Conclusions

A number of important points emerge from our examination of Ptolemy's model of the Sun and the planets. First, it is an excellent approximation to represent planetary orbits as eccentric circles. This follows because the eccentricity (i.e., the displacement of the Sun from the geometric center) of a Keplerian orbit scales as $e$ (the eccentricity), whereas the ellipticity scales as $e^2$, and $e\ll 1$ for all planets. Hence, Ptolemy's use of eccentric circles to represent planetary orbits in the Almagest is not a major source of error, as is commonly supposed. Second, the non-uniform rotation of a planet in a Keplerian orbit, which is mandated by Kepler's second law, can be represented (up to first-order in $e$) by an intuitively simple geometric model in which the radius vector connecting the planet to the so-called equant--a point diagrammatically opposite the Sun from the geometric center of the orbit--rotates uniformly. This, of course, is exactly what Ptolemy does in the Almagest. Third, the deferents and epicycles in Ptolemy's model all have simple physical interpretations. For the superior planets, the deferent represents the orbit of the planet itself around the Sun, and the epicycle represents the Earth's orbit. The opposite is true for the inferior planets. Thus, Ptolemy's representation of the apparent motion of a planet, as seen from Earth, as a combination of two circular motions is essentially correct, both from a mathematical and a physical point of view. Fourth, after the slight errors in Ptolemy's model are corrected, it reduces to a remarkably simple system consisting of just nine circles (four deferents and five epicycles). Moreover, this simple system is able to account for the motion of the Sun and the planets visible to the naked eye to an accuracy which is more than adequate for naked-eye observations. The popular myth that Ptolemy's model requires a plethora of epicycles in order to accurately represent the observational data is simply untrue.

Table 1: Orbital elements of the planets inferred using updated Almagest model. Here, $a$ is the semi-major radius, $T$ the orbital period, $e$ the eccentricity, $\omega$ the argument of the perihelion, $\theta$ the longitude of the ascending node, $i$ the inclination to the ecliptic, and $\phi _0$ the mean anomaly at 00:00:00 TDT on Jan. 1, 1995.

Planet	$a (AU)$	$T(days)$	$e$	$\omega(^\circ)$	$\theta(^\circ)$	$i(^\circ)$	$\phi_0(^\circ)$
Mercury	0.3776	87.978	0.216	25.86	50.32	7.01	87.98
Venus	0.7240	224.69	$6.400\times 10^{-3}$	49.65	76.66	3.39	9.403
Earth	1.000	365.24	$1.673\times 10^{-2}$	102.76	-	-	357.41
Mars	1.514	686.92	$9.130\times 10^{-2}$	289.05	48.45	1.83	140.45
Jupiter	5.196	4332.6	$4.853\times 10^{-2}$	275.55	100.36	1.30	226.67
Saturn	9.519	10759.1	$5.399\times 10^{-2}$	113.37	338.76	2.48	256.77

Table 2: True orbital elements of the planets for epoch J2000. See caption on Tab. 1. (Source: E.M. Standish JPL/Caltech).

Planet	$a (AU)$	$T(days)$	$e$	$\omega(^\circ)$	$\theta(^\circ)$	$i(^\circ)$
Mercury	0.3871	87.969	0.206	29.12	48.33	7.00
Venus	0.7233	224.70	$6.777\times 10^{-3}$	54.92	76.68	3.39
Earth	1.000	365.24	$1.671\times 10^{-2}$	102.94	-	-
Mars	1.524	686.96	$9.339\times 10^{-2}$	286.50	49.56	1.85
Jupiter	5.203	4332.6	$4.839\times 10^{-2}$	274.25	100.47	1.30
Saturn	9.537	10759.1	$5.386\times 10^{-2}$	113.66	338.94	2.49