148 EPICYCLIC THEORY OF ANCIENT INDIANS Let E represent the centre of the Earth (Fig.15).APM the Sun's circular orbit or concentric; let A and P be the apogee and the perigee respectively. From EA, cut off EC equal to the radius of the Sun's epicycle. With centre C and radius equal to EA describe the eccentric A'P'S cutting AP and AP produced at P' and A'. Here A' and P' are the real apogee and peri gee of the Sun's orbit. Let PM and P'S be any two equal arcs measured from P and P.* The idea is that the mean planet M and the apparent Sun S move simultaneously from P and P' in the counterclockwise direction along the concentric and the eccentric circles. They move with the same angular motion and arrive simultaneously at M and S. Here EM and CS are parallel and equal, hence MS is also equal and parallel to EC. Let SH be drawn perpendicular to EM. . The angle PEM is the mean anomaly and the angle P'ES the true anomaly; the angle SEM is the equation of the centre, is readily seen to be plus (+) from P' to A' and minus (-) from A to P.. Thus as regards the character of the equation, the eccentric circle is quite right. We now turn to exmine how far it is true as to the amount. Let the angle SEM denoted by E and the angle LPEM =/P'CS=0; EP-CP'=a; EC-MS-p, then tan E== SH HE It we now put P a p sin 0 a-p cos P a sin e- 2² 2zsin 20+ sin 3 0......... Now the true value of E in elliptic motion is given by E 2e-- € sin 0+² sin 20+ 132* 12 23 3a³ =28-- sin 30*;... 03 P , as a first appoximation- 4 a ➡2e. Hence -2e², which is greater than bye. In the že case of the Sun if the value of p be correctly taken the error in the coefficient of the second term becomes-+-3'; similarly in the case of the Moon, the corresponding error becomes+8'.
- Goditay's Astronomy, p. 149.