A history of Greek mathematics - Wilbourhall.org

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276 TRIGONOMETRY
bo 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 continued,
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 IV : The length of the
months and the theory of the moon. Book V : The construction
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 VI : Conjunctions and oppositions of sun
and moon, solar and lunar eclipses and their periods. Books
VII and VIII are about the fixed stars and the precession of
the equinoxes, and Books IX-XIII are devoted to the movements
of the planets.
Trigonometry in
Ptolemy.
What interests the historian of mathematics is the trigonometry
in Ptolemy. It is evident that no part of the trigonometry,
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 propositions,
so that we may not only have the lengths of the chords