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 develop 
ment of trigonometry. Even if he did not invent it, 
Hipparchus is the first person of whose systematic v^e of 
trigonometry we have documentary evidence. (1) Theon 
of Alexandria says on the Syntaxis of Ptolemy, à 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 (Sia 
rfjs eK tcûv ypayyan' p.e6o8iKrjs avrcov awrao-ecoy). The pro 
positions are famous, and it cannot be doubted that Hippar 
chus 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 hy means 
of numerical calculations (Si dpidycov) 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 Theo 
dosius 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. 9-18. 
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