History of Hindu Mathematics: A Source Book v1.c7.s7: 7. Brâhmî Numerals
History of Hindu Mathematics: A Source Book v1.c7.s7
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7. Brâhmî Numerals
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**Early Occurrence and Forms.** The Brâhmî inscriptions are found distributed all over India. The Brâhmî script was, thus, the national script of the ancient Hindus. It is undoubtedly an invention of the Brâhmaṇas. The early grammatical and phonetic researches seem to have resulted in the perfection of this script about 1,000 B.C. or earlier. The Brâhmî numerals are likewise a purely Indian invention. Attempts have been made by several writers of note to evolve a theory of a foreign origin of the numerals, but we are convinced that all those attempts were utter failures.¹ These theories will be dealt with at their proper places. Due to the lack of early documents, we are not in a position to say what exactly were the original forms of the Brâhmî symbols. Our knowledge of these symbols goes back to the time of King Aśoka (*c.* 300 B.C.) whose vast dominions included the whole of India and extended in the north upto Central Asia. The forms of these symbols are:
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¹ *Cf.* Langdon's opinion in *Mohenjo-daro and the Indus Valley Civilisation*, ch. xxiii.
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The next important inscription containing numerals is found in a cave on the top of the Nânâghât hill in Central India, about seventy-five miles from Poona. The cave was made as a resting place for travellers by order of a King named Vediśrî, a descendant of King Sâtavâhana. The inscription contains a list of gifts made on the occasion of the performance of several *yajñas* or religious sacrifices. It was first deciphered by Pandit Bhagavanlal Indraji who has given the interpretation of the numerical symbols.¹ These occur at about thirty places, and their forms are as below:
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¹ "On Ancient Nâgarî Numeration; from an inscription at Nânâghât," *Journ. of the Bombay Branch of the Royal Asiatic Society*, 1876, Vol. XII, p. 404.
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A number of inscriptions containing numerals and dating from the first or the second century A.D. are found in a cave in the district of Nasik in the Bombay presidency. These contain a fuller list of numerals. The forms² are as follows:
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² E. Senart, "The inscriptions in the caves at Nasik," *EI*, Vol. VIII, pp. 59-96; "The inscriptions in the cave at Karle," *EI*, Vol. VII, pp. 47-74.
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Even after the invention of the zero and the place-value system, the same numerical symbols from 1 to 9, continued to be employed with the zero to denote numbers. Thus the gradual development of these forms can be easily traced. This gradual change from the old system without place-value to the new system with the zero and the place-value is to be met with in India alone. All other nations of the world have given up their indigenous numerical symbols which they had used without place-value and have adopted the zero and a new set of symbols, which were never in use in those countries previously. This fact alone is a strong proof of the Hindu origin of the zero and the place-value system.
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The numbers 1, 2 and 3 of the Brâhmî notation were denoted by one, two and three horizontal¹ lines placed one below the other. These forms clearly distinguish the Brâhmî notation from the Kharoṣṭhî and the Semitic systems.
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¹ It has been incorrectly stated by Smith and Karpinski that the Nânâghât forms were vertical. See *Hindu-Arabic Numerals*, p. 28.
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It cannot be said why the strokes were horizontal in Brâhmî and vertical in Kharoṣṭhî and Semitic writings, just as it cannot be said why the writing proceeded from left to right in Brâhmî and from right to left in Kharoṣṭhî and Semitic writings. It appears to us that the Brâhmî and the Kharoṣṭhî (Semitic) numerals have always existed side by side and it cannot be definitely said which of these is the earlier. The difference in writing the symbols 1 to 3, seems to be due to the inherent difference between the two systems of writing. The principles upon which numerical signs are formed in the two systems are quite different.
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**Difference from other Notations.** In the Brâhmî there are separate signs for each of the numbers 1, 4 to 9 and 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 200, 300, . . . . . 1000, 2000, etc., while in the oldest Kharoṣṭhî and in the earliest Semitic writings, the Hieroglyphic and the Phoenician, the only symbols are those for 1, 10, 20 and 100.
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The Hieratic and the Demotic numerals, however, resemble the Brâhmî in having nineteen symbols for the numbers from 1 to 100, but the principle of formation of the numbers 200, 300, 400, 2,000, 3,000 and 4,000 are different, as will appear from Table II(*c*). The method of formation of intermediate and higher numbers is also different in the two systems. While the Brâhmî places the bigger numbers to the left, the arrangement is the reverse of this in the Kharoṣṭhî and Semitic writings. Thus the number 274, is written in Brâhmî with the help of the symbols for 200, 70 and 4 as (200) (70) (4), while in the Kharoṣṭhî and the Semitic numerals it is written as (4) (70) (200).¹
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¹ Compare the same number written in Kharoṣṭhî, p. 24.
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**Theories about their Origin.** Quite a large number of theories have been advanced to explain the origin of the Brâhmî numerals. Points of resemblance have been imagined between these numerals and those of other nations. Recourse has been taken by writers to the turning, twisting, adding on or cutting off of parts of the numerals of other nations to fit their pet theories. It is needless to say that each of these theories had its own supporters who were quite convinced of the correctness of their explanations. We give below the outlines of some of these theories:
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1. Cunningham² believed that writing had been known in India from the earliest known times, and that the earliest alphabet was pictographic. He suggested that the Brâhmî script was derived from the early pictographic writing. The theory is evidently capable of extension to the numerical signs. Later epigraphists, however, discarded the hypothesis as it appeared too fanciful to them. Cunningham's bold hypothesis regarding the antiquity of writing in India has been more than justified by the recent discovery of the use of a quasi-pictographic script on certain seals and in inscriptions belonging to the fourth millenium B.C. found amongst the excavations at Mohenjo-daro and Harappa. His theory has been revived by Langdon who is of opinion that the Brâhmî alphabet could be derived from the pictographs of Mohenjo-daro.¹ The theory is incomplete as the writings of Mohenjo-daro have not been completely deciphered as yet. It can be called a guess only. As regards the evolution of the Brâhmî numerals, it may be stated that it is at present extremely difficult to differentiate the numerical symbols from the Mohenjo-daro script. If the surmise that the figures, given on p. 19, are numerical symbols be correct, it will not be possible to develop a theory deriving the Brâhmî numerals from them.
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² Inscriptions of Aśoka, *Corpus Inscriptionum Indicarum*, Vol. I, p. 52.
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¹ *Mohenjo-daro etc.*, Chap. xii. This view is strongly supported by Hunter, *l.c.*, p. 490.
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2. Bayley² asserted that the principles of the Brâhmî system have been derived from the hieroglyphic notation of the Egyptians, and that the majority of the Indian symbols have been borrowed from Phoenician, Bactrian, and Akkadean figures or letters. As has been already remarked³ the principles of the Brâhmî and the hieroglyphic systems are entirely different and unconnected. The reader will find the hieroglyphic and the Brâhmî systems shown together in Tables II (*a*), (*b*), (*c*), and convince himself of the incorrectness of Bayley's assertion. Moreover, the assumption that the Hindus borrowed from four or five different, partly very ancient and partly more modern, sources, is extremely difficult to believe. Regarding the resemblance between the Bactrian and Akkadean numbers and the Brâhmî forms postulated by Bayley, Bühler¹ remarks that in four cases (four, six, seven and ten) the facts are absolutely against Bayley's hypothesis. Some writers have also criticized Bayley's drawings as being affected by his theory.² Under these circumstances his derivation has to be rejected.
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² *Journal of the Royal Asiatic Soc.*, XV, part I, reprint, London, 1882, pp. 12 and 17. The theory was supported by Taylor, *The Alphabet*, London, 1883, Vol. II, pp. 265-66.
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³ See pages 27-8.
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¹ Bühler, *On the Origin of the Indian Brâhma Alphabet*, Strassburg, 1898, pp. 52, 53 foot-note.
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² *Cf.* Smith and Karpinski, *Hindu Arabic Numerals*, pp. 30-1.
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3. Burnell³ pointed out the general agreement of the principles of the Indian system with those of the Demotic notation of the Egyptians. He asserted a resemblance between the Demotic signs for 1 to 9 and the corresponding Indian symbols, and put forward the theory that the Hindus borrowed these signs and later on modified them and converted them into *akṣaras* (letter forms).
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4. Bühler³ has put forward a modification of Burnell's theory. He states, "It seems to me probable that the *Brâhma* numerals are derived from the Egyptian Hieratic figures, and that the Hindus effected their transformation into *Akṣaras*, because they were already accustomed to express numerals by words."
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³ Bühler, *l.c.*, p. 82.
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The above theories like the one examined before are not well founded. Tables II (*a*), (*b*), (*c*), show the Hieratic and Demotic symbols together with those of the Brâhmî. An examination of the Tables will reveal that out of the nineteen symbols to represent the numbers from 1 to 100, only the nine of the Brâhmî resembles the corresponding symbol of the Demotic or the Hieratic. There is absolutely no resemblance between any of the others. To base the derivation on a resemblance between the Hieratic 5 and the Brâhmî 7, as is sought to be done, is absurd It will be observed that in the Hieratic system the sign for one thousand is not used in the formation of the other thousands. The similarity in principle, even if it were complete, would not force us to conclude that one of these nations copied the other. The use of nineteen signs afforded the easiest and probably the best method of denoting numbers. It is not beyond the limits of probability that what appeared easy to the Egyptians might have also independently occurred to the Hindus.
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There are on the other hand some considerations which make us suggest that the Egyptians borrowed the principles of the Hieratic and the Demotic systems from outside, and probably from India—a hypothesis which is not *a priori* impossible as it has been shown that the numeration system of the ancient Hindus based on nineteen signs might have been perfected about 1,000 B.C. It is known that the ancient Egyptian system employed only four signs, those for 1, 10, 20 and 100. Why should there be a sudden change from the old system to one containing nineteen signs cannot be adequately explained except on the hypothesis of foreign influence. Further, the cursive forms for the numbers 2, 3 and 4 are unsuited to the right to left Hieratic or Demotic script. Although these figures are connected with the earlier hieroglyphic and Phoenician figures, yet it is possible that the cursive combinations might have been formed to obtain the nineteen signs necessary for the new system, under the influence of a people with a left to right script. It may be, however, asserted that the hypothesis of an Indian origin of the Hieratic system is a mere suggestion. The two points noted above, by themselves, would not be enough, unless backed by other facts, to put forward a theory. It is expected that further discoveries will throw light on this point.
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**Relation with Letter Forms.** It was suggested by James Princep,¹ as early as 1838, that the numerals were formed after the initial letters of the number names. But knowing the pronunciation of the number names, we find this not to be the case. Other investigators The use of the letter system, the *akṣarapallî*, suggests that the old Brâhmî numerals also might have been fashioned after the letters or the syllables of the Brâhmî alphabet.
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A detailed examination of the numerals and the letter-forms has been made, and the result is tabulated below:¹
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¹ For the varying forms of the numerals see Tables III—XIII.
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The numerals 1, 2 and 3 which were represented by horizontal strokes do not resemble any letter. They may have been derived from the *mâtṛkâ* signs.
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4. The earliest form of this numeral in the inscriptions of Aśoka resembles the letter *ka*; the Nânâghât and Nâsik cave forms resemble the syllable *pka*. While this seems to have been the case with most of the later inscriptions, forms resembling the syllables *pna*, *lka*, *ṭka*, *pkṛ* are also to be met with.
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5. In most of the inscriptions it resembles the syllable *tṛ*, whilst forms resembling *ta*, *tâ*, *pu*, *hu*, *ru*, *tṛ*, *trâ*, *nâ*, *na*, *hṛ*, *hra*, and *ha* are also found.
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6. In most of the inscriptions it resembles *phra*, and in a few *phrâ*, *pha*, *phâ*, *ja* and *hâ*. The resemblance in this case is, however, not close enough.
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7. In most of the inscriptions it resembles *gra* or *gu*, whilst n some it is like *ga*.
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8. In most cases it resembles the syllable *hra* or *hrâ*. In some inscriptions the form of this numeral cannot be said to resemble any letter or syllable.
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9. The oldest forms of this numeral, those in the Nânâghât, the Kuṣâna and the Kṣatrapa inscriptions, cannot be said to resemble any letter or syllable. The later forms, however, resemble the letter *u* or *om*.
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10. The oldest forms of this too do not resemble any letter or syllable, although later forms may be said to resemble *rya*, *ha*, *hṛ*, *kha*, and *thu* or *tha*.¹
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¹ According to Bühler, *l.c.*, p. 80.
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20. In all cases it resembles the letter *tha*.
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30. Resembles the letter *la* in all cases.
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40. Resembles *pta* or *sa* in all cases.
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50. It may be said to resemble the *anunâsika*.
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60. Resembles *pu*, *pa* and *pra*.
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70. Resembles *pu*, *pta*, *pra*, *pna*, *pnâ*, or *hra*.
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80. Resembles the *upadhmânîya* sign.
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90. Resembles the *upadhmânîya* sign with a central cross.
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100. Resembles the syllable *su* in most of the inscriptions. In some it resembles *a*.
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**Indraji's Theory.** The above details show that,
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1. No phonetical value can be assigned to the ancient forms of the numerals 6 and 10, and 1, 2 and 3;
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2. There is a great deal of variation in the phonetical values of the other units, excepting the cases of 7 and 9; and
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3. The phonetical values of the tens are determinate excepting that of 70 which offers great variation.
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Thus leaving aside the symbols for 1, 2 and 3, we find that out of the 16 symbols, no resemblance to letter-forms can be satisfactorily shown in the case of two (*i.e.*, 6 and 10),² and that in the case of three others (*i.e.*, 7, 9 and 70) there are too many variations in the phonetical values, whilst in the case of the remaining 11 symbols, the phonetical values are more or less determinate. These facts led Pandit Bhagavanlal Indraji to put forward the hypothesis that the Brâhmî numerals are derived from the letters or syllables of the Brâhmî script. The Pandit, however, admitted his inability ascribes the invention of the Brâhmî script and the numeral notation to Brahmâ, the Creator, and thereby claims it as a national invention of the remotest antiquity.¹
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² The resemblance to *ja*, *sa*, *phra*, etc. and *thû*, *tha*, etc., respectively, stated by Bühler (*l. c.*, p. 80), does not appear to us to be close enough.
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¹ Bühler (*l.c.*, p. 1, foot-note,) quotes several authorities. Of these the *Nârada Smṛti* and the Jaina canonical work, the *Samavâyâṅga-sûtra*, belong to the fourth century B.C.
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**Period of Invention.** The invention of the system may be assigned to the period 1,000 B.C. to 600 B.C. As the Aśokan numerical figures indicate that the system was common all over India,² and that it has had a long history, the lower limit 1,000 B.C. is certainly not placed too early. On the other hand general considerations, such as the high development of the arts and the sciences, the mention of numerical signs and of 64 different scripts in ancient Buddhist literature,³ and the use of large numbers at a very early period, all point to the date of the invention of the system as being nearer to 1,000 B.C., if not earlier.
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² Megasthenes speaks of mile-stones indicating distances and the halting places on the roads. *Indika of Megasthenes*, pp. 125-126; Bühler, *l.c.*
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³ Related in the *Lalitavistara*, both in the Sanskrit text and the Chinese translation of 308 A.D. The Jaina *Samavâyâṅga-sûtra* (*c.* 300 B.C.) and *Pannavanâ-sûtra* (*c.* 168 B.C.) each gives a list of 18 scripts; see Weber, *Indische Studien*, 16, 280, 399.
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**Resume.** The strength of Pandit Indraji's hypothesis lies in the fact that out of the nineteen signs, eleven definitely resemble the letters or the signs of the Brâhmî alphabet. The resemblance is too striking to be entirely accidental. Moreover, it has been found that the numerical forms closely followed the changing forms of the letters from century to century. This is especially true in the case of the tens and shows that the writers of the ancient inscriptions knew the phonetical values of these symbols. The divergence from letter forms in the case of the signs for the units may be due to the fact that they were the first to be invented and were in more common use, so that they acquired special cursive forms and did not follow the changes in the forms of the corresponding letters. We may now summarize the discussion given in this section by saying that (1) the Brâhmî numerical forms were undoubtedly of Indian origin, (2) the form of the tens were derived from certain letters or signs of the alphabet, and (3) the origin of the forms of the units is doubtful. It is probable that they, too, were fashioned after the letters of the alphabet, but there appears to be no means of justifying this assertion unless the forms of these numerals anterior to Aśoka are discovered.
Linked texts (8)
- Nāradasmṛti — Nārada, Sanskrit
- Ἰνδική — Λούκιος Φλάουιος Ἀρριανός, Ancient Greek
- Harappā
- Mohenjo-daro
- Pannavaṇā — Śyāmārya, Ardhamāgadhī Prakrit
- Samavāyaṃga — Sudharman, Ardhamāgadhī Prakrit
- Nāsik
- Nānāghāṭ