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The Hindu-Arabic Numerals c7: Chapter III. Later Hindu Forms, with a Place Value

The Hindu-Arabic Numerals c7

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Chapter III. Later Hindu Forms, with a Place Value

c7.2

Before speaking of the perfected Hindu numerals with the zero and the place value, it is necessary to consider the third system mentioned on page 19, — the word and letter forms. The use of words with place value began at least as early as the 6th century of the Christian era. In many of the manuals of astronomy and mathematics, and often in other works in mentioning dates, numbers are represented by the names of certain objects or ideas. For example, zero is represented by "the void" (*śūnya*), or "heaven-space" (*ambara ākāśa*); one by "stick" (*rupa*), "moon" (*indu śaśin*), "earth" (*bhū*), "beginning" (*ādi*), "Brahma," or, in general, by anything markedly unique; two by "the twins" (*yama*), "hands" (*kara*), "eyes" (*nayana*), etc.; four by "oceans," five by "senses" (*viṣaya*) or "arrows" (the five arrows of Kāmadēva); six by "seasons" or "flavors"; seven by "mountain" (*aga*), and so on.¹ These names, accommodating themselves to the verse in which scientific works were written, had the additional advantage of not admitting, as did the figures, easy alteration, since any change would tend to disturb the meter.

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¹ Bühler, loc. cit., p. 80; J. F. Fleet, *Corpus inscriptionum Indicarum*, Vol. III, Calcutta, 1888. Lists of such words are given also by Al-Bīrūnī in his work *India*; by Burnell, loc. cit.; by E. Jacquet, "Mode d'expression symbolique des nombres employé par les Indiens, les Tibétains et les Javanais," *Journal Asiatique*, Vol. XVI, Paris, 1835.

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As an example of this system, the date "Śaka Saṃvat, 867" (A.D. 945 or 946), is given by "*giri-rasa-vasu*," meaning "the mountains" (seven), "the flavors" (six), and the gods "*Vasu*" of which there were eight. In reading the date these are read from right to left.¹ The period of invention of this system is uncertain. The first trace seems to be in the *Śrautasūtra* of Kātyāyana and Lāṭyāyana.² It was certainly known to Varāha-Mihira (d. 587),³ for he used it in the *Bṛhat-Saṃhitā*.⁴ It has also been asserted⁵ that Āryabhaṭa (c. 500 A.D.) was familiar with this system, but there is nothing to prove the statement.⁶ The earliest epigraphical examples of the system are found in the Bayang (Cambodia) inscriptions of 604 and 624 A.D.⁷

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¹ This date is given by Fleet, loc. cit., Vol. III, p. 73, as the earliest epigraphical instance of this usage in India proper.

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² Weber, *Indische Studien*, Vol. VIII, p. 166 seq.

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³ *Journal of the Royal Asiatic Society*, Vol. I (N.S.), p. 407.

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⁴ VIII, 20, 21.

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⁵ Th. H. Martin, *Les signes numéraux . . .*, Rome, 1864; Lassen, *Indische Alterthumskunde*, Vol. II, 2d ed., Leipzig and London, 1874, p. 1153.

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⁶ But see Burnell, loc. cit., and Thibaut, *Astronomie, Astrologie und Mathematik*, p. 71.

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⁷ A. Barth, "Inscriptions Sanscrites du Cambodge," in the *Notices et extraits des Mss. de la Bibliothèque nationale*, Vol. XXVII, Part I, pp. 1–180, 1885; see also numerous articles in *Journal Asiatique*, by Aymonier.

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Mention should also be made, in this connection, of a curious system of alphabetic numerals that sprang up in southern India. In this we have the numerals represented by the letters as given in the following table:

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| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 0 |
| k | kh | g | gh | ṅ | c | ch | j | jh | ñ |
| ṭ | ṭh | ḍ | ḍh | ṇ | t | th | d | dh | n |
| p | ph | b | bh | m | | | | | |
| y | r | l | v | ś | ṣ | s | h | ḷ | |

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By this plan a numeral might be represented by any one of several letters, as shown in the preceding table, and thus it could the more easily be formed into a word for mnemonic purposes. For example, the word

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{ccccccc} ² & ³ & ¹ & ⁵ & ⁶ & ⁵ & ¹ / kha & gont & yan & me & ṣa & mā & pa

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has the value 1,565,132, reading from right to left.¹ This, the oldest specimen (1184 A.D.) known of this notation, is given in a commentary on the Rigveda, representing the number of days that had elapsed from the beginning of the Kaliyuga. Burnell² states that this system is even yet in use for remembering rules to calculate horoscopes, and for astronomical tables.

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¹ Bühler, loc. cit., p. 82.

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² Loc. cit., p. 79.

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A second system of this kind is still used in the pagination of manuscripts in Ceylon, Siam, and Burma, having also had its rise in southern India. In this the thirty-four consonants when followed by *a* (as *ka* ... *la*) designate the numbers 1–34; by *ā* (as *kā* ... *lā*), those from 35 to 68; by *i* (*ki* ... *li*), those from 69 to 102, inclusive; and so on.³

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³ Bühler, loc. cit., p. 83. The Hindu astrologers still use an alphabetical system of numerals. [Burnell, loc. cit., p. 79.]

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As already stated, however, the Hindu system as thus far described was no improvement upon many others of the ancients, such as those used by the Greeks and the Hebrews. Having no zero, it was impracticable to designate the tens, hundreds, and other units of higher order by the same symbols used for the units from one to nine. In other words, there was no possibility of place value without some further improvement. So the Nānā Ghāt symbols required the writing of “thousand seven twenty-four” about like T 7, tw, 4 in modern symbols, instead of 7024, in which the seven of the thousands, the two of the tens (concealed in the word twenty, being originally “twain of tens,” the *-ty* signifying ten), and the four of the units are given as spoken and the order of the unit (tens, hundreds, etc.) is given by the place. To complete the system only the zero was needed; but it was probably eight centuries after the Nānā Ghāt inscriptions were cut, before this important symbol appeared; and not until a considerably later period did it become well known. Who it was to whom the invention is due, or where he lived, or even in what century, will probably always remain a mystery.¹ It is possible that one of the forms of ancient abacus suggested to some Hindu astronomer or mathematician the use of a symbol to stand for the vacant line when the counters were removed. It is well established that in different parts of India the names of the higher powers took different forms, even the order being interchanged.² Nevertheless, as the significance of the name of the unit was given by the order in reading, these variations did not lead to error. Indeed the variation itself may have necessitated the introduction of a word to signify a vacant place or lacking unit, with the ultimate introduction of a zero symbol for this word.

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¹ Well could Ramus say, “Quicunq; autem fuerit inventor decem notarum laudem magnam meruit.”

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² Al-Bīrūnī gives lists.

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To enable us to appreciate the force of this argument a large number, 8,443,682,155, may be considered as the Hindus wrote and read it, and then, by way of contrast, as the Greeks and Arabs would have read it.

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*Modern American reading*, 8 billion, 443 million, 682 thousand, 155.

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*Hindu*, 8 padmas, 4 vyarbudas, 4 kōṭis, 3 prayutas, 6 lakṣas, 8 ayutas, 2 sahasra, 1 śata, 5 daśan, 5.

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*Arabic and early German*, eight thousand thousand thousand and four hundred thousand thousand and forty-three thousand thousand, and six hundred thousand and eighty-two thousand and one hundred fifty-five (or five and fifty).

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*Greek*, eighty-four myriads of myriads and four thousand three hundred sixty-eight myriads and two thousand and one hundred fifty-five.

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As Woepcke¹ pointed out, the reading of numbers of this kind shows that the notation adopted by the Hindus tended to bring out the place idea. No other language than the Sanskrit has made such consistent application, in numeration, of the decimal system of numbers. The introduction of myriads as in the Greek, and thousands as in Arabic and in modern numeration, is really a step away from a decimal scheme. So in the numbers below one hundred, in English, eleven and twelve are out of harmony with the rest of the -teens, while the naming of all the numbers between ten and twenty is not analogous to the naming of the numbers above twenty. To conform to our written system we should have ten-one, ten-two, ten-three, and so on, as we have twenty-one, twenty-two, and the like. The Sanskrit is consistent, the units, however, preceding the tens and hundreds. Nor did any. other ancient people carry the numeration as far as did the Hindus.²

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¹ *Propagation*, loc. cit., p. 443.

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² See the quotation from *The Light of Asia* in Chapter II, p. 16.

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When the *aṅkapallī*,¹ the decimal-place system of writing numbers, was perfected, the tenth symbol was called the *śūnyabindu*, generally shortened to *śūnya* (the void). Brockhaus² has well said that if there was any invention for which the Hindus, by all their philosophy and religion, were well fitted, it was the invention of a symbol for zero. This making of nothingness the crux of a tremendous achievement was a step in complete harmony with the genius of the Hindu.

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¹ The nine ciphers were called *aṅka*.

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² "Zur Geschichte des indischen Ziffernsystems," *Zeitschrift für die Kunde des Morgenlandes*, Vol. IV, 1842, pp. 74–83.

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It is generally thought that this *śūnya* as a symbol was not used before about 500 a.d., although some writers have placed it earlier.³ Since Āryabhaṭa gives our common method of extracting roots, it would seem that he may have known a decimal notation,⁴ although he did not use the characters from which our numerals are derived.⁵ Moreover, he frequently speaks of the void.¹ If he refers to a symbol this would put the zero as far back as 500 A.D., but of course he may have referred merely to the concept of nothingness.

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³ It is found in the Bakhṣālī MS. of an elementary arithmetic which Hoernle placed, at first, about the beginning of our era, but the date is much in question. G. Thibaut, loc. cit., places it between 700 and 900 a.d.; Cantor places the body of the work about the third or fourth century a.d., *Geschichte der Mathematik*, Vol. I (3), p. 598.

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⁴ For the opposite side of the case see G. R. Kaye, "Notes on Indian Mathematics, No. 2.—Āryabhata," *Journ. and Proc. of the Asiatic Soc. of Bengal*, Vol. IV, 1908, pp. 111–141.

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⁵ He used one of the alphabetic systems explained above. This ran up to 10¹⁸ and was not difficult, beginning as follows: [Figure:] the same letter (*ka*) appearing in the successive consonant forms, *ka*, *kha*, *ga*, *gha*, etc. See C. I. Gerhardt, *Über die Entstehung und Ausbreitung des dekadischen Zahlensystems*, Programm, p. 17, Salzwedel, 1853, and *Études historiques sur l'arithmétique de position*, Programm, p. 24, Berlin, 1856; E. Jacquet, *Mode d'expression symbolique des nombres*, loc. cit., p. 97; L. Rodet, “Sur la véritable signification de la notation numérique inventée par Āryabhata,” *Journal Asiatique*, Vol. XVI (7), pp. 440–485. On the two Āryabhaṭas see Kaye, *Bibl. Math.*, Vol. X (3), p. 289.

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¹ Using *kha*, a synonym of *śūnya*. [Bayley, loc. cit., p. 22, and L. Rodet, *Journal Asiatique*, Vol. XVI (7), p. 443.]

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A little later, but also in the sixth century, Varāha-Mihira² wrote a work entitled *Bṛhat Saṃhitā*³ in which he frequently uses *śūnya* in speaking of numerals, so that it has been thought that he was referring to a definite symbol. This, of course, would add to the probability that Āryabhaṭa was doing the same.

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² Varāha-Mihira, *Pañcasiddhāntikā*, translated by G. Thibaut and M. S. Dvivedī, Benares, 1889; see Bühler, loc. cit., p. 78; Bayley, loc. cit., p. 23.

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³ *Bṛhat Saṃhitā*, translated by Kern, *Journal of the Royal Asiatic Society*, 1870–1875.

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It should also be mentioned as a matter of interest, and somewhat related to the question at issue, that Varāha-Mihira used the word-system with place value⁴ as explained above.

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⁴ It is stated by Bühler in a personal letter to Bayley (loc. cit., p. 65) that there are hundreds of instances of this usage in the *Bṛhat Saṃhitā*. The system was also used in the *Pañcasiddhāntikā* as early as 505 A.D. [Bühler, *Palaeographie*, p. 80, and Fleet, *Journal of the Royal Asiatic Society*, 1910, p. 819.]

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The first kind of alphabetic numerals and also the word-system (in both of which the place value is used) are plays upon, or variations of, position arithmetic, which would be most likely to occur in the country of its origin.⁵

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⁵ Cantor, *Geschichte der Mathematik*, Vol. I (3), p. 608.

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At the opening of the next century (c. 620 A.D.) Bāṇa⁶ wrote of Subandhus’s *Vāsavadattā* as a celebrated work, and mentioned that the stars dotting the sky are here compared with zeros, these being points as in the modern Arabic system. On the other hand, a strong argument against any Hindu knowledge of the symbol zero at this time is the fact that about 700 a.d. the Arabs overran the province of Sind and thus had an opportunity of knowing the common methods used there for writing numbers. And yet, when they received the complete system in 776 they looked upon it as something new.¹ Such evidence is not conclusive, but it tends to show that the complete system was probably not in common use in India at the beginning of the eighth century. On the other hand, we must bear in mind the fact that a traveler in Germany in the year 1700 would probably have heard or seen nothing of decimal fractions, although these were perfected a century before that date. The élite of the mathematicians may have known the zero even in Āryabhaṭa's time, while the merchants and the common people may not have grasped the significance of the novelty until a long time after. On the whole, the evidence seems to point to the west coast of India as the region where the complete system was first seen.² As mentioned above, traces of the numeral words with place value, which do not, however, absolutely require a decimal place-system of symbols, are found very early in Cambodia, as well as in India.

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⁶ Bühler, loc. cit., p. 78.

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¹ Bayley, p. 38.

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² Noviomagus, in his *De numeris libri duo*, Paris, 1539, confesses his ignorance as to the origin of the zero, but says: "D. Henricus Grauius, vir Graecè & Hebraicè eximè doctus, Hebraicam originem ostendit," adding that Valla "Indis Orientalibus gentibus inventionem tribuit."

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Concerning the earliest epigraphical instances of the use of the nine symbols, plus the zero, with place value, there is some question. Colebrooke¹ in 1807 warned against the possibility of forgery in many of the ancient copper-plate land grants. On this account Fleet, in the *Indian Antiquary*,² discusses at length this phase of the work of the epigraphists in India, holding that many of these forgeries were made about the end of the eleventh century. Colebrooke³ takes a more rational view of these forgeries than does Kaye, who seems to hold that they tend to invalidate the whole Indian hypothesis. "But even where that may be suspected, the historical uses of a monument fabricated so much nearer to the times to which it assumes to belong, will not be entirely superseded. The necessity of rendering the forged grant credible would compel a fabricator to adhere to history, and conform to established notions: and the tradition, which prevailed in his time, and by which he must be guided, would probably be so much nearer to the truth, as it was less remote from the period which it concerned."⁴ Bühler⁵ gives the copper-plate Gurjara inscription of Cedi-saṃvat 346 (595 A.D.) as the oldest epigraphical use of the numerals⁶ "in which the symbols correspond to the alphabet numerals of the period and the place." Vincent A. Smith⁷ quotes a stone inscription of 815 A.D., dated Saṃvat 872. So F. Kielhorn in the *Epigraphia Indica*⁸ gives a Pathari pillar inscription of Parabala, dated Vikrama-saṃvat 917, which corresponds to 861 A.D., and refers also to another copper-plate inscription dated Vikrama-saṃvat 813 (756 a.d.). The inscription quoted by V. A. Smith above is that given by D. R. Bhandarkar,¹ and another is given by the same writer as of date Saka-saṃvat 715 (798 a.d.), being incised on a pilaster. Kielhorn² also gives two copper-plate inscriptions of the time of Mahendrapala of Kanauj, Valhabī-saṃvat 574 (893 a.d.) and Vikrama-saṃvat 956 (899 a.d.). That there should be any inscriptions of date as early even as 750 a.d., would tend to show that the system was at least a century older. As will be shown in the further development, it was more than two centuries after the introduction of the numerals into Europe that they appeared there upon coins and inscriptions. While Thibaut³ does not consider it necessary to quote any specific instances of the use of the numerals, he states that traces are found from 590 a.d. on. "That the system now in use by all civilized nations is of Hindu origin cannot be doubted; no other nation has any claim upon its discovery, especially since the references to the origin of the system which are found in the nations of western Asia point unanimously towards India."⁴

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¹ See *Essays*, Vol. II, pp. 287 and 288.

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² Vol. XXX, p. 205 seqq.

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³ Loc. cit., p. 284 seqq.

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⁴ Colebrooke, loc. cit., p. 288.

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⁵ Loc. cit., p. 78.

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⁶ Hereafter, unless expressly stated to the contrary, we shall use the word "numerals" to mean numerals with place value.

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⁷ "The Gurjaras of Rājputāna and Kanauj," in *Journal of the Royal Asiatic Society*, January and April, 1909.

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⁸ Vol. IX, 1908, p. 248.

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¹ *Epigraphia Indica*, Vol. IX, pp. 193 and 198.

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² *Epigraphia Indica*, Vol. IX, p. 1.

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³ Loc. cit., p. 71.

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⁴ Thibaut, p. 71.

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The testimony and opinions of men like Bühler, Kielhorn, V. A. Smith, Bhandarkar, and Thibaut are entitled to the most serious consideration. As authorities on ancient Indian epigraphy no others rank higher. Their work is accepted by Indian scholars the world over, and their united judgment as to the rise of the system with a place value — that it took place in India as early as the sixth century a.d. — must stand unless new evidence of great weight can be submitted to the contrary.

c7.65

Many early writers remarked upon the diversity of Indian numeral forms. Al-Bīrūnī was probably the first; noteworthy is also Johannes Hispalensis,¹ who gives the variant forms for seven and four. We insert on p. 49 a table of numerals used with place value. While the chief authority for this is Bühler,² several specimens are given which are not found in his work and which are of unusual interest.

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¹ "Est autem in aliquibus figurarum istarum apud multos diuersitas. Quidam enim septimam hanc figuram representant," etc. [Boncompagni, *Trattati*, p. 28.] Eneström has shown that very likely this work is incorrectly attributed to Johannes Hispalensis. [*Bibliotheca Mathematica*, Vol. IX (3), p. 2.]

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² *Indische Palaeographie*, Tafel IX.

c7.68

The Śāradā forms given in the table use the circle as a symbol for 1 and the dot for zero. They are taken from the paging and text of *The Kashmirian Atharva-Veda*,³ of which the manuscript used is certainly four hundred years old. Similar forms are found in a manuscript belonging to the University of Tübingen. Two other series presented are from Tibetan books in the library of one of the authors.

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³ Edited by Bloomfield and Garbe, Baltimore, 1901, containing photographic reproductions of the manuscript.

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For purposes of comparison the modern Sanskrit and Arabic numeral forms are added.

c7.72

a Bakhṣālī MS. See page 43; Hoernle, R., *The Indian Antiquary*, Vol. XVII, pp. 33–48, 1 plate; Hoernle, *Verhandlungen des VII. Internationalen Orientalisten-Congresses, Arische Section*, Vienna, 1888, “On the Bakshālī Manuscript,” pp. 127–147, 3 plates; Bühler, loc. cit.

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b 3, 4, 6, from H. H. Dhruva, “Three Land-Grants from Sankheda,” *Epigraphia Indica*, Vol. II, pp. 19–24 with plates; date 595 A.D. 7, 1, 5,

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NUMERALS USED WITH PLACE VALUE

c7.76

from Bhandarkar, "Daulatabad Plates," *Epigraphia Indica*, Vol. IX, part V; date c. 798 a.d.

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c 8, 7, 2, from "Buckhala Inscription of Nagabhatta," Bhandarkar, *Epigraphia Indica*, Vol. IX, part V; date 815 a.d. 5 from "The Morbi Copper-Plate," Bhandarkar, *The Indian Antiquary*, Vol. II, pp. 257–258, with plate; date 804 a.d. See Bühler, *loc. cit.*

c7.78

d 8 from the above Morbi Copper-Plate. 4, 5, 7, 9, and 0, from "Asni Inscription of Mahīpala," *The Indian Antiquary*, Vol. XVI, pp. 174–175; inscription is on red sandstone, date 917 a.d. See Bühler.

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e 8, 9, 4, from "Rashtrakuta Grant of Amoghavarsha," J. F. Fleet, *The Indian Antiquary*, Vol. XII, pp. 263–272; copper-plate grant of date c. 972 a.d. See Bühler. 7, 3, 5, from "Torkhede Copper-Plate Grant of the Time of Govindaraja of Gujerat," Fleet, *Epigraphia Indica*, Vol. III, pp. 53–58. See Bühler.

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f From "A Copper-Plate Grant of King Tritochanapâla Chaulukya of Lâṭadeśa," H. H. Dhruva, *Indian Antiquary*, Vol. XII, pp. 196–205; date 1050 a.d. See Bühler.

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g Burnell, A. C., *South Indian Palæography*, plate XXIII, Telugu-Canarese numerals of the eleventh century. See Bühler.

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h and i From a manuscript of the second half of the thirteenth century, reproduced in "Della vita e delle opere di Leonardo Pisano," Baldassare Boncompagni, Rome, 1852, in *Atti dell’ Accademia Pontificia dei nuovi Lincei*, anno V.

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j and k From a fourteenth-century manuscript, as reproduced in *Della vita etc.*, Boncompagni, *loc. cit.*

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l From a Tibetan MS. in the library of D. E. Smith.

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m From a Tibetan block-book in the library of D. E. Smith.

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n *Śāradā* numerals from *The Kashmirian Atharva-Veda, reproduced by chromophotography from the manuscript in the University Library at Tübingen*, Bloomfield and Garbe, Baltimore, 1901. Somewhat similar forms are given under "Numération Cachemirienne," by Pihan, *Exposé etc.*, p. 84.

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