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A History of the Kerala School of Hindu Astronomy (in Perspective) c8: Chapter II. Anticipation of Modern Mathematical Discoveries by Kerala Astronomers

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

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English

Contents

Chapter II. Anticipation of Modern Mathematical Discoveries by Kerala Astronomers

1. The background

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In the light of the trends that developed in the study and practice in Kerala of the twin sciences of astronomy and mathematics, as indicated in the last chapter, it need not be surprising that some of the significant findings of modern mathematics are found anticipated in the astronomical works produced during the mediaeval times in that region. In their endeavour to make the results obtained by computation accord with observation, the Āryabhaṭan school of astronomers of this corner of India made substantial advancement in their computational skill. Working on the basic principles of algebra, geometry and limiting values, they produced, by dint of shrewd thinking and logical reasoning, coupled with a fund of intuition, significant results on such matters as the reduction to the ecliptic, summations and associated integrations leading up to the value of π etc. Especially, the achievements connected with π were brought to the notice of scholars by Charles M. Whish who presented a paper on the subject before the Royal Asiatic Society in 1832.¹ Among other things, Whish set out in this paper the several formulae enunciated in four Keralite works on astronomy and mathematics, viz., *Tantrasaṅgraha*, *Yuktibhāṣa*, *Karaṇapaddhati* and *Sadratnamālā*, for the circumference of a circle in terms of its diameter and transformations of the infinite series for π associated with the name of Leibnitz.

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¹ C.M. Whish, ‘On the Hindu quadrature of the circle, and the infinite series of the proportion of the circumference to the diameter exhibited in the four Śāstras, the *Tantrasamgraham*, *Yucti-bhasha*, *Carana Padhati* and *Sadratna-mala*’, *Trans. Royal Asiatic Soc. of Gr. Britain and Ireland*, 3 (1835) 509-23.

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The significance of the findings of Whish was hardly taken note of by historians of mathematics or of Indian literature, for more than a century. Instead, the view continued to prevail that Indian astronomy and mathematics were mostly borrowed from the West and that, at least since Bhāskara II, little original contribution was made by India to these disciplines.¹ The credit for launching, in the 1940-s, a detailed study of the discoveries of Whish in terms of modern mathematics goes to Prof. C.T. Rajagopal and his students and colleagues, including K. Mukunda Marar, A. Venkataraman, T.V. Vedamurthi Aiyar and K. Balagangadharan. The valuable notes added by Rama Varma Maru Thampuran and A.R. Akhileswara Aiyar to their excellent edition of the First Part of *Yuktibhāṣā*, (Trichur, Kerala, 1948), an expository work in Malayalam, new light on whose date and authorship has recently been thrown,² is another significant contribution in this direction. More recently, some other scholars like T. S. Kuppanna Sastri, T. A. Saraswathi and R.C. Gupta have been making valuable contributions to connected topics. The attempt made herein below to present, in a succinct form, the results of the investigations of the said scholars and those of the present writer which lie scattered in different books and periodicals, will, it is to be hoped, give a bird's-eye view of the significant contributions of Kerala savants to mathematics and astronomy.

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¹ *Vide*, for instance, G.R. Kaye: "After the time of Bhāskara (born A D. 1114), no Indian mathematical work of historical value or interest is known." (*Indian mathematics*, Calcutta, 1915, p. 24); A. B. Keith: "After Bhāskara, no progress can be recorded in Indian astronomy." (*A history of Indian literature*, Oxford, 1929, p. 523); A. A. Macdonell: "The last eminent Indian astronomer was Bhāskarācārya, born in A.D. 1114." (*A history of Sanskrit literature*, Indian rep., Delhi, 1962, p. 370); D. Arka Somayaji, "Bhāskara is the last name in the field. Barring a number of commentators, no name worthy of mention is there in India subsequent to Bhāskara." (*A critical study of the ancient Hindu astronomy*, Dharwar, 1971, p. 3). Standard historians of mathematics like Cantor, Cajori and D.E. Smith, too, hold the same view. The first Western historian of mathematics to take note of Whish's findings would appear to be A. P. Juschkewitsche in his *Geschichte der Mathematic Mittelalter*, (Leipzig, 1964), as pointed out to C. T. Rajagopal by D. T. Whiteside, the eminent Cambridge authority on Isacc Newton.

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² K. V. Sarma, 'Jyeṣṭhadeva and his identification as the author of *Yuktibhāṣā*', *Adyar Library Bulletin*, 22 (1958) 35-40.

2. Tycho Brahe's Reduction to the ecliptic

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In astronomical computations, the longitude of a planet is measured along the ecliptic, while, in fact, its motion takes place along its own orbit which, generally, deviates slightly from the ecliptic. For an accurate computation of the planet's position this deviation has to be taken into consideration. In the West, it was the Danish astronomer Tycho Brahe (1546-1601) who enunciated a formula for this ‘Reduction to the ecliptic’.¹

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¹ *Vide* his *Astronomiae instauratae Progymnasmata*, Vol. I, Prague, 1602.

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In India, a formula for this Reduction was enunciated, for the first time, by a contemporary Kerala astronomer Acyuta Piṣāraṭi (1550-1621), author of *Rāśigola-sphuṭānīti* (‘Computation of True longitude on the celestial sphere’)² and several other works on astronomy. Acyuta gives his formula for this reduction in the case of the moon in his work entitled *Sphuṭanirṇaya* (‘Computation of true longitudes of planets’) in the following verse :

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² K. V. Sarma, *The Rāśigolasphuṭānīti*, (On the calculation of true longitude on the celestial sphere and Reduction to the ecliptic in Indian astronomy), Critical ed. and Tr. with Introduction, Adyar Library and Res. Centre, Madras, 1955.

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*paṭonasya vidhos tu koṭibhujayor jīve mithas tāḍayet*
*antyakṣepaśarahatam vadham amum vikṣepakoṭyā haret |*
*labdham vyāsadaloddhṛtam himakare svarṇam, vipāte vidhau*
*yugmāyugmapadopage ; vidhur ayam spaṣṭo bhagole bhavet ||*

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‘Multiply the tabular cosine (*koṭijyā*) and sine (*bhujajyā*) of the moon-minus-node and the product by the tabular versine (*śara*) of the maximum latitude (*antya-kṣepa*) of the moon. Divide this by the tabular cosine of the latitude at the particular moment and the quotient is to be divided again by the tabular radius (*vyāsadala*). The result (will give the correction for longitude which) is to be added to or subtracted from the moon’s longitude, as the moon-minus-node is in an even or an odd quadrant, respectively. The True Moon measured on the ecliptic is thus got’.³

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³ This verse has been extracted by the author in his *Rāśigola-sphuṭānīti*, verse 47, *op. cit.*, p. 29.

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Acyuta’s formula may be expressed in terms of modern mathematics thus: ‘If *F* is the longitudinal difference between the node and the planet, *w* the maximum latitude and *y* the actual latitude, then, the correction *k*=sin *F*. cos *F* (1−cos *w*) / cos *y*.⁴

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⁴ For a demonstration of the correctness of this formula in terms of modern mathematics, see K.V. Sarma, *op. cit.*, Introduction, pp. 10-13.

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In another of his works, viz. *Uparāgakriyākrama* (‘Procedures in the computation eclipses’), Acyuta gives also a simplified version of the said formula.¹

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¹ For a demonstration of this formula, see *ibid.*, pp. 13-14.

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The actual date of composition of the *Sphuṭanirṇaya*, which is the earliest work to enunciate the reduction formula, is not known, but it is definitely before 1583 which is the date of composition of the *Uparāgakriyākrama* which contains the simplified formula.²

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² On the chronology of these works, see *ibid.*, pp. 14-15.

3. Newton-Gauss Interpolation formula

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It has been demonstrated in detail that Indian mathematics employed, in its search for better results, improved rules of interpolation, by using the second differences and that a particular case, upto the second order, of the more general Newton-Stirling interpolation formula was known to Brahmagupta (*c* A.D. 625).³ Following this line of thought, the Kerala astronomer Govindasvāmin (*c*. 800-850) has enunciated a set of rules for the computation of intermediary functional values, different formulae being laid down for different augmental intervals:

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³ R. C. Gupta, ‘Second order interpolation in Indian mathematics upto the fifteenth century,’ *Indian Jl. of Hist. of Sc.*, 4 (1969) 86-98.

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*gacchad-yāta-guṇāntaravapur yātaisya-diṣvāsanac-*
*chedabhyāsa-samūha-kārmukakṛti-prāptat tribhis taḍitat |*
*vedaiḥ ṣaḍbhir avāptam antyaguṇāje rāśyoḥ kramād antyabhe*
*gantavyāhata-vartamāna-guṇajāc cāpāptam ekādibhiḥ ||*

*antyād utkramataḥ krameṇa viṣamaiḥ saṅkhyāviśeṣaiḥ kṣiped*
*bhaṅktvāptam, yadi maurvikāvidhir ayam makhyāḥ kramād vartate*
*śodhyam vyutkramatas tathākṛtaphalām...*

(*Bhāṣya* on *Mahābhāskarīya*, 4.22)⁴

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⁴ *Mahābhāskarīya* of Bhāskarācārya, with the *Bhāṣya* of Govindasvāmin and the super-commentary *Siddhāntadīpikā* of Parameśvara, ed. by T.S. Kuppanna Sastri, Madras, 1957, pp. 201-2.

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‘Multiply the difference of the last and the current sine differences by the two parts of the elemental arc (made by any intermediary point on it) and divide by the square of the elemental arc and further multiply by three. Now divide the result so obtained by four in the first rāśi, or by six in the second rāśi. The final result thus obtained should be added to the portion of the current sine difference (got by linear proportion).

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‘In the last (third) rāśi, multiply the linearly proportional part of the current sine difference by the remaining part of the elemental arc and divide by the elemental arc. Now, divide the result (so obtained) by the odd numbers (1, 3, 5 etc.) according to the current sine difference (is first, second, third etc.), when counted from the end in the reverse order. Add the final result thus obtained to the portion of the current sine difference (got by ordinary proportion). These are the rules for computing true sine differences for (direct) sines. In the case of versed sines, apply the rules in the reverse order and the above corrections are to be subtracted from the respective differences (got by linear interpolation)’.¹

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¹ For the rationale of this formula, see R.C. Gupta, *ib.*, pp. 91-92.

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Using the general functional notation and finite difference operator, the rule for the second rāśi may be put as :

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$f(x+nh)=f(x)+nΔ f(x)+(n(n-1))/2Δ f(x)-Δ f(x-h)$

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which is a particular case (up to the second order) of the general Newton-Gauss interpolation formula.²

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.

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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.

7. Lhuiler’s formula for the Circum-radius of a Cyclic quadrilateral

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In Western mathematics, the eighteenth century mathematician Lhuiler is credited with the discovery, in 1782, of an expression for the circum-radius of a cyclic quadrilateral.² In India, however, we find the same formula enunciated by the Kerala astronomer Parameśvara (c. 1360-1455) in his commentary on the *Līlāvatī*, in the following lines :

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² *Vide* D.E. Smith, *History of mathematics*, Boston, 1925, vol. II, p. 286.

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*doṣṇām dvayor dvayor ghātayutānām tisṛṇām vadhāt |*
*ekaikonetarattraikyam catuṣkavdhabhajitam ||*
*labdhamūlena yadyṛttam viṣkambhardhena nirmitam |*
*sarvam caturbhujakṣetram tasminn evātiṣṭhate ||*

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‘The three sums of the products of the sides, taken two at a time, are to be multiplied together and divided by the product of the sums of the sides taken three at a time and diminished by the fourth. If a circle is drawn with the square root of this quantity as radius, the whole quadrilateral will be situated inside it.’

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Thus, if a, b, c, d are the sides and r the circum-radius,

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r = √(((ab+cd)(ac+bd)(ad+bc))/((a+b+c-d)(b+c+d-a)(c+d+a-b)(d+a+b-c)))

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The rationale of this formula has been given in the 16th cent. Kerala commentary *Kriyākramakarī* on the *Līlāvatī*.¹

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¹ On this, see T.A. Saraswathi, 'Development of mathematical ideas in India', *Indian Jl. of. His. of Sc.*, 4 (1969) 69.

8. Gregory and Leibnitz's Series for the Inverse tangent

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The power series for arc tan x, which was enunciated for the first time in the United Kingdom in 1671 by the Scottish mathematician James Gregory (1638-75) and in Europe in 1673 by the German mathematician and philosopher Gottfried Wilhelm Leibnitz (1646-1716), in the case of x=1, may be stated as follows : In the case of infinite series of powers of x representing an arc of a circle of unit radius which subtends at the centre of the circle an angle whose tangent (x) does not exceed unity,

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arc tan x = x - (x³)/3 + (x⁵)/5 - (|x| 1)

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In India, this series was enunciated by Mādhava of Saṅgamagrāma (1350-1410), nearly three centuries before it was discovered in the West. Mādhava's enunciation of the formula is contained in the following lines :

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*iṣṭajyā-trijyayor ghātāt koṭyāptam prathamam phalam |*
*jyāvargam guṇakam kṛtvā koṭivargam ca hārakam ||*
*prathamādiphalebhyo 'tha neya phalakṛtir muhuḥ |*
*eka-tryādy-ojasaṅkhyābhir bhakteṣv eteṣv anukramāt ||*
*ojānām saṃyutes tyaktvā yugmayogam dhanur bhavet |*
*doḥ-koṭyor alpam eveha kalpanīyam iha smṛtam |*
*labdhīnām avasānam syān na tathāpi muhuḥ kṛte ||*²

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² Quoted in the Keralite commentary *Kriyākramakarī* on Bhāskara II's *Līlāvatī*, kāṇḍa 2, vṛtta. 40. These lines have been quoted also in the *Yuktibhāṣā* where a geometrical proof for it is given (see *op. cit.*, pp. 113-16).

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'Obtain first the result of multiplying the *jyā* (of a given *dhanus*) by the *trijyā* and dividing the product by the *koṭi* (of the *dhanus*). Multiply this result by the square of the *jya* and divide the square by the *koti*. Thus we obtain a second result, (as also) a sequence of the further results by repeatedly multiplying by the square of the *jya* and dividing by the square of the *koti*. Divide the terms of the sequence in order by the odd numbers 1, 3, 5, etc.; after this, add all the odd terms and subtract from them all the even terms (without disturbing the order of the terms). Thus is obtained the *dhanus* whose two elements are the given *jya* and *koti*. Here, the smaller of the two elements should be taken as the *jya*, since, otherwise, the series obtained will be non-finite (in value).’

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According to the above formula, if R is the radius and *s* and *c* are the sine and cosine chords of the arc,

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the arc= sR/c - sR/3c · (s²)/(c²) + sR/5c · (s⁴)/(c⁴) -

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which is Gregory’s general series for arc tan *x*.

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Mādhava’s verses quoted above have been extracted and explained in the *Yuktibhāṣā* (*op. cit.*, pp. 113-16) in the traditional manner and its rationale demonstrated in terms of modern mathematics by C.T. Rajagopal and others.¹ Rajagopal’s demonstration throws up, *inter alia*, certain interesting anticipations of Kerala mathematicians. Thus, after giving the general derivation of Gregory’s series according to the *Yuktibhāṣā* and offering the proof therefor, Rajagopal observes: “There are two points which emerge from a consideration of the mathematical text presented here. In the first place, it employs relations which would appear not to have been noticed in Europe before modern forerunners and followers of the calculus started investigations.........Our second point is not unconnected with the first. The Hindu mathematicians achieved, without the aid of calculus, results which, for us, are treated best by means of the calculus.²

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¹ C. T. Rajagopal and T V. Vedamurthi Aiyar, ‘On the Hindu proof of Gregory’s series’, *Scripta Math.*, 17 (1951) 65-74; K. Mukunda Marar and C.T. Rajagopal, ‘On the quadrature of the circle’, *Jl. of the Bombay Br. of the RAS* (NS) 20 (1944) 65-82; T.A. Saraswathi, ‘Development of mathematical series’, *ibid.*, ‘Development of mathematical ideas’, *ibid.*

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² “This is not to gainsay the fact that (i) the Hindus’ proof of Gregory’s series shows their awareness of the principle of integration as we ordinarily use it nowadays; (ii) their intuitive perception of small quantities like O (1/n^p), n → ∞, in a certain role, is as good as a practical knowledge of differentiation.”

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But the Europeans who first explored the possibilities of expressing a circular arc as an infinite series, deliberately employed the calculus to guide them.”¹

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¹ Mukunda Marar and Rajagopal, ‘On the Hindu quadrature of the circle’, ibid., p. 68.

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9. Leibnitz’s Power series for π

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In modern mathematics, the irrationality of π was demonstrated, for the first time, by Lambert in a paper read before the Berlin Academy in 1671. It is interesting to note that nearly two centuries prior to this, Nīlakaṇṭha Somayājī (1444-1545) stressed the same point when he expressed himself in his *Bhāṣya* on the *Āryabhaṭīya* (Gaṇita. 10) to the effect that it is impossible to measure the circumference of a circle in exact terms of its diameter; it could be stated only approximately (*āsanna*).² What is more significant, is that, after elaborating the point further, he quotes anonymously an earlier authority to the same effect, which means that the fact had long been recognised in Indian mathematics.³

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² Cf.: *āsannaḥ... kutaḥ punaḥ vāstavīm saṅkhyam utsṛjya āsannā eva ihoktā ? ucyate. tasya vaktum aśakyatvāt. kutaḥ? yena mānena miyamāno vyāso niravayavaḥ syāt, tenaiva mānena miyamānaḥ paridhiḥ punaḥ sāvayava eva syāt ... niravayavatvam tu na labhyam iti bhāvaḥ.* (edn. Trivandrum 1930, Pt. I, pp. 41-42).

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³ Cf.: *ata evāhuḥ*—“*kṛtakānityavad vyāsa-paridhī niyatau mithaḥ*”, ibid., p. 42.

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The enunciation of the value of π as a particular case of the power series for arc tan *x*, due to Leibnitz (1673), is explicitly: π/4=1−1/3+1/5−……

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This formula has been anticipated by Mādhava of Saṅgamagrāma (1350-1410) while enunciating a method for the circumference of a circle, in the following verse:

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*vyāse vāridhi-nihate rūpahṛte vyāsasagarābhihate |*
*tri-śarādi-viṣamaśaṅkhyā-bhaktam ṛṇam svam pṛthak kramāt kuryāt ||*⁴

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⁴ Quoted in the *Kriyākramakarī* (op. cit.), with the prefatory statement; *atrāha Mādhavah ... ito’pi laghutaraparidhyānayanopāyaḥ amunaivoktam, yathā*, com. on *Līlāvatī*, kāṇḍa 2, vṛtta., 40).

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‘Multiply the diameter by 4. Subtract from it and add to it alternately the quotients obtained by dividing four times the diameter to the odd integers 3, 5 etc.’

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For a circumference C of a circle of diameter D, this gives the formula : C, (i.e., π D)=4D-4D/3+4D/5......or π/4=1+1/3-1/5......¹

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¹ For a detailed demonstration, see : *Yuktibhāṣā*, op. cit., pp. 99, 112-13 : C.T. Rajagopal, T.V. Vedamurth. Aiyar, K. Mukunda Marar. ibid., P.K. Koru, *Karaṇapaddhati*, op. cit., pp. 166-67.

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10. Approximations to the value of π

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Continuing his enunciation of the circumference of a circle, as given above, Mādhava goes on to give a rational approximation to its value and, through it, the value of π :

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yatsaṅkhyayā 'tra haraṇe kṛte nivṛttā hṛtis tu jāmitayā |
tasya urdhvagatayās samasaṅkhya taddalam guṇo 'nte syāt ||
tadvargai rūpahato hāro vyāsābdhighātatāḥ prāgvat |
tasyām aptam svamṛṇe kṛte dhane śodhanañ ca karaṇīyam |
sūkṣmaḥ paridhiḥ sā syāt bahukṛtvo haraṇato 'tisūkṣmaś ca ||

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‘Let the process stop at a certain stage, giving rise to a ‘finite sum’. Multiply four times the diameter by half the even integer subsequent to the last odd integer used as divisor and then divide by the square of the integer increased by unity. The result is the correction to be added to or subtracted from our ‘finite sum’, the choice of addition or subtraction depending on the sign of the last term in the sum. The final result is the circumference determined more accurately than (that obtained) by taking a large number of terms, *i.e.*, terms going beyond the stage at which we stopped.’

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The formula enunciated here, which gives the value of π to an advanced degree of accuracy, may be expressed thus :

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C 4D 1 - ⅓ + ⅕ - ± 1/n ((n+1)/2)/((n+1)²+1) . where n is large.²

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∴ (π)/4 = 1 - ⅓ + ⅕ - ± 1/n ((n+1)/2)/((n+1)²+1)

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² For a demonstration of this formula; see Rajagopal, Vedamurthi Aiyar, Mukunda Marar, ibid., P.K. Koru, ibid., pp. 176-79; *Yuktibhāṣā*, op. cit., pp. 120-23.

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A still better correction is suggested by Mādhava in the following lines :

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asmat sukṣmataro 'nyo vilikhyate kaścanāpi saṃskāraḥ |
ante samasaṅkhyādalavargas saiko guṇaḥ, sa eva punaḥ ||
yugaguṇito rūpayutaḥ samasaṅkhyādalahato bhaved hāraḥ |
triśarādiviṣamasaṅkhyāharaṇāt param etad eva vā kāryam ||¹

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¹ Quoted in the Kriyākramakarī on Līlāvatī, kāṇḍa 2, vṛtta., 40.

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‘A correction still more precise is being stated here. The multiplier is the square of half the even integer (next greater than the last odd-interger divisor) increased by unity. This multiplier multiplied by 4, then increased by unity, and then multiplied by half the even-integer (already defined), is the divisor. This correction may be applied after the division by the odd integers 3, 5 etc. (indicated in the previous method and the same method of calculation followed).’

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This formula which gives a still more accurate value of π may be expressd thus :

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$C = 4D 1 - ⅓ + ⅕ - ± 1/n (( (n+1)/2 )² + 1)/([ ( (n+1)/2 )² · 4 + 1 ] ( (n+1)/2 ))$

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$(π)/4 = 1 - ⅓ + ⅕ - ± 1/n (( (n+1)/2 )² + 1)/([ ( (n+1)/2 )² · 4 + 1 ] ( (n+1)/2 ))$

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where n is odd and large.²

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² For its demonstration, see Yuktibhāṣā, op. cit., pp. 136-39.

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Elsewhere, Mādhava gives a series for π, different from that of Leibnitz. Thus, in continuation of the formula for the inverse tangent given in the lines iṣṭajyātrījyayor ghātāt etc. (see above, p. 20), he mentions the measure of the circumference when the arc calculated is half of circumference, thus :

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vyāsavargād ravihatāt padam syāt prathamam phalam |
tadāditas triśaṅkhyāptam phalam syād uttarottaram ||

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*rupādyayugmasaṅkhyābhir hṛteṣv eṣu yathākramam |* *viṣamāṇam yutes tyaktvā samam hi paridhir bhavet ||*

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‘Multiply the square of the diameter by 12 and extract the square root of the product. That is the first term. Divide the first term by 3 to obtain the third and so on and thus get further terms. Divide the terms, in order, by the odd numbers 1, (3, 5) etc. Add the odd-order terms to and subtract the even-order terms from the preceding. The result will give the circumference’.¹

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The above enunciation gives the following formula for the circumference :

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C (i.e., π D) = √12D² - √12D² / 3·3 + √12D² / 5·3² - √12D² / 7·3³ ...

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= √12D { 1 - 1 / 3·3 + 1 / 5·3² - 1 / 7·3³ ...

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π = √12 { 1 - 1 / 3·3 + 1 / 5·3² - 1 / 7·3³ ...

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It would be interesting to derive from Mādhava’s above-noted value of π, the following relation between circumference and diameter given by him :

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*vibudha-netra-gaj-āhi-hutāśana-* *tri-guṇa-veda-bha-vāraṇa-bāhavāḥ |* *nava-nikharva-mite vṛtlvistare* *paridhi-mānam idam jagadur budhāḥ ||²*

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1. For an exposition and demonstration of these verses, see *Yuktibhāṣā*, op. cit., pp 114-19; Marar and Rajagopal, ‘On the Hindu quadrature of the circle’, ibid., p. 78; P.K. Koru, op. cit., pp. 203-6.

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2. Quoted in the *Kriyākramakarī*, oh *Līlāvatī*, kāṇḍa 2, vṛtta. 40, with the prefatory statement: *Madhavācāryaḥ punaḥ ato ‘pyāsannatamam paridhimānam uktavān*.

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That is, for a diameter of 9¹¹, the circumference is 28,27,43,33,88,233. The value of π from this, correct to eleven decimal places, would be 3.14159265359, which accords closely with the modern value of π, which is 3.14159265.

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Using the above relationship of the diameter and the circumference, Mādhava has derived the 24 mahājyās (accurate sign-chords) which he has depicted in the mnemonic beginning with śreṣṭham nāma variṣṭhanām (0°—224′—50″—22″).¹ It has also been verified that the value of the radian assumed by Mādhava in this evaluation corresponds to 3437′—44″—48″—22″, which is remarkably close to its modern approximation, viz., 3437′.74577978 = 3437′—44′—48″—22″, 49.²

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¹ For the full enunciation of these maha-jyās of Mādhava and their rationale, see Yuktibhāṣa, op. cit., pp 195-98; P.K. Koru, op, cit., pp. 203-6.

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² See Rajagopal and Venkataraman, ‘The sine and cosine power series in Hindu mathematics’, JRASB (Sc.), 15 (1949) 3.

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Rajagopal and his colleagues have drawn attention to and demonstrated, in terms of modern mathematics, a large number of further rational approximations of π which are found enunciated in later works like Yuktibhāṣa, Karaṇapaddhati and Sadratnamāla,³ for most of which there are no exact parallels in Western mathematics. These approximations had been obtained by the application of various corrections, presumably in the form of continued fractions, on partial sums of the infinite series for π/4. Indeed, the enunciation of these corrections speaks volumes about the fervour with which mediaeval mathematicians of Kerala attacked this fascinating problem and the success they achieved therein, some of their enunciations anticipating Western discoveries by nearly three hundred years. The same may be said with justification in the case of certain of their other mathematical contributions also. A competent and critical analysis, in terms of modern mathematics, of the writings of Kerala astronomers and mathematicians, the bulk of which still remains locked up in old palmleaf manuscripts, written in Malayalam script, may be expected to throw welcome light on the advances, down the centuries, made in these disciplines, in one remote corner of India.

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³ For these enunciations and their traditional expositions, see : Nīlakaṇṭha Somayājī’s Bhāṣya on the Āryabhaṭīya, Gaṇita, 9-12 (edn., Trivandrum, 1930), Pt. I, pp 40 ff.; Yuktibhāṣa, op. cit., pp. 72-142; Kriyākramakarī on the Līlāvatī 2, vṛtta. 40, and Karaṇapaddhati, op. cit., pp. 173-88. And, for their demonstration in modern terms see, Marar and Rajagopal and Vedamurthi Aiyar, ‘On the Hindu proof of Gregory’s series’, Scr. Math. 17 (1951) 65-74; Rajagopal and Vedamurthi Aiyar, ‘A Hindu approximation to Pi’, Scr. Math. 18 (1952) 25-31; T.A. Saraswathi, ‘The development of mathematical series in India’, Bul. of the National Inst. of Sc. of India, 21 (1962) 320-43.

11. Some other highlights

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In the context of the continuity of astronomical tradition and the diligence with which its traditional exponents cultivated its study, a reference might be made here to certain other significant methodologies evolved and discoveries made by Kerala astronomers and mathematicians. The introduction of the *Kaṭapayādi* notation and the expression of astronomical data through mnemonic tables and verses composed with the help of this facile notation has already been noticed. In his table of sine-differences enumerated in the verse *makhi-bhakhi* etc. (*Gītika* 10), *Āryabhaṭa* gives only their rough values, correct to the minute. It is interesting that Govindasvāmin (*c.* 800-50) developed their evaluation still more accurately, correct to the second order of sexagesimal fractions, viz., seconds and thirds, which he gives in his *Bhāṣya* on *Mahabhāskarīya*. He also gives a formula for the derivation of the value of the last sine-difference when the tabular interval is halved. Developing the topic further, he enunciates also a rule for computing, from a given last sine-difference, the other sine-differences lying between 60° and 90°.¹

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¹ See *Mahabhāskarīya* with Govindasvāmin's *Bhāṣya*, *op. cit.*, ch 4, verse 22, pp. 201-2. For an exposition of these rules and their rationale, see R.C. Gupta, 'Fractional parts of Āryabhaṭa's sines and certain rules found in Govindasvāmin's *Bhāṣya* on the *Mahabhāskarīya*', Indian JI. Hist. of Sc. 6 (1971) 51-59.

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The *Kuṭṭakāra-gaṇita*, which is a complex type of calculation involving the Rule of three, Indeterminate equations and Continued fractions, has been developed by Kerala mathematicians to a high degree. It enabled them to solve various problems which require the application of higher algebra according to modern working. *Śaṅkaranārāyaṇa* (*c.* 825-900) quotes in his commentary on the *Lahgubhāskarīya* (*op. cit.*, pp. 95-97) a short disquisition on the subject from *Govindakṛti*, a lost work of Govindasvāmin, while the later works like *Yuktibhāṣā* (16th cent.) give full accounts of the processes involved.¹ Nīlakaṇṭha Somayājī (1444-1545) has made significant contributions including logical proofs and demonstrations to geometrical theorems, a formula for the sum of an infinite, geometrically progressing convergent series, and the rationale for the formula for the sum of a geometrical progression. He has also offered geometrical demonstrations for the sum of an arithmetic progression, the sum of triangular numbers and the sums of squares and cubes of natural numbers.²

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¹ See *Yuktibhāṣā*, ch. 5, op. cit., pp. 54-71. For the rationale of *Kuṭṭakāra* according to modern mathematical terms, see *ibid.*, Appendix, pp. xli-lxi.

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² See for details, T.C. Saraswathi, 'The development of mathematical series in India', *Bul. of the National Sciences of India*, No. 21 (1962) 320-43.

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