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A History of the Kerala School of Hindu Astronomy (in Perspective) c8.s6: 8. Gregory and Leibnitz's Series for the Inverse tangent

A History of the Kerala School of Hindu Astronomy (in Perspective) c8.s6

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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.

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