History of Hindu Mathematics: A Source Book v2.c17.s10: 12. Simultaneous Quadratic Equations

History of Hindu Mathematics: A Source Book v2.c17.s10

Language: English

12. Simultaneous Quadratic Equations

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**Common Forms.** Various problems involving simultaneous quadratic equations of the following forms have been treated by Hindu writers :

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$.x - y &= d / xy &= b (i) .x + y &= a / xy &= b (ii)$ $.x² + y² &= c / xy &= b (iii) .x² + y² &= c / x + y &= a (iv)$

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For the solution of (*i*) Āryabhaṭa I (499) states the following rule:

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"The square-root of four times the product (of two quantities) added with the square of their difference, being added and diminished by their difference and halved gives the two multiplicands."

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*i.e.*, x = ½(√(d²+4b)+d), y = ½(√(d²+4b)-d).

"The square-root of the sum of the square of the difference of the residues and two squared times the product of the residues, being added and subtracted by the difference of the residues, and halved (gives) the desired residues severally."

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Nârâyaṇa (1357) writes :

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"The square-root of the square of the difference of two quantities plus four times their product is their sum."

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"The square of the difference of the quantities together with twice their product is equal to the sum of their squares. The square-root of this result plus twice the product is the sum."

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For the solution of (*ii*) the following rule is given by Mahâvîra (850):

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"Subtract four times the area (of a rectangle) from the square of the semi-perimeter; then by *saṅkramaṇa*⁵ between the square-root of that (remainder) and the semi-perimeter, the base and the upright are obtained."

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¹ *Ā*, ii. 24.

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³ *GK*, i. 35.

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⁴ *GK*, i. 36.

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⁵ Given *a* and *b*, the process of *saṅkramaṇa* is the finding of half their sum and difference, *i.e.*, (a+b)/2 and (a-b)/2 (see pp. 43f).

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⁶ *GSS*, vii. 129½.

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*i.e.*, x = ½(a + √(a² - 4b)), y = ½(a - √(a² - 4b)).

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Nârâyaṇa says : “The square-root of the square of the sum minus four times the product is the difference.”[¹]

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For (*iii*) Mahâvîra gives the rule : “Add to and subtract twice the area (of a rectangle) from the square of the diagonal and extract the square-roots. By *saṅkramaṇa* between the greater and lesser of these (roots), the side and upright (are found).”[²]

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*i.e.*, x = ½(√(c + 2b) + √(c - 2b)), y = ½(√(c + 2b) - √(c - 2b)).

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For equations (*iv*) Âryabhaṭa I writes : “From the square of the sum (of two quantities) subtract the sum of their squares. Half of the remainder is their product.”[³]

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The remaining operations will be similar to those for the equations (*ii*); so that

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x = ½(a + √(2c - a²)), y = ½(a - √(2c - a²)).

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² *GSS*, vii. 127½.

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Brahmagupta says : “Subtract the square of the sum from twice the sum of the squares ; the square-root of the remainder being added to and subtracted from the sum and halved, (gives) the desired residues.”[⁴]

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Mahâvîra,[⁵] Bhâskara II[⁶] and Nârâyaṇa[⁷] have also treated these equations.

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Nârâyaṇa has given two other forms of simultaneous quadratic equations, namely, $.x² + y² &= c / x - y &= d (v) .x² - y² &= m / xy &= b (vi)$

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¹ *GK*, i. 35.

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³ *Â*, ii. 23.

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⁴ *BrSpSi*, xviii. 98.

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⁵ *GSS*, vii. 125½.

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⁶ *L*, p. 39.

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⁷ *GK*, i. 37.

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For the solution of (v) he gives the rule :

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“The square-root of twice the sum of the squares decreased by the square of the difference is equal to the sum.”

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*i.e.*, $x + y = √(2c - d²).$ Therefore $x = ½(√(2c - d²) + d), y = ½(√(2c - d²) - d).$

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For (vi) Nârâyaṇa writes :

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“Suppose the square of the product as the product (of two quantities) and the difference of the squares as their difference. From them by *saṅkrama* will be obtained the (square) quantities. Their square-roots severally will give the quantities (required).”

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We have $.x² - y² &= m / x²y² &= b²$

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These are of the form (i). Therefore $x² = ½(√(m² + 4b²) + m), y² = ½(√(m² + 4b²) - m).$ Whence we get the values of x and y.

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² *GK*, i. 34.

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**Rule of Dissimilar Operations.** The process of solving the following two particular cases of simultaneous quadratic equations was distinguished by most Hindu mathematicians by the special designation *viṣamakarma* (dissimilar operation) :

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¹ *GK*, i. 33.

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³ The name *viṣama-karma* originated obviously in contradistinction to the name *saṅkramaṇa*. This is evident from the term *viṣama-saṅkramaṇa* used by Mahâvîra for *viṣama-karma*.

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$.x² - y² &= m / x - y &= n (i) .x² - y² &= m / x + y &= p (ii)$

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These equations are found to have been regarded by them as of fundamental importance. The solutions given are:

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for (i) x = ½(m/n + n), y = ½(m/n - n);

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for (ii) x = ½(p + m/p), y = ½(p - m/p).

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Thus Brahmagupta says:

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"The difference of the squares (of the unknowns) is divided by the difference (of the unknowns) and the quotient is increased and diminished by the difference and divided by two; (the results will be the two unknown quantities); (this is) dissimilar operation."¹

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¹ BrSpSi, xviii. 36.

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The same rule is restated by him on a different occasion in the course of solving a problem.

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"If then the difference of their squares, also the difference of them (are given): the difference of the squares is divided by the difference of them, and this (latter) is added to and subtracted from the quotient and then divided by two; (the results are) the residues; whence the number of elapsed days (can be found)."²

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² BrSpSi, xviii. 97.

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Mahâvîra states:

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"The *saṅkramaṇa* of the divisor and the quotient of the two quantities is dissimilar (operation); so it is called by those who have reached the end of the ocean of mathematics."³

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³ GSS, vi. 2.

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Similar rules are given also by other writers.⁴

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⁴ MSi, xv. 22; SiŚe, xiv. 13; L, pp. 13, 37; GK, i. 32.

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**Mahâvîra's Rules.** Mahâvîra (850) has treated certain problems involving the simultaneous quadratic equations : $u + x = a, & urw = α x, / u + y = b. & usw = α y.$ $Here r/s = x/y = (a - u)/(b - u).$ $Therefore u = (rb - sa)/(r - s).$ $Hence x = ((a - b)/(r - s))r, y = ((a - b)/(r - s))s, w = ((a - b)/(rb - sa))α.$

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In the above equations x, y are the interests accrued on the principal u in the periods r, s respectively and w is the rate of interest per α.

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Mahâvîra states the result thus :

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"The difference of the mixed sums [a, b] multiplied by each other's periods [r, s], being divided by the difference of the periods, the quotient is known as the principal [u]."^1

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Again, there are problems involving the equations: $u + x = p, & uxw = α m, / u + y = q. & uyw = α n.$ Where x, y are the periods for which the principal u is lent out at the rate of interest w per α and m, n are the respective interests. $Here m/n = x/y = (p - u)/(q - u).$ $Therefore u = (mq - np)/(m - n).$

Linked texts (7)

Gaṇitasārasaṅgraha — Mahāvīra (mathematician), Sanskrit
Līlāvaī — Kouhala, Māhārāṣṭrī Prakrit
Brāhmasphuṭasiddhānta — Brahmagupta, Sanskrit
Āryabhaṭīya — Āryabhaṭa, Sanskrit
Siddhāntaśekhara — Śrīpati, Sanskrit
Mahāsiddhānta (Āryasiddhānta) — Āryabhaṭa II, Sanskrit
Gaṇitakaumudī — Nārāyaṇa Paṇḍita (mathematician), Sanskrit