and in II. 4. 2 he says he will state summarily, about each of
the fixed stars, along with what sign of the zodiac it rises and
sets and from which degree to which degree of each sign it
rises or sets in the regions about Greece or wherever the
longest day is
14^ equinoctial hours, adding that he has given
special proofs in another work designed so that it is possible
in practically every place in the inhabited earth to follow
the differences between the concurrent risings and settings. 3
Where Hipparchus speaks of proofs ' by means of lines ', he
does not mean a merely graphical method, by construction
only, but theoretical determination by geometry, followed by
calculation, just as Ptolemy uses the expression e/c tS>v ypa/i-
/jlcov
of his calculation of chords and the expressions cr(paipiKal
SeigeLs and ypafifiLKal Seigeis of the fundamental proposition
in spherical trigonometry (Menelaus's theorem applied to the
sphere) and its various applications to particular cases. It
is significant that in the Syntaxis VIII. 5, where Ptolemy
applies the proposition to the very problem of finding the
times of concurrent rising, culmination and setting of the
fixed stars, he says that the times can be obtained ' by lines
only ' (8ia ixovoav tcov ypafifi(£>v).
A
Hence we may be certain
that, in the other books of his own to which Hipparchus refers
1
3
2
Ed. Manitius, pp. 148-50. lb., pp. 128. 5, 148. 20.
4
lb., pp. 182. 19-184. 5.
Syntaxis, vol. ii, p. 193.