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

CELESTIAL AND POLAR LONGITUDE 143
In order to find σN, he would multiply
σK by (R cos σKP) / R ; according to him,
σN = (σK × R cos σKP) / R + KH¹
This is a decided improvement on Brahmagupta's corres-
ponding rule. The declination σN obtained would be very
nearly accurate.
Having obtained σN, Bhāskara² then directs the finding of
NH, thus,
NH = (σK × R sin σKP) / (R cos σN)
He then directs the finding of MK on the ecliptic of which
NH is the projection by means of the times of rising of the signs
of the zodiac on the equator.
Thus, the Indian methods show a beginning and develop-
ment only. The Greek method as given by Ptolemy is mathe-
matically accurate.
Greek Method³
To transform the celestial longitude and celestial latitude to
right ascension and declination.
Let the great circle ΠσK meet the equator at Δ. Ptolemy
would then from the given value of γK, find γΔ and ΔK by
using his tables for the rising of signs of the zodiac on the equa-
tor. He then takes ΠPσ for the triangle and γNΔ Q for the
transversal. The Menelaus' Equation, then, is
(sin ΠQ / sin QP) × (sin PN / sin Nσ) × (sin σΔ / sin ΔΠ) = 1²
Here ΠQ = 90° + ω, QP = 90°, PN = 90°; σΔ = σK + KΔ.
ΠΔ = 90° + KΔ, whence Nσ is obtained.
He next takes PNQ for the triangle and ΠσΔ for the trans-
versal,