Back in March, when I last

wrote on this subject we saw that the Pre-SR astronomers wanted to predict the positions of the planets in the sky. But how did they go about it? Post-SR astronomers do it by using the tools of calculus, i.e. they write down the appropriate differential equations, apply some initial conditions, and integrate the combined system. In practice, given the uncertainties inherent in observations, rather than only one initial position and velocity, a large number of past observed positions are used to constrain the solutions of those equations so that the error in their post-diction of those observations is the minimum possible. Then things are integrated forward to get predictions of future positions, in three dimensions. For the state of the art visit the

Solar System Dynamics pages at

JPL. The underlying assumptions are, for the most part, Newton's Three Laws of motion, and his Law of Gravitation---General Relativity is required for Mercury's orbit. Comets suffer from additional forces due to their losing mass---but the principle is the same: Sum up all the forces acting on a body and this is its acceleration. Integrate the acceleration to get the velocity, and the velocity to get the position.

Pre-SR astronomers worked from a different set of assumptions. In their view, the planets moved exclusively on circles which turned at constant angular rates. This restriction posed something of a difficulty since the observed planets don't move at constant rates in the sky. The superior planets (Mars, Jupiter, and Saturn) even, at times near opposition, move backwards on the sky, which is hard to explain with constantly rotating circles. So, the Pre-SR astronomers had to employ a variety of schemes to get variable apparent motions from their constant true motions.

*Epicycles* were small circles containing the planet and were carried around but the large circle. The large circles,

*deferents*, themselves were placed off center (

*eccentric*) with respect to the Earth so that their apparent motions would be faster when they were closer to the Earth and slower when they were more distant. This works even better when the planet moves at a constant speed not as seen from the center, but from the

*equant point* placed symmetrically on the opposite side of the circle's centre from the Earth. This last device, however, represents a step away from a uniformly rotating circle and proved controversial for that reason.

What is striking about all this is just how non-geocentric this geocentric model really is. The Earth is contained within the region where the planet (including the Sun) travelled, but it is not at the centre of any of the circles of motion. In the case of the Sun, the centre of the deferent is offset by 1/24

^{th} of its radius. Further, different eccentricities are needed for different planets, so the deferents don't have a common centre about a fictitious point, never mind the Earth. About the only thing that is geocentric is the spherical annulus containing all this apparatus. All planets have their closest and furthest points from the Earth (

*perigee* and

*apogee*). The apparatus for a planet necessarily lies between a pair geocentric spheres with the radii of perigee and apogee for that planet. The motion of the planet itself is not geocentric at all.

In this period all the calculations of planetary motion were expressed in terms of angles on circles of relative sizes, since no one knew how far away any of the planets were. There were some inaccurate estimates as to the relative distances of the Moon and Sun, but that was about as far as things went. Even the order of the planets was a matter of conjecture. Indeed, this subject was about the only thing Ptolemy didn't settle in the

*Almagest*. He did produce an ordering in a later work called

*Planetary Hypotheses*, but there wasn't any observational evidence to support it. (Quite frankly, the whole subject was of secondary importance since it had no practical affect on their objectives.) Once an order has been chosen, and on the assumption that the spherical annuli were tightly nested, the relative distances of the planets, and the stars, immediately follow. But these distances are entirely arbitrary. Consequently, anyone wanting to resurrect a geocentric model of this type has his work cut out for him.

NASA can send spacecraft around the solar system on a complicated trajectories shaped by passages by several other planets along the way. Ptolomaic astronomers would be entirely incapable of doing this successfully in the event they had been able to even conceive of it. Let the modern geocentric astronomer produce a physical model able to replicate this feat, and we'll have something to talk about.