Pramāṇa · Search · About · API

History of Hindu Mathematics: A Source Book v1.c7.s5: 5. The Development of Numerical Symbolism

History of Hindu Mathematics: A Source Book v1.c7.s5

Language
English

Contents

5. The Development of Numerical Symbolism

v1.c7.s5.1

**Writing in Ancient India.** It is generally held that numerical symbols were invented after writing had been in use for some time, and that in the early stages the numbers were written out in full in words. This seems to be true for the bigger units, but the signs for the smaller units are as old as writing itself.

v1.c7.s5.2

Until quite recently historians were divided as to the date when writing was in use in India. There were some who stated that writing was known even in the Vedic age, but the majority following Weber, Taylor, Bühler and others were of opinion that writing was introduced into India from the West about the eighth century B.C. These writers built up theories deriving the ancient Indian script, as found in the inscriptions of Aśoka, from the more ancient writing discovered in Egypt and Mesopotamia. The Semitic origin was first suggested by Sir W. Jones, in the year 1806, and later on supported by Kopp (1821), Lespius (1834), and many others. The supporters of this theory, however, do not completely agree amongst themselves. For, whilst W. Deccke and I. Taylor derive the Indian script from a South-Semitic script, Weber and Bühler derive it from the Phoenician or a North-Semitic script.¹ Bühler rejects the derivation from a South-Semitic script, stating that the theory requires too many assumptions, and makes too many changes in the letter forms to be quite convincing. He, however, supports Weber's derivation from a North-Semitic script and has given details of the theory.² Ojha³ has examined Bühler's theory in detail and rejects it stating that it is fanciful and that the facts are against it. He states that only one out of the twenty-two letters of the Phoenician (North-Semitic) script resembles a phonetically similar Brâhmî letter. He supports his argument in a most convincing manner by a table of the two alphabets, with phonetically similar letters arranged in a line. He further shows that following Bühler's method of derivation almost any script could be proved to be the parent of another.⁴

v1.c7.s5.3

¹ For minor differences in the theories set up by different writers and also for several other theories, see Bühler, *Palaeography*, p. 9; the notes give the references.

v1.c7.s5.4

² Bühler, *l.c.*, pp. 9f.

v1.c7.s5.5

³ *PLM*, pp. 18-31.

v1.c7.s5.6

⁴ Recently several other eminent historians have expressed their disagreement with Bühler's derivation. See Bhandarkar, "Origin of the Indian Alphabet," *Sir Asutosh Mukerji Jubilee Volumes*, Vol. III, 1922, p. 493; H. C. Ray, "The Indian Alphabet," *IA*, III, 1924, p. 233; also *Mohenjo-daro and the Indus Valley Civilisation*, 1931, p. 424, where the following remark occurs: "I am convinced that all attempts to derive the Brâhmî alphabet from Semitic alphabets were complete failures."

v1.c7.s5.7

Other scholars, who held that writing was known in India as early as the Vedic age, based their conclusion upon literary evidence. The *Vasiṣṭha Dharmasûtra*, which originally belonged to a school of the *Ṛgveda* offers clear evidence of the use of writing in the Vedic period. Vasiṣṭha (xvi. 10, 14-15) mentions written documents as legal evidence, and the first of these sûtras is a quotation from an older work or from traditional lore. Another quotation from the *Ṛgveda* itself (x. 62. 7), which refers to the writing of the number eight is: *Sahasraṁ me dadato aṣṭakarṇyaḥ*, meaning "gave me a thousand cows on whose ears the number eight was written." The above interpretation, although doubted by some scholars, seems to be correct, as it is supported by Pâṇini.¹ Moreover, the practice of making marks on the ears of cows to denote their relation to their owners, seems to have been prevalent in ancient India.²

v1.c7.s5.8

¹ *Karṇo varṇa lakṣaṇât* (vi. 2. 112) and also (vi. 3. 115) support the interpretation.

v1.c7.s5.9

² *Atharvaveda* (vi. 141) mentions the method of making *mithuna* marks on the ears. In (xii. 4. 6) the practice is denounced. The *Maitrâyaṇî Saṁhitâ* has a chapter dealing with this topic. The method of making such marks is dealt with in iv. 2. 9.

v1.c7.s5.10

At another place in the *Ṛgveda* (x. 34), we find mention of a gambler lamenting his lot and saying that "having staked on one,³ he lost his faithful wife...." Again, in the *Atharvaveda* (vii. 50, (52), 5) we find the mention of the word "written amount".⁴ Pâṇini's grammar (*c.* 700 B.C.) contains the terms *yavanânî* ("Semitic writing") and the compounds *lipikâra* and *libikâra* (iii. 2. 21) (writer), which show that writing was known in his time. In addition to these passages, the Vedic works contain some technical terms, such as *akṣara* (a letter of the alphabet), *kâṇḍa* (chapter), *paṭala*, *grantha* (book), etc., which have been quoted as evidence of writing. These specific references to written documents when considered with the advanced state of Vedic civilisation, especially the high development of trade and complicated monetary transactions, the use of prose in the *Brâhmaṇas*, the collection, the methodical arrangement, the numeration, the analysis of the Vedic texts and the phonetic and lexicographic researches found in the *Vedâṅgas*, form sufficient grounds for assigning a very early date to the use of writing in India.¹ Although these arguments possess considerable weight, they were not generally recognised, as will always happen if an *argumentum ex impossibili* is used. R. Shamasastry (1906) has published a derivation based upon ancient Indian hieroglyphic pictures which he believes to be preserved in the *tântric* figures. His learned article has not attracted the attention it deserves.

v1.c7.s5.11

³ Here 'one' refers to the number stamped on the dice.

v1.c7.s5.12

⁴ *Ajaiṣaṁ tvâ saṁlikhitamajaiṣamuta saṁrudham*.

v1.c7.s5.13

¹ Cf. Bühler, *l.c.*, p. 3.

v1.c7.s5.14

Recent discoveries have, however, sounded the death knell of all theories deriving the Indian script from foreign sources. Pottery belonging to the Megalithic (*c.* 1,500 B.C.) and Neolithic (6,000 B.C.—3,000 B.C.) ages, preserved in the Madras Museum, has been found to be inscribed with writing. And according to Bhandarkar² five of these marks are identical with the Brâhmî characters of the time of Aśoka. The excavations at Mohenjo-daro and Harappa have also brought to light written documents, seals and inscriptions, dating from before 3,000 B.C. Thus it would be now absurd to trace the Brâhmî to any Semitic alphabet of the eighth or ninth century B.C.

v1.c7.s5.15

² *l.c.*

Earliest Numerals

v1.c7.s5.t1.1

The numerical figures contained in the seals and inscriptions of Mohenjo-daro, have not been completely deciphered as yet. The vertical stroke and combinations of vertical strokes arranged side by side, or one group below another, have been found. The numbers one to thirteen seem to have been written by means of vertical strokes, probably, as in the figures given below:³

v1.c7.s5.t1.2

³ Marshall, *l.c.*, pp. 450-52. See also "Mohenjo-daro—Indus Epigraphy" by G. R. Hunter (*JRAS*, April, 1932, pp. 470, 478ff.) who is more pronounced about the numerical values of some of the signs.

v1.c7.s5.t1.4

It is not yet quite certain whether there were special signs for greater numbers such as 20, 30, the hundreds and higher numbers. There are numerous other signs which are believed to represent such numbers, but there seems to be no means of finding out the true values of these signs at present.

v1.c7.s5.t1.5

Between the finds of Mohenjo-daro and the inscriptions of Aśoka, which contain numerals, there is a gap of 2,700 years or more. No written documents containing numerals and belonging to this intervening period have been so far discovered. The literary evidence, however, points to the use of numerical symbols at a very early date. The reference to the writing of the number eight in the *Ṛgveda* and the use of numerical denominations as big as 10¹² in the *Yajurveda Saṁhitā* and in several other Vedic works, quoted before, offer sufficient grounds for concluding that, even at that remote period, the Hindus must have possessed a well developed system of numerical symbols. The conclusion is supported by the fact that the Greek and the Roman numerical terminologies did not go beyond 10⁴, even after writing and a satisfactory numerical symbolism had been in use for several centuries.

v1.c7.s5.t1.6

The writings on the inscriptions of Aśoka show that in his time the use of numerical symbols in India was quite common.¹ The variations in the forms of the numerical signs suggest that the symbols had been in use for a long time.

v1.c7.s5.t1.7

¹ Megasthenes speaks of mile-stones indicating the distances and the halting places on the roads. The distances must have been written in numerical figures (Bühler, *l.c.*, p. 6; also *Indika of Megasthenes*, pp. 125-26). The complicated system of keeping accounts mentioned in the *Arthaśāstra* of Kauṭilya confirms the conclusion.

v1.c7.s5.t1.8

Most of the inscriptions of Aśoka and the following period are written in a script which has been called *Brāhmī*, whilst some are in a different script known as *Kharoṣṭhī*. The forms of the numerical symbols in the two scripts are different. We consider them separately.

Linked texts (8)