## A Modern Almagest: An updated version of Ptolemy's Almagest

The *Almagest* of Claudius Ptolemy is often unfairly disparaged by modern commentators for
a number of supposed failings. Firstly, it is often alleged that Ptolemy's adoption of circular
orbits is a major source of error in his model. This is untrue. Planetary orbits are highly
circular, due to their relatively small eccentricities (the departure from circularity scales
as the eccentricity, e, squared). It is far more important to get the eccentricities of these
orbits (i.e., the displacement of the Sun from the geometric center, which scales as e) correct, rather than the ellipticities. This is exactly what Ptolemy does in the Almagest.
Secondly, it is generally supposed that Ptolemy's introduction of epicycles into his model is
merely a clumsy way of retaining a geocentric model whilst allowing for retrograde motion. This is untrue.
The deferents and epicycles in the Almagest have a clear physical significance. For a
superior planet, the deferent represents the orbit of the planet around the Sun, whereas the
epicycle represents the Earth's orbit. The opposite is true for an inferior planet. Thirdly, it
is often alleged that Ptolemy's model requires epicycles on epicycles, and epicycles on
epicycles on epicycles, etc., in order to fit the observational data to any degree of accuracy. This is untrue. In the
first place, Ptolemy never proposed more than one level of epicycles. In the second place, a
(slightly corrected) version of Ptolemy's model containing only 10 circles (four deferents and
five epicycles) is capable of accounting for the apparent movements of the Sun and the five
visible planets to an accuracy which is more than adequate for naked-eye astronomy. It is worth
noting that Ptolemy's model contains fewer epicycles than that proposed by Copernicus, and is
also more accurate.

Misconceptions currently 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 rather primitive in nature, and,
consequently, somewhat unwieldy. Dates are a major stumbling block, since three different systems are used,
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 by modern
astronomers.

I have attempted to reconstruct Ptolemy's model of the solar system employing
modern mathematical methods, standard dates, and conventional astronomical terminology. I hope
that the result will make the full extent of Ptolemy's scientific
achievement more manifest to a modern audience.

This work is available in three formats:
- [PDF]
- A fully indexed and hyperlinked PDF file with embedded Type 1 fonts.
- [HTML]
- An HTML document.
- [BOOK]
- A profesionally-bound print-on-demand book.
- [GIT]
- A public Git repository of the Latex source.

Richard Fitzpatrick
Last modified: Tue Apr 16 09:15:04 CDT 2013