Introduction

The modern world inherited two major scientific treaties from the civilization of Ancient Greece.

The first, the Elements of Euclid, is a large compendium of mathematical theorems regarding 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 are all ultimately derivable 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, the Almagest¹ of Claudius Ptolemy, is an attempt to construct a mathematical model of the apparent motions of the Sun, the Moon, and the planets in the Earth's sky, against the background of the (approximately) fixed stars, using Euclidean geometry. On the basis of his own naked-eye observations, and 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 $23.4^\circ$ to the equator. Furthermore, the planets move on the rims of small circles called epicycles, whose centers revolve around the Earth on circles called deferents--see Figs. 8 and 9. The planetary deferents and epicycles also maintain fixed inclinations, which are all fairly close to $23.4^\circ$, to the 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 in circles--led him to construct an overcomplicated, unwieldy, and faintly ridiculous model of the solar system. As is well-known, this model was superceded in 1543 by the heliocentric model of Nicolaus Copernicus, in which the planets revolve about the Sun in circular orbits. The Copernican model was, in turn, superceded in the early 1500's by the, ultimately correct, model of Johannes Kepler, in which the planets revolve about the Sun in elliptical orbits.

The aim of this study is to re-examine the scientific merits of Ptolemy's Almagest. Section 2 is devoted to the model of solar and planetary motion outlined in this work. This model is first critically reviewed, and then compared with the corresponding models of Copernicus and Kepler. It is then demonstrated that Ptolemy's system of epicycles and deferents can be derived mathematically from Kepler's theory of planetary motion, via the use of a small eccentricity expansion. Finally, it is shown that when Ptolemy's model is combined with a few year's worth of positional data for the Sun and the (visible) planets, it is able to infer highly accurate values of the orbital elements of these heavenly bodies. Moreover, once the orbital elements have been determined, the subsequent predictions of the Almagest model are astonishingly close to the actual data. Section 3 is devoted to the motion of the Moon, which is considerably more complicated than that of the planets. Using observational data as a guide, an accurate theory of the Moon's motion is constructed along the lines suggested in the Almagest. Finally, in Sect. 4, it is demonstrated that the Almagest model is capable of correctly predicting the occurrence of solar and lunar eclipses. It is hoped that this study will help convince the reader of the high scientific value of Claudius Ptolemy's contribution to astronomy.