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 proposi
tions in spherical trigonometry, calculating arcs from others
which are given, by means of tables. (3) In the only work
of Ins 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 27g-° north of the equator describes above the horizon an
arc containing 3 minutes less than 15/2 4ths of the whole
circle 1 ; then, after some more inferences, he says, ‘For each
of the aforesaid facts is proved by means of lines {8La t5>v
ypaygoor) 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 {fa ne pi r<5 v awavaroXciv),
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 ex tw ypay-
ya>v of his calculation of chords and the expressions crfatpucal
Seigets and ypaypuKal 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 ’ (8ià pov(ûv tcûv ypaypcûv)} Hence we may be certain
that, in the other books of his own to which Hipparchus refers
1 Ed. Manitius, pp. 148-50.
? lb., pp. 182. 19-184. 5.
2 lb., pp. 128. 5, 148. 20.
4 Syntaxis, vol. ii, p. 193.