276 4 
TRIGONOMETRY 
be given here. Book I : Indispensable preliminaries to the 
study of the Ptolemaic system, general explanations of 
the different motions of the heavenly bodies in relation to 
the earth as centre, propositions required for the preparation 
of Tables of Chords, the Table itself, some propositions in 
spherical geometry leading to trigonometrical calculations of 
the relations of arcs of the equator, ecliptic, horizon and 
meridian, a ‘ Table of Obliquity for calculating declinations 
for each degree-point on the ecliptic, and finally a method of 
finding the right ascensions for arcs of the ecliptic equal to 
one-third of a sign or 10°. Book II: The same subject con 
tinued, i.e. problems on the sphere, with special reference to 
the differences between various latitudes, the length of the 
longest day at any degree of latitude, and the like. Book III : 
On the length of the year and the motion of the sun on the 
eccentric and epicycle hypotheses. Book IY : The length of the 
months and the theory of the moon. Book V : The construc 
tion of the astrolabe, and the theory of the moon continued, 
the diameters of the sun, the moon and the earth’s shadow, 
the distance of the sun and the dimensions of the sun, moon 
and earth. Book YI : Conjunctions and oppositions of sun 
and moon, solar and lunar eclipses and their periods. Books 
VII and YIII are about the fixed stars and the precession of 
the equinoxes, and Books IX-XIII are devoted to the move 
ments of the planets. 
Trigonometry in Ptolemy. 
What interests the historian of mathematics is the trigono 
metry in Ptolemy. It is evident that no part of the trigono 
metry, or of the matter preliminary to it, in Ptolemy was new. 
What he did was to abstract from earlier treatises, and to 
condense into the smallest possible space, the minimum of 
propositions necessary to establish the methods and formulae 
used. Thus at the beginning of the preliminaries to the 
Table of Chords in Book I he says : 
1 We will first show how we can establish a systematic and 
speedy method of obtaining the lengths of the chords based on 
the uniform use of the smallest possible number of proposi 
tions, so that we may not only have the lengths of the chords