Purpose of Treatise

As we have seen, misconceptions abound regarding the details of Ptolemy's model of the solar system, as well as its scientific merit. Part of the reason for this is that the Almagest is an extremely difficult book for a modern reader to comprehend. For instance, virtually all of its theoretical results are justified via lengthy and opaque geometric proofs. Moreover, the plane and spherical trigonometry employed by Ptolemy is of a rather primitive nature, and, consequently, somewhat unwieldy. Dates are also a major stumbling block, since three different systems are used in the Almagest, all of which are archaic, and essentially meaningless to the modern reader. Another difficulty is the unfamiliar, and far from optimal, Ancient Greek method of representing numbers and fractions. Finally, the terminology employed in the Almagest is, in many instances, significantly different to that used in modern astronomy textbooks.

The aim of this treatise is to reconstruct Ptolemy's model of the solar system employing modern mathematical methods, standard dates, and conventional astronomical terminology. It is hoped that the resulting model will enable the reader to comprehend the full extent of Ptolemy's scientific achievement. In fact, the model described in this work is a somewhat improved version of Ptolemy's, in that all of the previously mentioned deficiencies have been corrected. Furthermore, Ptolemy's equant scheme has been replaced by a Keplerian scheme, expanded to second-order in the planetary eccentricities. It should be noted, however, that these two schemes are essentially indistinguishable for small eccentricity orbits. Certain aspects of the Almagest have not been reproduced. For instance, it was not thought necessary to instruct the reader on how to construct trigonometric tables, or primitive astronomical instruments. Furthermore, no attempted has been made to derive any of the model parameters directly from observational data, since the orbital elements and physical properties of the sun, moon, and planets are, by now, extremely well established. Any detailed discussion of the fixed stars has also been omitted, because stellar positions are also very well established, and the apparent motion of the stars in the sky is comparatively straightforward compared to those of solar system objects. What remains is a mathematical model of the solar system which is surprisingly accurate (the maximum errors in the ecliptic longitudes of the sun, moon, Mercury, Venus, Mars, Jupiter, and Saturn during the years 1995-2006 CE are $0.7'$, $14'$, $28'$, $10'$, $14'$, $4'$, and $1'$, respectively), yet sufficiently simple that all of the necessary calculations can be performed by hand, with the aid of tables. The form of the calculations, as well as the layout of the tables, is, for the most part, fairly similar to those found in the Almagest. Many examples of the use of the tables are provided.