The first of these, the *Elements* of Euclid, is a
large compendium of mathematical theorems concerning geometry, proportion, and number
theory. These theorems were not necessarily discovered by Euclid
himself--being largely the work of earlier mathematicians,
such as Eudoxos of Cnidus, and Theaetetus of Athens--but were arranged
by him in a logical manner, so as to demonstrate that they can all ultimately
be derived from five simple axioms. The Elements is rightly regarded
as the first, largely succesful, attempt to construct an
axiomatic system in mathematics, and is still held in high esteem within
the scientific community.

The second treatise, the *Almagest*^{} of Claudius Ptolemy,
is an attempt to find a simple geometric explanation for the apparent motions of the sun,
the moon, and the five visible planets in the earth's sky. On the basis of his own naked-eye observations, combined with those of earlier astronomers such as Hipparchus of Nicaea, Ptolemy proposed a model of the solar system in which the earth is *stationary*.
According to this model, the sun moves in a circular orbit, (nearly) centered on the earth, which
maintains a fixed inclination of about to the terrestrial equator.
Furthermore, the planets move on the rims of small
circles called *epicycles*, whose centers revolve around the earth
on large eccentric circles called *deferents*--see Fig. 30. The
planetary deferents and epicycles also
maintain fixed inclinations,^{} which are all fairly close to ,
to the terrestrial equator.

The scientific reputation of the Almagest has not fared as well
as that of
Euclid's Elements. Nowadays, it is a commonly held belief, even amongst scientists, that Ptolemy's mistaken adherence to the
tenets of Aristotelian philosophy--in particular, the immovability of the earth, and
the necessity for heavenly bodies to move uniformly in circles--led him to
construct an overcomplicated, unwieldy, and faintly ridiculous model of planetary motion. As is well-known, this model was superseded in 1543 CE by the *heliocentric* model of
Nicolaus Copernicus, in which the planets revolve about the
sun in *circular* orbits.^{} The Copernican model was, in turn,
superseded in the early 1600's CE by the, ultimately correct, model
of Johannes Kepler, in which the planets revolve about the sun in eccentric
*elliptical* orbits.

The aim of this treatise is to re-examine the scientific merits of Ptolemy's Almagest.