In order to establish the Theory of Epicycles, it was necessary to assign the magnitudes, distances, and positions of the circles or spheres in which the heavenly bodies were moved, in such a manner as to account for their apparently irregular motions. We may best understand what was the problem to be solved, by calling to mind what we now know to be the real motions of the heavens. The true motion of the earth round the sun, and therefore the apparent annual motion of the sun, is performed, not in a circle of which the earth is the centre, but in an ellipse or oval, the earth being nearer to one end than to the other; and the motion is most rapid when the sun is at the nearer end of this oval. But instead of an oval, we may suppose the sun to move uniformly in a circle, the earth being now, not in the centre, but nearer to one side; for on this supposition, the sun will appear to move most quickly when he is nearest to the earth, or in his _Perigee_, as that point is called. Such an orbit is called an _Eccentric_, and the distance of the earth from the centre of the circle is called the _Eccentricity_. It may easily be shown by geometrical reasoning, that the inequality of apparent motion so produced, is exactly the same in {146} detail, as the inequality which follows from the hypothesis of a small _Epicycle_, turning uniformly on its axis, and carrying the sun in its circumference, while the centre of this epicycle moves uniformly in a circle of which the earth is the centre. This identity of the results of the hypothesis of the Eccentric and the Epicycle is proved by Ptolemy in the third book of the "Almagest." _The Sun's Eccentric._--When Hipparchus had clearly conceived these hypotheses, as _possible_ ways of accounting for the sun's motion, the task which he had to perform, in order to show that they deserved to be adopted, was to assign a place to the _Perigee_, a magnitude to the _Eccentricity_, and an _Epoch_ at which the sun was at the perigee; and to show that, in this way, he had produced a true representation of the motions of the sun. This, accordingly, he did; and having thus determined, with considerable exactness, both the law of the solar irregularities, and the numbers on which their amount depends, he was able to assign the motions and places of the sun for any moment of future time with corresponding exactness; he was able, in short, to construct _Solar Tables_, by means of which the sun's place with respect to the stars could be correctly found at any time. These tables (as they are given by Ptolemy)[63\3] give the _Anomaly_, or inequality of the sun's motion; and this they exhibit by means of the _Prosthapheresis_, the quantity of which, at any distance of the sun from the _Apogee_, it is requisite to add to or subtract from the arc, which he would have described if his motion had been equable. [Note 63\3: Syntax. 1. iii.]