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Mathematics in Ancient and Medieval India c5: Chapter II Arithmetic

Mathematics in Ancient and Medieval India c5

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Chapter II Arithmetic

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The necessity for the development of a systematic method of counting in an organised social life led to the birth of arithmetic in all countries at the dawn of civilization. The Babylonians developed a sexagesimal system of expressing numbers in cuneiform writing on clay tablets. The Egyptians developed a system of numeration, using hieroglyphic notation besides hieratic and demotic ones. Fractions were a subject of great difficulty to the Egyptians as well as to the Babylonians, Greeks and Romans. Indians were responsible for the introduction of the decimal place-value system and more or less modern form of numerals with the symbol for zero, for the development of simplified and straight forward methods of fundamental operations, and for the calculation by rule of three. The Indian numerals with zero, the principle of position (decimal place-value) and some of the fundamental operations found their way to Arabia by the efforts of Abbāsid Khalifs (8th to 10th century A.D.) at Bagdad.

Numbers

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The Sanskrit names for the counting of numbers used in Vedic India from one to nine are : *eka*, *dvi*, *tri*, *catur*, *pañca*, *ṣaṭ*, *sapta*, *aṣṭa*, *nava*. Besides 10, multiples of 10 upto 100, separate names for numbers were recognised and named as follows : *daśa* (10), *viṁśati* (2× 10), *triṁśat* (3× 10)...etc. For larger numbers, ten and multiples of ten had formed the basis of numeration. While the Greeks had no terminology for denomination above myriad ( 10⁴ ) and the Romans above the mill ( 10³ ), the ancient Indians dealt freely with upto twelve denominations. The *Yajurveda Saṃhita* thus gives : *eka* ( 1 ), *daśa* ( 10 ), *śata* (10²), *sahasra* (10³), *ayuta* (10⁴), *niyuta* (10⁵), *prayuta* ( 10⁶ ), *arbuda* ( 10⁷ ), *nyarbuda* ( 10⁸ ), *samudra* ( 10⁹ ), *madhya* ( 10¹⁰ ), *anta* ( 10¹¹ ) and *parārdha* ( 10¹² ). The same list occurs in the *Taittirīya Saṃhita*, *Maitrāyaṇī Saṃhita* and others.¹ For expressing compound numbers lying between 10 and multiples of 10 upto 100, addition of simple numbers (from 1 to 9) was made to the required multiples. For example, *ekādaśa* (=1+10=11) *saptaviṃśati* ( =7+2 × 10=27 ), *aṣṭatriṃśat* ( =8+3 × 10=38 ) etc. Compound numbers above 100 were expressed as follows : *ṣaṣṭiṃ sahasrā sapta śatāni navatiṃ nava* (=60 × 10³ +7 × 10²+9 × 10 +9=60799 ). In certain special cases, the principle of subtraction was also in evidence. Thus number 19, 29 etc. were expressed as *ekānnaviṃśati* ( =20 – 1=19 ), *ekānna triṃsat* ( 30 – 1=29 ) respectively. In the *Sūtra* period, *ekānna* was changed to *ekona*. The principles explained above were also followed in the *Śulba-sūtras* and later works. In one instance, 972 is expressed as *aṣṭaviṃśatyūnaṃ sahasram*=1000–28 ( *Āpastamba-śulba*, 5. 7 ). The Babylonians ( c. 1600 B. C. ) introduced earlier sexagesimal scale of notation.² There is no evidence that the decimal scale of the Indians was derived from the Babylonians, though the principle might be the same. The decimal scale, as it could be seen afterwards, is in fact far more convenient and simple and is a natural way of expressing numerical value which has been adopted subsequently all over the world as the ideal system.

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¹ *Yajurveda Saṃhitā*, 17. 2; *Taittirī ya Saṃhitā*, 4. 40. 11. 4 and 7. 2. 20. 1; *Maitrāyanī Saṃhitā*, 2. 8. 14; *Kāṭhaka Saṃhitā*, 17. 10; vide also *History of Hindu Mathematics* by Datta and Singh, Part I, Motilal Banarasi Das, Lahore, p. 9, 1935.

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² Cajori, F. *A History of Mathematics*, 2nd revised and enlarged edition, p. 4.

Classification of Numbers

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Classification of numbers into odd and even first appeared in the Vedic literature.¹ The classification also appeared in the Jaina works but the authors did not consider unity a number like the ancient Greeks. In the Jaina work, *Anuyogadvārasūtra*², the numbers were classified also as numerable ( *saṃkhyeya* ), innumerable ( *asaṃkhyeya* ) and infinite ( *ananta* ). The highest numerable number of the Jaina corresponds to Alef-zero³ of the modern mathematics and they also made an attempt to define number beyond Alef-zero. As regards innumerable number, names of several sub-classifications were mentioned but the idea was not very clear. Idea of infinity was expressed in connection with different dimensions namely infinity in one, two, three and infinite dimensions.

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¹ *Taittirīya Saṃhitā*, 7. 2. 12-13.

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² Datta, B. 'The Jaina School of Mathematics', *Bulletin of the Calcutta Math. Society*, 21, p. 136-38, 1929.

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³ The cardinal number of the aggregates of all the finite integers 1, 2, 3, ...... n is called Alef-Zero. The theory of such numbers was fully developed by G. Cantor in 1883 A. D.

Indices

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For representation of large numbers, different specifications seem to have been necessitated to indicate them correctly. For instance, the raising of a number to its own power was technically known as *varga saṃvargita*. First *varga saṃvargita* of a = aa = b, say; second *varga saṃvargita* of a = bb = (aa)aa = c, say. Third *varga saṃvargita* of a = cc etc. In this process third *varga saṃvargita* of 2 comes out to be (256)²⁵⁶, a number higher than the number of electrons in the universe. To express a large number, Jaina authors (c. 300 B.C.) employed powers of integers now known as laws of indices, examples like : $am × an = am+n, (am)n = amn$ are found in their texts.¹

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¹ *Bulletin of the Calcutta Mathematical Society*, 21, p. 136-38, 1929.

Logarithms

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The expression log ₂ 8 = 3 means the number 8 can be divided by 2 three times. This was defined by Napier (1550–1617) who has been given priority with regard to the publication of a table of logarithms and an account of their meaning and use. In the earliest centuries of the Christian era Jainas (vide *Dhavalā* commentary of Vīrasenācārya) conceived the idea of logarithms to the base 2, 3, 4, though no general use of the idea seems to have been made by them. They used *ardhaccheda*, *tṛkaccheda*, *caturthaccheda* of a number, as the number of times it can be divided by 2, 3, 4, etc. For instance *ardhaccheda* of x = log ₂ x, *tṛkaccheda* of x = log ₃ x etc. Moreover, there is evidence in the *Dhavalā*² commentary that the following rules were known to them.

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² Singh, A. N. ‘History of Mathematics in India from Jaina Sources’, *Jaina Antiquary*, 15, no. 2, 46-53, 1949.

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(1) log m/n=log m-log n

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(2) log (m. n)=log m+log n

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(3) log ₂ (2)m=m

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(4) log (xx)=2x log x

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(5) log log (xx)=log x+1+log x; the expression log log x is known as vargasalaka of x.

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(6) log (xxx)=xxlog xx etc.

Fraction

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Few fractions are mentioned in the Vedic literature. *Ardha*, *pāda*, *sapha* and *kala* denote ½, ¼, ⅛, 1/16 respectively. But only the first two are common. The word *pāda* was derived from four parts of a stanza; a stanza in Sanskrit literature is divided into four parts from the Vedic times, each of which is called a *pāda*. The idea of a fraction of 1/16 was derived from the increasing visibility of the moon from the new moon to fullmoon covering the period of sixteen days and thus the moon is said to possess 16 *kalas* or parts. Mention of the number of fractions are also found in the *Ṛgveda*, *Taittirīya Saṃhita*, *Taittirīya Brāhmaṇa* etc.¹ In the *śulbas*², the fraction is denoted by the term *bhāga*, e.g., *pañcamabhaga*=⅕, *daśamabhaga*=1/10 etc. A number followed by a fraction was denoted in the *śulba* as follows : *ardhacaturtha* =3½, *ardhapañcaśat*=49½ etc. Four fundamental arithmetical operations with elementary fractions were known to the *Śulbakāras*. Some expressions other than unit fractions were also used in the *Śulbasūtras*¹ (c. 600 B.C. ), *Arthaśāstra*² of Kauṭilya (3rd century B. C. ), *Vedāṅgajyotiṣa*³ (200 B.C. ), *Sūryaprajñapti*⁴ (200 B.C. ), etc. In the *Śulba-sūtras*, fractions were expressed in language. The *Bakhshālī Ms.* (4th century A.D. )⁵ first expressed operations of fraction in terms of symbols both with and without the dividing lines. This is followed in the *Brāhmasphuṭasiddhānta* of Brahmagupta (628 A.D.) and *Gaṇitasārasaṁgraha* of Mahāvīra (c. 850 A.D.) and in later works. For addition and subtraction of fractions, Brahmagupta suggested the operation of reduction to a common denominator. The works of Mahāvīra

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¹ Datta, B. and Singh, A. *History of Hindu Mathematics*, Part I, Motilal Banarasi Das, Lahore, p. 185, 1935.

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² Datta, B. *The Science of Śulba*, University of Calcutta, p. 212-16, 1932.

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¹ *caturtha-saviśeṣa saptama*=1/7(¼√2. (*Āpastambaśulba*, 19. 4; 19. 7); *caturdaśa prakramān triṁśca prakrama saptabhāgān*=14 3/7 prakramas (*Kātyāyanaśulba*, 6. 2) etc.

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² Shamasastry, R. *Kauṭilya's Arthaśāstra* (Eng. tr. ), fifth ed., p. 113–21, Mysore, 1956.

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³ *kalā daśa saviṁśa nāḍikā*=10 1/20 (*Vedāṅga Jyotiṣa*, v. 38).

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⁴ *do joyanāhaṁ addhaduyātvaliśate sitayabhāge*=2(42⅛)/183 (*Sūryaprajñapti*, sūtra 18).

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⁵ 11 / 1 yu 5 / 1 means 11+5; 2 / 1 / 2 represents 2½; 1 / 1 / 2 as 1½;

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|1 & 1 & 1 & 1 / 1 & 1 & 1 & 1 / 2+ & 3 & 4+ & 5| bha|36 / 1| means 36/((1-½)(1+⅓)(1-¼)(1+⅕))

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and Śrīdhara contain special rule for their reductions to proper fractions. The fraction was denoted as *bhinna* by Mahāvīra (c. 850 A. D.) and Śrīdhara (c. 903 A. D.). The numbers of the form $(a/b±c/d), (a/b of c/d), (a/b±c/d of a/b)$ were termed *jāti* (i.e., classes).

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Amongst other nations¹, Babylonians (c. 2000 B.C.) on their cuneiform clay tablets included some special symbols for ⅔, 2/18, 4/18 and 5/7 etc. Egyptians, Greeks as well as the Romans dealt with unit fraction. Chang Chiu-Chien (6th century A.D.) remarked in the preface of his *Arithmetical classic* that the chinese found it difficult to express a fractional number and to perform any operation with them. Even as late as 10th century A.D. Rabbi Saadia ben Joshep al-Fayyaumi, a Hebrew writer living in Egypt made wide use of the unit fraction in his computations.

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¹ Smith, D. E. *History of Mathematics*, 2, Dover Publication, p. 210-215.

Numerals

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The early symbolic numerals of the world were of various types. The characters varied to meet the linguistic conditions of the different cultural areas in different parts. Here main types of numerals used by the Ancient Babylonians, Egyptians, Chinese, Greeks, Romans, Indians and Arabians are tabulated below.

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TABLE OF NUMERALS

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TABLE

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( Contd. )

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| | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 20 | 30 | 50 | 60 | 70 | ... | 100 | 200 | 300 | ... | 1000 | 2000 | 4000 | 5000 | 6000 | 10000 |
| (d) *Greek* Geometric (5th century B. C.) Attice system | | | | | | | | | | | | | | | | | | | | | | | | | | | |
| Alphabetic system ( 300 B. C. ) | | α | β | γ | δ | ε | | ζ | η | θ | ι | κ | λ | | | | | ρ | σ | τ | | α | β | | | | δ |
| (e) *Roman* | | I | II | | | V | | | | | X | | | L | | | | C | | | | M | | | | | |
| (f) *Indian* Kharoṣṭhī ( 250 B. C. ) | | I | II | III | × | I× | II× | | | XX | 7 | 3 | | 733 | | | | λ | λII | | | | | | | | |
| Brāhmī-Aśokān ( 250 B. C. ) | | - | = | | + | | 6 | | | | | | | C | | | | H | | | | | | | | | |
| Nānāghāt ( 150 B. C. ) | | = | | | | | | 4 | 7 | | α | α | O | | | | | 21 | 21 | | | 4 | | | | | |
| Nāsik ( 100 A. D. ) | | = | | | | h | 4 | 7 | y | ? | α | | ν | | | | J | τ | F | | 9 | 9 | | | | |
| Bakhshālī Ms. ( 400 A. D. ) | • | ~ | 3 | 3 | y | 4 | | 7 | 9 | → | place | value. | | | | | | | | | | | | | | |
| Sanskrit form (10th century A. D.) | 0 | 9 | 2 | 3 | 8 | y | ξ | υ | 2 | ε | → | place | value. | | | | | | | | | | | | | | |
| (g) *Modern Arabic* (10th century A.D. | • | 1 | ۲ | ۳ | ε | 0 | 4 | V | ∧ | 9 | → | place | value. | | | | | | | | | | | | | | |

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(a) *Babylonians* : The Babylonians wrote with stylus on clay tablets the result being wedge-shaped characters. The tablets were then baked in the sun. The numbers were usually expressed with the help of a sexagesimal scale. Practically 59 symbols from one to fifty-nine (made by two basic symbols for one and ten) were used. Whenever a symbol for sixty was required, it was replaced by 1 of the next higher order as follows : 1.10 = 1 × 60 + 10 = 70 2.12 = 2 × 60 + 12 = 132

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In this way a big member was expressed by 1. 41. 33. 59. 3. 45. This obviously involves an idea of place-value. The numerical system is not very happy, because, a comparatively greater amounts of repetition is unavoidable for obtaining a number below fiftynine. In many cases, the numerical value of a symbol had to be determined from the context.¹ For example, in certain tablet 71 is represented by symbols of 1.10.1 i.e. 1 × 60 + 10 + 1 = 71. The number 171½ was represented by symbols of 2.50. 1. ½ i. e. 2 × 60 + 50 + 1 + ½ = 171½. The symbol for 1.12 may be expressed as 1.12.0 = 4320 or as 1.12 = 72 or as 1; 12 = 1⅕ etc. written sexagesimally.

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¹ Smith, D. E. *Ibid.*, 2, p. 37-38; Neugebauer, 0. *The Exact Sciences in Antiquity*, p. 20, 26, Copenhagen, 1951.

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(b) *Egyptians and the Semitic Groups* : The Egyptians used stone, papyrus, wood and pieces of pottery for writing. They in their Hieroglyphic system of writing possessed separate symbols for one, ten, hundred, thousand, ten thousand, hundred thousand and million. The Hieratic (religious) and Demotic (popular) system of writings had signs of symbols for one, five to ten, twenty, thirty,......to hundred. The Hieroglyphic and Hieratic numbers were commonly written from right to left and also from left to right and sometimes as in early inscriptions from top downwards. The numbers were expressed by repetition and addition of these symbols as necessary. As for example, the number 23 was represented by IIIUU i.e. 1+1+1+10+10=23; this was also written as UUIII. They therefore had little idea of place value.

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The Phoenician use symbols for one, ten, twenty and hundred; the Palmyrene for one, ten, twenty and hundred. There is no evidence that they had any idea of place value.

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(c) *Chinese* : Of the several systems used by the Chinese, those of Shang oracle bone form ( 14th to 11th century B.C. ), Bronze and Coin form ( 10th to 3rd century B.C. ) and the Counting rod forms ( 2nd century B.C. to 4th century A.D. ) deserve special notice in this connection.¹ Their symbolic numerals were written usually from top downwards with their values arranged in the descending order.

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¹ Needham, J. *The Science and Civilisation in China*, 3, p. 6-7, Tables 22 and 23, Cambridge, 1957.

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In the Shang oracle bone form, separate symbols for one, five, six, seven, eight, nine, ten, twenty, hundred, thousand etc. were used. For two to four, the symbol for one was repeated and other numbers were obtained by similar repetitions. For instance, 162 was expressed by the symbols of 100, 60 and 2 placed in the left to right order.

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In the Bronze and Coin form (10th century B. C. to 3rd century B. C. ) and other forms found on coins of Chou period (6th to 3rd century B.C.) more or less similar symbols as those of Shang oracle bone were employed. Neither improvement of method nor use of place-value was known to them.

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The Chinese in their Counting rod forms between 2nd century B.C. to 4th century A.D. used separate symbols for one, six, ten, sixty. Symbols for one was repeated to express numbers upto five and the values for seven to nine were represented by successive addition of symbols for one to that of six. For the numbers of the next higher order than 100, different symbols were used.

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(d) *Greeks* : The Greeks initially were deeply interested in numbers which were connected with geometric forms.¹ They had two other systems of notation—the Attice system and the Alphabetic system. The former, made of upright strokes and initials of several number names, was no better than the systems prevalent among the Egyptians and other contemporary civilisations. This system was replaced at about third century B.C. by the alphabetic system of numerals which were commonly known among the Hebrews, the Syrians, the Arabs, the Persians and other Semitic people. The alphabetic system had twenty-four letters in their common Ionic alphabet, to which they added three Phoenician alphabets to make it twenty-seven. These are divided into three groups, first nine alphabets denoting nine units, second nine alphabets denoting nine multiples of ten and the third nine alphabets denoting nine multiples of hundred. They wrote from left to right. In writing a compound number, the element of the higher denominations was put first. For examble, 43,678 was represented by the symbols of four myriad ( one myriad=ten thousands ), three thousand, six hundred, seventy, and eight maintaining an additive principle.

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¹ Smith, D. E. *History of Mathematics*, 2, p. 24-25, Dover Publication.

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(e) *Romans* : The Roman numbers were expressed in terms of symbols for one, five, ten, fifty, hundred, five hundreds, and thousand. The normal mode of writing was to put the greater elements in the beginning. For example, the number 1548 was written as MCCCCCXXXXVIII i.e. 1000+500+40+8. In certain cases, it was reversed to denote subtraction. But on a longer view, i.e. in a compound number of still greater value, it will be found that the normal mode has been preserved on the whole. The system had the advantage of using only few symbols for expressing any number in a very simple way. This is the reason why it received a considerable popularity and is still adopted occasionally in numbering the chapter and year of books.

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(f) *Indian* : It has already been discussed how the word numerals on a decimal scale were used by the Vedic Indians to express numbers. Regarding notational use, the earliest example of the use of symbols for the expression of numbers in India is provided by Kharoṣṭhī ( 250 B.C. ) and Brāhmī ( 250 B. C. ) inscriptions.¹

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¹ Bhau Daji, ‘The Ancient Sanscrit Inscriptions in the cave inscriptions’, *Journal of the Bombay Branch of the Royal Asiatic Society*, 8, p. 231, 1868; Smith, D. E. *History of Mathematics*, 2, p. 66, Dover Edition; Das, Sukumar Ranjan, ‘The Origin and Development of Numerals’, *Indian Historical Quarterly*, 3, p. 97-120, 356-57, Calcutta, 1927.

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The Kharoṣṭhī was particularly script of North-Western India (including Afganisthan and the Northern Punjab). It disappeared in the 3rd century A.D. In the Kharoṣṭhī numerals, there were separate symbols for one, four, ten, twenty and hundred. They were written from right to left with lower elements before the higher. For example, the number 122 was expressed by symbols of 2, 20, 100, and 274 by 4, 10, 20, 20, 20, 200. But in cases of the hundred and its multiples, the symbols for the smaller elements were placed after the symbol for hundred to denote multiplication. All these were common with the principles underlying other systems of Semitic numeral notation which we have discussed before.

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The Brāhmī numeral system consisted of separate signs for one and four to nine, two and three were denoted by repetitions of the symbols for one. It had also separate symbols for ten and its multiples as well as four hundreds upto seventy thousands. The plan was indeed nearly the same, as those of the Demotic, Hieratic and the Chinese. The symbols were written from left to right beginning with symbols of higher values. For example, 274 was written with the help of the symbols for 200, 70 and 4. The Nānāghāt and the Nāsik cave inscriptions amongst few other inscriptions namely Kuṣāṇa (150 A.D.), Kṣatrapa (200 A.D.) and Gupta (c. 300-450 A.D.) are noteworthy. Both these types (vide Table) bear a close relation and resemble each other. According to many foreign scholars, both Nānāghāt (150 B.C.) and the Nāsik (100 A.D.) types of numerals form the basis of our third or the modern numeral system.

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This third numeral system, (given in the *Bakhshālī MS.*—vide Table) which contains only ten symbols from one to nine and the symbol for zero could express any number however large on decimal place-value system. In this system the values of the notational places, *eka*, *daśa*, *śata* etc. increase either from right to left or from left to right. The numbers are always read from higher to lower order. This system is identical with the system which as we have seen appeared later amongst the Arabs during the tenth century of the Christian era. From there it went to Europe. And this system has now been adopted by all the civilized peoples of the world.

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(g) *Arabic*: Bīrūnī (973 1048 A. D.) has referred to two types of notation of numbers namely, the alphabetic (*abjd*) system as *Hurūf al-jummal* or *Hiṣāb al-jummal* and the modern numerals as *al-arqam al-hind* (*Indian numerals*). Numerous applications of alphabetic numerals are also available in his *Chronology of the Ancient Nations*.¹ Bīrūnī in his *Kitāb al-tafhim li-awā'il sinā'at at-tanjīm* (The Book of Instructions in the Art of Astrology) has given both these notation of numbers. The alphabetic system (*Hurūf al-jummal*), as used by Bīrūnī was apportioned in orderly sequence to 26 alphabets though initially there was no particular order in their use. It was based on the values of the alphabets without use of decimal place order. This notation seems to have been used exclusively by Arabian astronomers. The other system (*al-arqam al-hind*) given by Bīrūnī and others is shown in the table (vide p. 60). In the tenth and a few following centuries it is found that several mathematical works were composed by the Arab scholars having titles containing the words, *al-hind*, *al-hindi*, *al-takht*, *al-ghobar*.¹ These contained methods and systems very closely resembling those of the Indians.

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¹ Sachau, E. C. *The Chronology of the Ancient Nation*, London, 1879; the work is a English translation ofthe Arabic original.

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¹ Datta, B. 'Testimony of Foreign Scholars on the Origin of the Decimal Place-value Numerals', *Bulletin of the Calcutta Mathematical Society*, 24, p. 193, 1932; Saidan, A. S. 'The Developement of Hindu Arabic Arithmetic', *Islamic Culture*, 39, p. 209-21, 1965.

Symbol for Zero and Decimal Place-value System

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² Bag, A. K. 'Symbol for Zero in Mathematical notation in India', *Boletin de la Academia Nacional de Ciencias*, Tome 48, Primer Congreso Argentino de Historia de la Ciencia, Primera Parte, p. 274-54, 1970.

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The concept of 'nothingness' or 'void' which is inherent and instinctive to man of all ages found its expression in the philosophical and literary works of all ancient civilised nations. The feeling of necessity to give a symbolical expression of this concept in mathematical calculation which would permit the representation of a number of only high order without any limit led ultimately to discovery of a symbol for what is now known as zero.

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The concept of 'nothingness' or 'void' in mathematical system was first introduced by the Babylonians ( 14th century B.C. ) who left an empty space in between two other numeral symbols along with their sexagesimal place-value system.³ But this gave rise to much confusion for giving any quantitative expression of the numerical value involved as there can be no fixed standard for an empty space to allow for its unambiguous repetitions. It therefore, helped very little in the symbolical expression of mathematical figures.

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³ Neugebauer, O. *The Exact Sciences in Antiquity*, p. 20, Copenhagen, 1951.

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In certain Greek astronomical Papyri Mss. ( 100 B.C. ) as well as Ptolemy's table of chords ( 150 A.D. ), symbols resembling that of modern zero were used to indicate the ending of a sentence or a marginal gap.¹ According to some scholars, this served as the origin of the introduction of the symbol for zero in mathematical notation. There is no justification of such a view as their application in mathematical calculation could not be traced in any of the ancient contemporary Greek mathematical works or thereafter.

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¹ Neugebauer, O. *Ibid.*, p. 10, 13-14.

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In India, the application of zero as a part of the numeral system, is characterised by its two types of uses as : A ) word-numerals, B ) symbolic-use.

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(A) *Word-numerals* : The word-numerals were based on a decimal scale of notation namely *eka*, *daśa*, *śata*, *sahasra*......upto 12 places from the time of *Saṃhitas* (1500 B.C.).² A large number like, *ṣaṣṭhīm sahasrā sapta śatāni navatim nava* ( 60, 799 ) was expressed by placing the word numerals of higher order to be followed by one of lower order in the scale. This successive placing of, *sahasra*, *śata*, *daśa*, *eka* for the expression of a large number in left to right order of gradually decreasing values obviously shows that the idea of decimal place-value was implied.¹ The scale is undoubtedly more convenient than the Babylonian sexagesimal scale. The *Śatapatha Brāhmaṇa*, *Taittirīya Brāhmaṇa*, *Chāndogya Upaniṣad*, *Vedāṅga Jyotiṣa* etc. had similar uses of numerals. The idea of decimal place-value developed when the decimal scale was associated with the value of the places in a right to left order (or left to right) in the scale with the introduction of symbol for zero. That means, the symbols or word numerals took values × 1, × 10, × 100.........when placed in 1st, 2nd, 3rd places......respectively. From the beginning of the Christian era, the word-numerals with their various synonyms began to be used to avoid repetition of the same word and to keep rythm of ślokas.

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² *Yajurveda Saṃhitā*, 17. 2; *Taittirīya Saṃhitā*, 4. 40. 11. 4 and 7. 2. 20. 1; *Maitrāyaṇī Saṃhitā*, 2. 8. 14; *Kāṭhaka Saṃhitā*, 17. 10; vide also *Hist. of Hindu Mathematics* by Datta and Singh, Part I, Motilal Banarasi Das, Lahore, p. 9, 1935.

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¹ Datta, B. ‘Vedic Mathematics’, *The Cultural Heritage of India*, old series, 3, p. 395-97, Ramkrishna Centenary Memorial, Belur Math, Calcutta.

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A few synonyms of 0, 1, 2, 3, etc. may be given as follows :

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0 *śunya, kha, ākāśa, ambara, vyoma, nabha, pūrṇa* etc. 1 *kṣiti, dharā, pṛthvī, bhū, bhūmi, indu, candra, abja, vidhu, śaśāṅka* etc. 2 *yama, aśvin, dasra, akṣi, netra, nayana, bāhu, kara* etc. 3 *rāma, guṇa, agni, bhuvana, loka* etc.

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The application of nine word numerals on the decimal place-value scale was found in Sanskrit works and incriptions from 2nd century A.D. onwards. Several examples are given below :—

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(1) *Pauliśasiddhānta* (200 A.D.) *kha-kha-rūpa-aṣṭa*¹ 0 0 1 8 =8100.

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¹ *Pañcasiddhāntika* of Varāha, ed. by Thibaut, ch. 6, Verse 9.

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(2) *Sūryasiddānta* (400 A.D.)
(i) *nava-vasu-sapta-aṣṭa-kha-nava-aśvi*² 9 8 7 8 0 9 2 =2908789.
(ii) *svara-eka-pakṣa-ambara-svara-ṛtu*³ 7 1 2 0 7 6 =670217.

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² Vide *Pañcasiddhāntikā* of Varāhamihira (ed. by G. Thibaut) Ch. 9, v. 3.

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³ *Ibid*, Ch. 9, v. 2.

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(3) *Pañcasiddhāntika* (505 A.D.)
(i) *śunya-dvi-pañca-yama*⁴ 0 2 5 2 =2520
(ii) *muṇi-yama-yama-dvi* 7 2 2 2 =2227

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⁴ Ch. 1, v. 17. (Thibaut's edition).

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(4) *Inscription at Cambodia*⁵ (604 A.D.) *rasa-dasra-śarai śakendra varṣe* 6 2 5 =526 śaka year.

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⁵ Coedès, G. ‘A propos de l'origine des chiffres arabes’, *Bulletin of the School of Oriental Studies*, 6, p. 352, 1930-32.

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(5) *Inscription at Champa*⁶ (731 A.D.) *rāma-artha-ṣaṭkai śaka* 3 5 6 =653 śaka year

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⁶ Huber, E. ‘Études Indo-Chinoises’, *Bulletin de l'école Française, d'extrent orient*, 11, p. 266-67, 1911.

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(6) *Inscription at Java*⁷ (732 A.D.) *śruti-indriya-rasa śaka* 4 5 6 =654 śaka Such uses of word-numerals (i. e. same or synonymous words taking different values when placed at different places) are to be found in the *Brāhmasphuṭasiddhānta* and *Khaṇḍakhādyaka* of Brahmagupta (628 A D.) and in almost all later mathematical and astronomical works. The use of the word-numeral system is also found to occur in many of the inscriptions of Cambodia, Java, Champa, Malay etc. at about 7th century A. D.

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⁷ Coedès, G. *Ibid.*, p. 326.

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(B) *Symbolic-use* : A good number of inscription using numeral symbols are also known. Some of these inscriptions used a point-symbol (*śunya-bindu*) as well as circular symbol for zero. The earliest evidence of the symbol for zero is found in the *Bakhshālī Ms.*¹ (400 A.D.). A chronological list of the appearance of the symbols of numerals from one to nine along with or without that of zero fitted in a decimal place-value scale, is given below :

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¹ Vide Hoernlé, A.F.R. ‘The Bakhshālī Ms.’, *Indian Antiquary*, *17*, p. 33-48, 1888; Kaye, G. R. The *Bakhshālī Ms.*, Parts I & II, New Imperial Series No. 43, Calcutta, 1927; Datta, B. ‘The *Bakhshālī* Mathematics’, *Bulletin of the Calcutta Mathematical Society*, *21*, p. 1-60, 1929. The Bakhshālī Ms. was composed in the second century A. D. (Hoernlé, and Datta) and the present Ms. was probably a commentary.

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(1) *Bakhshālī* Ms. (400 A.D.)

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| *Symbolic expression* | | *Number* |
| (i) [Figure:] | (17 verso) | 330 |
| (ii) [Figure:] | (22 recto) | 157500 |
| (iii) [Figure:] | (22 recto) | 947 |
| (iv) [Figure:] | (56 recto) | 846720 |

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(2) *Gurjara Grant inscription*¹ ( 595 A.D. )

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¹ Epigraphica Indica, 2, p. 20, 1894.

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[Figure:] 346

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(3) *Khmere inscription at Sambor*² ( 683 A.D. )

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² Coedès, G., *Ibid.*, 6, p. 326, Plate IV.

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[Figure:] ( śaka ) 605 ( śaka )

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(4) *Malay inscription at Palembang*³ ( 684 A.D. )

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³ Westenenk, L. C. *Djāwā* periodical, *1*, no. 1. p. 10, vide plate I, vide also *Acta orientalia*, 2, p. 12-21, 1924.

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[Figure:] ( śaka ) 606 ( śaka ) [Figure:] ( śaka ) 60 ( śaka )

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(5) *Malay inscription at Kotakapur*⁴ ( 716 A.D. )

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⁴ Kern, H. Inscriptie van Kota Kapur, *Bidragen tot de Taal-. Land-en Volkenkunde van Nederlandische...*, 67, p. 393-400, 1913; Kern, H. *Verspreide Geschriften*, 7, p. 208-209, 1917; Coedès, G. *Ibid.*, plate IV.

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[Figure:] ( śaka ) 608 ( śaka )

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(6) *Dinaya Sanskrit inscription at Java*⁵ ( 760 A.D. )

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⁵ Bosch, F. D. K. ‘De Sanskrit inscriptie op den steen van Dinaja (682 śaka)’, *Tijdschrift*, 57, p. 411, 1916; vide also 64, p. 227-29, 1924.

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[Figure:] ( śaka ) 2 & 8 & 6 / nayana-vasu-rase & śake =682 śaka

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(7) *Inscription at Po-nagar Champa*⁶ (813 A. D.)

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⁶ Finot, L. ‘Les inscriptions de Jaya Paramesvaravarman I, roi du Champa’, *Bulletin de l'école Française d'extrent orient*, 15, p. 47, 1915.

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(i) [Figure:] (śaka) 735 (śaka)
(ii) [Figure:] (śaka) 735 (śaka)

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(8) *Inscription at Bakul*¹ (839 A. D.)

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¹ Finot, L. *Ibid.*, p. 47.

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[Figure:] (śaka) 751 (śaka)

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We find here the use of word numerals with right to left order and symbolic numerals with a left to right order. The use of word numerals with left to right order has also been found in the *Mahāsiddhanta* of Āryabhaṭa II. These orders were so used for no obvious reason except possibly for emphasizing the idea that the decimal place-value concept remains unchanged in whatever order (right or left) the numerals are arranged when the number is always expressed by reading from higher to lower order in the scale. The symbolic use of zero is found to occur in the *Bakhshālī* Ms. (400 A. D.) of India as well as in number of inscriptions found at Sambor, Malay, Java, used in a decimal place-value system with their dates expressed in Śaka era of Indian origin. Some of these symbols including that of zero found in these inscriptions bear more or less close resemblance with those of the corresponding numbers in the *Bakhshālī* Ms. . The *Dinaya Sanskrit inscription* (see p. 72) found at Java (760 A. D.) however contained both word-numerals and the numeral symbols. This seems to suggest strongly a common origin of all these symbols obviously in India. Moreover, the evidence of the association of decimal scale e. g. *eka*, *daśa*, *śata* etc. with places is found in the works of Āryabhaṭa I (b. 476 A. D.) onwards. He writes that the value of the places are ten times of the preceding (*sthanāt-sthānaṃ daśaguṇaṃ syāt*, *Āryabhaṭīya*, Gaṇita, verse 2) in a numerical notation thereby expressing the idea of decimal place-value. That the decimal place-value concept was current before seventh century A.D., has also been attested by Vyāsa (before 700 A. D.) in his philosophical work, *Yogadarśanabhāṣya* (Ch 3, Sū. 13) thus : *yathaikā rekhā śatasthāne śataṃ daśasthāne daśa ekaṃ caikasthāne* i. e. "a (numerical) sign denotes hundred in the *śata* place, ten in the *daśa* place, and one in the *eka* place." The very similar language is used by Śaṅkarācārya (c. 800 A. D.) in his *bhāṣya* on *Vedānta Sūtra* (II. 2. 17). These illustrations establish without doubt that by the time of compositions of these two works, the decimal place-value concept must have been well established and was generally known. This system has appeared amongst the Arabs in the tenth century A. D. From there it went to Europe and it has now been adopted by all the civilised peoples of the world.

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The Chinese (8th century A. D.) left a gap or some vacant space similar to the Babylonians where a zero was required. This is found on some Thang Mss. of the Tunhuang cave-temples. One of the Ms. rolls, namely *Li-Chhêng Suan Ching*¹ contains the number 405 both in written form and in rod numerals like 𝄀𝄀𝄀𝄀 𝄀𝄀𝄀𝄀𝄀. A symbol for zero in the usual circular form appeared only in 1247 A. D. in a work *Su Shu Chiu Chang* of Chhin Chiu-Shao.² Thus it gives : | ≡ ╥ 000 as 147000. Needham, from the consideration of amalgamation of Indian, Chinese and of the local residence of Java, Champa, Malay etc. due to colonisation, concluded that the origin of the symbol for zero should be attributed to the mixed culture of the Indians and the Chinese. But such an argument is quite untenable in view of the fact that the use of the symbol for zero fitted in a decimal place-value scale appears in China only in Ms. of the 13th century A. D. Furthermore, there is no evidence that the use of word-numerals including that of zero in a decimal place-value scale was ever made by the Chinese or even by any other nations except by the Indian prior to their symbolic use. This has also been clearly recognised by the French scholar Coedès.¹

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¹ Needham, J. *Science and Civilisation in China*, 3, p. 9, Cambridge, 1959.

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² Smith, D. E. *Hist. of Mathematics*, 2, p. 42, Dover Edition.

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¹ Coedès, G. *Ibid.*, 6, p. 325.

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There is no doubt that the Arabs derived their knowledge of decimal place-value from India. Al-Bīrūnī (1030 A. D.), who stayed in India quite a number of years and is very much well known for his encyclopaedical knowledge, made the following observations in his *India* (Eng. tr. Sachau, *1*, p. 174), “the numerical signs which we use are derived from the finest of the Hindu signs.” The earliest use of zero in Arabia by a circular symbol was found in a 10th century A. D. manuscript.² In other Mss. during the same period, the forms such as ᴑ, ᵂ and ⎴ were used for zero. The dot symbol (•), similar to that of the Indians, was also used by Bīrūnī in his table. The modern Arabic symbol 5 is represented by a symbol that looks somewhat like the modern zero and the zero is denoted by a dot. The Indian name *śunya* was taken over by the Arabs as *aṣ-ṣifr* or *ṣifr*. This was subsequently changed to zephirum (1202 A. D. - Fibonacci), *tziphra* (1340 A. D. - Planudes) and *zenero*, *zepiro* (c. 16th century A. D. Italy).³

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² Smith, D. E. *History of Math.*, 2, p. 74, Table Column III, Dover Edition.

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³ Smith, D. E. *Ibid.*, p. 71.

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The zero in its circular symbol appeared in Europe in a 13th century A. D. manuscript, reproduced in *Della vita e delle opere di Leonardo Pisaro* (Rome, 1852). Next, it is found to have been used in European Manuscripts from 14th century A. D. onwards.¹

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¹ Smith, D. E. *Ibid.*, p. 71.

Eight Fundamental Operations

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The eight fundamental operations of Indian arithmetic after the invention of the decimal place value system of numeration are : addition (*samkalita*), subtraction ( *vyavakalita* or *vyutkalita* ), multiplication (*guṇana*), division (*bhāgahāra*), square (*varga*), square-root ( *varga-mūla* ), cube ( *ghana* ), cube-root ( *ghana-mūla*).

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These operations were carried out on a dust computing board (*pāṭī*). The method required the rubbing out of digits in every operation. According to Arabian scholars Āl-Khwārizmī (825 A. D.), Al-Uqlidisi (952 A. D.) Hindu arithmetic entered Islam with dust abacus as an intrinsic tool of it.² The latter author replaced the dust abacus to suit paper and ink. He used the Arabic word *takht* for *pāṭī*.

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² Datta, B. 'Testimony of early Arab writers on the origin of our numerals', *Bulletin of the Calcutta Mathematical Society*, 24, p. 193-218, 1932; Saidan, A. S. 'The development of Hindu Arabic Arithmetic', *Islamic Culture*, 39, p. 209-221, 1965.

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The ancient Indian mathematical works have not devoted much space to the methods of addition and subtraction in view of simple and self evident character, though such methods were described in details in the *Līlāvatī* of Bhāskara II ( 1150 A. D. ) and in two commentaries of *Līlāvatī* namely *Gaṇitāmṛta* of Sūryadāsa (1538 A. D.) and *Manorañjanā* of Rāmakṛṣṇa. Several methods of multiplication namely *kapāṭasandhi*, *gomūtrikā*, *khaṇḍa*, *bheda* and *iṣṭa* etc. were known. Datta and Singh have described this method fully in their *History of Hindu Mathematics*¹ as it appeared in the works of Mahāvīra (c. 850 A.D.), Śrīdhara (900 A.D.) and Śrīpati (1039 A.D.). The main feature of the method of *kapāṭasandhi* was that it was carried out in two stages : (1) the relative positions of the multiplicand and the multiplier followed by, (2) the rubbing out of figures of the multiplicand and substitution of the figures of the product in their places. The *kapāṭa* and *sandhi* owe their origin from the first and second stages respectively. The terms *hanana*, *vadha* or *kṣaya* which mean killing' or ‘destroying’ have been used in the works of Brahmagupta² (628 A. D.) in describing his process of multiplication. It seems to suggest that this method of *kapāṭasandhi* was known to India in 7th century A.D. This method reappeared in works of the Arabian scholars³ Al-Nasabī (c. 1025 A. D.), Al-Ḥaṣṣār (c. 1175 A. D.), Al-Kalasādi (c. 1475 A. D.).

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¹ *I*, p. 134-149, 1935.

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² Brahmagupta mentions four methods of multiplication viz. *gomūtrikā*, *khaṇḍa*, *bheda* and *iṣṭa* (vide Datta and Singh, *Ibid.*, *1*, p. 135).

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³ Datta and Singh, *Ibid.*, *1*, p. 143.

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The method of division in India was given by Mahāvīra (850 A. D.), Śrīdhara (900 A. D.), Āryabhaṭa II (950 A. D.), Bhāskara II (1150 A. D.), Nārāyaṇa (1356 A. D.) and others.⁴ The partial divisions were done and partial quotients were written in a separate line on the dust board and the figures of these operations were obliterated. If the figures were not obliterated and the successive steps were written one below the other, this ancient process of division would become the modern method of long division. Āryabhaṭa I (499 A. D.) gave the method of extracting square root and cube root which depend on the method of division. This seems to suggest that the method originated in India about the 4th century A. D. It reappeared in the Arabian works¹ from 9th century onwards. From Arabia, the method travelled to Europe where it came to be known as the galley (*galea*, *batello*) method. Fundamental operations for square and cube as developed by ancient and medieval Indian workers have been discussed fully by Datta and Singh.²

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⁴ Datta and Singh, *Ibid.*, *1*, p. 150-154.

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¹ Smith, D. E. *History of Mathematics*, 2, p. 138-139, Dover Publication.

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² Datta and Singh, *Ibid.*, 1, p. 155-169.

Square root and Cube root

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Special credit is given to Āryabhaṭa I (c. 499 A.D.) for his rules for the extraction of the square and cube roots developed on the knowledge of decimal place value concept. His rule for square root extraction runs thus:

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*bhāgaṃ haredavargānnityaṃ dviguṇena vargamūlena |*
*vargādvarge śuddhe labdhaṃ sthānāntare mūlam ||*
(*Āryabhaṭīya* ii. 4)

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“Divide always the non-square (*avarga*) places by twice the square-root (up to the preceding odd place); after having subtracted the square (of the quotient) from the square place (*varga*), then the quotient placed at the separate place gives the root.”

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The other Indian scholars namely, Mahāvīra, Śrīdhara, Āryabhaṭa II, Bhāskara II, Kamalākara etc. have given the same method of extracting square root.¹ Āryabhaṭa I divides the number whose square-root is to be determined into *varga* (square) and *avarga* (non-square) places-respectively from right, while the other scholars have divided it into *sama* and *viṣama* places from right. The rule can be explained from the following illustration by putting ☐ and - marks against square and non-square places. Modern method of square root extraction is also given side by side for the purpose of comparison.

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¹ Singh, A. N. ‘On the Indian Method of Root Extraction,’ *Bulletin of the Calcutta Mathematical Society*, 18, 123-40, 1927.

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| | *Āryabhaṭa's method* | *Modern Method* |
| Subtract the square 2²= | □ - □ - □ / 5 4 7 5 6 ( root 2 / 4 ... (1) | · · · | / 5 4 7 5 6|234 / 4 - - (1) |
| Divide by twice the root i. e., 2×2= Subtract square of the quotient 3²=9 | .r@l 4) & 1 4 (3...(a) / & 1 2 / & 2 7 / & 9......(b)root 23 / (2) | r@l 43) & 1 4 7 / & 1 2 9...... (2) / 464 & 1 8 5 6 / & 1 8 5 6..(3) |
| Divide by twice the root i. e., 2×23=46 Subtract square of the quotient i. e., 4²=16 | .l \phantom) 1 8 5 (4......(a') / \phantom) 1 8 4 / \phantom)1 8\phantom41 6 / \phantom)1 8\phantom41 6......(b')(3) \thereforethe square-root is 234 | |

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This shows that present method of extracting square root is same as that of Āryabhaṭa I (c. 499 A. D. ) with the only difference that stages numbered (2) and (3) in the working of the modern method were carried out in two distinct steps.

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The rule for the cube root extraction runs as follows :

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*aghanādbhājeddvitīyāt trighanasya mūlavargeṇa |*
*vargastripūrvaṁ guṇitaśśodhyaḥ prathamādghanaśca ghanāt ||*
( *Āryabhaṭīya*, ii. 5 )

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"After having subtracted the cube (of the quotient) from the *ghana* place, divide the second *aghana* place by thrice the square of the root, and subtract the square of the quotient multiplied by thrice the previous root from the next ( *aghana* place ) and the cube ( of the quotient ) from the cubic place ( next *aghana* place )."

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The very same method is given by other Indian scholars like Brahmagupta, Mahāvīra, Śrīdhara, Āryabhaṭa II, Bhāskara II etc. Here also the number whose cube root is to be found is marked as—*ghana*, first *aghana*, and second *aghana*, from right.¹ For illustration, they may be denoted by the symbols say ⸬ , —, — respectively. Āryabhaṭa's method of extraction is illustrated² in the following example; the modern method is also given side by side.

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¹ *prathama sthānaṁ ghanasaṁjñaṁ dvitīye tṛtīye aghanasamjñe !* *caturthaghanasaṁjñaṁ, pañcamaṣaṣṭhe aghanasamjñā !!* (Parameśvara's Comm.)

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² The division is to be carried to such an extent that the steps (b) and (c) can be carried out ( vide, p. 81 ).

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| | *Āryabhaṭa I's method* | *Modern method* |
| Subtract the cube 1³ = 1 | 1 9 5 3 1 2 5 (cube root 1 1 ..... | 1953125 | 125 1 |
| | 9 | 953 |
| Divide by thrice the square of the root i e. 3 × 1² | 3) 6 (2....... (a) 3 5 | 1² × 300 & / = 300 & / 1 × 30·2 & / = 60 & / 2² = 4 & / 364 & |
| Subt. the square of the quotient multiplied by thrice the previous root i.e , 2² × 3 × 1 = 12 | 1 2 .......(b) | 728 |
| | 2 3 3 | 225125 |
| Subt. the cube of the quotient i.e., 2³ = 8 | 8....... (c) | 12² × 300 & / = 43200 & / 12 × 30 × 5 & / = 1800 & / 5² = 25 & / 45025 & |
| | 2 2 5 1 | 225125 |
| Divide by thrice the square of the quotient i.e., 3 × 12² = 432 | 432) 2 1 6 0 (5...(a') | |
| Subt. sq. of the quotient multiplied by thrice the previous root i.e., 5² × 3 × 12 = 900 | 9 1 2 9 0 0 ....(b') | |
| | 1 2 5 | |
| Subt. cube of the quotient i.e , 5³ = 125 | 1 2 5....(c') | |
| | ∴ cube root of 1953125 = 125. | |

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From the above it is clear that the modern cube root method of extraction is simply a contraction of Āryabhaṭa I's method.

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Methods of extraction of square roots (*khai fang*) and cubic roots (*khai li fang*) with the help of abacus or counting boards appear in China in the *Chiu-Chang suan shu* (later half of the 1st century A. D.).¹ Yang Hui (c. 1261 A. D.) explained the methods of root extraction as found in the *Chiu-Chang suan shu* with an example of finding the square root of 71824 in his book, *Hsiang Chieh chiu chang suan fa* (A Detailed Analysis of the Mathematical Methods in the "Nine Chapters"). This gives the various stages on the counting board with the help of a diagram and illustrates the geometrical basis of the method.² In the cube-root method, Liu Hui and Li Shun-feng, commentators in the *Chiu Chang suan shu* both hinted at the use of cubical blocks to demonstrate the process. In Europe the derivation of the square-root method was done possibly from the Euclidean concept (a+b)² = a²+2ab+b².

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¹ Needham, J. *Science and Civilisation in China*, 3, p. 65-68, 1959.

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² Yong, Lam Lay. "The Geometrical Basis of the Ancient Chinese Square Root Method", *Isis*, 61, No. 206, p. 92-101, 1970.

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In the 4th century A. D. Theon Alexandria, explained Ptolemy's method of extracting square roots and found the square-root of 4500 degrees to be 67º 4' 55" in sexagesimal unit. This method of extraction was purely geometrical and depended on Euclidean concept.³ In another place he explained 144 = 10² + 2.10.2 + 2² = (10 + 2)² with the help of a diagram. This figure is two-dimensional and no attempt was made to construct a three dimensional figure in order to extend the square-root method to find the cube-root. In the sixteenth century A. D. both square-root and cube-root methods were given by Cataneo, which are exactly the same as those of Āryabhaṭa I¹. Hence it is quite likely that the Indian methods of square-root and cube-root went to Europe through Arab intermediaries.

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³ Heath, T. L. *A History of Greek Mathematics*, 1, p. 60-63, Oxford, 1921; Yong, Lam Lay, Ibid, p. 100-101.

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¹ Smith, D. E. *History of Mathematics*, 2, Dover Publication, p. 148.

Numbers and Symbolism

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Bhāskara I (600 A. D.) freely used integral and fractional numbers and their operations in his *Āryabhaṭīya-bhāṣya*. He has referred to in this work the names of mathematicians like Maskarī Pūrāṇa, Mudgala Patana and others. In his Āryabhaṭīya-bhāṣya², he has given ample illustrations of these operations as follows :

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² Shukla, K. S. 'Hindu Mathematics in the seventh century as found in Bhāskara I's, commentary on the *Āryabhaṭīya*', *Gaṇita*, 22, No. 2, p. 61-63, 1971.

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i) x ± (y/z) = ((xz ± y))/z

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ii) x/y ± z/u = (xu ± yz)/yu

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iii) x/y × z/u = xz/yu

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iv) x/y ÷ z/u = xu/yz $v) (x/y)² = (x²)/(y²)$ $vi) (x/y)³ = (x³)/(y³)$ $vii) √(x/y) = (√x)/(√y)$ $viii) [3]x/y = ([3]x)/([3]y)$

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It is not clearly known to what extent he was indebted to these scholars since their works are lost.

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Bhāskara I wrote integral numbers within square or circular cells and sometimes without use of any cells The numbers 4, 6 and 7 without circular are written in any one of the following forms.

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The fractional numbers are sometimes enclosed within rectangular cells or brackets without sticking to any uniform pattern. Thus the number 4 ⅚ was written in any one of the following way.

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The enclosures were perhaps meant to avoid confusion and mixing up of numbers with the written matter.

Symbol for Minus Sign

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In the *Bakhśālī Ms*, the modern plus sign ( + ) has been used to represent minus sign. Bhāskara I has made use of a little circle ( • ) on the right of the numbers to be subtracted. For example,

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( ½ ) - ( ⅙ ) is written as ½ ⅙•

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The plus sign, to denote negative sign, is also used in Jaina work *Dhavalā*, a commentary on the *Ṣaṭkaṇḍāgama* (vol. 10, p. 151). The first letter *ri* of the word *rina* was written in Brāhmī script as [Figure:]. Perhaps the sign [Figure:] was changed later into + to denote negative sign. But how little circle for negative sign came to be used, is not known. Another manuscript,¹ a commentary on the *Pāṭīgaṇita* of *Śrīdhara*, available at Kashmir, has used the + symbol for the minus sign. The symbol + is generally used after the number affected, but sometimes it was also used before the number affected. Thus, - 3 is written as 3 +, and sometimes as + 3.

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¹ Ed. by K. S. Shukla, Lucknow, 1959.

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This Kashmirian manuscript has used dot ( . ) as well as ( o ) for zero. The dot symbol was also found to have been used in the *Bakhśālī Ms.*² (c. 400 A. D.). The dot as a symbol for zero is found to be more ancient than the small circle, and this shows that the mathematicians of north-west region of India had not yet given up the use of dot, the alternative symbol for zero.

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² The absence of the use of a small circle as a symbol for zero testifies that it is a much older work than the present commentary.

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*Rule of Three and its Inverse, Five, Seven, etc,*

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The term used in ancient India for rule of three is *trairāśika*, for three things were involved in the problems. It has appeared in the *Bakhshālī Ms.* (400 A. D.), *Āryabhaṭīya* of Āryabhaṭa I and in all other later mathematical works.¹ The method gives the solution of the problem like : if p yields f, what will i yield ? where p = *pramāṇa* (argument), f = *phala* (fruit) and i = *icchā* (requisition). According to Śrīdhara, the *pramāṇa* and *icchā*, which are of the same demominations should be set down in the first and last places; the *phala* which is of different denominations, be set down in the middle as follows :

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¹ *Āryabhaṭīya*, ii, 26; *Brahmasphuṭa-Siddhānta*, xii. 10, *Gaṇitasārasaṃgraha*, v. 2 (i); *Mahāsiddhānta*, xv. 24–25; *Gaṇitatilaka*, v. 86, p-68; *Siddhānta Śekhara*, xiii, 14; *Gaṇitakumudī* I, v. 60, p. 47; *Pāṭīsāra* of Muniśvara, v. 83.

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| p | f | i |

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For required result the middle quantity is to be multiplied by the last quantity and divided by the first quantity. i.e. required result = (f)/p

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*Example* "A certain lame person goes to a distance of ⅛ of a yojana in ⅕ of a day. Say in how much time will he go to a distance of 100 yojanas ?".² Here p = ⅛ of a yojana. f = ⅕ of a day. i = 100 yojanas.

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² *Gaṇitasārasaṃgraha*, v. 3, 4; *Gaṇitatilaka* v. 93, p. 72; *Pāṭīgaṇita*, Ex. 30, p. 24 (Eng. tr.)

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As per direction in the rule, we write

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| ⅛ | ⅓ | 100 |

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required result = (1/3 × 100)/(1/8) &= 266 \ 2/3 days / &= 8 months and 26 2/3 days

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In the Renaissance period, this rule was highly appreciated to solve commercial problems by Widman (1489 A. D.). Tonstall (1522 A. D.), Gamma Frisius (1540 A. D.), Recorde (c. 1542 A. D.) etc. Recorde calls this rule, ‘the rule of proportion’, sometimes ‘Golden Rule’ for its excellency in arithmetical computation.¹

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¹ Smith, D. E. *History of Mathematics*, Dover Publication, 2, p. 484.

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For inverse rule of three² (*vyasta-trairāśika*), there is change in the units of measurment. The required result in this case is obtained when the middle quantity is multiplied by the first quantity and divided by the last quantity i.e. (f)/i. Al-Bīrūnī wrote a treatise, *Fi raśikat al-hind* (‘rule of three or more terms of the Hindus’). In his *India* (1, p. 313), he has used an example of *vyasta-trairāśika* (Inverse Rule of Three). The rule was known to the Arabs from the eighth century onward. It is quite likely that the rule went to Europe through the Arabs where it was known later as ‘the Golden Rule’.

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