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

HIPPARCHUS 257
First systematic use of Trigonometry.
We come now to what is the most important from the
point of view of this work, Hipparchus's share in the development
of trigonometry. Even if he did not invent it,
Hipparchus is the first person of whose systematic use of
trigonometry we have documentary evidence. (1) Theon
of Alexandria says on the Syntaxis of Ptolemy, a propos of
Ptolemy's Table of Chords in a circle (equivalent to sines),
that Hipparchus, too, wrote a treatise in twelve books on
straight lines (i.e. chords) in a circle, while another in six
books was written by Menelaus. 1 In the Syntaxis I. 10
Ptolemy gives the necessary explanations as to the notation
used in his Table. The circumference of the circle is divided
into 360 parts or degrees; the diameter is also divided into
120 parts, and one of such parts is the unit of length in terms
of which the length of each chord is expressed ; each part,
whether of the circumference or diameter, is divided into 60
parts, each of these again into 60, and so on, according to the
system of sexagesimal fractions. Ptolemy then sets out the
minimum number of propositions in plane geometry upon
which the calculation of the chords in the Table is based (8ia
rijs e/c tcov ypafifioou fjieOoSiKrjs ccvtcov avo-Tcicrecos). The propositions
are famous, and it
cannot be doubted that Hipparchus
used a set of propositions of the same kind, though his
exposition probably ran to much greater length. As Ptolemy
definitely set himself to give the necessary propositions in the
shortest form possible, it will be better to give them under
Ptolemy rather than here. (2) Pappus, in speaking of Euclid's
propositions about the inequality of the times which equal arcs
of the zodiac take to rise, observes that ' Hipparchus in his book
On the rising of the twelve signs of the zodiac shows by means
of numerical calc%dations (6Y dpiOfioou) that equal arcs of the
semicircle beginning with Cancer which set in times having
a certain relation to one another do not everywhere show the
same relation between the times in which they rise ', 2 and so
on. We have seen that Euclid, Autolycus, and even Theodosius
could only prove that the said times are greater or less
1
Theon, Comm. on Syntaxis, p. 110, ed. Halma.
2
Pappus, vi, p. 600. 9-13.
1523.2 S
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
Ed. Manitius, pp. 148-50.
2 lb., pp. 128. 5, 148. 20.
lb., pp. 182. 19-184. 5.
4
Syntaxis, vol. ii, p. 193.