History of Hindu Mathematics: A Source Book v2.c17.s21: 23. Linear Functions Made Squares or Cubes

History of Hindu Mathematics: A Source Book v2.c17.s21

Language: English

23. Linear Functions Made Squares or Cubes

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**Square-pulveriser.** The indeterminate equation of the type

$bx + c = y²$ is called *varga-kuṭṭaka* or the "Square-pulveriser,"¹ inasmuch as, when expressed in the form $(y²-c)/b = x,$ the problem reduces to finding a square (*varga*) which being diminished by c will be exactly divisible by b, which closely resembles the problem solved by the method of the pulveriser (*kuṭṭaka*).

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² *BBi*, p. 103 ; also *vide infra*, p. 280.

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For the solution in *integers* of an equation of this type, the method of the earlier writers appears to have been to assume suitable arbitrary values for y and then to solve the equation for x. Brahmagupta gives the following problems:

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“The residue of the sun on Thursday is lessened and then multiplied by 5, or by 10. Making this (result) an exact square, within a year, a person becomes a mathematician.”²

“The residue of any optional revolution lessened by 92 and then multiplied by 83 becomes together with unity a square. A person solving this within a year is a mathematician.”³

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² *BrSpSi*, xviii. 76.

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³ *BrSpSi*, xviii. 79.

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That is to say, we are to solve the equations: (1) & 5x - 25 = y², / (2) & 10x - 100 = y², / (3) & 83x - 7635 = y².

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Pṛthûdakasvâmî solves them thus:

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(1.¹) Suppose y = 10; then x = 125. Or, put y = 5; then x = 10. (2.1) Suppose y = 10; then x = 20. (3.1) Assume y = 1; then x = 92.

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¹ *BBi*, p. 122.

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He then remarks that by virtue of the multiplicity of suppositions there will be an infinitude of solutions in every case. But no method has been given either by Brahmagupta or by his commentator Pṛthûdakasvâmî to obtain the general solution.

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The above method is remainder should be increased by so many times the divisor as will make a square. If this is not possible, then the problem is not soluble."¹

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¹ *BBi*, p. 121.

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*Case i.* Let *c* be a square, equal to β², say. Then we have to solve $bx + β² = y².$ The rule says, find *p* such that $p² = bq, 2pβ = br.$ Then assume $y = pu + β ;$ whence we get $x = qu² + ru.$

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Bhâskara II prefers the assumption $y = bv + β,$ so that we have $x = bv² + 2β v.$

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*Case ii.* If *c* is not a square, suppose c = bm + n. Then find *s* such that $n + sb = r².$ Now assume $y = bu ± r.$

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Substituting this value in the equation bx + c = y², we get bx + c &= (bu ± r)² / &= b²u² ± 2bru + r², / or bx + c - r² &= b²u² ± 2bru, / or bx + b(m - s) &= b²u² ± 2bru. / \therefore x &= bu² ± 2ru - (m - s).

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*Example* from Bhâskara II :² $7x + 30 = y².$

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On dividing 30 by 7 the remainder is found to be 2; we also know that 2 + 7.2 = 4². Therefore, we assume in accordance with the above rule $y = 7u ± 4;$ whence we get $x = 7u² ± 8u - 2,$ which is the general solution.

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² *BBi*, pp. 120, 121.

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**Cube-pulveriser.** The indeterminate equation of the type $bx + c = y³$ is called the *ghana-kuṭṭaka* or the "Cube-pulveriser."¹ For its solution in integers Bhâskara II says :

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¹ *BBi*, p. 122.

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"A method similar to the above may be applied also in the case of a cube thus : (find) a number whose cube is exactly divisible by the divisor, as also its product by thrice the cube-root of the absolute term. An unknown multiplied by that number and superadded by the cube-root of the absolute term should be assumed. If there be no cube-root of the absolute term, then after dividing it by the divisor, so many times the divisor should be added to the remainder as will make a cube. The cube-root of that will be the root of the absolute number. If there cannot be found a cube, even by so doing, that problem will be insoluble."²

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² *BBi*, p. 121.

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*Case i.* Let c = β³. Then we shall have to find p such that p³ = bq, 3pβ = br. Now assume y = pv + β. Substituting in the equation bx + β³ = y³ we get bx + \beta³ &= (pv + \beta)³ / &= p³v³ + 3pv\beta(pv + \beta) + \beta³, / or \phantom{bx + \beta³} bx &= bqv³ + brv(pv + \beta). / \therefore \phantom{bx + \beta³} x &= qv³ + rv(pv + \beta).

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Or, if we assume y = bv + β, we shall have $x = b²v³ + 3β v(bv + β).$

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*Case ii.* c ≠ a cube. Suppose c = bm + n; then find s such that $n + sb = r³.$ Now assume y = bv + r, whence we get $x = b²v³ + 3rv(bv + r) - (m - s),$ as the general solution.

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*Example from Bhâskara II :*¹ $5x + 6 = y³.$ Since 6 = 5.1 + 1 and 1 + 43.5 = 6³, we assume y = 5v + 6. Therefore x = 25v³ + 18v(5v + 6) + 42, is the general solution.

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¹ *BBi*, p. 122.

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**Equation bx ± c = ay².** To solve an equation of the type $ay² = bx ± c,$ Bhâskara II says :

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"Where the first side of the equation yields a root on being multiplied or divided² (by a number), there also the divisor will be as stated in the problem but the absolute term will be as modified by the operations."³

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² The printed text has *hitvâ kṣiptâ* (subtracting or adding). After collating with several copies Colebrooke accepted the reading *hatvâ kṣiptâ* (multiplying or adding). But we think that the correct reading should be *hatvâ hṛtvâ* (multiplying or dividing). For in his gloss Bhâskara II has employed the terms *guṇito vibhakto va* (multiplied or divided). Our emendation is further supported by the rationale of the rule.

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³ *BBi*, p. 121.

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What is implied is this: Multiplying both sided of the given equation by *a*, we get $a²y² = abx ± ac,$ Put u = ay, v = ax. Then the equation reduces to $u² = bv ± ac,$ which can be solved in the way described before.

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We take the following illustrative example with its solution from Bhâskara II :¹ 5y² + 3 = 16x. Multiplying by 5, and putting u = 5y, v = 5x, we get u² = 16v - 15. The solution of this is u = 8w ± 1, v = 4w² ± w + 1. Therefore, we have (1) 5y &= 8w + 1, / or (2) 5y &= 8w - 1. Now, solving by the method of the pulveriser, we get the solution of (1) as y = 8t + 5, w = 5t + 3; and that of (2) as y = 8t + 3, w = 5t + 2; where *t* is any rational number.

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¹ *BBi*, p. 121.

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**Equation bx ± c = ayn.** After describing the above methods Bhâskara II observes, *ityagre'pi yojyamiti śeṣaḥ* or "the same method can be applied further on (*i.e.*, to the cases of higher powers).”¹ Again at the end of the section he has added *evaṁ buddhimadbhiranyadapi yathâsambhavaṁ yojyam*, *i.e.*, “similar devices should be applied by the intelligent to further cases as far as practicable.”² What is implied is as follows :

Linked texts (2)

Brāhmasphuṭasiddhānta — Brahmagupta, Sanskrit
Bījagaṇita — Bhāskara II, Sanskrit