The details of the heavenly motions were for the mathematicians only, who dropped the orbs, and only took such circles out of them as were necessary in the explanation of the motions.
Without entering into all the details connected with this explana tion, which are rather complicated, and require besides some knowledge of the actual inequalities of the planetary motions, we shall take the two leading circumstances of those motions, namely, their not being uniform, and their being sometimes direct, or according to the order of the signs of the zodiac, and sometimes retrograde. One way of explaining the simple irregularity of motion was by supposing the orb of the planet to be a sphere, and revolving uniformly, hut not con centric with the earth. Let the earth be at E, and let the circle P Q revolve uniformly round the centre o, or let the planet revolve uniformly in that circle. Consequently, the nearer the planet is to r, the faster it will appear to move, and the contrary, that is, a spectator at r. will see the planet moving most slowly when at r, from whence the apparent motion will be accelerated until it arrives at n, and retarded while it returns to r on the other side. The circle r Q n is called an eccentric. This hypothesis was a tolerably good representa tion of the motion of the sun, when r o was taken in a certain specified proportion to the radius c n ; and if the sun had been placed at r, it would have made a sufficient representation of the motions of the planets, at least for the earlier periods of observation. But it must be remembered that though an acceleration and retardation would thus be established, it would not be precisely that of the planets, though sufficiently near, as remarked, to represent the results of rough observation.
The mode of obtaining the alternate progressive and retrograde motion is as follows :—Let r be the centre of a circle called an epicycle, and let r revolve uniformly round the earth at c, while the epicycle revolves uniformly round its centre, carrying the planet on its circum ference, or while the planet revolves uniformly round the epicycle.
Let the epicycle Itself move In the direction P Q n, and let the planet on the epicycle move in the same direction, or A n D. If then tiro times of revolution of the planet in the epicycle bo sufficiently great mintstresi with that of the epicycle itself, its retrogradation at D will more than compensate for the progression of the epicycle itself ; that ie, the planet will appear to a spectator at c to move In a retrograde direction, when it is in the lower part of the epicycle. But in the higher part, at A, both motions conspire to make it appear to move directly. There must consequently be an intermediate point at which the direct motion ceases and the retrograde begins; and near this leint the planet will appear stationary.
It is by a complicates( use of these methods that Ptolemy succeeds in giving a tokvable account of the angular motions known in his time ; but they fail in placing the planet at the right distance from the earth, though they may place it nearly in the right longitude. We imagine that in modern times very few persons have taken the trouble to make themselves acquainted with the details of this system. These may be learnt, with some trouble, from Dclambre's account of the Syntaxis Hist. Astron. Ana:), but they are explained with much more clearness in Mr. Narrien's Origin and Progress of Astronomy,' London, 1833. Those who would have Ptolemy's own explanation at less expense of time than hi necessary to find it in the Syntaxis, altould look at his short tract, wtpl Inro8fams wAaretalnes, which was published with the Sphere of Proclus, by Dr. Bainbridge, in 1620. The physical views may be collected, as well as the mathematical ones, from the a Philolaus: the first work of Bouillaud, Amsterdam, 1639. [Tnocuornar, Ctinvr.s.)