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.