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Mathematics in Ancient and Medieval India c7: Chapter IV Algebra

Mathematics in Ancient and Medieval India c7

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Chapter IV Algebra

c7.2

Early Indian and Chinese geometrical problems involved algebraical equations and their solutions similar to those of the Greeks who solved many comparatively difficult algebraic problems in a purely geometrical way. While the Greek algebra was developed by Diophantus in his *Arithmatica*, in the 3rd century A. D., the algebra in Babylon was developed much earlier in a more advanced form including problems on cubic and biquadratic equations, as recently shown by Neugebauer and others. One can not help wondering whether this Babylonian algebra could have been transmitted in seminal forms to lay the foundation of Indian and Chinese algebra on the one hand and for the Hellenistic development on the other. During the decay of Western Science in the early Middle Ages, the algebra of the Diophantine period was forgotten and when the great Arab Scientific Movement took place, Arabic algebra very probably derived its inspiration from India rather than from Greece.

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In India, the geometrical methods of solving algebraic problems, have been traced to the various *Śulbasūtras* of Baudhāyana, Āpastamba, Kātyāyana, Mānava and a few others. These problems, involving solutions of linear, simultaneous and even indeterminate equations, arose in connection with the construction of different types of sacrificial altars and arrangements for laying bricks into them. In the development of early mathematics, when the symbols for operation began to be used in the computations, a new branch evolved being separated from arithmetic and geometry which is known as algebra. The differentiation of algebra as a distinct branch from mathematics in general took place, from about the time of Brahmagupta (628 A. D.), following the technique of indeterminate analysis (*kuṭṭaka*). In fact, Brahmagupta used the terms *kuṭṭaka*, *kuṭṭa-gaṇita* to singnify algebra. The term *bījagaṇita* meaning science of calculation with elements or unknown quantities (*bīja*) was hinted by Pṛthudakasvāmī (850 A. D.) and used with definition by Bhāskara II (1150 A. D.).

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*Symbols of Operations*

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Addition was represented by simple juxtaposition of the terms or placing *yu* (*yu* means *yuta* or addition) in between the terms and subtraction by placing a plus sign (+) after the terms which is to be subtracted or a dot at its top; multiplication by placing the abbreviation *gu* (*guṇa* or *guṇita*) after the last term; division by putting the abbreviation *bhā* (*bhāga* or *bhājita*) in between the divisor and the dividend or by placing the divisor beneath the dividend; square-root by writing *mū* or *ka* (*mūla* or *karaṇī*) respectively after the terms and any one unknown quantity by the term *yāvattāvat* (*Sthānāṅga-Sūtra*), *yadr̥cchā*, *vāñchā*, *kāmika* (*Bakhshālī Ms.*) *gulikā* (Āryabhaṭa I) and *avyakta* (Brahmagupta, Śrīpati and Bhāskara II). Representation of several unknown terms involved in an equation was made by Brahmagupta (628 A. D.) by using abbreviation of names of several colours like, *kā* (*for kālaka* meaning black), *nī* (*for nīlaka* meaning blue) etc. For sign of equality, the terms *dṛśya* ( visible ) sometimes *rūpa* with used.

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Equations ( *samakaraṇa*, *samīkaraṇa*, *sadṛśīkaraṇa* etc. ) seem to have been classified in the *Sthānāṅgasūtra* ( c. 300 B. C. ) according to the powers of unknown quantity e.g. *yāvat-tāvat* ( simple ), *varga* ( quadratic ), *ghana* ( cubic ), *varga-varga* ( biquadratic ) etc. But such classification was not maintained. Brahmagupta ( 628 A. D. ) gave the following classifications : ( 1 ) *eka-varṇa samīkaraṇa*—equations in one unknown comprising linear and quadratic equations, ( ii ) *aneka-varṇa samīkaraṇa*— equations in many unknowns, ( iii ) *bhāvita* — equations involving products of unknown. This classification received further elaboration at the hands of Pṛthūdakāsvāmī ( 860 A. D. ) and Bhāskara II ( 1150 A. D. ).

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*Methods and Operations*

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This primitive method of solving simple linear equations of the type ax + b = 0 by subtituting guess values g₁, g₂ etc.¹ was in extensive use among the Arab and European mathematicians of the middle ages. The problem of this type dealt with in the *Sthānāṅgasūtra* using the term *yāvat-tāvat* for the unknown quantities has been discussed thoroughly by Datta². A solution to this problem also occurs in the *Bakhshālī Ms* ( 4th century A. D. ).

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¹ Smith, D. E. *History of Mathematics*, 2, p. 437-8, Dover Publication.

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² Datta, B. “The Jaina School of Mathematics', *Bulletin of the Calcutta Mathematical Society*, 21, p. 122, 1929.

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³ Datta, B, 'The Bakhshali Mathematics', *Bulletin of the Calcutta Mathematical Society*, 21, p. 31, 1929.

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Āryabhaṭa I¹ might have obtained the solution x = (b - a)/(m - n) from the relation m x + a = nx + b and tackled the problem of inverse operation e.g. ax = pq and ax ± b = p by giving the solution as x = pq/a and ax = p b respectively and so on. Bhāskara I (600 A. D.), the earlist exponent of Āryabhaṭa I has used a technical term, *yāvakarana* in his *Āryabhaṭiyabhāṣya*. The word *yāva* (*yu* + *ghañ*) means ‘to mix’ or ‘to separate’. It is also synonymous with *varga* or *kṛti*. The operation on *yāva* has been preserved by Pṛthudaka (850 A. D.) in his commentary on the *Brāhmasphuṭasiddhānta* as such

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¹ Āryabhaṭīya, v. 30, 28.

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*yāva* 0 *yā* 10 *rū* 8.
*yāva* 1 *yā* 0 *rū* 1. Here *yā* = *yāvattāvat* = unknown = x (say), *yāva* = square of the unknown quantity, *rū* = absolute number. The above can be written as 0.x² + 10.x - 8 = 1. x² + 0.x + 1, or, 10x - 8 = x² + 1.

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The word *yāva* having the idea ‘to mix’ or ‘to separate’ has some affinity with that of *aljabr*. The process on the operations (*al-Jabr*) and equations (*al-Muqābala*) has been used by al-Khwārizmī (c. 825 A. D.) in his work *Ḥiṣāb al-Jabr w'al Muqābalah* like,

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The process of *al-Jabr* bx + 2q = x² + bx - q
or bx + 2q + q = x² + bx

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The process of *al-Muqābalāh*
bx + 2q = x² + bx - q or 3q = x²

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Both these process contain the idea of inverse operation similar to Hindu method.

Quadratic Equation

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The *Śulbasūtras* contain problems involving quadratic equations of the type : ax² = c, ax² + bx = c but give no solution. The *Bakhshālī Ms.* gives the solution of a problem in the form which reduces to $x = (√(b² - 4ac) - b)/2a$

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Āryabhaṭa I (499 A. D.) and Brahmagupta (628 A. D.) clearly indicate their knowledge of quadratic equations and their solutions. In connection with an interest prob'em, Āryabhaṭa I gave the solution which in symbo s runs as follows : $x = (-p + √(p² + 4tpq))/2t$ where p = principal, t = time, q = sum of interest on principal and interest on interest in time t; x = interest on principal in unit time.

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A similar quadratic solution arising out of an interest problem is given by Brahmagupta. He has also given a solution of the problem of quadratic equations in connection with the determination of the number of terms (n) in an A. P.

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The method of transforming the left hand side of the quadratic equation ax² + bx = c into a whole square by multiplying both sides by 4a, adding b² and taking the square root, is given by Śrīdhara (c. 991 A.D.) in his algebra which is lost but preserved in the quotations of Bhāskara II (c. 1150 A. D. ), Jñānarāja and Sūryadāsa. Datta and Singh discussed in detail all these Indian methods of solving the quadratic equation in their work, *History of Hindu Mathematics*.¹

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¹ Datta, B. and *Singh, A. N. History of Hindu Mathematics*, 2, p. 59-87, Lahore, 1938.

Progressive Series

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The usual term for the series is *śreḍhī*. This term is used in almost all mathematical literatures from 5th century A. D. onwards. The other terms used in the *Bakhshālī Ms.*² are *saṃkalita*, *varga*³, *rupona*⁴. Another peculiar term found in the *Bakhshālī Ms.* is *pārtha*. In the *Bṛhaddhārā*⁵, the term *dhārā* ( ladder ) is used. The initial term in an A. P. is *ādi*, *mukha*, *badan* or *baktra* all meaning face. *Antya* or *anta* used to denote the last term; *caya*, *pracaya* or *uttara* stands for the common difference, *gaccha* and *pada* denotes the number of terms, the middle term is *madhya*; the sum of progression is *sarvadhana*, *śreḍhīphala*, *guṇita* or *citi*.

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² Folio 23 Recto; folio 4 verso etc.

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³ Here it means ‘group’.

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⁴ Hoernle ( *Indian Antiquary*, 17, p. 47 ) and Kaye ( vide his edition of *Bakhshālī Ms.*, p. 73 ) think that the origin of the term lies in the fact that ‘the rule in question began with the term *rupona* which corresponds to the ( t-1 ) of the formula’.

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⁵ The *Bṛhaddhārā-Parikarma*, a treatise on series is extant now. Certain portion of the work appears in the *Trilokasāra* of Nemicandra.

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The term used for the sum of G. P. in the *Trilokasāra*⁶ ( 978 A. D. ) is *guṇa-gaṇita* or *guṇa-saṃkalita*. The terms *śreḍhī*, *ādi*, *gaccha*, *uttara*, *guṇita* were standardised long ago in the Jain canons (2nd century A. D. ).

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⁶ Rule 231,

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In Indian treatises, three types of series occur viz. arithmetic, complex and geometric series. The classification appeared late in the 9th century A. D. in the *Gaṇitasārasaṃgraha* of Mahāvīra (850 A. D. ).

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*Arithmetic Progression*: Indian interest in the series began quite early in the Vedic age. A large number of Vedic series are mentioned in the *Taittirīya Saṃhitā*¹, *Vājasaneya Saṃhitā*² and *Pañcaviṃsa Brāhmaṇa*³. Some of the few such series are found in the *Bṛhaddevatā*⁴, *Śatapatha Brāhmaṇa*⁵, *Baudhāyana Śulba*⁶, Buddhist work *Dīgha Nikāya*⁷ and a Jain work *Antagoḍa Dasāo*⁸. The general rule for the summation of an A. P. : S = a + (a + b) + (a + 2b) + + [a + (n - 1)b] where S denotes the sum, a = first term, b = common difference and n = number of terms, is found in the *Bakhshālī Ms* ⁹ as follows: S = n [a + (n - 1) b/2 ]. Āryabhaṭa I¹⁰ (476 A. D. ) gave the above form as well as the form : S = n/2 (a + b). The same rule appears in the works of Brahmagupta¹ (628 A. D.), Mahāvīra² (850 A. D.), Śrīdhara³, Nemicandra⁴ and Bhāskara II⁵ (1150 A. D.). Āryabhaṭa I⁶ and all the above writers with the exception of Nemicandra gave also the sum of the following particular case :

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¹ 7.2. 11–17.

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² 18. 24.25.

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³ 18.3.

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⁴ Vide Macdonall's English translation, 1904.

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⁵ 10.5.4.7.

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⁶ Datta, B. *The Science of the Śulba*, Calcutta University, p. 217-18, 1932.

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⁷ David, T. W. R. *Dialogue of the Buddha*, 3, p. 70–72, 1928.

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⁸ Ed. by L. D. Barnett, p. 102-6, 1907.

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⁹ *Bulletin of the Calcutta Mathematical Society*, 21, p. 1–60.

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¹⁰ *Āryabhaṭīya-gaṇita*, v. 19.

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¹ Colebrooke, H. T, *Hindu Algebra*, p. 290.

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³ *Triśatikā*, Sudhākara's edition, rule 39.

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⁴ *Trilokasāra*, rule 164.

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⁵ Colebrooke's *Hindu Algebra*, p. 53.

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⁶ *Āryabhaṭīya-gaṇita*, v. 22; *Brāhmasphuṭasiddhānta* of Brahmagupta, ch. 12, v. 20.

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$1+2+3+......+n = n/2(n+1).$

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Āryabhaṭa I calculated the value of unknown number of terms n for an A. P. series when its sum S, first term a, and commn difference b, are known with the help of quadratic equation as follows :

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$n = ½[(√(8bS+(2a-b)²)-2a)/b+1]$

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² *Gaṇitasārasaṃgraha*, ch. 2, sl. 61, 62.

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The same was also given by Brahmagupta⁷ (628 A. D.), Mahāvīra⁸ (850 A. D.) and Bhāskara II⁹ (1150 A. D.).

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⁷ Vide Colebrooke's *Hindu Algebra*, p. 291.

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⁸ *Gaṇitasārasaṃgraha*, ch. 2, v. 69.

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⁹ *Līlāvatī*, rule 12 vide H. C. Banerjee's edition .

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*Complex series* : (a) Āryabhaṭa I¹⁰ (499 A. D.), Brahmagupta¹¹, Mahāvīra¹² and also Bhāskara II¹³ gave the summation of the series of squares and cubes of n natural numbers as follows :

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¹⁰ *Āryabhaṭīya-gaṇita*, v. 22, vide Clark's ed.

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¹¹ Vide Colebrooke's *Hindu Algebra*, p. 291.

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¹² *Gaṇitasārasaṃgraha*, ch. 6, v. 296, 301.

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¹³ Vide Colebrooke's *Hindu Algebra*, p. 53.

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$1² + 2² + 3² + ...... + n² = (n (n+1) (2n+1))/6$

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$1³ + 2³ + 3³ + ..... + n³ = [(n (n+1))/2]²$

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The formula for the sum of the cubes of the n natural numbers appears in *Codex Arcerianus*¹ (6th century A.D.). The Arabic Scholar āl-Karkhī² (c. 1020 A.D.) gave and supplied proofs to the summation to the above two series.

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¹ Smith, D. E. *History of Hindu Mathematics*, 2, p. 504, Dover Publication.

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Reference may be made here that the Jain mathematician Mahāvīra (850 A.D.) also gave the results of the generalised (algebraic) summation of the square and cube of the series a, a+b, a+2b,......corresponding to Āryabhaṭa I's work in natural number as follows : $a² + (a+b)² + (a+2b)² + .........to n terms$ $= n [((2n-1) b²)/b + ab (n-1) + a²]$ and a³ + (a+b)³ + (a+2b)³ + ......to n terms $= S²b + Sa (a-b) if a > b$ $= S²b - Sa (a-b) if a < b$ where S = a + (a+b) + (a+2b) + ......to n terms.

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² Cajori, F. *A History of Mathematics*, 2nd revised enlarged edition, p. 100.

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(b) Āryabhaṭa I³ (476 A.D.) gave the summation of the sums of n times the 1st term of n natural numbers, (n-1) times the 2nd term of n natural numbers, (n-2) times the 3rd term of natural numbers……etc. Thus if \begin{align*} S_1 &= n.1 \\ S_2 &= (n-1).2 \\ S_3 &= (n-2).3 \\ & \dots\dots\dots\dots\dots\dots \\ & \dots\dots\dots\dots\dots\dots \\ S_{n} &= [ n-(n-1) ].n \end{align*} then according to Āryabhaṭa I, \begin{align*} & S_1 + S*2 + \dots \dots + S*{n} \\ = \sum_{n=1}n S_{n} & \\ = n.1 + (n-1).2 & + (n-2).3 + \dots\dots + \\ & [n-(n-1)].n \\ = 1 + (1+2) & + (1+2+3) + \dots + (1+2+\dots + n) \\ = (n(n+1)(n+2))/6 & \end{align*}

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³ *Āryabhaṭīya-gaṇita*, v. 21.

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Nārāyaṇa ( 1356 A. D. ) calls this as repeated sum or *Vārasaṁkalita*. According to Nārāyaṇa,

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*Vārasaṁkalita* of 1st order of n natural numbers \begin{align*} &= 1 + 2 + 3 + \dots\dots + n \\ &= (n(n+1))/2 \end{align*}

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*Vārasaṁkalita* of 2nd order of n natural numbers \begin{align*} &= 1 + (1+2) + (1+2+3) + \dots n terms \\ &= \Sigma \ (n(n+1))/2 = (n(n+1)\ (n+2))/6 \end{align*}

c7.s2.49

Āryabhaṭa I gave the result on'y upto *Vārasaṁkalita* of 2nd order, but Nārāyaṇa proceeded further.

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His result for the *Vārasaṃkalita* of rth order of a natural numbers¹ is :

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¹ *ekādhikavāramitāḥ padādirupottarāḥ pṛthaktem'śāḥ ।* *ekādyekacayahaśāstaddhāto vārasaṃkalitam ॥* ( *Gaṇitakaumudī*, Part II, p. 411 ). "The number beginning with the number of terms in the series increasing by one and equal in number to one more than the number representing the order of summation separately form the numerators. The corresponding denominators are the natural number beginning with the product of these ( fractions ) is the *Vārasaṃkalita*".

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$(n(n + 1) (n + 2) (n + r))/(1.².³.4 (r + 1 ))$ $= (n(n + 1) (n + 2) (n + r))/((r + 1) !)$

c7.s2.53

In a similar manner Mahāvīra² ( 850 A. D. ) also has given an incorrect result of *Vārasaṃkalita* of 2nd order using the series a, a + b, a + 2b, in place of the series in n natural numbers. The result is

c7.s2.54

$n/2 [ ((2n-1)b²)/6 + b/2 + ab ](n + 1) + a(a + 1)$

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² *Gaṇitasārasaṃgraha*, ch. 6, rule 305½.

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The correct result is (n(n + 1))/2 [ a + (b(n - 1))/3 ]

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Nārāyaṇa³ calculated the *Vārasaṃkalita* of rth order by employing the general algebraic progressive series like a, a + b, a + 2b. The given result is :

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³ *Gaṇitakaumudī*, Part I', p. 124. Vide editor's note. For edition, see *General Bibliography*.

c7.s2.59

$a. n-1S_r. (r+1)/(n-1) + b. n-1S_r$ when a, b and n are the 1st term, common difference and number of terms of an A. P. Nārāyaṇa represented the parallel sides of a trapezia by a-b/2 and nb+a-b/2 and the altitude by n. Arithmetic Progression with sum=0 are diagrammatically represented by joining two equal isosceles inverted triangles with their vertices.

c7.s2.60

( c ) In the *Bakhshālī Ms.*¹ certain elementary cases of complex series without sum are mentioned. These series are classified into two types viz. ( i ) *yutivarga-krama* and ( ii ) *yutagaṇita-yutakrama*. Mahāvīra² ( 850 A. D. ) gave the sum of the following complex series which may be considered as one of his highest contribution in this branch : $a+(ar)+[ (ar)r ]+$ $[ (ar)r r ]+ n terms$ $=S ±(((S′)/a-n )m)/(r-1)--$ $where S=a+ar+ar²+ n terms.$

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*Geometrical Progression* : Earliest expression of an idea representing G. P. occurs in the *Chandaḥ-sūtra*³ ( rule of metres ) of Piṅgala ( c. 200 B. C. ) where the series : 1, 2¹, 2², 2³...is found. Mahāvīra⁴ ( 850

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¹ Datta, B. 'The Bakhshālī Mathematics', *Bulletin of the Calcutta Mathematical Society*, 21, p. 30-31, 1929.

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² *Gaṇitasārasaṃgraha*, ch. 2, rule 314.

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³ *Chandaḥ-sūtra* of Piṅgala, rules 8, 28, 29, 30, 31, 32.

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⁴ *Gaṇitasārasaṃgraha*, ch. 2, rules 94 and 93.

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A. D. ) gives the generalised result of the sum of the n terms of the G. P. as follows : $a + ar + ar² + ...... to n terms = (a(rn - 1))/(r - 1)$

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This formula reappears also in the works of Pṛthudakasvāmī¹ ( 864 A.D. ) and Nemicandra² (978 A. D.).

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¹ Colebrooke, *Hindu Algebra*. p. 291.

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² *Trilokasāra*, rule 231.

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Mahāvīra gave also the form S = (rarn-1 - a)/(r-1)

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The rule appears amongst the Arabians in the work of āl-Birūnī's *India* in connection with a chess-board problem which he might have learnt during his tour in India³.

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³ Smith, D. E. *Ibid.*, 2, p. 502, Dover Publication.

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From the above, Mahāvīra deduced three expressions⁴ also for finding the common ratio (r), first term ( a ) and number of terms (n).

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⁴ *Gaṇitasārasaṃgraha*, ch. 2, rules 97, 98 and 101.

Permutation and Combination

c7.s3.1

Smith⁵ opined that "the Hindus seem to have given no attention to permutation and combination until Bhāskara II took it up in his *Līlāvatī*". Chakravarty⁶ has shown that the subject of permutations and combinations is a noteworthy contribution of the Indians and its interest originated before the Christian era. A summary is also given by Srinivasienger.¹ In the early Jain canonical literature, permutation was termed *Vikalpa-gaṇita* and combination *Bhaṅga*. The Jain *Bhagavatī-sūtra* (300 B. C.) calculated the number of combinations of n fundamental categories taken two at a time, three at a time, and so on.² Suśruta (200 B. C.) correctly gave the sum of the combinations of six tastes taken one at a time, two at a time etc. upto all at a time.³ Varāhamihira⁴ (505 A. D.) has stated that "an immense number of perfumes can be made from sixteen substances taken in one, two, three or four proportions and has correctly given the number of perfumes resulting from 16 ingredients (being mixed in all proportions) as 174720". The results given agree with the modern formulae though there is no mention of any such formula in his text.

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⁵ Smith, D. E. *History of Mathematics*, 2, p. 525, Dover Publication.

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⁶ Chakravarty, G. 'Growth and Development of Permutations and Combinations in India', *Bulletin of the Calcutta Mathematical Society*, 24, p. 79-88, 1932.

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¹ Srinivasiengar, C. N. *The History of Ancient Indian Mathematics*, p. 26-28, 73, 83-84, Calcutta, 1967.

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² Sūtra, 314; vide also *Bulletin of the Calcutta Mathematical Society*, 21, p. 133-136, 1929.

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³ *Suśrutasaṃhitā*, ch. 63. (*Rasabhedavikalpādhyāya*).

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⁴ *Bṛhatsaṃhitā*, ch. 77, sl. 13, 14, 17 (for the translation vide Kern's work).

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The great astronomer and encyclopaedist Varāhamihira applied the same principle in his astrological work *Bṛhajjātaka* in connection with planetary conjunctions.⁵

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⁵ *Bṛhajjātaka*, rule 3.

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Mahāvīra⁶ (c. 850 A.D.) first gave the general formula for nc_r. Bhāskara II (c. 1150 A. D.) gave the name *ankapāśa* to the subject of permutation and combination. He also gave the general formulae for nc_r and np_r. Bhāskara II made some other valuable contributions to the subject of permutation and combination. He gave the rules¹ to find :

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⁶ *Gaṇitasārasaṃgraha*, ch. 3, v. 218.

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i ) the variation of the number of permutation of a number consisting of n digits of which p number of digits are to be alike and q also are to be alike, the result is (n !)/(p!q!) ;

c7.s3.13

ii ) the variation of numbers each of which contains n digits with a sum equal to S, zero should not be counted as a digit. The result of the problem is given by : S⁻¹c_n-1. This result holds good when 9+n>S.

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¹ *Līlāvatī*, rules 267, 272, also compare with 274.

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Bhāskara II gave no proof. Banerjee supplied the plausible proofs to the latter.²

Pascal Triangle and the Binomial Theorem

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The credit of expansion of Binomial theorem :

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(a+b)n=an+nc_₁ an-1b+nc_₂ an-2 b²+..........+bn in its simplest form i.e., when n is a positive integer, goes to India and it is the Indian scholar Halāyudha ( 10th century A. D. ) who established it much before the Chinese and European scholars could do anything about it³ It is based on a concept which developed in India in association with the problem of making verses of different metres ( *chandas* ) in a poetical composition. The *Varṇasaṅgīta* (the music of sound variations) of the Vedic and post Vedic composers depended only on the variation of two sounds namely *guru* (long) and *laghu* (short) sounds. Piṅgala (200 B. C.) in his *Chandaḥsūtra*¹ described clearly a rule of finding the variation of sounds of syllables in a particular metre. For example, in a three syllabic *madhyā chanda* based on *guru* and *laghu*, the variations of *guru* and *laghu* will be — three *guru* sounds will occur once, two *guru* and one *laghu* sound will occur thrice, one *guru* and two *laghu* sounds occur thrice, three *laghu* sounds will occur once. If we take *guru* as *a* and *laghu* as *b*, then it gives (a+b)=1.a³+3a²b+3ab²+1.b³. Similarly for a four syllabic *Pratiṭhā chanda*, it uses (a+b)⁴=1.a⁴+4a³b+6a²b²+4ab³+1.b⁴. The technique of finding the number of variation of sounds was actually practised by Piṅgala and other Vedic composers² to detect the quality as well as shortcomings of the metres.

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² Vide Haran Chandra Banerjee's edition of *Līlāvatī*, p. 197-200.

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³ Bag, A. K. 'Binomial Theorem in Ancient India', *Indian Journal of History of Science*, 1, No. 1, p. 68-74, 1966.

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Halāyudha (10-11th century A.D.) in his commentary on the Piṅgala *Chandaḥsūtra*³ (ch. 8, verse 34) introduced a *meruprastāra* (pyramidal scheme) technique with the coefficients of the different variations in the expansion of (a+b)¹, (a+b)², (a+b)³………to minimise the trouble of computing the different layers which run thus :

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¹ *Chandaḥ sūtra* of Piṅgalācārya, edited with the commentary of Halāyudha by Sitanath Sarman, Calcutta.

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² Vide *Jayadaman*, edited by Velāṅkar.

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³ Ch. 8, Verse 34.

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“Draw a square on the top, two squares are drawn below (side by side) so that half of each is extended

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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 symbo' for one is to be marked. Then in each of the two squares of the second line figure one is to be marked. 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 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 ) in a one syllabic metre ; the third line the same for two syllables, the fourth line for three syllables *meruprastāra* and so on.” [ vide 1 Fig. 17 (a), 17 (b) and 17 (c). ]

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Fig. 17 (a).

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Fig. 17 (b).

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In modern notation this gives to :

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Fig. 17 (c).

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In this way, the general expansion of $(a+b)n = an + n_c₁ an-1 b + + bn was$ readily obtained for a metre of n syllables.

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The exactly same technique of arriving at the triangular array appeared in China (1303 A.D.) in the work *Ssu Yuan Yü Chien*¹ ('the Old Method Chart of Seven Multiplying Squares' ) of Chu Shih Chieh and in Europe in the publication of Pascal's *Traité du Triangle Arithmetic* (1665 A.D.) The Triangle, however, appeared in Europe more than hundred years before the *Traité* on the title page of the arithmetic of Apianus (1527 A.D.), and in the work of Stifel (1544 A D.), Scheubel (1545 A.D.), Tartaglia (1556 A.D.), Bombelli (1572 A.D.) and others¹, but about four centuries later before it actually appeared in India.

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¹ Mikamī, Y. *Mathematics in China and Japan*, New York, p. 90; Smith, D. E. *History of Mathematics*, 2, p. 508, Dover Publication; Needham, J. *Science and Civilisation in China*, 3, p. 134.

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¹ Smith, D. E. *History of Mathematics*, 2, p. 508-11, Dover Publication.

Indeterminate Equations of the First Degree

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The subject of indeterminate analysis (first degree) is designated in Indian mathematics by the term *kuṭṭaka*. *Kuṭṭ* means 'to break' or 'to pulverize'. The name has been given on account of the process of continued division that is adopted for the solution. Indian interest in the indeterminate problems is first found in the *Śulbasūtras*, Detailed rules of solutions of the type : by = ax ± c were given by scholars from Āryabhaṭa I (476 A. D.) onwards Colebrooke², Chasles³ and Sen⁴ gave credit to Indians for attaining a general solution of the problem. Some scholars favoured Greeks and the Chinese for the origin. They are of opinion that Indians got inspiration in this subject from the foreign countries. In the foregoing discussion it can be seen that while the career of indeterminate (first degree) is doubtful in other cultural areas, in India it is based on strong and solid foundation.

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² Colebrooke, H. T. *Algebra with Arithmetic and Mensuration from the Sanskrit of Brahmagupte, and Bhāscara*, London, 1817.

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³ Chasles, M. *Apercu historique sur l'origine et le developpement des methodes en geometric*, 2e ed. Paris, 1875.

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⁴ Sen, S. N. 'Study of Indeterminate analysis in Ancient India' *Bulletin of History of science Congress*, held at Ithaca p. 493-97, 1962.

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The problems of altar construction in the *Śulbasūtras* necessitated the evolution of simultaneous indeterminate equations of the type : i ) x + y = 21 x/(p²) + y/(q²) = 1 ii ) x + y + z + u = 200 x/m + y/n + z/p + u/q = 7½ etc.

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A few arbitrary solutions were given by Baudhāyana (c. 600 B. C.), Āpastamba (c. 550 B. C.), and their commentators. The rational solutions of such equations were given by Datta¹ and Kamalamma.²

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¹ Datta, B. *The Science in the Śulba*, p. 180-86, Calcutta University, 1932.

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² Bulletin of the Calcutta Math. Society (BCMS), **40**, p. 140-44, 1948.

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Āryabhaṭa I (b. 476 A. D.) first gave a rule for obtaining the general solution of the problem by = ax ± c. His rule was obviously intended for the solution of the following problems :

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(1) To find the number (N), which when divided by a given number (a) will leave a remainder (r₁) and when divided by another given number (b) will leave a remainder (r₂).

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(2) To find a number (N) which being divided severally by the given numbers a₁, a₂, a₃, _n leaves remainders as r₁, r₂, r₃_n respectively.

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In modern notation, this can be written as, ( 1 ) N = ax + r₁ = by + r₂ ( 2 ) N = a₁x₁ + r₁ = a₂x₂ + r₂ = a₃x₃ + r₃ = = a_nx_n + r_n.

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*Nature of Solution ( Āryabhaṭa I )* Āryabhaṭa I keeps c = r₁ - r₂ always positive, consequently his rule is directed to solve an equation of the form : i ) y = (ax + c)/b if r₁ > r₂ and ii ) x = (by + c)/a if r₂ > r₁. Where a, b = divisors, *bhāgahāra*, *bhājaka*, *cheda* etc. r₁, r₂ = remainders, *agra*, *śeṣa* etc.

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Āryabhaṭa I's rule¹ : *adhikāgra bhāgahāraṃ chindyādūnāgra bhāgahāreṇa ।* *śeṣaparasparabhaktaṃ matiguṇamagrāntare kṣiptam ॥* *adhaupariguṇitamantyayugūnāgracchedabhājite śeṣam ।* *adhikāgracchedaguṇaṃ dvicchedāgramadhikāgrayutam ॥*

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¹ *Āryabhaṭīya*, Gaṇita, vss. 32-33.

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*English Translaton :* We follow Datta who based his translation on that of Bhāskara I ( 600 A. D. ), the first commentator on Āryabhaṭa I's work. The translation runs as follows : "Divide the divisor corresponding to the greater remainder by the divisor corresponding to the smaller remainder, being mutually divided, the last residue should be multiplied by an integer at our choice such that the product on being added to ( if the number of quotients in the division process is odd ) or subtracted ( if the number of quotients is even ) by the difference of the remainders, will be exactly divisible by the penultimate remainder. Place the quotients of the mutual division successively one below the other in a column, below them the optional multiplier and below it the quotient just obtained. The penultimate is multiplied by the one just above it and added to that just below it. Repeating this process, divide the last number obtained by the divisor corresponding to the smaller remainder. Then multiply the residue by the divisor corresponding to the greater remainder and add greater remainder. The result will be a number corresponding to the divisors.'¹

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¹ *Bulletin of the Calcutta Mathematical Society*, 24,p.19–36,1933.

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*Explanation of the Rule* :

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i ) Solution of y=(ax+c)/b.........(1) in positive integers when a and b are prime to each other.

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Repeated divisions following H. C. F. process may be placed as follows :—

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{rll} b) a & (a_1 & / ba_1 & & a = ba_1 + r_1 / r_1) & b (a_2 & b = r_1a_2 + r_2 / & r_1a_2 & r_1 = r_2a_3 + r_3 / & r_2) & r_1 (a_3 & r_2 = r_3a_4 + r_4 / & & r_2a_3 & and so on. / & & r_3) r_2 (a_4 & / & & r_3a_4 & / & & r_4 & / & & \dots & / & & \dots & / & & \dots & / & & r*{n-1}) r*{n-2} (a_n / & & r_{n-1}a_n / & & r_n

c7.s5.22

$then a/b = a₁ + 1/(a₂ +) 1/(a₃ +) 1/(a₄ +) 1/(a_n-1 +) 1/a_n$

c7.s5.23

$Let p₁/q₁, p₂/q₂, p₃/q₃, p_n/q_n be the succe-$ $ssive convergents of a/b, then :$

c7.s5.24

$p₁/q₁ = a₁$ $p₂/q₂ = a₁ + 1/a₂ = (a₁a₂ + 1)/a₂$ $p₃/q₃ = a₁ + 1/(a₂ +) 1/a₃$ $= a₁ + 1/(a₂ + 1/a₃)$ $= a₁ + a₃/(a₂a₃ + 1) = (a₁(a₂a₃ + 1) + a₃)/(a₂a₃ + 1)$ $p₄/q₄ = a₁ + 1/(a₂ +) 1/(a₃ +) 1/a₄$ $= a₁ + 1/(a₂ + 1/(a₃ + 1/a₄))$ $= a₁ + 1/(a₂ + a₄/(a₃a₄ + 1)) = a₁ + (a₃a₄ + 1)/(a₂(a₃a₄ + 1) + a₄)$ $= (a₁a₂(a₃a₄ + 1) + a₄ + a₃a₄ + 1)/(a₂(a₃a₄ + 1) + a₄)$ $and so on, and the last convergent p_n/q_n = a/b$

c7.s5.25

Evidently, p_n. q_n-1 - q_n. p_n-1 = ± 1 according as n = even or odd.

c7.s5.26

This result was perhaps known to Āryabhaṭa I for he always used the result p_n-q_n. u=-c (whether n is even or odd), the two numbers u and v were obtained by following an ingenious technique discussed later.

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*Case I : When n=even i.e. the number of partial quotients for finding u and v is odd* (vide Table I)

c7.s5.28

The cryptic Sanskrit terms *agrāntare kṣiptam* used by Āryabhaṭa I in the rule has been explained by his commentator Bhāskara I (c.600 A.D.) in his *Āryabhaṭīyabhāṣya* as *sameṣu kṣiptam, visameṣu, śodhyam*, i.e. add (the *kṣepa* quantity, c) when n=even, and subtract when n=odd." Since the ratio u/v was always used to determind the (n-1) th convergent, the *kṣepa* quantity c was to be added or subtracted according as the number of partial quotents arranged in Table I & II are odd and even. Suppose the process of division was stopped at remainder r₃

c7.s5.29

then (r₃t+c)/r₂=q₁ say, where t=(r₂ q₁-c)/r₃

c7.s5.30

The quotients a₁, a₂, a₃ with *mati* t, and final quotient q₁ are placed one after another in the first column (vide Table I) and the final numbers u and v were obtained as follows:—

c7.s5.31

*Table 1*

c7.s5.32

\begin{array}{c|c|c|c} a_1 & a_1 & a_1 & a_1 v+\alpha_1= \\ a_2 & a_2 & a_2 \alpha_1+t=v, say & u, say \\ a_3 & a_3 t+q_1=\alpha_1, say & \alpha_1 & \\ t & t & & \\ q_1 & & &

c7.s5.33

Now α₁ = a₃ t + q₁ {ll} = a_3 ( (r_2\ q_1 - c)/r_3 ) + q_1 & \because t = (r_2\ q_1 - c)/r_3 / = (q_1\ (\ r_2\ a_3 + r_3\ ) - a_3\ c)/r_3 / = (q_1\ r_1 - a_3\ c)/r_3 & ( p. 196 ) v = a₂ α₁ + t {ll} = a_2\ {\ (q_1\ r_1 - a_3\ c)/r_3\ } + (r_2\ q_1 - c)/r_3 / = (q_1\ (a_2 r_1 + r_2) - c\ (a_2 a_3 + 1))/r_3 / = (q_1 b - c\ (a_2 a_3 + 1))/r_3 & ( p. 196 ) u = a₁ v + α₁ {l} = a_1\ {\ (q_1\ b - c\ (a_2\ a_3 + 1))/r_3\ } + (q_1\ r_1 - a_3\ c)/r_3 / = (q_1\ (a_1\ b + r_1) - c\ {a_1\ (a_2\ a_3 + 1) + a_3\ })/r_3 / = (q_1\ a - c\ {a_1\ (a_2\ a_3 + 1) + a_3})/r_3 {ll} u/v & = (q_1\ a - c\ {a_1\ (a_2\ a_3 + 1) + a_3})/(q_1\ b - c\ (a_2\ a_3 + 1)) / & = (q_1\ a - cp_3)/(q_1\ b - cq_3) If there be (n-1) number of partial quotients in the Table I, then u/v = (q₁ a - p_n-1)/(q₁ b - q_n-1) The last n th convergent is always a/b hence p_n/q_n = a/b.

c7.s5.34

Āryabhaṭa 1 knew, $p_n. v - q_n. u = -c [ n = even] (2)$ For. p_n (q₁ b - cq_n-1) - q_n (q₁ a - c p_n-1) $= -c (p_n q_n-1 - q_n p_n-1) ( p_n = a, q_n = b)$ $= - c$

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*Case II : When n = odd i.e, the number of partial qutient*¹ *used for finding u and v is even.*

c7.s5.36

¹ Shukla, Kripa Shankar, *Mahābhāskarīya*, edited and translated into English with notes, p. 30–33, Lucknow University, 1960.

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Suppose the repeated division was stopped at the remainder r₄. Following rule, let us put, $(r₄ t - c)/r₃ = q₂, where t = (r₃ q₂ + c)/r₄$

c7.s5.38

Now we place below the quotients a₁, a₂, a₃, a₄, then *mati*, t and c and obtain u and v following a process of multiplication and addition as directed in the rule.

c7.s5.39

*Table II*

c7.s5.40

| a₁ | a₁ | a₁ | a₁ | & a₁ v + β₂ = u / & ( say ) |
| a₂ | a₂ | a₃ | a₂ β₂ + β₁ = v | |
| a₃ | a₂ | a₃ β₁ + t = β₂ | β₂ | |
| a₄ | a₄ t + q₂ = β₁ | β₁ | | |
| t | t | | | |
| q₂ | | | | |

c7.s5.41

If we calculate β₁, β₂, v, and u proceeding as before we will find, u/v = (q₂ a + c p₄)/(q₂ b + c q₄)

c7.s5.42

If there be (n-1) number of partial quotients in the Table II, then : $u/v = (q ₂a + cp_n-1)/(q ₂b + cq_n-1)$ Āryabhaṭa I knew that : $p_n · v - q_n · u = -c [n = odd] (3)$ For p_n · v - q_n · u $= p_n (q₂b + cq_n-1) - q_n (q ₂a + cp_n-1)$ $= c (p_nq_n-1 - q_n p_n-1) [ p_n = a, q_n = b]$ $= -c [n = odd]$

c7.s5.43

The equations (2) & (3) are identical. This was purposely done by Āryabhaṭa I, by selecting (r_nt ± c)/(r_n-1) = q₁ or q₂ according as the number of partial quotients in the Table are odd or even (chosen by Āryabhaṭa I.)

c7.s5.44

From (3), q_n u = p_n · v + c or bu = av + c (4) Evidently y = u and x = v when u = am + y, and v = bm + x, the equation (4) reduces to $by₁ = ax₁ + c$ ∴ y y₁ ( mod a ) and x = x₁ ( mod b ) hence ( x₁, y₁ ) is the least solution of (1)

c7.s5.45

ii) Solution of x = (by + c)/a in positive integer when a and b are prime to each other.

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The method of solution is exactly similar, only select (r_n t ± c)/(r_n-1) = q₁ or q₂ according as the number of partial quotients are odd or even.

c7.s5.47

then x = u = bm + x₁ or x ≡ x₁ ( mod b ). and y = v = am + y₁ ≡ y₁ ( mod a )

c7.s5.48

( 2 ) *Solution of linear simultaneous indeterminate equation* To solve N = a₁x + r₁ = a₂y + r₂ = a₃z + r₃ = ... = ant + rₙ.

c7.s5.49

Then the least value of N = a₁α + r₁

c7.s5.50

General value = a₁ a₂n + a₁α + r₁ where n is any integer. Thus we find that a₁α + r₁ is the remainder left on dividing N by the product a₁ and a₂. This is exactly what has been stated by Āryabhaṭa I as follows : "( The result will be ) the residue corresponding to the product of the two divisors".

c7.s5.51

Let N' = a₁ a₂n + (a₁α + r₁) = a₃z + r₃

c7.s5.52

then N' = a₁ a₂ a₃t + a₁α + r₁ which shows that N' when divided severally by a₁ a₂ and a₃ leaves as remainder a₁ α and r₃ respectively and repeat the process and so on.

c7.s5.53

*Illustrations :*

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(1) 45x + 7 = 29y Now, 29) 45 (1 29
16) 29 (1 16
13) 16 (1 13
3) 13 (4 12 1

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Here no. of quotients omitting the first is odd.

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Therefore, (1 × 10 - 7)/3 = 1

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| | 1 | 1 | 1 | 1 | 143 |
| Now | 1 | 1 | 1 | 92 | 92 |
| | 1 | 1 | 51 | 51 | |
| | 4 | 41 | 41 | | |
| | 10 (*mati*) | 10 | | | |
| | 1 | | | | |

c7.s5.58

Now y = 143 = 45 × 3 + 8 = 8 ( mod 45 ) and x = 5 ( mod 29 )......

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( 2 ) N = 12x + 5 = 31y + 7 To find x, y, and N. Now 31y + 2 = 12x

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{r@{ }l} 12)\phantom31 & (2 / 24 & / 7)\phantom12 & (1 / \phantom7 & / 5)\phantom7 & (1 / 5 & / 2 &

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Here the no. of quotients excepting the first = even.

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$Here (2 × 4 + 2)/5 = 2$

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| *Table* | 2 | 2 | 2 | 26 |
| | 1 | 1 | 10 | 10 |
| | 1 | 6 | 6 | |
| | 4 (*mati*) | 4 | | |
| | 2 | | | |

c7.s5.64

Now x = 26, y = 10 ∴ N = 31y + 7 = 317

c7.s5.65

*Bhāskara I*: Bhāskara I (600A.D.) stated clearly that a and b should be prime to each other. He put the equation always in the form (ax - c)/b = y which is same as Āryabhaṭa I’s (by + c)/a = x. His rule is similar to that of Āryabhaṭa I and was given in his *Mahāsiddhānta* (ch. 1, v. 41–44). He for the first time used this equation to astronomy to compute the *ahargaṇas* and revolutions of the planets. He gave also the following results :

c7.s5.66

( i ) If (x₁, y₁) be solutions of ax + 1 = by, then (cx₁, cy₁) is solution of ax + c = by. (*Mahābhāskarīya*, ch. 1, v. 45–46i ).

c7.s5.67

( ii ) If x = α, y = β be a solution of Ax - C = by, then x = α, y = ma + β be a solution of (ax - C)/b = y, where a = mb + A. (*Mahābhāskarīya*, ch. 1. v. 47).

c7.s5.68

( iii ) If cx₁ x'₁ ( mod b ), cy₁ y'₁ ( mod a ), then (x'₁ y'₁) is the solution of ax + C = by. (*Mahābhāskarīya* ch. 1, v. 50).

c7.s5.69

Various problems were given by Bhāskara I, a few of which are given here :

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( 1 ) “What is that number which being divided by 12, leaves the remainder 5; again when divided by 31 has the remainder 7 ?” (*Āryabhaṭīya bhāṣya*). *Hints* : To solve N = 12x + 5 = 31y + 7. Here N = 317, x = 31, y = 10.

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( 2 ) "Find what is that number which divided by 8 is known to leave a remainder 5; by 9 a remainder 4; and by 7 a remainder 1". ( *Āryabhaṭīya bhāṣya* ).

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