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

98 UTTARAKHAṆḌA KHĀDYAKA one-sixteenth.¹ This stanza says that in 499 A.D., Mars's aphelion point had a longitude of 127⁰ ; of Jupiter the longitude of the aphelion was 170⁰. ( KK.II. 6²) According to Newcomb's rule, the longitude of the aphelion point of Mars in 499 A. D. works out to have been = 128⁰28'12''. According to the Conn. des. Temps' rule, the same was 128⁰27'51''. Hence Brahmagupta's determination of Mars's aphelion is correct within 1⁰30' and is therefore, quite satisfactory. According to the Khaṇḍakhādyaka proper it was 110⁰, and according to the Āryabhaṭīya 118⁰. Of planets, beginning with Mars, the degrees of longitude of the apogees are respectively 11, 22, 16, 8 and 24, each multiplied by 10. (KK .II. 6) Thus the longitudes of apogees of Mars = 110 (3 signs 20⁰); of Mercury = 220⁰ (7 signs 10⁰) ; of Jupiter = 160⁰ (5 signs 10⁰); of Venus = 80⁰ (2 signs 20⁰) and of Saturn = 240⁰ (8 signs). Compare these values with those given in the Pañcasiddhāntikā XVII. 2 (the Sūrya-siddhānta). Again according to this stanza Jupiter's aphelion had a longitude of 170⁰ in 499 A.D. According to Conn. des. Temps' rule the same was 170⁰25'. Thus here too, Brahmagupta is very accurate. According to the Khaṇḍakhādyaka proper, Jupiter's aphelion had a longitude of 160⁰ (KK. II.6) and according to the Āryabhaṭīya, the value was 180⁰. Brahmagupta First to Use Second Differences All these illustrations reproduced here very well establish the point that the great Indian astronomers from Āryabhaṭa I to Brahmagupta were aware of the methods of separating the two distinct planetary inequalities, viz., that of the apsis and of conjunction in the cases of the five ‘star’ planets (PSi. Intro- duction Lii). In the Khaṇḍakhādyaka, Brahmagupta having given the “sines” and the equations of the Sun and the Moon

1. सप्तदशांशैरधिकं भौमस्योच्चं गुरोर्दशभिरंशैः । सितशीघ्रात् कृतमुनयो लिप्ताः शोध्याः शनेः फलं मान्द्यम् ।। पञ्चांशोनं शैघ्यं षोडशभागाधिकं बुधस्य फलम् ।। —UKK. IX. 11
2. मन्दांशा दशगुणिता रुद्रा द्वियमाश्व षोडशाष्टजिनाः । —KK. II. 6