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No. 18.]
TRUE LONGITUDE OF THE SUN IN HINDU ASTRONOMY.
No. 18. THE TRUE LONGITUDE OF THE SUN IN HINDU ASTRONOMY. THE SIDDHANTA-SIROMANI.
241
BY ROBERT SEWELL (I.C.S., RETIRED). (Continued from Epig. Ind. XIV. p. 66.)
257. In my last article I have given Tables for finding the longitude of the sun, both mean and true, at any time of any year according to two of the great Indian astronomical authorities, the First Arya-Siddhanta or Aryabhatiya of Aryabhata (A.D. 499) and the Present Surya-Siddhanta (exact date unknown, introduce.l about A.D. 1100). The present Table affords similar information for the Siddhanta-Siromani (12th century).
As soon as I obtain definite assurance as to the value assigned by Brahmagupta to each of the twenty-four base-sines of angles, I shall be able to provide a similar Table for his BrahmaSiddhanta (A.D. 623). For the present this is not possible. We know that the sine-values given in detail in the Surya-Siddhanta (ii, 15-22) were the same as those used by Aryabhata six centuries earlier (see Arya-Siddhanta, gitika-pada, v. 10, and the Hindi Commentary by Udayanarayan Singh-Mozaffarpur, 1906-with list of differences between the sines); but according to the printed Benares edition of the Brahma-Siddhinta, with which one MS. copy in the India Office agrees (see II, Spashta-adhikara, vv. 2-5), Brahmagupta used a totally different set of sine-values, and these actually erroneous ones, while the values used a century before his time and 500 years later were as nearly as possible mathematically correct.1
Seeing no reason why Brahmagupta should have made his calculations by a set of sines that may be condemned as positively wrong, in opposition to the correct set in use before his day, I have instituted enquiries in order to ascertain whether perhaps the Benares edition of his Siddhanta may have followed a MS. which by some mischance contained a copyist's error, equally the source of error in the MS. in the India Office. Unfortunately two other MSS. in the India Office and one in the Bibliothéque Nationale in Paris are wanting in the passage which contains the sine-values.
Oxford possesses no copy. I have sought for information from India, but this is not yet to hand.
I hope to be allowed hereafter to publish a Table for work by the Siddhanta-Siromani assimilated to Table I of the Indian Calendar, which will enable us to ascertain the tithi, yoga, nakshatra and solar month according to that authority as easily as we have boen enabled to do according to the Arya- and Surya-Siddhantas; and this will, I hope, be followed by similar Tables for work by the Arya and possibly by the Brahma-Siddhanta.
Epigraphists will then be in a better position than heretofore to judge of the authenticity of inscription dates.
In case my Tables should be considered over-minute in detail, running as the entries do to several decimal points, I would ask readers to remember that they are designed as standard Tables for the settlement of the closest possible cases. Suoh a case as is mentioned in my former paper (above, Vol. XIII, 2, §§ 206, 207, on the cycle of Jupiter) proves that permanent reference Tables can hardly be too accurate. I have found other cases somewhat similar in
1 Sin. 90° radius. With (ratio of diam. to circumf.) 3.14159 the radius 3437-74967. The SuryaSiddhanta and Arya-Siddhanta radius is 3438. The Brahma-Siddhanta value of Sin. 90° or radius is 3270", which implies a ratio - 3.303. The ratio according to Archimedes (B. C. 250) was 3-14286. If the ratio 1: 10 mentioned in the Surya-Siddhanta should be adopted, we should have the ratio 3'16228. Brahmagupta's implied ratio, 3-303, is quite different from any of these. 2 K
