A history of Greek mathematics - Wilbourhall.org

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