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V. Aryabhata's System of expressing Numbers

Published online by Cambridge University Press:  15 March 2011

J. F. Fleet
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SEVERAL systems of expressing numbers, used in India, have been explained by Professor Bühler in §§ 33 to 35 of his work on Indian palaeography. There is a system, a highly interesting one, which was not noticed by him, because it has not been found used in inscriptions or in the pagination of literary works; namely, that of the astronomer Āryabhaṭa. It has been mentioned briefly by various other writers. And it was considered in some detail by Mr. C. M. Whish in 1820, and at more length by M. Léon Rodet in 1880. Those two treatments of it, however, scarcely suffice to do justice to it; particularly from lacking any table to make its details clear. And it deserves a full exposition, because it is of special interest in connection with two topics which have been reopened lately by Mr. G. R. Kaye; namely, the early use of the abacus in India, and the development of the decimal notation, that is, of the system of the nine significant digits 1 to 9, with the zero, cipher, or naught, used with place-value so that any particular sign denotes units, tens, thousands, etc., or the absence of them, according to its position as written in a row of figures. I propose, therefore, to consider it exhaustively here, but without venturing at present to offer any opinion on the two topics which Mr. Kaye has reopened: I only seek to exhibit fully, with a few introductory remarks about Āryabhaṭa himself, a system of numeration which must certainly be regarded as an important factor in considering them.

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Articles
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Journal of the Royal Asiatic Society , Volume 43 , Issue 1 , January 1911 , pp. 109 - 126
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Copyright © The Royal Asiatic Society 1911

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References

page 109 note 1 Grundriss der Indo-Arischen Philologie und Altertumskunde, vol. 1, part 11: English version in the Indian Antiquary, vol. 33 (1904)Google Scholar, appendix.

page 109 note 2 In the course of his article “On the Alphabetical Notation of the Hindus” published in the Transactions of the Literary Society of Madras, part 1 (1827), p. 54 ff.

page 109 note 3 In his article La Notation Numérique inventée par Âryabhata” published in the Journal Asiatique, series 7, vol. 16 (1880, part 2), p. 440 ff.Google Scholar

page 109 note 4 A translation of MrWhish's, article was given in the Journal Asiatique, 1835, 2. 116 ff.Google Scholar, and was accompanied by a large “paradigme synoptique”, which, however, only shows the 297 combinations with single letters and the values of them from one to a trillion (British): it does not illustrate the principles of the system.

page 109 note 5 See his articles “Notes on Indian Mathematics: Arithmetical Notation” in JASB, 1907. 475 ff., and “The Use of the Abacus in Ancient India”, id., 1908. 293 ff. He has noticed this system of expressing numbers in id., 1908. 117–8, in the course of a third article, “Notes on Indian Mathematics: No. 2: Āryabhaṭa”: but he, again, has not given a table.

page 110 note 1 Gaṇitapāda, verse 1.

page 110 note 2 More technically, “the troup of the nakshatras”.

page 111 note 1 Kālakriyāpāda, verse 10.

page 111 note 2 The commentator explains iha by vartamānē=shṭāviṁśē chaturyugē, to which there seems no objection: for iha in the sense of ‘now, at present’, see the St. Petersburg Dictionary. If, however, because only Vēdic references are given, we prefer to say “here (at this place)”, it will not affect the bearing of the verse.

page 111 note 3 Moreover, it is questionable whether this cycle was in use in Āryabhaṭa's time: at any rate, he has not mentioned it; he has given only the twelve-years cycle.

page 112 note 1 Kālakriyāpāda, verse 12.

page 112 note 2 For the term exeligmos, frequently a very convenient one to use, we are indebted to Dr. Burgess, who brought it to the front from Gemīnos and Ptolemy: see this Journal, 1893. 721. It answers to the Roman annus magnus or mundanus, and denotes a period of evolution and revolution in the course of which a given order of things is completed, as, for instance, by the sun, the moon, and the planets returning to a state of conjunction from which they have started.

page 113 note 1 Brāhma-Sphuṭa-Siddhānta, ed. Sudhakara Dvivedi, p. 407, verses 7, 8.

page 113 note 2 Thus, Varāhamihira (died a.d. 587) says in his Pañchasiddhāntikā, 15. 20, that Āryabhaṭa taught in one place that the day begins at midnight, and in another place that it begins at sunrise: but only the latter doctrine is found in the Āryabhaṭīya. Again, Brahmagupta (wrote a.d. 628) says in his Siddhānta, ed. cit., p. 148, verse 5, that Āryabhaṭa laid down in one place a number of civil days in his exeligmos which exceeded by three hundred the number taught by him elsewhere: but no such two statements are found in the Āryabhaṭīya.

page 113 note 3 Or, as the name occurs thus only in verse, should we rather say “Paramēśvara”?: especially since the colophons style the commentary “Pāramēśvarikā Bhaṭadīpikā”.

page 114 note 1 Op. cit., p. 149, verse 8.

page 114 note 2 The words are:—Āryabhaṭīyaṁ nāmnā pūrvaṁ Svāyaṁbhuvaṁ sadā sad=yat; “the foregoing (work) by name Āryabhaṭīya, which is derived from the Self-existent (Brahman) (and) is always good.” They are in the nomin. sing, neuter: and we are left to supply sūtram, tantram, or any other suitable word.

page 114 note 3 See his remarks under that verse and under Gaṇitapāda, verse 1; and the third of his introductory verses to his commentary.

page 115 note 1 See, e. g., the verse referred to in note 1 on p. 114 above.

page 116 note 1 It seems more conformable to general ideas to use in the sequel the term ‘place’ rather than ‘space’: and Āryabhaṭa himself in some words quoted farther on above has substituted sthāna for the kha which is used here. But the proper literal translation of kha in this verse seems to be ‘space’: the word is used in that sense in the Kālakriyāpāda, verse 15, where the earth is described as kha-madhya-sthā, “situated in the middle of space”; and in the Daśagītikasūtra, verse 4, which, with a view to deducing the orbits and distances of the planets and the nakshatras, teaches the measure of the circumference of kha in the sense of space, the visible universe, figured (according to the commentary) as the central section of the brahmāṇḍa or cosmic egg.

page 116 note 2 The highest such number is that of the rotations of the earth on its axis in his exeligmos; namely, 1,582,237,500 (verse 1). Verse 4 teaches the number of yōjanas in the circumference of space, which runs to fourteen places: but it does not state the number; it only shows how it is to be arrived at.

page 117 note 1 Līlāvatī, ed. Sudhakara Dvivedi, p. 2: a precisely similar list, except in substituting the synonymous mahāmbuja for mahāpadma, No. 13, and vārdhi for jaladhi, No. 15, is given by Hēmachandra in his Abhidhāna chintāmaṇi, verses 873, 874. I should have preferred to use some older list, giving all the eighteen names and at the same time agreeing exactly with Āryabhaṭa in respect of the first ten: but I have not been able to find any such.

As regards the time to which this scheme of numbers, or its embryo, can be traced back, it may be observed that the Śatapatha-Brāhmaṇa, 9. 1. 2. 16, 17, mentions two high quantities called by it anta and parārdha: but they are not necessarily the antya and parārdha of the list given above.

page 120 note 1 As a matter of fact, however, we find that the only long vowel actually used in the Daśagītikasūtra is ā; ten times, in verses 3, 5, 7, 9, 10. Combinations of two consonants, etc., are frequent: but the only combinations of three letters are hlya in verse 7, and chsga as a various reading in verse 10.

page 120 note 2 Compare ashṭam - āntya, ‘the ninth; immediately following the eighth’: see the St. Petersburg Dictionary, under antya. The commentary explains antya by ūrdhva-gata, ‘gone above, higher’.

page 120 note 3 Trans. Sachau, vol. 1, p. 175.

page 121 note 1 Or, by the means suggested by the commentator, the number thus arrived at might be expressed by kaṁ.

page 121 note 2 Is it possible that the vā at the end of the verse is a corrupt reading for hau? In that case we might translate:—“hau (stands) in the square at the end of the nine (pairs of spaces).”

page 122 note 1 Daśagītikasūtra, verse 1. In the third syllable the published text has shu, by a misprint for bu: the mistake is shown by examination of details, as well as by the commentary.

page 122 note 2 See Ind. Ant., vol. 17, pp. 36–8.

page 123 note 1 The rule in question is:—Aṅkānāṁ vāmatō gatiḥ. More may be said about it on some other occasion.

page 123 note 2 The svaki of the published text and commentary is either a misprint or a corrupt reading.

page 123 note 3 The metre is faulty here: it is set right by the various reading of another commentary; dhaha hach=sga instead of dhāhā sta sga.

page 124 note 1 I mean in a direct manner, as in cases given above; not to the indefinite extent to which it becomes possible when it is found convenient to break totals up into somewhat unusual components, illustrated by the following instances: in verse 10, for 106 we have sta = 90 + 16, instead of chaki = 6 + 100 or kicha or hcha both = 100 + 6; and for 37 we have pta = 21 + 16, instead of chhya = 7 + 30 or ychha = 30 + 7.