सिद्धान्तशिरोमणि: गणिताध्याय (भास्कराचार्य - ग्रहगणित, मध्यमाधिकार व स्पष्टाधिकार सटीक)

490
Comm. This is a correction to be made in the longi-
tude of the Moon to depict graphically the phase of the
Moon. Bhāskara says that many of the prior astronomers
took that the phase of the Moon was in direct proportion
to the elongation of the Moon. Taking the radius of the
Moon's disc to be six angulas or units, and assuming that
when the elongation is 180°, the entire disc being illumi-
nated, it was thought that 12 units of the diameter
correspond to 180° of elongation so that the Śukla of the
disc measured by the central width of the illuminated disc
increases at the rate of 1 unit for 15° of elongation. (The
word Śukla may be taken to correspond to the modern
word phase. Śukla is expressed in angulas, taking the
diameter of the disc to be 12 angulas. The measure of the
portion of the diameter of the Moon's disc perpendicular
to the diameter which is the join of the extremities of
the cusps, covered by the illuminated part of the disc,
(which may be defined as the maximum width of the
illuminated part of the disc) measured in angulas is said
to be the Śukla. The word 'phase' is used to signify the
ratio of the above width to the diameter. Śukla is
expressed in angulas, whereas phase is expressed as a ratio.
The Śukla is equal to twelve times phase). Bhāskara
rightly argues that this method of measuring the Śukla
is approximate because he says that six angulas of Śukla
is had when the elongation is not 90°, but only 85°-45'.
This may be substantiated as follows from fig. 116. Let
E, M, S stand for the earth, Moon and the Sun. Let
^
EMS = 90° so that it is a moment of dichotomy ie. the
moment when the phase is half and the Śukla 6 angulas.
Let θ be then the elongation of the Moon, so that
cos θ = EM / ES . Taking the average values given for EM
and ES by Bhāskara,
cos θ = 51566 / 689377 = 19 / 254 (obtaining a convergent) = .0748.