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Mathematics in Ancient and Medieval India c8.s2: Value of π

Mathematics in Ancient and Medieval India c8.s2

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Value of π

c8.s2.1

Historians have devoted much attention to the efforts of the old mathematicians to arrive at approximations to the value of the ratio between the circumference and the diameter of a circle. There is evidence¹ that the ancient Egyptians and old Babylonians had values such as 3.1604 and 3.125, the commonest practice in ancient civilisations was to take the ratio simply as 3. The ancient Indians also used the values varying from 3.1416(=3927/1250), 3.14136(=21600/6876)....., 3.14285(=22/7)........., 3.16227........, 3.0883........, 3.0885……., and 3. Of these the nearest approximation (3.1416) to the correct value is given by Āryabhaṭa I in the 5th century A. D. None of these values were determined from the ratio of the circumference of the earth to its diameter. On the other hand, they used the π values for determining the earth’s diameter from its circumference.¹ Nīlakaṇṭha ( 1500 A. D. ) gave the value 355/113 in his *Tantrasaṃgraha* ( TSS 188, ch. 2, vs 7 ). Mādhava ( 1400 A. D. ) used the circumference of a circle as 2827433388233 when its diameter was 9 × 10¹¹, which gives π = 3 14159265359 correct to 11 places of decimals.

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¹ Gow, J. *A Short History of Greek Mathematics*, Cambridge, p. 127, 1884; Smith, D. E. *History of Mathematics*, 2, p. 270, Dover Publication; Neugebauer, O. *The Exact Sciences in Antiquity*, Princeton University Press, p. 46, 53, 1952.

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¹ Datta, B. ‘Hindu Values of π’, *Journal of the Asiatic Society of Bengal*, 22, p. 25–42, 1928.

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The Greek value of π before Archimedes ( c. 250 B. C. ) was not satisfactory. Archimedes' values lie between the fraction 223/71 ( = 3.1408... ) and 22/7 ( = 3.1428... ) which were obtained by the use of 96 sided polygon.² Ptolemy³ ( 2nd century A. D. ) gave an expression for π in sexagesimal unit which is 3°8'30″. This when expressed in fraction gives out 377/120 ( = 3.14166... ) which differs only slightly from Āryabhaṭa I's value of π. It has therefore been suggested by many Western scholars like Rodet¹, Heath² and others³ that Āryabhaṭa I's value was borrowed from Greece. G. R. Kaye⁴, on the basis of Al-Bīrūnī's statement⁵ says that the value given by Āryabhaṭa I occurs in the *Puliśasiddhānta* which was admittedly composed under the Greek influence, supported the above view. The fact that the π values derived from the ratio of the circumference of the earth and its diameter differ substantially from the given values, is also regarded by many of these scholars as a strong argument in favour of the Greek influence on the Āryabhaṭa I's value of π. It should however be pointed out here that the *Puliśasiddhānta* as compiled by Varāhamihira (c. 505 A. D. ), the only available text of the said treatise does not make any mention of Āryabhaṭa I's value, though the other values are given.⁶ Al-Bīrūnī's statement might therefore be based on a recasted edition of *Puliśasiddhānta* whose authenticity cannot be relied upon in preference to that of the Varāhamihira. Then again as already stated above, the Indian workers never calculated their values from the ratio of the circumference of the earth and its diameter. There is therefore no justification for the assumption that Āryabhaṭa I's value was borrowed from Greeks. Needham is also of the same opinion.¹

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² Sarton, G. *Introduction to the History of Science*, 1, p. 169.

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³ Heath, T. L. *History of Mathematics*, 1, p. 253; Smith, D. E. *History of Mathematics*, 2, p. 308, Dover Publication ( Ptolemy accepted also the less correct value ).

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¹ Rodet, ‘Le’cons de Calcul d’Āryabhaṭa,’ *Journal Asiatique*, 7th Series, 13, p. 411, 1878.

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² *History of Greek Mathematics*, 1, p. 234, 1921 (The word *ayuta* used by Āryabhaṭa I is used earlier than that of the Greek word ‘myriad’. It occurs many times in Vedic and Post-vedic literature (vide Macdonell and Keith, *Vedic Index*, 1, p. 343).

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³ Smith, D. E. *History of Mathematics*, 2, p. 318, Dover Publication.

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⁴ *Journal of the Asiatic Society of Bengal*, 8, p. 126, 1908.

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⁵ Al-Bīrūnī’s *India*—translated by Sachau, 1, p. 168; 2, p. 72.

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⁶ *Pañcasiddhāntikā* of Varāhamihira, ch. 4, v. 1.

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¹ Needham, J. *The Science and Civilization in China*, 3, p. 101, see also foot-note.

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As for contemporary development elsewhere, a Chinese scholar Tsu Chhung Chih² (430–501 A. D.) gave also the value 355/113. Whether Indian scholar Nārāyaṇa was influenced by this result is yet to be found.

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² Needham, J. *Ibid.*, 3, p. 101.

c8.s2.16

*Later Approximation*—After Bhāskara II (c. 1150 A. D.), it is Nīlakaṇṭha who began to realize the irrationality of π. The reason for irrational value of π, according to Nīlakaṇṭha is given as follows : “If the diameter, measured with respect to a particular unit of measurement, is commensurable with respect to that same unit of measurement, the circumference is in-commensurable; and if with respect to any unit the circumference is commensurable, then, with respect to the same unit, the diameter is incommensurable”.³ He therefore proceeded to calculate the value of π in terms of series.

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³ *yena mānena mīyamāno vyāso niravayavaḥ syāt, tenaiva mīyamānaḥ paridhiḥ punaḥ sāvayava eva syāt yena ca mīyamānaḥ paridhirniravayavastenaiva mīyamāno vyāso’pi sāvayava eva ।* (*Āryabhaṭīyabhāṣya* of Nīlakaṇṭha, under *Gaṇita*, v. 10)

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