पञ्चसिद्धान्तिका (आचार्य वराहमिहिर - सूर्य, रोमक, पौलिश, वासिष्ठ एवं पितामह सिद्धान्त)

IV. 44 IV. THREE PROBLEMS 117
We have mentioned that the author’s rough procedure is indicated only when the Sun is in the
six signs from Meṣa, because only then have we to deduct the degrees of half-cara, and the question,
what is to be done when the half-cara is greater, arises. As for the subtraction of sine half-cara in the
six signs from Tulā, that will always be less, and the question cannot arise.
TS have not understood the author here, and say something unconnected and useless. (See their
commentary p.25, and English Translation, pp.34-35).
The rules, (for the Sun in the northern hemisphere) can be deriyed thus: (see fig. 14.)
Z = Zenith
P = North pole
E = East point
S = Sun
DD′ = Diurnal Circle
AS = Altitude of the Sun
ZS = zenith distance of the Sun.
EP = Unmaṇḍalam
Sin AS = Sin altitude of the Sun
= Śaṅkuliptās (or Mahā Śaṅku
or Great gnomon)/120
[Fig. IV 14]
By the well-known formula of the spherical triangle,
Sin altitude of the Sun = sin AS = cos ZS
= cos S P. cos Z P + sin S P. sin Z P. cos PZ.
Here, using the tabular sines,
sin S P = dyujyā/120 = diameter of the diurnal circle/240.
sin Z P = sin co-latitude = lambajyā/120.
cos S P = sin (90° − SP) = sin declination of the Sun = krāntijyā/120.
cos Z P = sin latitude = akṣajyā/120.
cos S PZ = sin S PE = sin (D PS − D PE) = sin (degrees of the taken time − degrees of half-cara)
∴ Śaṅkuliptās (i.e. Great gnomon) = sin declination × sin latitude ÷ 120 + day-diameter × sin
co-latitude × sin (degrees of taken time − degrees of half-cara) ÷ 28,800.
= day-diameter × sin colatitude × {sin (degrees of taken time − degrees of half-cara) + sin decli-
nation × sin latitude × 240 ÷ (day-diameter × sin colatitude)} ÷ 28,800
= day-diameter × sin colat. {sin (degrees of taken time − degrees of half-cara) + sin half-cara}
÷ 28,800
= rule (i) applied to Sun in the northern hemisphere.
In the same manner, the rule can be proved for the Sun in the southern hemisphere, but here
the degrees of half-cara is first to be added (instead of being subtracted) to the degrees of time, and
sin half-cara is to be subtracted instead of being added, because here, S PE = D PS + D PE, and
these changes have to be made accordingly.