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

'
PTOLEMY'S SYNTAXIS
275
276 TRIGONOMETRY
(died 901), is extant in part, as well as the version by Nasiraddin
at-TusI (1201-74).
The first edition to be published was the Latin translation
made by Gherard of Cremona from the Arabic, which was
finished in 1175 but was not published till 1515, when it was
brought out, without the author's name, by Peter Liechtenstein
at Venice. A translation from the Greek had been made
about 1160 by an unknown writer for a certain Henricus
Aristippus, Archdeacon of Catania, who, having been sent by
William I, King of Sicily, on a mission to the Byzantine
Emperor Manuel I. Comnenus in 1158, brought back with
him a Greek manuscript of the Syntaxis as a present; this
translation, however, exists only in manuscripts in the Vatican
and at Florence. The first Latin translation from the Greek
to be published was that made by Georgius of Trebizond ' ' for
Pope Nicolas V in 1451 ;
this was revised and published by
Lucas Gauricus at Venice in 1528. The editio princeps of the
Greek text was brought out by Grynaeus at Basel in 1538.
The next complete edition was that of Halma published
1813-16, which is now rare. All the more welcome, therefore,
is the definitive Greek text of the astronomical works
of Ptolemy edited by Heiberg (1899-1907), to which is now
added, so far as the Syntaxis is concerned, a most valuable
supplement in the German translation (with notes) by Manitius
(Teubner, 1912-13).
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.
Summary of Contents.
Trigonometry in
Ptolemy.
The Syntaxis is most valuable for the reason that it contains
very full particulars of observations and investigations
by Hipparchus, as well as of the earlier observations recorded
by him, e.g. that of a lunar eclipse in 721 B.C. Ptolemy
based himself very largely upon Hipparchus, e.g. in the
preparation of a Table of Chords (equivalent to sines), the
theory of eccentrics and epicycles, &c. ; and it is questionable
whether he himself contributed anything of great value except
a definite theory of the motion of the five planets, for which
Hipparchus had only collected material in the shape of observations
made by his predecessors and himself. A very short
indication of the subjects of the different Books is all that can
T 2
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