ब्राह्मस्फुटसिद्धान्त (ब्रह्मगुप्त - शून्य, कुट्टक, बीजगणित एवं सम्पूर्ण २१ अध्याय सान्वय सटीक)

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 perigee 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 ∠PEM
=∠P'CS=θ; EP=CP'=a; EC=MS=p, then
tan E = SH / HE = p sin θ / (a - p cos θ)
∴ E = (p / a) sin θ - (p² / 2a²) sin 2θ + (p³ / 3a³) sin 3 θ.........
Now the true value of E in elliptic motion is given by
E = ( 2e - e³/4 ) sin θ + (5/4) e² sin 2θ + (13e³ / 12) sin 3θ*;...
It we now put p / a = 2e - e³/4, as a first approximation p / a = 2e.
Hence p² / 2a² = 2e², which is greater than 5/4 e² by 3/4 e². In the
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'.
*Godfray's Astronomy, p. 149.