$ 39) Hipparchus 43 



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 ACS 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 neces- 

 sary 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 t-, where s denotes the number of feet fallen, and 

 / 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 / = nt + 2 e sin nt, where / is the 

 distance from a fixed point in the orbit, t the time, and n t e certain 

 numerical quantities. 



