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History of Hindu Mathematics: A Source Book v1.c7: Chapter I. Numeral Notation

History of Hindu Mathematics: A Source Book v1.c7

Language
English

Contents

Chapter I. Numeral Notation

1. A Glimpse of Ancient India

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The student of ancient Indian History is struck by the marvellous attainments of the Hindus, both in the Arts and the Sciences, at a very early period. The discoveries at Mohenjo-daro reveal that as early as 3,000 B.C. the inhabitants of the land of the Sindhu—the Hindus—built brick houses, planned cities, used metals such as gold, silver, copper and bronze, and lived a highly organised life. In fact, they were far in advance of any other people of that period. The earliest works available, the *Vedas* (*c.* 3,000 B.C. or probably much earlier), although consisting mainly of hymns of praise and poems of worship, show a high state of civilisation. The *Brâhmaṇa* literature (*c.* 2,000 B.C.) which follows the *Vedas*, is partly ritualistic and partly philosophical. In these works are to be found well-developed systems of metaphysical, social and religious philosophy, as well as the germs of most of the sciences and arts which have helped to make up the modern civilisation. It is here that we find the beginnings of the science of mathematics (arithmetic, geometry, algebra, etc.) and astronomy. This *Brâhmaṇa* period was followed by more than two thousand years of continuous progress and brilliant achievements. Although during this period there were several foreign invasions as well as internal wars and many great kingdoms rose and fell, yet the continuity of intellectual progress was maintained. The constitution of Hindu society was mainly responsible for this. The foreign invaders, instead of being a hindrance, contributed to progress and the strengthening of Hindu society by bringing in new blood. They settled in the land, adopted the religion and customs of the conquered and were completely absorbed into Hindu society. There were a class of people—the Brâhmaṇas—who took the vow of poverty, and devoted themselves, from one generation to another, to the cultivation of the sciences and arts, religion and philosophy. The Brâhmaṇas, thanks to their selflessness and intellectual attainments, were highly respected by the kings and the people alike. They were the law-givers and advisers of the kings. In fact, this body of selfless thinkers and learned men were the real rulers of the land.

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The great Epic, the *Râmâyaṇa*, was composed by Vâlmîki, the father of Sanskrit poetry, about 1,000 B.C., Pâṇini, the grammarian, perfected Sanskrit grammar about 700 B.C. and Suśruta wrote on the sciences of medicine and surgery about 600 B.C.¹ A century later, Mahâvîra and Buddha taught their unique systems of religious and moral philosophy, and the doctrine of *Nirvâṇa*. With the spread of these religions evolved the Jaina and Buddhist literatures. Some of the earlier *Purâṇas* and *Dharma-śâstras* were written about this time. The period 400 B.C. to 400 A.D., however, seems to have been a period of great activity and progress. During this period flourished the great Jaina metaphysician Umâsvâti, Patañjali, the grammarian and philosopher, Kauṭilya, the celebrated politician, Nâgârjuna, the chemist, Caraka, the physician, and the immortal poets Aśvaghoṣa, Bhâsa and Kâlidâsa. The great astronomical *Siddhântas*, the *Sûrya*, the *Pitâmaha*, the *Vaśiṣṭha*, the *Parâśara* and others were written during this period and the decimal place-value notation was perfected.

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¹ There is considerable divergence of opinion regarding the dates of the pre-historic works and personalities mentioned in this section. We have given those dates that appear most plausible.

2. Hindus and Mathematics

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**Appreciation of Mathematics.** It is said that in ancient India no science did ever attain an independent existence and was cultivated for its own sake. Whatever of any science is found in Vedic India is supposed to have originated and grown as the handmaid of one or the other of the six "members of the Veda," and consequently with the primary object of helping the Vedic rituals. It is also supposed, sometimes, that any further culture of the science was somewhat discouraged by the Vedic Hindus in suspicion that it might prove a hindrance to their great quest of the knowledge of the Supreme by diverting the mind to other external channels. That is not, indeed, a correct view on the whole. It is perhaps true that in the earlier Vedic Age, sciences grew as help to religion. But it is generally found that the interest of people in a particular branch of knowledge, in all climes and times, has always been aroused and guided by specific reasons. Religion being the prime avocation of the earlier Hindus, it is not unnatural that the culture of other branches of knowledge grew as help to it and was kept subsidiary. But there is enough evidence to show that in course of time all the sciences outgrew their original purposes and were cultivated for their own sake. A new orientation had indeed set in in the latter part of the Vedic Age.

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There is a story in the *Chândogya Upaniṣad*¹ whose value in support of our view cannot be over-estimated. It is said that once upon a time Nârada approached the sage Sanatkumâra and begged of him the *Brahma-vidyâ* or the supreme knowledge. Sanatkumâra asked Nârada to state what sciences and arts he had already studied so that he (Sanatkumâra) might judge what still remained to be learnt by him. Thereupon Nârada enumerated the various sciences and arts studied by him. This list included astronomy (*nakṣatra-vidyâ*) and arithmetic (*râśi-vidyâ*). Thus the culture of the science of mathematics or of any other branch of secular knowledge, was not considered to be a hindrance to spiritual knowledge. In fact, *Aparâ-vidyâ* ("secular knowledge") was then considered to be a helpful adjunct to *Parâ-vidyâ* ("spiritual knowledge").¹

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¹ *Chândogya Upaniṣad*, vii. 1, 2, 4.

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¹ *Muṇḍakopaniṣad*, i. 1, 3-5.

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Importance to the culture of *Gaṇita* (mathematics) is also given by the Jainas. Their religious literature is generally classified into four branches, called *anuyoga* ("exposition of principles"). One of them is *gaṇitânuyoga* ("the exposition of the principles of mathematics"). The knowledge of *Saṁkhyâna* (literally, "the science of numbers," meaning arithmetic and astronomy) is stated to be one of the principal accomplishments of the Jaina priest.² In Buddhist literature too, arithmetic (*gaṇanâ*, *saṁkhyâna*) is regarded as the first and the noblest of the arts.³ All these will give a fair idea of the importance and value set upon the culture of *gaṇita* in ancient India.

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² *Bhagavatî-sûtra*, Sûtra 90; *Uttarâdhyayana-sûtra*, xxv. 7, 8, 38.

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³ *Vinaya Piṭaka*, ed. Oldenberg, Vol. IV, p. 7; *Majjhima Nikâya*, Vol. I, p. 85; *Cullaniddesa*, p. 199.

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The following appreciation of mathematics, although belonging to a much later date, will be found to be interesting, especially, as it comes from the pen of Mahāvîra (850 A.D.), one of the best mathematicians of his time:

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“In all transactions which relate to worldly, Vedic or other similar religious affairs calculation is of use. In the science of love, in the science of wealth, in music and in drama, in the art of cooking, in medicine, in architecture, in prosody, in poetics and poetry, in logic and grammar and such other things, and in relation to all that constitutes the peculiar value of the arts, the science of calculation (*gaṇita*) is held in high esteem. In relation to the movements of the sun and other heavenly bodies, in connection with eclipses and conjunctions of planets, and in connection with the *tripraśna* (direction, position and time) and the course of the moon—indeed in all these it is utilised. The number, the diameter and the perimeter of islands, oceans and mountains; the extensive dimensions of the rows of habitations and halls belonging to the inhabitants of the world, of the interspace between the worlds, of the world of light, of the world of the gods and of the dwellers in hell, and other miscellaneous measurements of all sorts—all these are made out by the help of *gaṇita*. The configuration of living beings therein, the length of their lives, their eight attributes, and other similar things; their progress and other such things, their staying together, etc.—all these are dependent upon *gaṇita* (for their due comprehension). What is the good of saying much? Whatever there is in all the three worlds, which are possessed of moving and non-moving beings, cannot exist as apart from *gaṇita* (measurement and calculation).

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“With the help of the accomplished holy sages, who are worthy to be worshipped by the lords of the world, and of their disciples and disciples’ disciples, who constitute the well-known series of preceptors,

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I glean from the great ocean of the knowledge of numbers a little of its essence, in the manner in which gems are picked from the sea, gold is from the stony rock and pearl from the oyster shell; and give out according to the power of my intelligence, the *Sâra-saṁgraha*, a small work on *gaṇita*, which is (however) not small in value."¹

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¹ *GSS*, i. 9-19.

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**Mathematics in Hindu Education.** The elementary stage in Hindu education lasted from the age of five till the age of twelve. This period slightly differed in the case of sons of kings and noblemen. The main subjects of study were *lipi* or *lekhâ* (alphabets, reading and writing), *rûpa* (drawing and geometry) and *gaṇanâ* (arithmetic). It is said in the *Arthaśâstra* of Kauṭilya (400 B.C.) that having undergone the ceremony of tonsure, the student shall learn the alphabets (*lipi*) and arithmetic (*saṁkhyâna*).² We find in the Hâthîgumphâ Inscription³ that king Khâravela (163 B.C.) of Kaliṅga spent nine years (from the age of sixteen to the age of 25) in learning *lekhâ*, *rûpa* and *gaṇanâ*. Prince Gautama began his education when he was eight years of age "firstly (with) writing and then arithmetic as the most important of the 72 sciences and arts."⁴ Mention of *lekhâ*, *rûpa* and *gaṇanâ* is also found in the Jaina canonical works.⁵

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² *Arthaśâstra*, ed. by R. Shamasastri, i. 5, 2; Eng. trans., p. 10.

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³ *Hathigumpha and three other inscriptions*, ed. by Bhagavanlal Indraji, p. 22.

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⁴ *Antagaḍa-dasâo and Anuttarovavâiya-dasâo*, Eng. trans. by L. D. Barnett, 1907, p. 30; cf. *Kalpasûtra* of Bhadrabâhu, Sûtra 211.

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⁵ E.g., *Samavâyâṅga-sûtra*, Sûtra 72.

3. Scope and Development of Hindu Mathematics

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*Gaṇita* literally means "the science of calculation" and is the Hindu name for mathematics. The term is a very ancient one and occurs copiously in Vedic literature. The *Vedâṅga Jyotiṣa* (*c.* 1,200 B.C.) gives it the highest place of honour among the sciences which form the *Vedâṅga*: "As the crests on the heads of peacocks, as the gems on the hoods of snakes, so is *gaṇita* at the top of the sciences known as the *Vedâṅga*."¹ In ancient Buddhist literature we find mention of three classes of *gaṇita*: (1) *mudrâ* ("finger arithmetic"), (2) *gaṇanâ* ("mental arithmetic") and (3) *saṁkhyâna* ("higher arithmetic in general"). One of the earliest enumerations of these three classes occurs in the *Dîgha Nikâya*,² and it is also found in the *Vinaya Piṭaka*,³ *Divyâvadâna*⁴ and *Milindapañho*.⁵ The word *saṁkhyâna* has been used for *gaṇita* in several old works.⁶ At this remote period *gaṇita* included astronomy, but geometry (*kṣetra-gaṇita*) belonged to a different group of sciences known as *Kalpasûtra*.

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¹ "Yathâ śikhâ mayurâṇâṁ nâgânâṁ maṇayo yathâ Tadvadvedâṅgaśâstrâṇâṁ gaṇitaṁ mûrdhani sthitam." *Vedâṅga Jyotiṣa*, 4.

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² I, p. 51.

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³ IV, p. 7.

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⁴ *Divyâvadâna*, ed. by E. B. Cowell and R. A. Neil, Cambridge, 1886, pp. 3, 26 and 88.

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⁵ *Milindapañho*, Eng. trans. by Rhys Davids, Oxford, 1890, p. 91.

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⁶ E.g., *Kalpasûtra* of Bhadrabâhu, ed. by H. Jacobi, Leipzig, 1897; *Bhagavatî-sûtra*, Bombay, 1918, p. 112; *Arthaśâstra*, i. 5. 2.

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It is believed that some time before the beginning of the Christian era, there was a renaissance of Hindu *Gaṇita*.⁷ The effect of this revival on the scope of *gaṇita* was great. Astronomy (*jyotiṣa*) became a separate subject and geometry (*kṣetra-gaṇita*) came to be included within its scope. The subjects treated in the Hindu *Gaṇita* of the early renaissance period consisted of the following:¹ *Parikarma* ("fundamental operations"), *Vyavahâra* ("determinations"), *Rajju* ("rope," meaning geometry), *Râśi* ("rule of three"), *Kalâsavarṇa* ("operations with fractions"), *Yâvat tâvat* ("as many as," meaning simple equations), *Varga* ("Square," meaning quadratic equations), *Ghana* ("Cube", meaning cubic equations), *Varga-varga* (biquadratic equations) and *Vikalpa* ("permutations and combinations").

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⁷ Bibhutibhusan Datta, "The scope and development of Hindu Gaṇita," *Indian Historical Quarterly*, V, 1929, pp. 479-512.

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¹ "Parikammaṁ vavahâro rajju râsî kalâsavanne ya । Jâvantâvati vaggo ghano tataha vaggavaggo vikappo ta ॥" *Sthânâṅgasûtra*, Sûtra 747.

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Thus *gaṇita* came to mean mathematics in general, while 'finger arithmetic' as well as 'mental arithmetic' were excluded from the scope of its meaning For the calculations involved in *gaṇita*, the use of some writing material was essential. The calculations were performed on a board (*pâṭî*) with a piece of chalk or on sand (*dhûlî*) spread on the ground or on the *pâṭî*. Thus the terms *pâṭî-gaṇita* ("science of calculation on the board") or *dhûlî-karma* ("dust-work"), came to be used for higher mathematics. Later on the section of *gaṇita* dealing with algebra was given the name *Bîja-gaṇita*. The first to effect this separation was Brahmagupta (628), but he did not use the term *Bîja-gaṇita*. The chapter dealing with algebra in his *Brâhma-sphuṭa-siddhânta* is called *Kuṭṭaka*. Śrîdharâcârya (750) regarded *Pâṭî-gaṇita* and *Bîja-gaṇita* as separate and wrote separate treatises on each. This distinction between *Pâṭîgaṇita* and *Bîjagaṇita* has been preserved by later writers.

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Having given a brief survey of the position and scope of mathematics in Ancient India, we turn to the purpose in hand—that of giving a connected account of the development and growth of the different branches of mathematics. The numeration system of the Hindus will engage our attention first.

4. Numeral Terminology

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**Scale of Notation.** We can definitely say that from the very earliest known times, ten has formed the basis of numeration in India.¹ In fact, there is absolutely no trace of the extensive use of any other base of numeration in the whole of Sanskrit literature. It is also characteristic of India that there should be found at a very early date long series of number names for very high numerals. While the Greeks had no terminology for denominations above the *myriad* (10⁴), and the Romans above the *mille* (10³), the ancient Hindus dealt freely with no less than eighteen denominations. In modern times also, the numeral language of no other nation is as scientific and perfect as that of the Hindus.

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¹ Various instances are to be found in the *Ṛgveda*; noted by Macdonell and Keith, *Vedic Index*, Vol. I, p. 343.

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In the *Yajurveda Saṃhitā* (*Vājasaneyī*)² the following list of numeral denominations is given: *Eka* (1), *daśa* (10), *śata* (100), *sahasra* (1000), *ayuta* (10,000), *niyuta* (100,000), *prayuta* (1,000,000), *arbuda* (10,000,000), *nyarbuda* (100,000,000), *samudra* (1,000,000,000), *madhya* (10,000,000,000), *anta* (100,000,000,000), *parārdha* (1,000,000,000,000). The same list occurs at two places in the *Taittirīya Saṃhitā*.³ The *Maitrāyaṇī*⁴ and *Kâṭhaka*¹ *Saṁhitâs* contain the same list with slight alterations. The *Pañcaviṁśa Brâhmaṇa* has the *Yajurveda* list upto *nyarbuda* inclusive, and then follow *nikharva*, *vâḍava*, *akṣiti*, etc. The *Śâṅkhyâyana Śrauta Sûtra* continues the series after *nyarbuda* with *nikharva*, *samudra*, *salila*, *antya*, *ananta* (= 10 billions). Each of these denominations is 10 times the preceding, so that they were aptly called *daśaguṇottara saṁjñâ*² ("decuple terms").

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² *Yajurveda Saṃhitā*, xvii. 2.

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³ iv. 40. 11. 4; and vii. 2. 20. 1.

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⁴ ii. 8. 14; the list has *ayuta*, *prayuta*, then again *ayuta*, then *nyarbuda*, *samudra*, *madhya*, *anta*, *parārdha*.

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¹ xvii. 10; the list is the same with the exception that *niyuta* and *prayuta* change places. In xxxix. 6, after *nyarbuda* a new term *vâḍava* intervenes.

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² Cf. Bhâskara II, *L*, p. 2.

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Coming to later times, *i.e.*, about the 5th century B.C., we find successful attempts made to continue the series of number names based on the centesimal scale.³ We quote below from the *Lalitavistara*,⁴ a well-known Buddhist work of the first century B.C., the dialogue between Arjuna, the mathematician, and Prince Gautama (Bodhisattva):

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³ *Śatottara gaṇanâ* or *Śatottara saṁjñâ* (names on the centesimal scale).

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⁴ *Lalitavistara*, ed. by Rajendra Lal Mitra, Calcutta, 1877, p. 168.

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"The mathematician Arjuna asked the Bodhisattva, 'O young man, do you know the counting which goes beyond the *koṭi* on the centesimal scale?

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Bodhisattva: I know.

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Arjuna: How does the counting proceed beyond the *koṭi* on the centesimal scale?

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Bodhisattva: Hundred *koṭis* are called *ayuta*, hundred *ayutas* *niyuta*, hundred *niyutas* *kaṅkara*, hundred *kaṅkaras* *vivara*, hundred *vivaras* *kṣobhya*, hundred *kṣobhyas* *vivâha*, hundred *vivâhas* *utsaṅga*, hundred *utsaṅgas* *bahula*, hundred *bahulas* *nâgabala*, hundred *nâgabalas* *tiṭi-* *lambha*, hundred *titilambhas* *vyavasthâna-prajñapti*, hundred *vyavasthâna-prajñaptis* *hetuhila*, hundred *hetuhilas* *karahu*, hundred *karahus* *hetvindriya*, hundred *hetvindriyas* *samâpta-lambha*, hundred *samâpta-lambhas* *gaṇanâgati*, hundred *gaṇanâgatis* *niravadya*, hundred *niravadyas* *mudrâ-bala*, hundred *mudrâ-balas* *sarva-bala*, hundred *sarva-balas* *visamjñâ-gati*, hundred *visamjñâ-gatis* *sarvajñâ*, hundred *sarvajñâs* *vibhutamgamâ*, hundred *vibhutamgamâs* *tallakṣaṇa*.¹”

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¹ Thus *tallakṣaṇa*=10⁵³. This and the following show that the Hindus anticipated Archimedes by several centuries in the matter of evolving a series of number names which “are sufficient to exceed not only the number of a sand-heap as large as the whole earth, but one as large as the universe.” *Cf.* ‘*De harenae numero*’ in the 1544 edition of the *Opera* of Archimedes; quoted by Smith and Karpinski, *Hindu Arabic Numerals*, Boston, 1911, p. 16.

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Another interesting series of number names increasing by multiples of 10 millions is found in Kâccâyana's Pali Grammar.² “For example: *dasa* (10) multiplied by *dasa* (10) becomes *sata* (100), *sata* (100) multiplied by ten becomes *sahassa* (1,000), *sahassa* multiplied by ten becomes *dasa* *sahassa* (10,000), *dasa* *sahassa* multiplied by ten becomes *sata* *sahassa*³ (100,000), *sata* *sahassa* multiplied by ten becomes *dasa* *sata* *sahassa* (1,000,000), *dasa* *sata* *sahassa* multiplied by ten becomes *koṭi* (10,000,000). Hundred-hundred-thousand *koṭis* give *pakoṭi*.⁴ In this manner the further terms are formed. What are their names? .......... hundred hundred-thousands is *koṭi*, hundred-hundredthousand *koṭis* is *pakoṭi*, hundred-hundred-thousand *pakoṭis* is *koṭippakoṭi*, hundred-hundred-thousand *koṭippakoṭis* is *nahuta*, hundred-hundred-thousand *nahutas* is *ninnahuta*, hundred-hundred-thousand *ninnahutas* is *akkhobhiṇi*; similarly we have *bindu*, *abbuda*, *nirabbuda*, *ahaha*, *ababa*, *atata*, *sogandhika*, *uppala*, *kumuda*, *puṇḍarîka* *paduma*, *kathâna*, *mahâkathâna*, *asaṅkhyeya*.”

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² “Grammaire Pâlie de Kâccâyana,” *Journ. Asiatique*, Sixieme Serie, XVII, 1871, p. 411. The explanations to sūtras 51 and 52 are quoted here.

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³ Also called *lakkha* (*lakṣa*).

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⁴ Also called *koṭi-koṭi*, *i.e.*, (10,000,000)²=10¹⁴. The following numbers are in the denomination *koṭi-koṭi*. Compare the *Anuyogadvâra-sūtra*, Sūtra 142.

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In the *Anuyogadvâra-sûtra*² (*c.* 100 B.C.), a Jaina canonical work written before the commencement of the Christian era, the total number of human beings in the world is given thus: “a number which when expressed in terms of the denominations, *koṭi-koṭi*, etc., occupies twenty-nine places (*sthâna*), or it is beyond the 24th place and within the 32nd place, or it is a number obtained by multiplying sixth square (of two) by (its) fifth square, (*i.e.*, 2⁹⁶), or it is a number which can be divided (by two) ninety-six times.” Another big number that occurs in the Jaina works is the number representing the period of time known as *Śîrṣaprahelikâ*. According to the commentator Hema Candra (b. 1089)³, this number is so large as to occupy 194 notational places (*aṅka-sthânehi*). It is also stated to be (8,400,000)²⁸.

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² *Sûtra* 142.

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³ The figures within brackets after the names of authors or works denote dates after Christ.

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**Notational Places.** Later on, when the idea of place-value was developed, the denominations (number names) were used to denote the places which unity would occupy in order to represent them (denominations) in writing a number on the decimal scale. For instance, according to Âryabhaṭa I (499) the denominations are the names of ‘places’. He says: “*Eka* (unit) *daśa* (ten), *śata* (hundred), *sahasra* (thousand), *ayuta* (ten thousand), *niyuta* (hundred thousand), *prayuta* (million), *koṭi* (ten million), *arbuda* (hundred million), and *vṛnda* (thousand million) are respectively from *place to place* each ten times the preceding.”¹ The first use of the word ‘place’ for the denomination is met with in the Jaina work quoted above.

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¹ Thus *asaṅkhyeya* is (10)¹⁴⁰ = (10,000,000)²⁰.

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¹ *Ā*, ii. 2.

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In most of the mathematical works, the denominations are called “names of places,” and eighteen of these are generally enumerated. Śrīdhara (750) gives the following names:² *eka*, *daśa*, *śata*, *sahasra*, *ayuta*, *lakṣa*, *prayuta*, *koṭi*, *arbuda*, *abja*, *kharva*, *nikharva*, *mahāsaroja*, *śaṅku*, *saritā-pati*, *antya*, *madhya*, *parārdha*, and adds that the decuple names proceed even beyond this. Mahāvīra (850) gives twenty-four notational places:³ *eka*, *daśa*, *śata*, *sahasra*, *daśa-sahasra*, *lakṣa*, *daśa-lakṣa*, *koṭi*, *daśa-koṭi*, *śata-koṭi*, *arbuda*, *nyarbuda*, *kharva*, *mahākharva*, *padma*, *mahā-padma*, *kṣoṇi*, *mahā-kṣoṇi*, *śaṅkha*, *mahā-śaṅkha*, *kṣiti*, *mahā-kṣiti*, *kṣobha*, *mahākṣobha*.

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² *Triś*, R. 2-3; the term used is *daśaguṇāḥ saṁjñāḥ*, i.e., “decuple names.”

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³ *GSS*, i. 63-68; “The first place is what is known as *eka*; the second is *daśa*” etc.

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Bhāskara II's (1150) list agrees with that of Śrīdhara except for *mahāsaroja* and *saritāpati* which are replaced by their synonyms *mahāpadma* and *jaladhi* respectively. He remarks that the names of places have been assigned for practical use by ancient writers.⁴

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⁴ *L*, p. 2.

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Nārāyaṇa (1356) gives a similar list in which *abja*, *mahāsaroja* and *saritāpati* are replaced by their synonyms *saroja*, *mahābja* and *pārāvāra* respectively.

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**Numerals in Spoken Language.** The Sanskrit names for the numbers from one to nine are: *eka*, *dvi*, *tri*, *catur*, *pañca* *ṣaṭ*, *sapta*, *aṣṭa*, *nava*. These with the numerical denominations already mentioned suffice to express any required number. In an additive system it is immaterial how the elements of different denominations, of which a number is composed, are spoken. Thus one-ten or ten-one would mean the same. But it has become the usual custom from times immemorial to adhere to a definite mode of arrangement, instead of speaking in a haphazard manner.

v1.c7.s4.34

In the Sanskrit language the arrangement is that when a number expression is composed of the first two denominations only, the smaller element is spoken first, but when it is composed also of higher denominations, the bigger elements precede the smaller ones, the order of the first two denominations remaining as before. Thus, if a number expression contains the first four denominations, the normal mode of expression would be to say the thousands first, then hundreds, then units and then tens. It will be observed that there is a sudden change of order in the process of formation of the number expression when we go beyond hundred. The change of order, however, is common to most of the important languages of the world.¹ Nothing definite appears to be known as to the cause of this sudden change.

v1.c7.s4.35

¹ Only in very few languages is the order continuously descending. In English the smaller elements are spoken first in the case of numbers upto twenty only.

v1.c7.s4.36

The numbers 19, 29, 39, 49, etc., offer us instances of the use of the *subtractive* principle in the spoken language. In Vedic times we find the use² of the terms *ekânna-viṁśati* (one-less-twenty) and *ekânna-catvâriṁśat* (one-less-forty) for nineteen and thirty-nine respectively. In later times (Sûtra period) the *ekânna* was changed to *ekona*, and occasionally even the prefix *eka* was deleted and we have *ûna-vimśati*, *ûna-trimśat*, etc.—forms which are used upto the present day. The alternative expressions *nava-daśa* (nine-ten), *nava-vimśati* (nine-twenty), etc., were also sometimes used.¹

v1.c7.s4.37

² *Taittirîya Saṁhitâ*, vii. 2. 11.

v1.c7.s4.38

¹ 19 = nava-daśa (*Vâjasaneyî Saṁhitâ*, xiv. 23; *Taittirîya Saṁhitâ*, xiv. 23. 30). 29 = nava-viṁśati (*Vâjasaneyî Saṁhitâ*, xiv. 31). 99 = nava-navati (*Ṛgveda*, i. 84. 13).

v1.c7.s4.39

Practically the whole of Sanskrit literature is in verse, so that for the sake of metrical convenience, various devices were resorted to in the formation of number expressions, the most common being the use of the additive² method. The following are a few examples of common occurrence taken from mathematical works :

v1.c7.s4.40

² 3339 = trîṇi śatâni trîsahasrâṇi triṁśa ca nava ca, i.e., “three hundreds and three thousands and thirty and nine.” (*Ṛgveda*, iii. 9. 9; also x. 52. 6.)

v1.c7.s4.41

*Subtractive:*
(1) the number 139 is expressed as 40+100-1;³
(2) 297 is expressed as 300-3.⁴

v1.c7.s4.42

³ *GSS*, i. 4: *cattvâriṁśascaikona śatâdhika* (“forty increased by one-less-hundred”).

v1.c7.s4.43

⁴ *L*, p. 4, Ex. 1: *Trihînasya śata-trayasya* (“three less three hundred”).

v1.c7.s4.44

*Multiplicative:*
(1) the number eighteen is expressed as 2 × 9;⁵
(2) twenty-seven is expressed as 3 × 9 and 12 as 2 × 6;⁶
(3) 28,483 is expressed as 83+400+(4000 × 7).⁷

v1.c7.s4.45

⁵ *Â*, ii. 3: *dvi-navaka*.

v1.c7.s4.46

⁶ *Triś*, Ex. 43: *tri-navaka* (“three nines”), *dvi-ṣaṭ* (“two sixes”).

v1.c7.s4.47

⁷ *GSS*, i. 28: *tryaśîtimiśrâṇi catuśśatâni catussahasraghna nagânvitâni* (“eighty-three combined with four hundred and four thousand multiplied by seven”).

v1.c7.s4.48

The expression of the number 12345654321 in the form "beginning with 1 upto 6 and then diminishing in order" is rather interesting.¹

v1.c7.s4.49

¹ *GSS*, i. 27: *ekâdiṣaḍantâni krameṇa hînâni*.

v1.c7.s4.50

What are known as alphabetic and word numerals were generally employed for the expression of large numbers. A detailed account of these numerals will be given later on.

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.

6. Kharoṣṭhī Numerals

v1.c7.s6.1

**Early Occurrence.** The *Kharoṣṭhī lipi* is a script written from right to left. The majority of the Kharoṣṭhī inscriptions have been found in the ancient province of Gāndhāra, the modern eastern Afghanistan and the northern Punjab. It was a popular script meant for clerks and men of business. The period during which it seems to have been used in India extends from the fourth century B.C. to the third century A.D. In the Kharoṣṭhī inscriptions of Aśoka only four numerals have been found. These are the primitive vertical marks for one, two, four, and five, thus:

v1.c7.s6.3

More developed forms of these numerals are found in the inscriptions of the Śakas, of the Pārthians and of the Kuṣânas, of the 1st century B.C. and the 1st and 2nd centuries A.D., as well as in other probably later documents. The following are some of the numerals of this period:

v1.c7.s6.5

**Forms and their Origin.** It cannot be satisfactorily explained why the number four, which was previously represented by four vertical lines came to be represented by a cross later on. The representation of the numbers five to eight follows the additive principle, with four as the base. This method of writing the numbers 4 to 8 is not met with in the early records of the Semites. We do not know how the number nine was written. It is very probable that it was written as [Figure:] , *i.e.*, 4+4+1 (reading from right to left, the order being the same as that of the script). The number 10 has an entirely new sign. The question why it was not written as [Figure:] , or why the base [Figure:] (4) was abandoned cannot be satisfactorily answered.

v1.c7.s6.6

It is accepted by all that the Kharoṣṭhî is a foreign script brought into India from the west. The exact period at which it was imported is unknown. It might have been introduced at the time of the conquest of the Punjab by Darius (*c.* 500 B.C.) or earlier.¹ The numerals given above undoubtedly belong to this script as they proceed from right to left.

v1.c7.s6.7

¹ The theory of the foreign origin of the script has to be revised in the light of the discoveries at Mohenjo-daro and Harappa, especially in view of the fact that the Mohenjo-daro alphabet ran from right to left.

v1.c7.s6.8

The old symbols of the inscriptions of Aśoka, however, seem to have undergone modification in India, especially the numbers from 4 to 19. The symbols for four and ten seem to have been coined in India, in order to introduce simplification and also to bring the Kharoṣṭhî numeral system in line with the Brâhmî notation already in extensive use. The symbol [Figure:] seems to have been derived by turning the Brâhmî symbol [Figure:] which represents 4 in the inscriptions of Aśoka. The inclined cross to represent 4 is found in the Nabatean numerals in use in the earlier centuries of the Christian Era.² The Nabatean numerals resemble the Kharoṣṭhî also in the use of the scale of twenty and in the method of formation of the hundreds. It is possible that the Semites might have borrowed the Kharoṣṭhî symbol for 4, although it is not unlikely, as Bühler thinks, that the symbol might have been invented independently by both nations.

v1.c7.s6.9

² J. Euting, *Nabatäische Inschriften aus Arabien*, Berlin, 1885, pp. 96-97.

v1.c7.s6.10

The numeral [Figure:] (10) closely resembles the letter *a* of the Brâhmî alphabet. The symbol for twenty [Figure:] appears to be a cursive combination of two tens. It resembles one of the early Phoenician forms found in the papyrus Blacas¹ (5th century B.C.). The mode of expressing the numbers 30, 40, etc., by the help of the symbols for 10 and 20, is the same as amongst the early Phoenicians and Aramaeans.

v1.c7.s6.11

¹ Bühler, *Palaeography*, p. 77; Ojha, *l.c.*, p. 128; see Table II(*b*).

v1.c7.s6.12

The symbol for 100 resembles the letter *ta* or *tra* of the Brâhmî script, to the right of which stands a vertical stroke.

v1.c7.s6.13

The symbols for 200, 300, etc., are formed by writing the symbols for 2, 3, etc., respectively to the right of the symbol for 100. This evidently is the use of the multiplicative principle, as is found amongst the early Phoenicians.²

v1.c7.s6.14

² See Table II(*c*).

v1.c7.s6.15

The formation of other numbers may be illustrated by the number 274 which is written with the help of the symbols for 2, 100, 20, 10 and 4 arranged as

v1.c7.s6.17

in the right to left order. The 2 on the right of 100 multiplies 100, whilst the numbers written to the left are added, thus giving 274.

v1.c7.s6.18

The ancient Kharoṣṭhî numerals are given in Table I.

7. Brâhmî Numerals

v1.c7.s7.1

**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:

v1.c7.s7.2

¹ *Cf.* Langdon's opinion in *Mohenjo-daro and the Indus Valley Civilisation*, ch. xxiii.

v1.c7.s7.4

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:

v1.c7.s7.5

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.

v1.c7.s7.7

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:

v1.c7.s7.8

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.

v1.c7.s7.10

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.

v1.c7.s7.11

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.

v1.c7.s7.12

¹ It has been incorrectly stated by Smith and Karpinski that the Nânâghât forms were vertical. See *Hindu-Arabic Numerals*, p. 28.

v1.c7.s7.13

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.

v1.c7.s7.14

**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.

v1.c7.s7.15

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

v1.c7.s7.16

¹ Compare the same number written in Kharoṣṭhî, p. 24.

v1.c7.s7.17

**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:

v1.c7.s7.18

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.

v1.c7.s7.19

² Inscriptions of Aśoka, *Corpus Inscriptionum Indicarum*, Vol. I, p. 52.

v1.c7.s7.20

¹ *Mohenjo-daro etc.*, Chap. xii. This view is strongly supported by Hunter, *l.c.*, p. 490.

v1.c7.s7.21

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.

v1.c7.s7.22

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.

v1.c7.s7.24

¹ Bühler, *On the Origin of the Indian Brâhma Alphabet*, Strassburg, 1898, pp. 52, 53 foot-note.

v1.c7.s7.25

² *Cf.* Smith and Karpinski, *Hindu Arabic Numerals*, pp. 30-1.

v1.c7.s7.26

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).

v1.c7.s7.27

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."

v1.c7.s7.28

³ Bühler, *l.c.*, p. 82.

v1.c7.s7.29

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.

v1.c7.s7.31

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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