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A Concise History of Science in India c7: 3 Mathematics

A Concise History of Science in India c7

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English

Contents

3 Mathematics

c7.1

THE history of the development of mathematics in India is as old as the civilization of its people itself. It begins with the rudiments of metrology and computations in prehistoric times, of which some fragmentary evidence has survived to this day. The sacred literature of the Vedic Hindus—the *Saṃhitās*, the *Kalpasūtras* and the *Vedāṅgas*—contain enough materials, albeit scattered, to help form a good idea of the mathematical ability during the time of development of this class of literature. The *Śulba-sūtras* which form part of the *Kalpasūtras* are a veritable storehouse of information concerning enumeration, arithmetical operations, fractions, properties of rectilinear figures, the so-called Pythagorean Theorem, surds, irrational numbers, quadratic and indeterminate equations and related matters. The *Brāhmaṇas* and some *sūtras* contain interesting materials concerning progressive series, permutations and combinations.

c7.2

Of the various religious sects that attained prominence in the closing phase of the Vedic period, the Jainas deserve special notice for their interest in, and cultivation of, mathematics. Their canonical literature lays great emphasis on mathematics and enumerates various topics such as number reckoning, fundamental operations, geometry, mensuration, fractions, equations, permutations and combinations. In some departments they made advances further than where the Vedic Hindus had left off.

c7.3

Indian mathematics received a new lease of life during the first few centuries of the Christian era. The demands of astronomy, particularly the need for more accurate computations of planetary motions, eclipses, etc., were in no small measure responsible for this. From now on we meet with a succession of mathematicians and astronomers and a fairly steady growth of mathematical-astronomical literature. The mathematical contents of this literature and its voluminous commentaries have for obvious reasons occupied the greater part of the discussion in the following pages.

c7.4

The medieval period witnessed the growth of a sizeable mathematical literature in Arabic and Persian and presented an opportunity for cross-fertilization of the efforts of two distinct cultures. How far that opportunity was seized has been discussed.

c7.5

Situated at the cross-roads of many cultures, it was not given to Indian mathematics to develop in isolation. Although what happened in the most ancient times must remain a matter of speculation, records of late ancient and medieval times show that Indian mathematics stimulated mathematical endeavours abroad and itself received inspiration from neighbouring and distant culture areas. Such stimuli have been noticed even in the case of the complex and many-sided phenomenon of European Renaissance of the sixteenth century.

Metrology and Computations in Prehistoric Times

c7.s1.1

It is reasonable to believe that the builders of the Indus civilization developed a good degree of skill in their measuring and computational techniques. Without these it is difficult to comprehend their town-planning and architectural proficiencies and various other aspects of their civilization. As to metrology, we have the evidence of seals with pictographic inscriptions and numerous stone weights unearthed at Mohenjo-daro and Harappa. The pictographic inscriptions have not yet been deciphered, but by measuring the large number of weight specimens in modern units one can guess what their weight system possibly was. From the published masses of these weights, frequency charts have been constructed, which clearly show that these weights were intended to be fractions or multiples of the *uncia*.a Thus 288 specimens of small cubes of chert have been shown to fall under seven distinct groups. By assuming 1 *uncia* equal to 27·2 gm., these groups are as follows:

c7.s1.2

| 1/16 uncia = 1·7 gm. | 1 uncia = 27·2 gm. |
| ⅛ ,, = 3·4 ,, | 2 ,, = 54·4 ,, |
| ¼ ,, = 6·8 ,, | 5 ,, = 136 ,, |
| ½ ,, = 13·6 ,, | |

c7.s1.3

The discovery in large numbers from both the sites of the weight specimens of mass 13·6 gm. has led others to suggest this as the probable unit. The existence of a developed weighing system is further borne out by the discovery from these sites of the remains of quite a few metal scales used probably for weighing light and precious substances.

c7.s1.4

FIG. 3.1. Sketch of an Indus scale.

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An interesting specimen of the Indus scale has come down to us among the finds from Mohenjo-daro in the form of the fragment of a shell measuring 6·62 cm. in length by 0·62 cm. The scale shows nine parallel lines cut with a fine saw. One of these lines is marked by a fine circle and the sixth line from it by a dot (Fig. 3.1). The measurement of these five intervals shows 1·32 inches making the distance between two consecutive lines equal to 0·264 inch. If the length between the circle and the dot can be called the Indus inch, as Berriman did, this becomes exactly equal to 2 Sumerian *shushi*. Another significance of the Indus scale is that 25 Indus inches equal 33 inches which again equal the yard prevalent throughout north India in Akbar’s time.

c7.s1.6

a Berriman, pp. 33–35.

c7.s1.7

Mackaya conjectured that the aforesaid scale was a part of the whole, probably 13·2 inches long, and hinted at a decimal mode of dividing and measuring lengths such as was characteristic of ancient Egypt and Elam. It is then easy to think that such a decimal scale appeared in some original centre of diffusion in West Asia and then spread to other centres of culture. This does not preclude the possibility of independent development in different centres.

c7.s1.8

Indus seals and inscriptions, as found in Mohenjo-daro, show that numbers were represented by vertical strokes arranged side by side or in lateral groups. This might probably be some kind of rod numerals, of which we have many examples in other ancient culture areas, notably among the Chinese and the Mayans.

c7.s1.10

How far they went with such rod numerals in representing higher numbers like 10, 20, 30 . . . or 100, 200 and so on, it is not possible to say. Some of the vertical or horizontal strokes, as we shall see later, survived in the Kharoṣṭhī and Brāhmī numerals.

Mathematical Knowledge as Revealed in the Saṃhitās, the Brāhmaṇas, the Vedāṅgas and Related Literature

Survey of Literature

c7.s2.t1.1

In the chapter on survey of source materials, we have noticed briefly the various sections of the Vedic literature, e.g. the *Saṃhitās*, the *Brāhmaṇas* and the *Vedāṅgas*. This vast sacred literature contains enough material to help form a good idea of the mathematical ability of the Vedic Hindus. These materials are mostly scattered and diffused in the *Saṃhitās* and the *Brāhmaṇas*. We are somewhat more fortunate with regard to the *Vedāṅgas*, for two of them, the *Kalpasūtras* (ritual) and the *Jyotiṣa* (astronomy), directly concern a large body of mathematical knowledge without which neither the construction of various kinds of sacrificial altars nor the reckoning of time for calendrical purposes would have been possible. This debt to mathematics or the science of calculation (*gaṇanā*, *rāśi-vidyā*) was freely acknowledged and the study of mathematics was always given a pride of place in the various branches of learning. When Sanat Kumāra wanted to know from Nārada the various sciences and arts the latter had studied, he gave a list which included astronomy and mathematics. In the *Vedāṅga Jyotiṣa*, the science of calculation has been likened to the crest on the peacock's head or the proverbial gem on the snake's hood. Mathematics, along with astronomy, was also held in high esteem by the Jaina and the Buddhist monks. One of the four branches of the religious literature of the Jainas, e.g. the *gaṇitānuyoga*, was concerned with the exposition of mathematical principles. The *Vinaya Piṭaka*, *Dīgha Nikāya*, *Divyāvadāna* and other Buddhist texts also expatiate on the importance of the study of *gaṇita* or *saṃkhyāna* (science of higher calculations).

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a Mackay, pp. 348, 404.

c7.s2.t1.3

We have referred to the *Kalpasūtras* as an important source of Vedic mathematics. One class of this ritual literature is the *Śrauta-sūtra* which deals with directions for the laying of sacrificial fires for *agnihotra*, the new and the full moon, the seasonal, the *soma* and other sacrifices. Of special importance for our purpose are sections called *Śulba-sūtras* which are directly attached to these *Śrauta-sūtras*. These sections deal with rules for the measurement and construction of the various sacrificial altars and consequently involve geometrical propositions and problems relating to rectilinear figures, their combinations and transformation, squaring the circle, circling the square as well as arithmetical and algebraic solutions of problems arising out of such measurements and constructions. The word *śulba* (also spelt as *śulva*) means a 'cord', a 'rope' or a 'string', and its root *śulb* signifies 'measuring' or 'act of measurement'.a Therefore, works entitled *śulba-sūtras* may be literally taken to mean a collection or compendium of rules concerning measurements with the help of a cord of various linear, spatial or three-dimensional figures. Quite appropriately, the *śulba-sūtras* represented the *Brāhmaṇa* geometry or mensuration, the *śulba-vijñāna*, as mentioned specifically in the *Mānava* and other *śulbas*, and the *śulbavid*, the expert geometer, was held in high esteem in the learned priestly circles.

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One would expect a *śulba* section attached to each *Śrauta-sūtra*, and there are *Śrauta-sūtras* belonging to all the four *Saṃhitās*. But what we possess today are a small number of *śulba-sūtras* attached to the *Śrautas* belonging only to the various schools of the *Yajurveda*. Of them the Black Yajurvedins of the *Taittirīya* school were the most active, and produced the most comprehensive *śulbas*, e.g. those of Baudhāyana, Vādhula, Āpastamba and Hiraṇyakeśin. There are two *śulbas* by Mānava and Vārāha of the *Maitrāyaṇī* and one by Laugākṣi of the *Kāṭhaka-Kapiṣṭhala* school. Of the White Yajurvedins, Kātyāyana, the prolific *sūtrakāra*, is credited with a small but more scientifically executed *śulba* work. The initiative of the Yajurvedins in producing works of this kind is not surprising when we bear in mind that they were the principal custodians of the knowledge of sacrificial formulas and specialized in the techniques of performing various sacrifices.

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a Datta (4), p. 8.

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The *Baudhāyana Śulba-sūtra* is the oldest (600–500 b.c.) and the most voluminous work of this class. It comprises 525 *sūtras* divided into three chapters. The first chapter of 116 *sūtras* gives geometrical propositions required for the construction of altars and deals with their relative positions and spatial magnitudes; the second chapter of 86 *sūtras* describes the various fire-altars (*agnis*) and their spatial relationships; the third chapter containing 323 *sūtras* is devoted to *kāmya agnis* or sacrificial altars designed to attain desired ends.

c7.s2.t1.7

The *Mānava Śulba-sūtra* (posterior to Baudhāyana), a part of the *Śrauta-sūtra* by the same author, is a small compilation and gives specifications of the measuring tape, gnomon and methods of finding the cardinal points not given by Baudhāyana or Āpastamba. It contains descriptions of a number of altars, e.g. the *suparṇa-citi*, the *pākayājñikī*, the *marūti* and the *vāruṇī*, not found in other works.

c7.s2.t1.8

The *Āpastamba Śulba-sūtra* (500–400 b.c.) comprises 223 *sūtras* distributed among six chapters. Āpastamba covers the same ground and gives the same rules as was done by Baudhāyana.

c7.s2.t1.9

The *Kātyāyana Śulba-sūtra* (400–300 b.c.), also known as *Kātīya-śulbapariśiṣṭa*, is a comparatively small tract of 102 *sūtras* arranged in six chapters. Kātyāyana’s treatment is more succinct and systematic. Like Mānava, he also gives specifications of the measuring tape, discusses the gnomon and its application in determining the cardinal points. Kātyāyana gives solutions of rational triangles and states the Pythagorean Theorem in a generalized form. In the *śulba*, Kātyāyana deals with different *agnis* and their spatial relations, but not with the *kāmya agnis*, because the latter is discussed in a separate chapter of his *Śrauta-sūtra*.

c7.s2.t1.10

Other *śulba-sūtras* referred to are minor works and do not throw any additional light. Hiraṇyakeśin’s work is no longer extant, but is known from references and occasional extracts in other *śulba-sūtras*.

c7.s2.t1.11

We have already noticed the *Vedāṅga* and the *Ārca Jyotiṣa*. Although these are astronomical works, these offer examples of application of elementary mathematics in astronomical computations, divisions of the lunar zodiac, calculation of *tithis*, etc.

Arithmetic

c7.s2.t2.1

*Number*

c7.s2.t2.2

Like the Egyptians, the Vedic Hindus adopted 10 as the basis of numeration and developed a great interest in thinking out and naming large numbers. The Greeks, on the other hand, fought shy of large numbers and their terminology hardly took them beyond the ‘myriad’ (10⁴). The Egyptians freely dealt with large numbers on the decimal scale, such as hundred, thousand, million, tens of million and so on and had meaningful Hieroglyphic symbols to express them. The same is true of the Vedic Hindus. The various recensions of the *Yajurveda Saṃhitā* give names to numbers as large as 10¹². The *Taittirīya Saṃhitā*a gives the following denominations: *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 is repeated in the *Pañcaviṃśa Brāhmaṇa* with further extensions. The following passage from the *Pañcaviṃśa Brāhmaṇa*b will give an idea of the context and the manner in which large numbers were introduced:

c7.s2.t2.3

‘By sacrificing with the *agnihotra*, he reaches in one day ten “House-lords” (i.e. he becomes equal to, gains the merits of ten H.); by sacrificing with the *agnihotra* during ten years, he becomes equal to one who (regularly) performs the sacrifices of full and new moon, he becomes equal to one who performs the sacrifices of *soma*. By offering the *agniṣṭoma* sacrifices, he becomes equal to one who performs a sacrifice of a thousand cows as sacrificial fee. By offering ten of these, he becomes equal to one who performs a sacrifice with ten thousand *dakṣiṇās*. By offering ten of these, he becomes equal to one who sacrifices with a sacrifice of a hundred thousand *dakṣiṇās*. By offering ten of these, he becomes equal to one who sacrifices with a sacrifice of a million *dakṣiṇās*. By offering ten of these, he becomes equal to one who sacrifices with a sacrifice of 10 million *dakṣiṇās*. By offering ten of these, he becomes equal to one who sacrifices with a sacrifice of 100 million *dakṣiṇās*. By offering ten of these, he becomes equal to one who sacrifices with a sacrifice of 1,000 million *dakṣiṇās*. By offering ten of these, he becomes equal to one who sacrifices with a sacrifice of 10,000 million *dakṣiṇās*. By offering ten of these, he becomes equal to one who sacrifices with a sacrifice of 100,000 million *dakṣiṇās*. By offering ten of these he becomes the cow.’

c7.s2.t2.4

The method of obtaining higher and higher numbers in multiples of 10, described later on as *daśaguṇottara*, is clearly indicated. The number names in this decimal scale in ascending order are: *eka*, *daśa*, *śata*, *sahasra*, etc., the same as given in the *Taittirīya Saṃhitā*. It appears that the thinking out and the naming of such large numbers were a favourite pastime of the ancient Indian mathematicians. The same tendency is noticed in the Buddhist literature where we find a centesimal scale (*śatottara gaṇanā*) and the name *tallakṣaṇa* for the number 10⁵⁸. The Jaina work, the *Anuyo-gadvāra-sūtra* (c. 100 B.C.), called the places in powers of 10 as *gaṇanā-sthāna* and mentioned large numbers up to 29 places and beyond. The Jainas are further credited with the conception of a time-scale called *śīrṣa-prahelikā* (8,400,000) and the suggestion of building up fantastically large numbers in ascending powers of this figure.

c7.s2.t2.5

a *Taitt. S.*, 4.4.11.4.; 7.2.20.1.

c7.s2.t2.6

b *Pañc. Br.*, xvii. 14.1.2; Eng. trans. by Caland.

c7.s2.t2.7

The Vedic Hindus showed the same proficiency in developing a scientific vocabulary of number names, in which the principles of addition, subtraction and multiplication were conveniently used. The system required the naming of (1) the first nine digits, e.g. *eka*, *dvi*, *tri*, *catur*, *pañca*, *ṣaṭ*, *sapta*, *aṣṭa* and *nava*; (2) the second group of nine numbers obtained by multiplying each of the above digits by 10, e.g. *daśa*, *viṃśati*, *triṃśat*, *catvāriṃśat*, *pañcāśat*, *ṣaṣṭi*, *saptati*, *aśīti* and *navati*; and (3) the third group of 11 numbers beginning with 100 and followed by its multiples of 10 and its higher powers, e.g. *śata*, *sahasra*, *ayuta*, . . . *parārdha* (see above). The multiplicative principle is already indicated in forming the numbers of the second and third group. Other examples are *ṣaṣṭiṃ sahasrāṇi*, *pañcāśat sahasram*, *dvā-saptatiḥ sahasrāṇi*. The additive principle is generally used in naming a number in which the numbers of the first and second group participate, e.g. *eka-daśa* (11), *sapta-viṃśati* (27), *aṣṭa-triṃśat* (38). Additive and multiplicative principles are simultaneously used when, in the number concerned, the members of the third group participate along with those of the second or the first, e.g. *sapta śatāni viṃśatiḥ* (720), *ṣaṣṭiṃ-sahasra navatim nava* (60,099).a These principles were widely used in constructing number names in other culture areas, as witness such terms as *pente-deka* (50), *pente-hekaton* (500) among the Greeks.

c7.s2.t2.8

Of greater importance, however, is the infrequent application of the subtractive principle for which the Etruscans are generally given the credit.b Proceeding from right to left, they wrote 27 as XIIIXX [20+(10−3)]; 38 as XIIXXX [30+(10−2)] and so on. The same subtractive principle, in the number nomenclature, was applied by the Vedic Hindus in the coinage of such terms as *ekānna-śata* (100−1), *ekānna-viṃśati* (20−1), *ekānna-catvāriṃśat*,c etc. The word *ekānna*, meaning ‘one less’, was later on shortened into *ekona* and *ūna*. In a limited way, the Babylonians also used this principle in forming their inauspicious number 19 as <<ᵞ< (20−1), where ᵞ stands for −1.

c7.s2.t2.9

Fundamental Operations

c7.s2.t2.10

The four fundamental arithmetical operations, e.g. addition, subtraction, multiplication and division, are nowhere spelled out in the Vedic literature, as these were obviously taken for granted being commonplace operations. One, however, meets with the interesting case of dividing one thousand into three equal parts in the *Ṛgveda* and the *Brāhmaṇas*. In the *Ṛgveda*, the division of the thousand is not explicitly mentioned, but it is done so in the *Tattirīya Saṃhitā*d which, while incorporating the *ṛc*, modifies the last line as ‘Ye did divide the thousand into three’. The full explanation of the arithmetical feat of Indra and Viṣṇu in the proposed division is given in the *Śatapatha Brāhmaṇa* as follows: 'For when Indra and Viṣṇu divided a thousand (cows) into three parts, there was one left, and here they caused to propagate herself in three kinds; and hence, even now, if anyone were to divide a thousand by three, one would remain over.'a

c7.s2.t2.11

a *RV.* I. 164.11; I. 53.9.

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b Smith (D. E.), II, p. 58.

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c *Taitt. S.*, vii. 2.11.

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d *Taitt. S.*, vi. 1.6.

c7.s2.t2.15

*Fractions*

c7.s2.t2.16

Acquaintance with the fundamental arithmetical operations with elementary fractions is clearly indicated in the Vedic texts and their appendages. The *Ṛgveda* gives names of a number of simple fractions such as *ardha* (½), *tripāda* (¾) and the *Maitrāyaṇī Saṃhitā* mentions *pāda* (¼), *śapha* (⅛), *kuṣṭha* (1/12), *kalā* (1/16). From the *Śulba-sūtras* onwards we meet with terms such as *aṃśa*, *bhāga* to denote fractions in general. These terms are used in combination with cardinal or ordinal number names; in the case of the latter the word *aṃśa* or *bhāga* is often omitted. A few examples follow:

c7.s2.t2.17

{ll} tribhāga, tryaṃśa & = 1/3 / pañcama-bhāga, pañcama & = 1/5 / dvādaśa-bhāga or dvādaśa & = 1/12 / pañcadaśa-bhāga & = 1/15 / tri-aṣṭama, tryaṣṭa & = 3/8 / dvi-saptama & = 2/7 / trayastryaṣṭa & = 3 3/8 / pañcamasya caturviṃśa & = 1/24 of 1/5

c7.s2.t2.18

There are a number of other peculiar modes of expressing fractions, e.g. *ardhāṣṭama* (7½), *ardhanavama* (8½), *ardhadaśama* (9½)b or *dvi-guṇa* (½), *tri-guṇa* (⅓), *caturguṇa* (¼).c

c7.s2.t2.19

The *Śulba-sūtras* contain several instances of addition, subtraction, multiplication, division and the squaring of fractions. Consider the following example from Baudhāyana:d

c7.s2.t2.20

'Then he measures the area of this square-shaped *citi*, whose side is three *puruṣas* less one-third. At the western side of this square is the handle whose east-west length is half a *puruṣa* plus ten *aṅgulas* (one-twelfth *puruṣa*) and north-south breadth one *puruṣa* less one-third. This makes the (area of the) *agni* sevenfold plus two *aratnis* and the *prādeśa*.'

c7.s2.t2.21

c *Mśl.*, v. 5.

c7.s2.t2.22

Two *aratnis* and the *prādeśa* equal 60 *aṅgulas* or ½ *puruṣa*. Expressed in figures, the above means:

c7.s2.t2.23

b *Bśl.*, ii. 1–3.

c7.s2.t2.24

d *Bśl.*, iii. 219–24.

c7.s2.t2.25

The area of the *agni* =(3-⅓)²+(½+1/12)(1-⅓)=7½.

c7.s2.t2.26

a *Śat. Br.*, iii. 3.1.13.

c7.s2.t2.27

The results of a few more statements of this nature are given below:

c7.s2.t2.28

(1) Number of bricks each 1/25 sq. *puruṣa* to cover 7½ sq. *puruṣa*
= 7½ ÷ 1/25 = 187½
(2) Number of bricks each 1/15 of ½ sq. *puruṣa* to cover 7½ sq. *puruṣa*
= 7½ ÷ 1/15 of ½ = 15/2 ÷ 1/30 = 15/2 × 30 = 225.

c7.s2.t2.29

*Progressive Series*

c7.s2.t2.30

Indian interest in progressive series has been traced to the *Saṃhitās*. The *Taittirīya Saṃhitā* indicates the following arithmetical series in which the odd (*ayugma*) and the even (*yugma*) ones are noticeable:

c7.s2.t2.31

1, 3, 5, ... 19, 29, 39, ... 99
2, 4, 6, 20
4, 8, 12, 20
5, 10, 15, 100
10, 20, 30, 100

c7.s2.t2.32

The *Pañcaviṃśa Brāhmaṇa*a prescribes *dakṣiṇās* in gold measures (*mānas*) to be given in geometrical progressions, depending on the nature and the time of offerings, as follows:

c7.s2.t2.33

12 (gold *mānas*), 24, 48, 96, 192 ... 3072, 6144 ... 49152, 98304, 196608, 393216.

c7.s2.t2.34

In the above examples, the summations of such series are not given. This is correctly done in the *Śatapatha Brāhmaṇa*b as follows: ‘... there are seven of these meters, increasing by four syllables; and the triplets of these make seven hundred and twenty syllables and thirty-six in addition thereto.’

c7.s2.t2.35

In other words, it is proposed to compute

c7.s2.t2.36

3(24+28+32+ 7 terms) = 3 × 7/22× 24+(7-1)× 4 = 756.

c7.s2.t2.37

It is true that no general rule for finding the summation as indicated above is given in the *Śatapatha Brāhmaṇa*. But the recurrence of both arithmetical and geometrical series, with the correct statement of the results of their summation, strongly suggests that the Vedic Hindus probably possessed some method of finding the summation of such series. Thus the *Bṛhaddevatā* (400 B.C.) gives the summation

c7.s2.t2.38

2+3+4+ +1000 = 500499

c7.s2.t2.39

and Bhadrabāhu’s *Kalpasūtra* (300 B.C.), a Jaina canonical work, the summation of the geometrical series

c7.s2.t2.40

b *Śat. Br.*, x. 5.4.7

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1+2+4+ +8192 = 16383.

c7.s2.t2.42

The series played an important part in prosody. Piṅgala, in his *Chandaḥsūtra* (200 B.C.), used a geometrical series 2, 2², 2³

c7.s2.t2.43

a *Pañc. Br.*, xviii. 3.

c7.s2.t2.44

It is noteworthy that the summation of progressive series has ever since continued to occupy the attention of subsequent Indian mathematicians such as Mahāvīra, Bhāskara II, Nārāyaṇa and others who have all given general formulas for obtaining the sum. In the *Sthānāṅgasūtra*, progressive series is included in the mathematical topic *Vyavahāra*, as explained by the commentator Abhayadeva Sūrī (A.D. 1050). Later on, it became the subject of special treatises, *e.g.* *Bṛhaddhāra Parikarma*, referred to by Nemicandra.

Geometry

c7.s2.t3.1

*Postulates*

c7.s2.t3.2

We have already stated that the various *śulba-sūtras* that have come down to us as parts of the *Śrauta-sūtras* are Brāhmaṇic geometrical manuals for the construction of sacrificial altars. In the various rules given, certain assumptions resembling Euclidean postulates are taken for granted. Thus a straight line is capable of division into an infinite number of equal parts; a circle can be divided into any number of parts by drawing diameters; each diagonal bisects a rectangle and both diagonals bisect each other, dividing the rectangle into four parts with the vertically opposite ones equal in all respects; a straight line joining the vertex with the middle point of the opposite side divides an isosceles triangle into two equal halves; parallelograms drawn on the same base and between the same parallels are equal to one another; a square inscribed within a circle and touching the circumference is the maximum, and so on. The texts give several rules as to how to construct a straight line perpendicular to another straight line, a square with a given side, a rectangle with given sides, an isosceles trapezium of a

c7.s2.t3.4

FIG. 3.2.

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given altitude, face and base. For constructing a square, Baudhāyana gives, among others, the following rule:

c7.s2.t3.6

‘If you intend to draw a square, take a cord of length equal to the side desired of the square. Make a tie at each end of the cord and mark its middle point. Draw the east-west line (*EW* of Fig. 3.2) and fix a pole at its middle (*O*). After fastening the two ties at that pole draw a circle with the mid-point of the cord; fix a pole at each end of the diameter (along the east-west line). Fasten one tie at the eastern pole and draw a circle with the other tie. Likewise, draw a circle round the western pole. Join the points of intersections of these circles to obtain the second (north-south) diameter; fix a pole at each end of this diameter. Fasten both ties at the east pole and describe a circle with the mid-point (of the cord). Draw similar circles around the southern, the western and the northern poles. The exterior points of intersection of these four circles will give the four corners (*ABCD*) of the required square.’a

c7.s2.t3.7

*Combinations and Transformations of Rectilinear Figures*

c7.s2.t3.8

Of greater importance are the rules given in the *śulbas* for the combination and transformation of rectilinear figures, specially the squares and the rectangles. In a series of rules for transforming a square into a rectangle, the simplest one is to draw the diagonal *AC* of the square *ABCD* to divide it into two equal parts, to divide again one of these parts into two equal parts by joining the vertex *D* with the mid-point *E* of the diagonal and then to fit them suitably one to each side *AB* and *BC* of the square. *AFGC* is then the required rectangle. In the *paitṛkīvedi*, it is often required to find one square double of another given square. This is done by joining the mid-points of the sides of a square. The thing is achieved by drawing on the other side of the diagonal in Fig. 3.3 a similar rectangle *AHIC*.

c7.s2.t3.9

When one side of a rectangle is given, the rule of transforming a square into a rectangle of the given side is given by Āpastamba as follows:b

c7.s2.t3.10

‘If you desire to make a rectangle out of a square (cut off from it a rectangle) of a side of the desired length. The excess remaining is to be added suitably as to fit (the rectangle cut off).’

c7.s2.t3.11

Baudhāyana also gives a similar rule.c The directions are incomplete but, according to interpretations given of the above rule by the commentators Sundararāja and Dvārakānātha Yajvā, the true process appears to be as illustrated in Fig. 3.4. *ABCD* is the given square. Its side *AB* is extended to *E*, so that *EB* equals the given side of the rectangle into which the square is to be transformed. Join *EC* cutting *AD* at *O*. Complete the rectangles *EFCB* and *GFCH*. Then *GFCH* is the required rectangle. From the postulate that a rectangle is bisected by its diagonal, it is easy to see that the rectangles *AOHB*, *GODF* are equal in areas and therefore the square *ABCD* equals the rectangle *GFCH*.

c7.s2.t3.12

a *Bśl.*, i. 22–28.

c7.s2.t3.13

b *Āśl.*, iii. 1.

c7.s2.t3.14

c *Bśl.*, i. 53.

c7.s2.t3.15

Baudhāyana, Āpastamba and Kātyāyana all teach methods of constructing a square equal to the sum of two different squares. The method

c7.s2.t3.16

FIG. 3.3.

c7.s2.t3.17

FIG. 3.4.

c7.s2.t3.18

FIG. 3.5.

c7.s2.t3.19

is to cut off from the larger of the two given squares, *ABCD*, a rectangular portion *ABEF* of which the side *BE* is equal to the side of the given smaller square. Then the diagonal *AE* is the side of the square equal to the sum of the two given squares. The proof will follow from Fig. 3.5, in which *AEGH* is the required square, and *ABCD* and *CKGI* are the given squares.a In other words, the proposed construction is based on what has generally come to be known as the theorem of Pythagoras.

c7.s2.t3.20

*Pythagorean Theorem*

c7.s2.t3.21

The theorem of the square of the diagonal is more explicitly given in more or less the same language in all the *śulba-sūtras* we know of. Here is Baudhāyana’s definition:b

c7.s2.t3.22

*dīrghacaturasrasyākṣṇayārajjuḥ pārśvamānī tiryaṅmānī ca*
*yatpṛthagbhūte kurutastadubhayaṁ karoti* |
‘The diagonal of a rectangle produces by itself both (the areas) produced separately by its two sides.’

c7.s2.t3.23

The same definition is given by Āpastamba. Kātyāyana’s definition given in the same language is characterized by the explicit mention of the knowledge of plane figures:

c7.s2.t3.24

*dīrghacaturasrasyākṣṇayārajjustiryaṅmānī pārśvamānī ca yat*
*pṛthagbhūte kurutastadubhayaṁ karotīti kṣetrajñānam* |

c7.s2.t3.25

Note the word *iti kṣetrajñānam* at the end, meaning ‘this is the knowledge of plane figures’.

c7.s2.t3.26

A question has often been asked whether such a definition resulted from empirical guesswork or was based on a proof of some kind. Although it is too much to expect elaboration of a proof in manuals of this type there is hardly any doubt that the Vedic *śulbavids* at this distant date possessed a valid proof of the theorem, of which the texts themselves provide reliable indications. For example, Baudhāyana, in the very next rule,a says that the rectangles for which the above is true have their sides 3 and 4, 12 and 5, 15 and 8, 7 and 24, 12 and 35, 15 and 36. The sum of squares of these numbers is also a square for each pair.

c7.s2.t3.27

a Datta (4), p. 77.

c7.s2.t3.28

b *Bśl.*, i. 48.

c7.s2.t3.29

3²+4² &= 5² / 12²+5² &= 13² / 15²+8² &= 17² / 7²+24² &= 25² / 12²+35² &= 37² / 15²+36² &= 39²

c7.s2.t3.30

FIG. 3.6.

c7.s2.t3.31

The above relationships between sides and diagonals or hypotenuse for rational rectangles or triangles have been freely used for finding the perpendicular directions (north-south) at the east or the west point of the east-west line in altar settings. The method is to take a cord of suitable length greater than the distance between the two poles one each at the east and the west point, fasten a loop at either end of the cord, fix it with the two poles by these loops and then stretch it by the *nirañchana* mark. The *nirañchana* mark divides the cord in two portions *x* and *y* such that a²+x²=y², where *a* is the distance between the east-west pole (Fig. 3.6). This finding of the properties of rational triangles is a strong argument in favour of the Vedic Hindus possessing proofs of the theorem of the square of diagonal or the hypotenuse. It is possible that by actually drawing the squares on the diagonal and the sides of a rational rectangle, dividing them into elementary unit squares and then counting them, they might have arrived at the truth of this theorem (Fig. 3.7).

c7.s2.t3.32

FIG. 3.7.

c7.s2.t3.33

*Question of Origin*

c7.s2.t3.34

The question of the Greek origin of the theorem and also whether Pythagoras himself was the discoverer of it and its proof has by no means been solved. The tradition attributing the theorem to Pythagoras is due to Cicero (*c.* 50 B.C.), Diogenes Laertius (second century A.D.), Athenaeus (*c.* A.D. 300), Heron (third century A.D.), and Proclus (*c.* A.D. 460), and therefore started about five centuries after the death of Pythagoras. Junge pointed out that the Greek literature of the first five centuries after Pythagoras contained no mention of the discovery of this or any other important geometrical theorem by the great philosopher and furthermore emphasized uncertainties in the statements of Plutarch and Proclus. Although various attempts have been made to justify the tradition and trace the proof to Pythagoras, no record of proof has come down to us earlier than that given by Euclid (Theorem 47, BK I). As to the relation 4²+3²=5² from which the theorem of rational triangle is derivable, very ancient Egyptian knowledge is attested by the Kahun papyrus of the twelfth dynasty (*c.* 2000 B.C.), but its association with rational triangles does not seem indicated in this or other Egyptian papyrii.a It is interesting to note that among the Egyptians, geometry of surveying was considered to be the science of the ‘rope-stretchers’ (*harpedonap'tae*) who thus appear to be the Egyptian counterpart of the Indian *śulbavids*. As to the antiquity of Pythagorean theorem in China, it is stated, though not proved, in the arithmetical classic *Chou Pei Suan Ching* (third or fourth century B.C.); the numerical relationship 4, 3 and 5 between the sides and the diagonal of a rational rectangle is also given in this text.

c7.s2.t3.35

a *Bśl.*, i. 49.

c7.s2.t3.36

*Areas and Volumes*

c7.s2.t3.37

Regarding other areas of geometry, the areas and volumes of a number of plane and solid figures have been given as follows:

c7.s2.t3.38

Area of a triangle = ½(base)×(altitude)
Area of a parallelogram = (base)×(altitude)
Area of a trapezium = ½(sum of two parallel sides)×(altitude)
Volume of a prism or cylinder = (base)×(height)
Volume of the frustum of a pyramid = ((a+a'))/2((b+b'))/2h,

c7.s2.t3.39

where a, b are the length and breadth of the rectangular base, a', b' the corresponding ones of the parallel face, and h is the height.

c7.s2.t3.40

*Squaring the Circle; π*

c7.s2.t3.41

Another type of problem which interested the *śulba* geometers was the circling of the square or its converse the squaring of the circle. Through these exercises, the Vedic Hindus were led to finding approximate values of π. For circling a square, Baudhāyana’s rule is first to draw the arc of a circle about the centre O of the given square ABCD with a radius equal to half the diagonal, intercepting the east-west line at P. Then the circle drawn about the centre O with a radius equal to half the side of the square plus one-third of the portion between the (previous) circle and the square (OQ=OR+QR where QR=PR/3) answers for the problem. If the side of the given square be 2a, the radius of the required circle works out to a × 1·1380718, taking the value of √2 = 1·4142156 as given in the *śulba* (*vide infra*). That the area of the circle 4·068987a² is somewhat higher than 4a² is fully reflected in the *śulba-sūtras*.

c7.s2.t3.42

a Heath, I, p. 352.

c7.s2.t3.43

FIG. 3.8.

c7.s2.t3.44

For squaring a circle, the same *śulba-sūtra* advises the diameter to be first divided into eight equal parts. The last 1/8th part is further divided into 29 parts and then 28 of these subdivisions are rejected (from this 1/8th part). Furthermore, from it is rejected 1/6th of the preceding subdivision (1/(8·29)) minus 1/8th part of the last subdivision. If d be the diameter of the circle, the side of the required square is given by $7d/8 + d/8 - 28d/(8·29) - (d/(8·29·6) - d/(8·29·6·8)) = 1224/1393 d (approx.).$

c7.s2.t3.45

From the foregoing operations, it is possible to calculate the value of π. In the example of circling of the square, the radius r in terms of a, half the side of the square is given by $r = a + a/3 (√2-1),$ where $π = (4a²)/([a+a/3(√2-1)]²) = 4/([1+⅓(√2-1)]²) = 3·0883$ accepting the *śulba* value of √2. Other results are 3·0885, 3·004. Baudhāyana's use of 3 as the value of π is therefore quite understandable. A more accurate value of π, e.g. 3·16049, is inferable from the data given in the *Mānava Śulba-sūtra*.

Algebra

c7.s2.t4.1

The altar geometry of the *Śulba-sūtras* does not fail to give us a glimpse of the beginnings of algebraic notions among the Vedic Hindus. These notions include quadratic equations, indeterminate equations, surds, conception of irrational numbers and determinations of their approximate values.

c7.s2.t4.2

FIG. 3.9a. *Mahāvedi.*

c7.s2.t4.3

Quadratic Equation

c7.s2.t4.4

The sacrificial ritual often necessitated enlargement or reduction of the altar in accordance with a number of plans. According to one plan, the shape is to remain the same, all sides being proportionally increased or decreased. The standard *mahāvedi*, for example, has the shape of an isosceles trapezium having for its face, base and altitude 24, 30 and 36 units respectively (area 972 sq. units). The standard *śyena-cit* (the falcon-shaped altar) consists of a central trunk of 4 sq. units, two rectangular wings each measuring 1 by 1⅕ or 1⅕ sq. units and a rectangular tail (*prādeśa*) 1 by 1 1/10 or 1 1/10 sq. units and has a total area of 7½ sq. units. By the first plan, the measures are left undisturbed but the unit of length is changed as required. According to another plan, some parts of the altar are proportionately increased while others are left undisturbed. Thus, in the case of *śyena-cit*, while the units for the complete square are proportionately changed, that of the fractional parts for the two *aratnis* and one *prādeśa* are left unchanged. If p represents the modified unit and α the change in area, we obtain the following quadratic equations for the two types of *vedis* mentioned above:

c7.s2.t4.5

FIG. 3.9b. *Śyena-cit.*

c7.s2.t4.6

Plan I: *Mahāvedi*: 36p× ((30p+24p))/2=(36×(30+24))/2+α

c7.s2.t4.7

972p²=α+972

c7.s2.t4.8

$Śyena-cit: 2p × 2p + 2[p(p+p/5)] + p(p+p/10) = 15/2 + α$ $15p² = 2α + 15$ $Plan II: Śyena-cit: 2p × 2p + 2[p(p+⅕)] + p[p+1/10] = 15/2 + α$ $14p² + p = 2α + 15$

c7.s2.t4.9

Specific examples given in the *śulba-sūtras* of Baudhāyana, Āpastamba and Kātyāyana agree with the solutions of these equations.

c7.s2.t4.10

Indeterminate Equations

c7.s2.t4.11

FIG. 3.10.

c7.s2.t4.12

The *śulbas* contain rules for the construction of a square *n* times a given square. The rule involves the application of the diagonal square theorem, for it says that the diagonal of the square produces the double square, the diagonal of the rectangle formed of the previous diagonal and the side of the given square gives a square thrice as large, the diagonal of the rectangle formed of the previous diagonal and the side of the given square produces a fourfold square and so on. Kātyāyana gives a general rule for finding a square *n* times a given square. According to this rule, the required square will be given by the altitude of an isosceles triangle whose base is (*n*−1) times and the sum of the two equal sides (*n*+1) times the side of the given square. That is, in Fig. 3.10, $BC = (n-1)a, AB = AC = ((n+1))/2a,$ where *a* is the side of the given square. It is clear that $AD² = [((n+1)a)/2]² - [((n-1)a)/2]² = na².$ If we put n = m², the above leads to $m²a² + ((m²-1)/2)² a² = ((m²+1)/2)² a²$ which is a solution of indeterminate equation of the second degree $x² + y² = z².$

c7.s2.t4.13

The questions relating to finding the sizes and number of bricks required for building the different layers of the altars envisage solutions of simultaneous indeterminate equations of the first degree. The *Baudhāyana Śulba-sūtra* prescribes that a *gārhapatya vedi* should be constructed with five layers of bricks, each layer containing 21 bricks so arranged that their rift in two consecutive layers do not coincide. For a square altar in which square bricks are used, Baudhāyana says that three kinds of bricks having their sides 1/6, 1/4 and 1/3 of the side (*vyāyāma*) of the altar should answer for the construction. How were the sizes as well as the number of bricks determined? Since 21 is not a square number and the rifts of consecutive layers must not coincide, square bricks of at least three different sides must be used. Suppose *a* to be the side of the square altar and l₁, l₂, l₃ the side of each of the three square bricks so that l₁p₁=l₂p₂=l₃p₃=a, where p₁, p₂, p₃ are rational integers. If *x* and *y* be the number of bricks of sides l₁ and l₂ for the first layer, and *x'* and *y'* those of sides l₃ and l₁ for the second layer, we have

c7.s2.t4.14

for the first layer, $x+y = 21$ $xl₁²+yl₂² = a²$ $x/(p₁²)+y/(p₂²) = 1$

c7.s2.t4.15

for the second layer, $x'+y' = 21$ $(x')/(p₃²)+(y')/(p₁²) = 1.$

c7.s2.t4.16

In other words, the problem leads to the following simultaneous indeterminate equations of the first degree: $x+y = 21$ $x/(m²)+y/(n²) = 1$

c7.s2.t4.17

where *m* and *n* are integral parts of *a*. Although the method of solution is not indicated, Baudhāyana gives correctly the values of *m* and *n* as 6 and 4 for x = 9, y = 12 and again as 3 and 6 for x = 5 and y = 16.

c7.s2.t4.18

Surds

c7.s2.t4.19

The technical expressions used in the *śulbas* for such quantities as √2, √3, √4, √18, etc., are *dvi-karaṇī*, *tṛ-karaṇī*, *catuṣkaraṇī*, *aṣṭādaśa-karaṇī*, etc. By using words for ordinal numbers along with *karaṇī*, quantities √(⅓), √(1/7) are expressed as *tṛtīya-karaṇī*, *saptama-karaṇī*, etc. Thus *karaṇī* has been used in the same sense as root or surd. Elementary operations with surds are clearly indicated in the various places of the texts. To draw a *sautramaṇikī vedi*, an isosceles trapezium of the same shape as, but one-third the area of, the *mahāvedi*, Āpastamba uses the surds as follows: 'For a unit *prakrama* 1/√3 of a *prakrama* is to be substituted;

c7.s2.t4.20

alternatively, the transverse sides should be 8 and 10 times √3 and the east-west line 12 times √3.'a Expressed in figures, $the first direction gives the area = 36/(√3) · ½(24/(√3) + 30/(√3))$ $= 324 sq. units;$ $the second direction gives the area = 12√3×½(8√3+10√3)$ $= 324 sq. units.$

c7.s2.t4.21

*Irrational Numbers and their Approximate Values*

c7.s2.t4.22

The Vedic Hindus have been credited further with the notion of irrationality of the quantities √2, √3. They have even given the values of such irrational numbers up to a high degree of approximation. For the value of √2, the exact wordings of some of the *śulbakāras* are quoted below:

c7.s2.t4.23

*pramāṇaṃ tṛtīyena vardhayettacca caturthenātmacatus-*
*triṃśonena | saviśeṣaḥ |*b
*karaṇīm tṛtīyena vardhayettacca svacathurthenātma-*
*catustriṃśonena saviśeṣa iti viśeṣaḥ*c

c7.s2.t4.24

Both versions having used practically the same wordings, these may be rendered as follows:

c7.s2.t4.25

'Increase the measure by its third and this third by its own fourth less the thirty-fourth part of that fourth. This is the value with a special quantity in excess.'d

c7.s2.t4.26

If we take 1 for the measure of a square side, the above formula gives the diagonal as follows: $√2 = 1 + ⅓ + 1/3.4 - 1/3.4.34 = 1·4142156.$

c7.s2.t4.27

In the same manner, the value of *tṛ-karaṇī* is given by $√3 = 1 + ⅔ + 1/3.5 - 1/3.5.52 = 1·7320513.$

c7.s2.t4.28

The method by which the values of these irrationals were obtained is not indicated in the texts. They did not obviously know how to extract the square root up to so many decimal places at this distant time. Thibaut, Bürk, Müller, Datta and others have reconstructed possible methods out of the very elements and procedures dealt with in the *śulbas*. For √2, one possible method appears to be as follows:

c7.s2.t4.29

b *Bśl.*, i. 61–62; *Āśl.*, i. 6.

c7.s2.t4.30

Two squares of unit side are taken. One of them is divided into three equal rectangular strips, of which the first and the second are marked 1 and 2. The third strip is subdivided into three squares of which the first is marked 3. The remaining two squares are each divided into four equal strips marked by 4, 5, 6, 7 and 8, 9, 10, 11. These eleven strips are added to the other square in the manner shown in Fig. 3.11 to obtain a square less a small square (shaded) at the corner. The side of the square equals 1+⅓+1/3.4 and the area of the shaded square (1/3.4)², so that the former is greater than the sum of the two original squares by the amount (1/3.4)². To find the approximate value, suppose x be the thickness of the smallest strips, 4, 5, ... 11. Then $2x(1+⅓+1/3.4)-x²=(1/3.4)²$ or $x=1/3.4.34, neglecting x² which is too small.$

c7.s2.t4.31

a *Āśl.*, v. 8.

c7.s2.t4.32

c *Kśl.*, ii. 13.

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d Thibaut (1), p. 21; Datta (4), pp. 188 ff.

c7.s2.t4.34

FIG. 3.11.

c7.s2.t4.35

The side of the square equalling the sum of the two original squares or the diagonal of each of the original square is given by $√ 2=1+⅓+1/3.4-1/3.4.34.$

c7.s2.t4.36

One of the cuneiform tablets from the old Babylonian times (1600 B.C.), now in the Yale Babylonian collection, shows a square with its two diagonals and values in sexagesimal system, from which √ 2 in the same system works out as $√ 2=1; 24, 51, 10.$

c7.s2.t4.37

Neugebauera has shown that the sexagesimal equivalent of the value of √ 2 as given in the *Sulba-sūtras* is 1; 24, 51, 10, 37, He has further hazarded a guess that both the main term and the subtractive correction of the Hindu value were based ultimately on Balylonian approximations. The more ancient character of the Babylonian cuneiform texts cannot, however, be doubted. Serious objections have been raised in accepting any date for the earliest of the *Sulba-sūtras* beyond 500 B.C. But that itself does not appear to be a reliable ground for making such a suggestion in view of a more complete and clear textual statement and several indications of the derivations of approximate values, embedded in the very texts themselves.

c7.s2.t4.38

a Neugebauer (1), p. 34.

c7.s2.t4.39

Permutations, Combinations and Pascal Triangle

c7.s2.t4.40

Another favourite mathematical pursuit of the Vedic Hindus was in the field of permutations and combinations, which was also very popular among the ancient Jainas about whose mathematical interests we shall speak presently. This interest was undoubtedly activated by the considerations of the Vedic meters and their variations. There are several Vedic meters, e.g. *Gāyatri*, *Anuṣṭubh*, *Bṛhatī*, *Triṣṭubh*, *Jagatī*, to mention a few, with 6, 8, 9, 11, 12 syllables. The Vedic meter specialists were concerned with the problem of producing different possible types of meters from those of varying syllables by changing the long and short sounds within each syllable group. In this effort, they were led invariably to laying the foundation of the mathematics of permutations and combinations. Emerging from the intricacies of the Vedic meters, these rules found immediate applications, e.g. finding possible combinations of the six tastes taking one, two, three, etc., at a time,a possible numbers of philosophical categories through combinations of *n* fundamental categories taken one, two, three, etc., at a time,b total number of perfumes that can be made from sixteen different substances taken one, two, three or four substances at a time,c and so on. In all these examples, the results are given correctly.

c7.s2.t4.41

Special importance attaches to Piṅgala’s *Chandaḥ-sūtra* (200 B.C.) which contains a method called *meru-prastāra* for finding the number of combinations of *n* syllables taken 1, 2, 3 . . . *n* at a time. The method, as explained by the commentator Halāyudha (tenth century A.D.), is as follows:

c7.s2.t4.42

‘After drawing a square on the top, two squares are drawn below (side by side) so that half of each is extended on either side. Below it three squares, below it (again) four squares are drawn and the process is repeated till the desired pyramid is attained. In the (topmost) first square the symbol for one is to be marked. Then in each of the two squares of the second line figure one is to be placed. Then in the third line figure one is to be placed on each of the two extreme squares. In the middle square (of the third line) the sum of the figures in the two squares immediately above is to be placed; this is the meaning of the term *pūrṇa*. In the fourth line one is to be placed in each of the two extreme squares. In each of the two middle squares, the sum of the figures in the two squares immediately above, that is three, is placed. Subsequent squares are filled in this way. Thus the second line gives the expansion of combinations of (short and long sounds forming) one syllable; the third line the same for two syllables, the fourth line for three syllables, and so on.’d

c7.s2.t4.43

a *SS. Rasabheda-vikalpādhyāya*, 63.

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b *Bhag. Sū.*, 314.

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c *Bṛh. S.*, lxxvii. 13–14.

c7.s2.t4.46

d Bag, pp. 72–73.

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The *meru-prastāra* is the same as the triangular array known in Europe as Pascal's triangle. The triangle first appeared in Europe on the title page of the arithmetic of Apianus (A.D. 1527) and subsequently in the works of Stifel (A.D. 1544), Scheubel (A.D. 1545), Tartaglia (A.D. 1556),

c7.s2.t4.48

Fig. 3.12.

c7.s2.t4.49

Bombelli (A.D. 1572) and other Renaissance methematicians and in Pascal's posthumous work *Traité du triangle arithmétique* (1665). Apart from giving a quick method of finding the number of combinations, the method in its association with meters clearly envisaged the computation of the binomial terms an, an-1b, . . . abn-1, bn of the expression (a+b)n.

Jaina Mathematics

c7.s3.1

Despite great interest attached to mathematics by the Jainas of ancient India, very little of their mathematical effort has unfortunately survived to this day. We have of course the mid-ninth century mathematical classic *Gaṇita-sāra-saṃgraha* by Mahāvīra (*c.* A.D. 850), but no such comparable work belonging to the pre-Christian centuries is extant. We only have a few fragments or insertions in canonical or other types of non-mathematical literature from which to judge of their early achievements. That a good body of mathematical literature must have existed at one time seems indicated in Mahāvīra's own statement in which he described himself as a mere compiler from the great ocean of the knowledge of mathematics from which long lines of holy sages had skilfully gathered many precious mathematical gems. About the first century B.C. the *Sthānāṅga-sūtra*, a Jaina canonical work, listed several mathematical topics which used to be cultivated at that time. These topics are: *saṃkhyāyana* (science of numbers), *parikarma* (fundamental operations), *vyavahāra* (subjects of treatment), *rajju* (geometry, like *śulba*), *rāśi* (heap, solid mensuration), *kalāsavarṇa* (fractions), *yāvat-tāvat* (equations, algebra), *varga* (square, quadratic equations), *ghana* (cube, cubic equations), *varga-varga* (biquadratic equations) and *vikalpa* (permutations and combinations). The Sanskrit version of the above enumeration in Pali runs as follows:

c7.s3.2

*parikarma rajjuḥ rāśiḥ vyavahārastathā kalāsavarṇaśca* |
*pudgalāḥ yāvattāvat bhavanti ghana ghanamūlaṁ vargaḥ vargamūlam* ||

c7.s3.3

These subjects and the various technical terms used by the Jainas passed later on into the mathematical works of scholars irrespective of their religious beliefs and adherences. It is thus quite reasonable to believe that in the period intervening the literature of the *Brāhmaṇas* and the *Sūtras* of the various Vedic schools and the period of specialization and Siddhāntic astronomy from about the fourth or fifth century A.D. the Jaina mathematicians played a significant role. The disappearance of their works might be due to (*a*) supersession of their simple processes and methods by better and more sophisticated ones due to Āryabhaṭa, Brahmagupta, Mahāvīra, Śrīpati, Śrīdhara, Bhāskara and others and (*b*) progressive deterioration of the culture of mathematics in their religious order.

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