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A History of the Kerala School of Hindu Astronomy (in Perspective) c8.s4: 4. Taylor series for Sine and Cosine functions

A History of the Kerala School of Hindu Astronomy (in Perspective) c8.s4

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4. Taylor series for Sine and Cosine functions

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The approximations for sine and cosine functions up to the second power of small quantities, following from a well-known series expansion due to the British mathematician Brook Taylor (1685-1731)³ may be expressed as :

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³ W.W.R. Ball, *A short account of the history of mathematics*, Dover reprint, New York, 1960, pp. 380-81. But see C. B. Boyer, *History of mathematics*, Wiley, 1968, p. 422.

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$f(x+θ)=f(x)+θ f'(x)+(θ²)/(2!)f″(x)$ (approximately, when θ is small)

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² E. Whittaker and G. Robinson, *Calculus of observations*, London, 1965, p. 37.

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This approximation has been anticipated, in its particular cases, f(x)=sin x and f(x)=cos x, by more than three hundred years, by Mādhava of Saṅgamagrāma (c. 1340-1425)¹ in the following verses :

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¹ On Mādhava, his date and works, see K. V. Sarma, Introduction to his edition of the Veṇvāroha of Saṅgamagrāma Mādhava, (Sanskrit College Trippunithura, Kerala, 1965), and his paper on 'The date of Mādhava, a little-known Indian astronomer Quarterly Jl. of the Mythic Soc., 49 (1958) 183-86.

c8.s4.7

iṣṭa-doḥkoṭidhanuṣoḥ svasamīpasamīrite |
jye dve sāvayave nyasya kuryād ūnādhikam dhanuḥ ||
dvighna-talliptikāṭaikasarasailasikhindavaḥ |
nyasyācchedāya ca mithas tatsaṃskaravidhitsayā ||
chitvaikāṃ prakṣipej jahyāt taddhanuṣyadhikonake |
anyasyāṃ atha tām dvighnāṃ tathā ’syām iti saṃskṛtiḥ |
iti te kṛtasaṃskāre svaguṇau dhanuṣas tayoḥ ||²

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² Quoted by Nīlakaṇṭha Somayājin in his commentary on the Āryabhaṭīya Gaṇita., 12, (edn. Trivandrum, 1930, pp. 54-55), with the prefatory statement : tatrāha Mādhavah.

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'Placing the (sine and cosine) chords nearest to the arc whose sine and cosine chords are required, get the arc difference to be subtracted or added. For making the correction, 13,751 should be divided by twice the arc difference in minutes and the quotient is to be placed as the divisor. Divide the one, (say sine), by this (divisor) and add to or subtract from the other (i.e., cosine), according as the arc difference is to be added or subtracted. Double this (result) and do as before (i. e., divide by the divisor). Add or subtract the result (so obtained) to or from the first sine or cosine to get the desired sine or cosine chords'.³

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³ For this translation and its rationale in terms of modern mathematics, see R.C. Gupta, 'Second order of interpolation in Indian mathematics up to the fifteenth century', Indian Jl. of Hist. of Sc., 4 (1969) 92-94. For a traditional elucidation, see commentary by Śaṅkara Vāriyar on Nīlakaṇṭha's Tantra-saṅgraha, II, 10-13 (edn. Trivandrum, 1953, pp. 19-20) and Yuktibhāṣā, op. cit., pp. 168-71.

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5. Newton's Power series for the Sine and Cosine

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In Western mathematics, Newton (1642-1727) is credited with the enunciation, in about 1670,¹ of the sine and cosine power series, which might be stated as :

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¹ Moritz Cantor, *Vorlesungen uber Geschichte der Mathematik*, 2nd edn., Vol. III, cited by C.T. Rajagopal and A. Venkataraman in their paper on 'The sine and cosine power series in Hindu mathematics', *Jl. of the Royal As. Soc. of Bengal*, (Science), 15 (1949) 1-13, fn. on p. 1..

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sin x = x - (x³)/(3!) + (x⁵)/(5!) -

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cos x = 1 - (x²)/(2!) + (x⁴)/(4!) -

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These formulae are implied in the following verses of Mādhava, depicting the derivation of the series of sine and tabular versine (*śara*) values of the arc correct to 1/3600 of a degree :

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nihatya cāpavargeṇa cāpam tattatphalāni ca |
haret samūlayugvargais trijyāvargahataiḥ kramāt ||
cāpam phalāni cādhodho nyasyoparyupari tyajet |
jīvāptyai, saṅgraho 'syaiva vidvān-ityādinā kṛtaḥ ||
nihatya cāpavargeṇa rūpam tattatphalāni ca |
hared vimūlayugvargais trijyāvargahataiḥ kramāt ||
kintu vyāsadalenaiva dvighnenādyam vibhajyatām |
phalāny adhodhaḥ kramaśo nyasyoparyupari tyajet ||
śarāptyai, saṅgraho 'syaiva stenasīrī-tyādinā kṛtaḥ |

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'Multiply repeatedly the arc by its square and divide by the square of the even numbers (2, 4 etc.) increased by that number and then multiplied by the square of the radius. Place the arc and the results (of the above operation) one below the other and subtract each from what is above it. (This is the method) to derive the arcs, which are collected (and stated, in order, in the mnemonic verse) beginning with the expression vidvān (i.e., 0° 0' 0" 44" stated in the kaṭapayādi notation).

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'Multiply repeatedly the unit measurement, (which is the radius), by the square of the arc and divide by the square of the even numbers (2, 4 etc.) decreased by that number and then multiplied by the square of the radius; the first is, however, to be divided by twice the radius. Place the results one below the other and subtract each from the one above it. (This is the method) to derive the *śara-s* (tabular versines of the arc) which are collected (and stated, in order in the mnemonic verse) beginning with the expression *stena* (i.e., 0° 0′ 0″ 6″′).¹

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¹ For an exposition of these verses and a demonstration of the procedure described, see *Yuktibhāṣā*, *op. cit.*, Notes on pp. 183-93. See also *Karaṇapaddhati*, ed. with com. in Malayalam by P. K Koru. (Cherp, Kerala, 1953), pp. 199-207. For a detailed exposition of the rationale of the method, see C.T. Rajagopal and A. Venkataraman, *ibid.*; T. A. Saraswathi, 'Development of mathematical ideas in India', *Indian Jl. of Hist of Sc.*, 4 (1969) 75-76.

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These verses, taken with certain statements occurring in the *Yuktibhāṣā* (pp. 183-90) and in the *Karaṇapaddhati* (IV. 12-13) lead one directly to Newton's sine and cosine formulae. Though positive proofs are still wanting about the authorship of the above verses which explain the derivation of the sine and *śara* values enumerated in the mnemonical verses beginning with *vidvān* and *stena* which are definitely known to be composed by Mādhava, it is very likely that he himself is the author of these verses also.² But whatever that be, since he he has derived the values *vidvān* etc., it but logical to presume that he knew also the method therefor.

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² While quoting these verses in their Notes, the editors of *Yuktibhāṣā* give the source as *Tantrasaṅgraha*, but without any further reference (see *op. cit.*, p. 190), and other scholars have followed this lead (see, e g., C. T. Rajagopal, *ib.*, p. 2). But these verses, as also several others similarly quoted by the editors of the *Yuktibhāṣā*, do not occur in the *Tantrasaṅgraha* (vide edn. of the work, Trivandrum, 1958). T. A. Saraswathi says that "the *Tantrasaṅgraha* published from the Trivandrum University (sic) seems to contain one part only of the text. The editors of the *Yuktibhāṣā* had access to the complete MS. in the Trippunithura Skt. College Library, I am informed." (Fn. 9 on p. 231 of the *Bul.* of the National Inst. of Sc., No. 21 (1962). This information is not likely to be correct. The Trivandrum edition of *Tantrasaṅgraha* is, in fact, complete in itself. The second part of the Trippunithura MS. is likely to be another astronomical work written in continuation of *Tantrasaṅgraha*.

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6. Infinite G.P. Convergent series

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The credit for enunciating in India, for the first time, a formula for the sum of an infinite convergent geometrical progression, goes to Nīlakaṇṭha Somayājī (born 1444). This he gives in his *Āryabhaṭīya-Bhāṣya*, while explaining the process of deriving the arc of a circle in terms of the chord by means of a computation which involves the summing up of an infinite convergent G.P. series : *evam yas tulyaccheda-paramebhaga-paramparayā anantāya api saṃyogaḥ, tasya anantānām api kalpyamānasya yogasya adyāvayavinaḥ paraspar-aṃsacchedād ekacchedāṃśasādhyam sarvatrāpi samānam eva.* (Bhāṣya on *Āryabhaṭīya*, Gaṇita. 17, edn., Trivandrum, 1930, p. 106). ‘Thus the sum of an infinite series, whose later terms (after the first) are got by diminishing the preceding one by the same divisor, is always equal to the first term divided by one less than the common mutual divisor.’

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In continuation of this enunciation, Nīlakaṇṭha elaborately demonstrates it for a finite G.P. and also for an infinite decreasing G.P.¹

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¹ On this, see T.C. Saraswathi, ‘The development of mathematical series in India’, *Bul. National Inst. of Sc. of India*, No. 21 (1962) 320 ff.; see esp. pp. 325-26.

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