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

MOON'S EQUATIONS 113 vation was up to the time of Brahmagupta, restricted to the time of eclipses perhaps also of syzygies. The modern form of the Moon's equations is =377' sin (nt—a)+13' sin 2(nt—a)+......... +76' sin [2(nt—θ)—(nt—a)]+40' sin 2(nt—θ)..............¹ where nt=mean longitude of the Moon, a the longitude of the perigee, θ=longitude of the Sun. Here the first two terms, viz., 377' sin (nt—a) + 13' sin 2(nt—a), are due to elliptic motion about the Earth in one focus; the term 76' sin [2(nt—θ)—(nt—a)] is known as the evec- tion. We combine a part of the first term with the evection term and the expression for the equation of centre becomes =301' sin (nt—a)+13' sin 2(nt—a)+............+152 sin (nt—θ) cos (θ—a)+40' sin 2(nt—θ). Now at syzygies and eclipses sin (nt—θ) and sin 2(nt—θ) will very nearly vanish. Hence according to modern astronomy at the syzygies and eclipses, the chief term of the Moon's equa- tion=301' sin(nt—a). This according to the Āryabhaṭīya =300' 15" sin (nt—a) ,, ,, Khaṇḍakhādyaka =296' sin (nt—a) ,, ,, Uttara Khaṇḍakhādyaka =301'.7 sin (nt—a) ,, ,, Brāhmasphuṭasiddhānta =293' 31" sin (nt—a) ,, ,, Greek astronomy =300' 15" sin (nt—a) very nearly. Hence both the Greek and the ancient Indian astronomers were very near the true value of the Moon's equation at the syzy- gies and eclipses. Godfray in his Lunar Theory, page 107, observes, "the hypothesis of an excentric, whose apse has a pro- gressive motion as conceived by Hipparchus served to calculate with considerable accuracy the circumstances of eclipses; and observations of eclipses, requiring no instruments, were then the only ones which could be made with sufficient exactness to test

1. The accurate values of the coefficients appear to be 377' 19".06, 12' 57".11, 76' 26" and 39' 30".