Pramāṇa · Search · About · API

Indian Epigraphy: A Guide to the Study of Inscriptions in Sanskrit, Prakrit, and the Other Indo-Aryan Languages c7.s4: 2.4 Numbers and Numerical Notation

Indian Epigraphy: A Guide to the Study of Inscriptions in Sanskrit, Prakrit, and the Other Indo-Aryan Languages c7.s4

Language
English

Contents

2.4 Numbers and Numerical Notation

c7.s4.1

BIP 76–87; IC II.683, 702–9; IIEP 125–8; BPLM 103–29; S. L. Gokhale, *Indian Numerals*; SAI 168–76.

2.4.1 Numerical notation in Brāhmī and the derived scripts

c7.s4.t1.1

2.4.1.1 The old additive/multiplicative system

c7.s4.t1.2

Numerical figures are found in Indic inscriptions from the earliest times, especially in the recording of dates but also in the details of financial or other transactions, the numbering of verses in metrical inscriptions (see 2.5.2.3), and in various other connections. Numbers in all periods have also been commonly expressed in words, either directly or (in later centuries) by means of chronograms and other indirect expressions (see 5.4.2.1.2).

c7.s4.t1.3

The original Brāhmī numerical system (see table 2.6), as first found in the Aśokan and other early inscriptions, is essentially additive/multiplicative in principle. It had a full array of separate symbols, not only for the digits from 1 to 9 but also for the decades from 10 to 90, as well as separate signs for 100 and 1,000; the system thus comprised an array of twenty basic signs.¹⁷⁹ The figures for 100 and 1,000 were treated multiplicatively; for example, 400 would be represented as a ligature of 𑀥 4 and 𑀰 100, that is, 𑀰𑀥. But multiples of 2 and 3 were indicated by the addition of one or two horizontal ticks, respectively, at the right of the sign for 100 or 1,000; thus 𑀰- = 200 and 𑀰= = 300.

c7.s4.t1.4

¹⁷⁹ On the notation of fractions in the old system, see Dines Chandra Sircar, “Fractions in an Early Inscription,” *Journal of the University of Gauhati* 1, 1950, 133–6.

c7.s4.t1.5

2.4.1.2 Origin and history of the old Brāhmī numerical system

c7.s4.t1.6

Early specimens of Brāhmī numerals are found in the Aśokan inscriptions (only 4, 6, 50, and 200), in the cave inscriptions of the first century B.C. from Nānāghāṭ (SI I.192–7) and of the first century A.D. from Nāsik (EI 8, 59–96), and in the inscriptions of the Kuṣāṇas. A complete set of the figures is first found in the unique series of dated coins issued by the Western Kṣatrapas in the second to fourth centuries A.D., which provided important clues for the decipherment of the numbers. Besides the limited body of early materials, the analysis of the old numerical system is further complicated by the fact that the symbols were subject from the earliest times to considerable graphic variability, which increased over the centuries and eventually resulted in the notoriously diverse character of the numerical symbols in the different Brāhmī-derived scripts. Fortunately, in many inscriptions, for example at Nāsik, numbers are expressed in both words and numerical characters, and these instances, together with valuable hints furnished by the dates of the Kṣatrapa coins, were instrumental in the decipherment of the Brāhmī numerals.¹⁸⁰

c7.s4.t1.7

¹⁸⁰ See section 6.2.3 for details.

c7.s4.t1.8

Although the system itself is nowadays fully understood, the historical origins of the numerical characters remain controversial. Several theories on this question are based on their apparent resemblance to certain letters of Brāhmī and related scripts. The first of these was Prinsep’s hypothesis¹⁸¹ that the number signs are derived from the characters for the initial letter of the Sanskrit word for each number. This theory, though accepted by F. Woepcke,¹⁸² is obviously unsatisfactory. In 1877, Bhagvānlāl Indrajī¹⁸³ declared that “all of [the numerals] … except the first three express letters or groups of letters”; thus the sign for 6 is interpreted as *phra* or *phrā*, 7 is *gra* or *grā*, and so on. In his postscript to Bhagvānlāl’s article (47–8; see also BIP 77–81), Bühler accepted the theory, though he admitted that the actual “origin” of these numerals remained obscure, since no rationale could be discerned for the particular phonetic values attributed to the numeral signs.¹⁸⁴

c7.s4.t1.9

¹⁸¹ JASB 7, 1838, 352–3 = *Essays on Indian Antiquities* II.77.

c7.s4.t1.10

¹⁸² “Mémoire sur la propagation des chiffres indiens,” *JA*, ser. 6, vol. 1, 1863, 27–79, 234–90, and 442–529 (esp. 70–3).

c7.s4.t1.11

¹⁸³ “On the Ancient Nāgarī Numerals,” IA 6, 42–8 (quoted 43).

c7.s4.t1.12

¹⁸⁴ As will shortly be shown, Bühler did not feel this theory to be incompatible with his belief in an ultimate Egyptian origin of the Brāhmī numerals, though he did not completely clarify his ideas as to the relationship between the two factors.

c7.s4.t1.13

TABLE 2.6. Numerical Notation in Brāhmī and Kharoṣṭhī

c7.s4.t1.14

| Value | Brāhmī | Kharoṣṭhī |
| 1 | 𑁒 | 𐩀 |
| 2 | 𑁓 | 𐩁 , 𐩂 |
| 3 | 𑁔 | 𐩃 , 𐩂𐩀 |
| 4 | 𑁕 | 𐩄 |
| 5 | 𑁖 | 𐩀𐩄 |
| 6 | 𑁗 | 𐩁𐩄 |
| 7 | 𑁘 | 𐩃𐩄 |
| 8 | 𑁙 | 𐩄𐩄 |
| 9 | 𑁚 | 𐩀𐩄𐩄 |
| 10 | 𑁛 | 𐩅 |
| 20 | 𑁜 | 𐩆 |
| 30 | 𑁝 | 𐩅𐩆 |
| 40 | 𑁞 | 𐩆𐩆 |
| 50 | 𑁟 | 𐩅𐩆𐩆 |
| 60 | 𑁠 | 𐩆𐩆𐩆 |
| 70 | 𑁡 | 𐩅𐩆𐩆𐩆 |
| 80 | 𑁢 | 𐩆𐩆𐩆𐩆 |
| 90 | 𑁣 | 𐩅𐩆𐩆𐩆𐩆 |
| 100 | 𑁤 | 𐩇 , 𐩇𐩂 |
| 200 | 𑁥 | 𐩇𐩁 |
| 300 | 𑁦 | 𐩇𐩃 |
| 1,000 | 𑁧 | 𐩈 |
| 2,000 | 𑁨 | (not attested) |
| 4,000 | 𑁪 | (not attested) |

c7.s4.t1.15

Representative Brāhmī forms are shown, mostly from Western Kṣatrapa coins; in practice, number signs in Brāhmī show much variation. The Kharoṣṭhī forms are from various inscriptions, with alternate forms from central Asian documents.

c7.s4.t1.16

For this and other reasons¹⁸⁵ the idea that the numerical symbols are actually Brāhmī letters is not widely accepted.¹⁸⁶ While it is true that some of the numerical symbols in Brāhmī and several connections between them involves considerable manipulation. In a few cases, notably the signs for 80 and 90, Bühler admits, "There is no actual resemblance between the Egyptian and Indian forms" (118). Another problem with the Egyptian derivation is historical, since, as Bühler admits (OIBA 119), the evidence for contact between India and Egypt in very early times is weak (Bayley's attempts [op. cit., 361–3] to establish such contacts notwithstanding), and in the absence of such corroboration the derivation can hardly be considered authoritative.

c7.s4.t1.17

Recently, Falk (SAI 175–6) has proposed a possible influence from the early Chinese system of numerical notation, but the similarities are in fact less striking than those with Egyptian, and the historical and geographical arguments are even less cogent.

c7.s4.t1.18

Thus some authors, unconvinced by the arguments for borrowing, hold out for an indigenous origin of the Brāhmī numbers. Ojhā, for example (BPLM 114), thinks that they are "independent creations of the Indo-Aryans" (*bhāratīya āryoṃ ke svatantra nirmāṇa*). Sircar, too (IIEP 126), says that they "appear to have developed out of certain signs of the pre-historic writing of India"; but neither author offers any direct proof for the presumed indigenous origin. A. H. Dani¹⁹² has recently presented a scheme for an internal derivation of the Brāhmī numerical symbols from a (hypothetical) basic sign for 10, but this purely formal hypothesis is not compelling.

c7.s4.t1.19

¹⁹² Preface to the second edition (1986) of DIP, ix–xvii.

c7.s4.t1.20

In conclusion, the problem of the origin of the Brāhmī numerals is roughly analogous to that of the Brāhmī script itself. We have, on the one hand, a proposal by Bühler et al. for a foreign origin, which is convincing to some extent but falls considerably short of real proof, on both formal and historical grounds;¹⁹³ and on the other hand, a camp which rejects Bühler's and other theories of borrowing and holds out for an indigenous origin, but without concrete evidence. However, it must be noted that the development of numerical notation figures does not necessarily follow the same principles as linguistic notation, nor is it necessarily linked to it. For whereas phonetic signs are normally completely arbitrary in form, numerical signs can often be derived as cursive reductions of collocations of counting strokes, as in the case of the figures for 1, 2, and 3, which resemble each other in many systems around the world, and, more significantly, in the hieratic Egyptian figures for the decades which are clearly derived from combinatory groups of the old hieroglyphic sign for 10. Thus, it is possible for independently derived systems of numerical notation to independently develop similar principles and even similar forms, so that it may well be that the old Indian numerical system was entirely indigenous in origin despite its apparent resemblance to the Egyptian system. This numerical system could also have been in origin entirely separate from, and perhaps older than, the Brāhmī script with which it came to be associated,¹⁹⁴ and this would explain the persistent failure of all attempts to interpret the numerical signs in terms of phonetic values derived from apparent resemblances to characters of Brāhmī or other scripts.

c7.s4.t1.21

¹⁹³ It should be kept in mind that the formal derivation of numeral signs does not necessarily follow the same principles as those which apply to the derivation of phonetic characters. For while the latter are (effectively) purely arbitrary symbols, the former are often derivable, at least as far as the lower numbers are concerned, from combinations of simple counting strokes; so that the numbers from 1 to 3 or 4, and sometimes higher ones as well, may often resemble each other in widely separated and unrelated systems. For this reason, purely formal similarities between numerical symbols are less significant than those of phonetic signs.

c7.s4.t1.22

¹⁹⁴ Cf. G. R. Kaye, "The Old Indian Numerical Symbols," IA 40, 1911, 49–55 (esp. 50, 54).

c7.s4.t1.23

2.4.1.3 The development of place-value notation

c7.s4.t1.24

Despite outward changes in form, the old additive system of Brāhmī numerals remained stable until about the seventh century A.D., at which point the modern system of decimal place-value notation, which can represent any number of any size with only the nine digit symbols plus a sign for zero, began to come into use. There are, however, several problems involved in determining exactly what is the earliest epigraphic attestation of the new, or “place-value,”¹⁹⁵ system. First of all, there is the problem of spurious inscriptions; several inscriptions which were previously thought to provide early specimens of place-value numbers have proven to be spurious or at least of doubtful authenticity.¹⁹⁶ Most important among these is the Māṅkaṇī (formerly called Saṅkheḍā) copper plate inscription (CII 4.1, 161–5), bearing in place-value characters the date 346, which is presumably attributable to the Kalacuri era and hence equivalent to A.D. 594/95 (if a current year) or A.D. 595/96 (if expired). Although this inscription is widely cited both in specialized publications (e.g., BIP 83 and IIEP 127) and in general works on the history of mathematics¹⁹⁷ as providing the earliest specimen of a place-value date, cogent historical and contextual arguments have been presented by V. V. Mirashi¹⁹⁸ to show that this is very likely a spurious record, and hence does not prove the use of the new system in the sixth century A.D.

c7.s4.t1.25

¹⁹⁵ In many publications on the subject of Indian numerals, the old and new styles are routinely referred to as the “numerical” and “decimal” systems, respectively. But this terminology is inaccurate, since in fact *both* systems are both numerical and decimal. I therefore prefer the terms “additive” and “place-value,” or simply “old” and “new.”

c7.s4.t1.26

¹⁹⁶ This problem is not merely coincidental; for it would only be expected that forgers of copper plate grants in the centuries following the transition from the old to the new notation system would predate the forged documents in the new style to which they were accustomed, thus giving rise to a body of documents which purport to give dates in new-style numbers before they were actually in use (see also 5.3).

c7.s4.t1.27

¹⁹⁷ E.g., Georges Ifrah, *From One to Zero: A Universal History of Numbers*, tr. Lowell Bair (New York: Viking, 1985), 437–40. Ifrah, in accepting the evidence of the Saṅkheḍā (i.e., Māṅkaṇī) plates, states that “[t]o the best of my knowledge, no serious reason for questioning the authenticity of the Indian copperplate deeds has ever been stated” (440). But the objections raised by Mirashi, as cited shortly, are indeed “serious” reasons for doubt.

c7.s4.t1.28

¹⁹⁸ CII 4.1, 161–3; see also “A Note on the Māṅkaṇi Grant of Taralasvāmin,” *Journal of the Ganganatha Jha Research Institute* 1, 1944, 389–94.

c7.s4.t1.29

Second, there is the problem of the great variability and consequent difficulty in interpreting the numerical figures. A case in point here is the Sakrāī stone inscription, dated, according to B. Ch. Chhabra (EI 27, 27–33), in [Vikrama] 699 = A.D. 643, which would make it the earliest new-system date (disregarding the Māṅkaṇī/Saṅkheḍā plates). But the reading of the date is controversial, and the first figure might be 8 rather than 6 (op. cit., 30), so that this too cannot be definitely considered the earliest place-value date.

c7.s4.t1.30

Actually, the earliest unquestionable inscriptional dates in the new system are not from India proper but from Southeast Asia. G. Coedès, in his important article “À propos de l'origine des chiffres arabes,”¹⁹⁹ pointed out several early place-value dates in the Śaka era on indubitably authentic Southeast Asian stone inscriptions, including two inscriptions from Indonesia and Cambodia dated in Śaka 605 = A.D. 683. Coedès thus concludes that this system therefore must have been in use by the late seventh century, and probably earlier, in Southeast Asia and presumably also in India.²⁰⁰

c7.s4.t1.31

¹⁹⁹ BSOAS 6, 1930–32, 323–8.

c7.s4.t1.32

²⁰⁰ Although both of the Southeast Asian inscriptions are in local languages, rather than Sanskrit, Coedès is no doubt correct in stating that the data do not justify the "étrange opinion" (324) of G. R. Kaye ("Notes on Indian Mathematics—Arithmetical Notation," *JASB*, n.s. 3, 1907, 475–508) that the symbolic number system originated in Southeast Asia and was imported thence into India, rather than vice versa. The early Sanskrit inscriptions of Southeast Asia generally give the dates in chronogram form, which presupposes the use of place-value numeration (see n. 203).

c7.s4.t1.33

Returning to India proper, the earliest definite specimen of pure place-value notation (as opposed to mixed or partial place-value notation, discussed later) now seems to be in the Siddhāntam plates of the Eastern Gaṅga king Devendravarman (EI 13, 212–6) dated (see Hultzsch, EI 18, 308) in the [Gaṅga] year 195, equivalent to approximately *A.D.* 693, or just ten years later than the Southeast Asian examples mentioned earlier. Another specimen from the same period is the Sudava plates (EI 26, 65–8) of Anantavarman, son of the aforementioned Devendravarman, dated [Gaṅga] 204 = *A.D.* 702. But undisputed cases of place-value notation continue to be somewhat scarce through the eighth century *A.D.*²⁰¹ Examples from the ninth century, such as the Tōrkhēḍē plates of [Śaka] 735 = *A.D.* 812 (EI 3, 53–8) and the Buckalā stone inscription of [Vikrama] 872 = *A.D.* 815 (EI 9, 198–200; see esp. 199 n. 1) are more secure.

c7.s4.t1.34

²⁰¹ The Dhiniki inscription of Vikrama 794 = *A.D.* 738 (IA 12, 151–6) is now generally agreed to be spurious, but the Sāmaṅgaḍ inscription (IA 11, 108–15) of the Rāṣṭrakūṭa king Dantidurga, dated Śaka 675 = *A.D.* 753/4, is a more likely instance, though its authenticity too has been called into question by V. S. Sukthankar in "Palaeographic Notes" (see n. 115), 313–7, mainly on paleographic grounds. The remarks of G. R. Kaye (*op. cit.*, 484) on this point are misleading, as he quotes Fleet out of context.

c7.s4.t1.35

After this time, that is, from about the middle of the ninth century onward, epigraphic notation of dates by the place-value system becomes standard. But the old system also continued to be used through the eighth century and sometimes even into the ninth, for instance, in the Barah copper plate of [Vikrama] 893 = *A.D.* 836 (EI 19, 15–19). An interesting illustration of the process of transition is provided by the Āhār stone inscription (EI 19, 52–62). This is a composite record of ten separate documents of different dates; the two earliest documents (nos. II and I), dated in [Harṣa] 258 and 259 = *A.D.* 864 and 865, respectively, use the old additive notation (cf. the editor's note, 58 n. 2), while all the later dates, from [Harṣa] 261 = *A.D.* 867 (no. IX) on, are in the new place-value system. This document thus allows us to precisely date the transition from old to new style in the area concerned (Bulandshahr Dist. in central northern India) to around *A.D.* 866.

c7.s4.t1.36

We also find during the transitional period in and around the seventh century not a few cases of a curious hybrid system which combines features of the old and new systems. Several cases of this have been found, interestingly enough, among the copper plate inscriptions of the Eastern Gaṅga kings in whose later records, as mentioned previously, were found the earliest definite cases of the place-value system. Thus, as early as ca. *A.D.* 578 we find in the Urlām plates of Hastivarman (EI 17, 330–4; corrected reading of the date in EI 18, 308) the [Gaṅga] year 80 written with the old-style sign for 80 plus the new-style sign for 0, and in the Purlē plates of Indravarman (EI 14, 360–3; see also EI 18, 308) the Gaṅga year 137 = *ca. A.D.* 635 written as 100-3-7 and the day 20 as 20-0.²⁰² It thus appears that the transition from the old additive system to the new place-value system, at least as far as epigraphic usage (which presumably mirrors popular usage) is concerned, was a gradual process. The displacement of the old style evidently took place slowly over a period of at least a century or so, and the new system itself underwent a process of gradual evolution in the seventh century, during which it was sometimes used in combination with features of the old style.²⁰³ This matter has broader implications for the important question of the origin of the zero sign and the now universal decimal place-value system of numerical notation, which, according to some scholars (see, e.g., Ifrah, op. cit. [n. 197], 428ff.) is ultimately of Indian origin; but the literature on this question is vast and goes far beyond the scope of the present book.

c7.s4.t1.37

²⁰² For further references and examples see E. Hultzsch, EI 18, 1925–26, 307–8; IIEP 127; G. S. Gai, "Two Epigraphic Notes," *Journal of the Ganganatha Jha Research Institute* 6, 1949, 306–7; Subrata Kumar Acharya, "The Transition from the Numerical to the Decimal System in the Inscriptions of Orissa," JESI 19, 1993, 52–62; and B. N. Mukherjee, "The Early Use of Decimal Notation in Indian Epigraphs," JESI 19, 1993, 80–3.

c7.s4.t1.38

²⁰³ Such is the picture to be derived from strictly epigraphic evidence; but if literary sources are taken into account, it would seem that the actual origins of the decimal place-value notation system must have been considerably earlier. Walter Eugene Clarke ("Hindu-Arabic Numerals," in *Indian Studies in Honor of Charles Rockwell Lanman* [Cambridge, Mass.: Harvard University Press, 1929], 217–36 [esp. 225–8]) noted indications from literary and scientific texts that place-value notation must have been in use well before the late seventh century; particularly noteworthy is the allusion to the *śūnyabindu*, or zero-sign, in Subandhu's *Vāsavadattā*, which must be earlier than the seventh century (Clarke, 225). Also significant is the attestation as early as Śaka 526 = A.D. 604 of the chronogram or numerical word system, which seems to presuppose the place-value system, in Sanskrit inscriptions from Southeast Asia (Clarke, 226; Coedès, op. cit. [n. 199], 325), though such chronograms are not attested until considerably later in India itself (Kaye, op. cit. [n. 200], 479–80; BIP 86).

c7.s4.t1.39

2.4.1.4 Later developments of the numerical notation system

c7.s4.t1.40

Once the new system came into universal use in or around the ninth century A.D., the subsequent history of numerical notation in the Brāhmī-derived scripts is essentially only a matter of the developments of outward forms of the symbols. These developments, however, are quite complex and not very well documented. Being a purely paleographic matter, they will not be presented here in detail; for further information the reader is referred to the standard works on paleography, especially BPLM, which presents this material in detail (103–27 and charts 71–76 and 84).²⁰⁴ The later phases of development of the numerals in the regional scripts in particular have not yet been adequately studied.

c7.s4.t1.41

²⁰⁴ Several of the standard paleographic studies treat numbers only briefly or not at all. Dani's DIP, e.g., did not cover the numerical symbols (cf. Gokhale, *Indian Numerals*, preface), although he has added some material on this subject in the preface to the second edition (see 2.4.1.2).

2.4.2 Numerical notation in Kharoṣṭhī

c7.s4.t2.1

The numerical notation of Kharoṣṭhī script (see table 2.6) follows an additive/multiplicative method essentially similar to that of the old Brāhmī system but considerably different from it in detail. Unlike the latter, which has separate signs for each of the units and decades, Kharoṣṭhī has symbols only for 1, 4 (a later development; see next paragraph), 10, 20, 100, and 1,000. All other numbers are indicated additively, somewhat as in Roman numerals; thus, for example, 76 would be written with the figures 20-20-20-10-4-1-1 (or in the earlier script, 20-20-20-10-1-1-1-1-1-1). The figures for 100 and 1,000 were subject to multiplicative treatment, as in Brāhmī; for example, 300 would be written as 3-100, and 2,000 as 2-1000.²⁰⁵

c7.s4.t2.2

²⁰⁵ On the decipherment of the Kharoṣṭhī numerical system, see section 6.3, p. 218.

c7.s4.t2.3

Like the Kharoṣṭhī script itself, its numerical system did not change greatly in the course of its history, though some significant developments did take place. The sign for 4, shaped like a Roman x, was apparently a secondary development, since in Aśokan Kharoṣṭhī 4 and 5 are written as 1-1-1-1(-1), while in all subsequent Kharoṣṭhī texts the special sign for 4 is used. The sign for 1,000 to date is known only from central Asian Kharoṣṭhī documents, though it probably existed in India as well. As shown by H. W. Bailey,²⁰⁶ it is derived from an Aramaic ligature ‘LP (i.e., *‘alap*, ‘thousand’). Also in the central Asian documents, we find cursively written combinations of the sign for 1 which have evidently developed into distinct characters for 2 (𐩽) and 3 (𐩾).

c7.s4.t2.4

²⁰⁶ “A Problem of the Kharoṣṭhī Script,” in *Essays and Studies Presented to Stanley Arthur Cook . . .*, 121–3 (see n. 138).

c7.s4.t2.5

The origin of the Kharoṣṭhī numerical system presents no difficulties, as its derivation from Aramaic, like that of the script itself, is clear. Not only the system²⁰⁷ but also all of the individual symbols²⁰⁸ can be easily associated with the Aramaic, as represented, for instance, in the papyri from Elephantine (Egypt).²⁰⁹

c7.s4.t2.6

²⁰⁷ The Aramaic system represented the digits up to 9 by simple combinations of the vertical stroke representing 1; this confirms that the later Kharoṣṭhī sign for 4 is a separate Indian development.

c7.s4.t2.7

²⁰⁸ The Kharoṣṭhī symbol for 20 is often described (e.g., in IIEP 125) as a doubling of the Kharoṣṭhī 10, but comparison with the Aramaic prototype indicates that it is rather a direct borrowing of the Aramaic 20.

c7.s4.t2.8

²⁰⁹ See Emil G. Kraeling, *The Brooklyn Museum Aramaic Papyri: New Documents of the Fifth Century B.C. from the Jewish Colony at Elephantine* (London: Yale University Press for the Brooklyn Museum, 1953). Papyrus 4 (168–9) has several numerical figures which are almost identical to the corresponding ones in Kharoṣṭhī.

Linked texts (5)