representation of the required variations in the sun's motion in the ecliptic, a method of representation which is in some respects more intelligible and vivid than the use of algebra, but which becomes unmanageable in complicated cases. It runs moreover the risk of being taken for a mechanism. The circle, being the simplest curve known, would naturally be thought of, and as any motion other than a uniform motion would itself require a special representation, the idea of Apollonius, adopted by Hipparchus, was to devise a proper combination of uniform circular motions.
39. The simplest device that was found to be satisfactory in the case of the sun was the use of the eccentric, i.e. a circle the centre of which (c) does not coincide with the position of the observer on the earth (e). If in fig. 17 a point, s, describes the eccentric circle a f g b uniformly, so that it always passes over equal arcs of the circle in equal times and the angle a c s increases uniformly, then it is evident that the angle a e s, or the apparent distance of s from a, does not increase uniformly. When s is near the point a, which is farthest from the earth and hence called the apogee, it appears on account of its greater distance from the observer to move more slowly than when near f or g; and it appears to move fastest when near b, the point nearest to e, hence called the perigee. Thus the motion of s varies in the same sort of way as the motion of the sun as actually observed. Before, however, the eccentric could be considered as satisfactory, it was necessary to show that it was possible to choose the direction of the line b e c a (the line of apses) which determines the positions of the sun when moving fastest and when moving most slowly, and the magnitude of the ratio of e c to the radius c a of the circle (the eccentricity), so as to make the calculated positions of the sun in various parts of its path differ from the observed positions at the corresponding
facts more simply and in a way more satisfactory to the mind by the formula s = 16 t2, where s denotes the number of feet fallen, and t the number of seconds. By giving t any assigned value, the corresponding space fallen through is at once obtained. Similarly the motion of the sun can be represented approximately by the more complicated formula l = nt + 2 e sin nt, where l is the distance from a fixed point in the orbit, t the time, and n, e certain numerical quantities.
