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.