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

BRAHMAGUPTA AND SINE RULE 101
Brahmagupta First to Introduce
Sine Rule in Indian Plane Trigonometry
In this connection, we shall reproduce the following verse
from the Khaṇḍakhādyaka :
Multiply the ‘sine’ of the (Śīghra) anomaly by the ‘sine’
of the maximum Śīghra equation and divide by the
‘sine’ of the corresponding Śīghra equation, the result
is the ‘Śīghra hypotenuse’ when the (Śīghra) anomaly
is half a circle, this śīghra hypotenuse is equal to the
radius diminished by the ‘sine’ of the maximum
equation ; when the anomaly is equal to the whole
circle, the same is equal to the radius increased by the
same ‘sine’ of the maximum equation.¹
Let S, E and P be the positions of the Sun, the Earth and the
planet, say Mars, respecti-
vely. Complete the paralle-
logram SEMP ; with M as
centre and MP as radius,
describe a circle. This circle
is the epicycle of conjunction
of Mars. Produce EM to cut
this circle at K. The ∠PMK
= ∠S′SP (the point S′ is on
[Fig. 4] ES produced ), the angle
gained by the Earth over Mars since the preceding conjunction.
The ∠PMK is called the śīghra anomaly or anomaly of conjunc-
tion. We take EM=360, and MP=234. The ∠PEM, which is
equal to ∠PES, the annual parallax of Mars, is called the śīghra
equation. The ∠MPE is equal to the ∠SEP, the elongation.
The ∠PMK is given, and PM and ME are also given. Hence in
the triangle MPE, we have
tan ½ (P−E) = (EM−MP)/(EM+MP) tan ½ PMK
= 126/594 tan ½ PMK
∴ L tan ½(P−E) = log [126/594] + L tan ½ PMK
We have also ½ (P+E) = ½ ∠PMK

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1. केन्द्रज्याऽन्त्यफलज्या गुणिता फलजीवया हृताकर्णः ।
   त्रिज्यास्य फलज्योना चक्रार्द्धे संयुता चक्रे ॥ KK. VI. 1.