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Jyotihshastra: Astral and Mathematical Literature c5.s2: Mathematics in Astronomy

Jyotiḥśāstra: Astral and Mathematical Literature c5.s2

Language
English

Mathematics in Astronomy

c5.s2.1

The mathematics utilized in astronomical texts is most developed in the areas of analemmata and of trigonometry—as well as, of course, the application of the pulverizer to astronomical problems that we have already considered. Some of the most significant mathematical work undertaken in India was in connection with trigonometry.⁶¹ Briefly, after the initial success in constructing tables of Sines and Versines and in inventing a rule for finding the approximate Sine of any angle in the fifth through seventh centuries, the next advance was the discovery by the second Bhāskara of the formula for the addition of Sines. The more advanced trigonometry of the seventeenth and eighteenth century astronomers of northern India seems to be due to the influence of al-Kāshī and Ulugh Beg.

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⁶¹ See, for example, A. N. SINGH, “Hindu Trigonometry,” PBMS, NS 1 (1939), 77–92; and the following articles, among others, by R. C. GUPTA: “Early Indians on Second Order Sine Differences,” IJHS 7 (1972), 81–86; “Bhāskara I’s Approximation to Sine,” IJHS 2 (1967), 121–136; “Addition and Subtraction Theorems for the Sine and the Cosine in Medieval India,” IJHS 9 (1974), 164–177; “Sine of Eighteen Degrees in India up to the Eighteenth Century,” IJHS 11 (1976), 1–10; and “Sines of Submultiple Arcs as found in the Siddhānta-tattvaviveka,” RUMJ 5 (1974), 21–27.

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But in Kerala truly extraordinary advances were made by a school of astronomer-mathematicians building on the pioneering work of Mādhava of Saṅgamagrāma in the late fourteenth and early fifteenth century. He evidently discovered what is called “Gregory’s Series” for π and the power series for sines and cosines associated with Newton and Leibniz.⁶² Mādhava’s work was continued by, and is largely known from the works of, Nīlakaṇṭha, Jyeṣṭhadeva, and Śaṅkara in the sixteenth century, and Putumana in the eighteenth.

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⁶² After the work of C. M. WHISH, “On the Hindú Quadrature of the Circle and the Infinite Series of the Proportion of the Circumference to the Diameter in Four Śāstras, the Tantrasangraham, Yucti Bhāshá, Carana Paddhati, and Sadratnamálá,” TRAS 3 (1830), 509–523, the investigation of this topic was revived by C. T. RAJAGOPAL; see K. M. MARAR and C. T. RAJAGOPAL, “On the Hindu Quadrature of the Circle,” JBBRAS, NS 20 (1944), 65–82, and “Gregory’s Series in the Mathematical Literature of Kerala,” MS 13 (1945), 92–98; C. T. RAJAGOPAL and A. VENKATARAMAN, “The Sine and Cosine Power-series in Hindu Mathematics,” JRASB/S 15 (1949), 1–13; C. T. RAJAGOPAL, “A Neglected Chapter of Hindu Mathematics,” SM 15 (1949), 201–209; C. T. RAJAGOPAL and T. V. V. AIYAR, “In the Hindu Proof of Gregory’s Series,” SM 17 (1951), 65–74, and “A Hindu Approximation to Pi,” SM 18 (1952), 25–30; and C. T. RAJAGOPAL and M. S. RANGACHARI, “On an Untapped Source of Medieval Keralese Mathematics,” AHES 18 (1978), 89–102. His investigations have been continued by T. A. SARASWATHI, “The Development of Mathematical Series in India after Bhāskara II,” BNISI 21 (1963), 320–343; by

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A. K. BAG, “Trigonometrical Series in the Karaṇapaddhati and the Probable Date of the Text,” IJHS 1 (1966), 98–106, and “Mādhava’s Sine and Cosine Series,” IJHS 11 (1976), 54–57; and by R. C. GUPTA, “The Mādhava-Gregory Series,” ME 7 (1973), B 67–70, “An Indian Form of Third Order Taylor Series Approximation of the Sine,” HM 1 (1974), 287–289, and “Mādhava’s and Other Medieval Indian Values of Pi,” ME 9 (1975), B 45–48.

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