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History of Hindu Mathematics: A Source Book v2.c17.s26: 28. Solution of axy=bx+cy+d

History of Hindu Mathematics: A Source Book v2.c17.s26

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28. Solution of axy=bx+cy+d

v2.c17.s26.1

**Bakhshâlî Treatise.** The earliest instance of a quadratic indeterminate equation of the type axy = bx + cy + d, in Hindu mathematics occurs in the Bakhshâlî Treatise (*c.* 200).¹ The text is very mutilated. But the example that is preserved is xy = 3x + 4y 1, of which the solutions preserved are x & = (3 · 4 - 1)/1 + 4 = 15, / y & = 1 + 3 = 4; and x & = 1 + 4 = 5, / y & = (3 · 4 + 1)/1 + 3 = 16.

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¹ *BMs*, Folio 27, recto; compare also Kaye's Introduction § 82.

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Hence, in general, the solutions of the equation xy = bx + cy + d, which appear to have been given are : .x & = (bc+d)/m + c, / y & = m + b;} or .x & = m + c, / y & = (bc+d)/m + b;} where m is an arbitrary number.

v2.c17.s26.4

**An Unknown Author's Rule.** Brahmagupta (628) has described the following method taken from an author who is not known now.²

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² Pṛthûdakasvâmî (860) says that the method is due to a writer other than Brahmagupta. This is further corroborated by Brahmagupta's strictures on it (*vide infra*, p. 299).

v2.c17.s26.6

“The product of the coefficient of the factum and the absolute number together with the product of the coefficients of the unknowns is divided by an optional number. Of the optional number and the quotient obtained, the greater is added to the lesser (of the coefficients of the unknowns) and the lesser to the greater (of the coefficients), and (the sums) are divided by the coefficient of the factum. (The results will be values of the unknowns) in the reverse order.”\1

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\1 *BrSpSi*, xviii, 60.

v2.c17.s26.8

As has been observed by Pṛthûdakasvâmî, this rule is to be applied to an equation containing the factum after it has been prepared by transposing the factum term to one side and the absolute term together with the simple unknown terms to the other. Then the solutions will be, *m* being an arbitrary rational number, $x = 1/a(m + c),$ $y = 1/a((ad + bc)/m + b),$ if b > c and m > (ad + bc)/m. If these conditions be reversed then x and y will have their values interchanged.

v2.c17.s26.9

The *rationale* of the above solutions can be easily shown to be as follows : &axy = bx + cy + d, / or &a²xy - abx - acy = ad, / or &(ax - c)(ay - b) = ad + bc. Suppose ax - c = m, a rational number ; then ay - b = (ad + bc)/m.

v2.c17.s26.10

Therefore [t] x &= 1/a(m + c), / y &= 1/a((ad + bc)/m + b).

v2.c17.s26.11

Or, we may put ay - b = m; then we shall have ax - c = (ad + bc)/m; whence [t] x &= 1/a((ad + bc)/m + c), / y &= 1/a(m + b).

v2.c17.s26.12

It will thus be found that the restrictive condition of adding the greater and lesser of the numbers *m* and (*ad* + *bc*)/*m* to the lesser and greater of the numbers *b* and *c* respectively as adumbrated in the above rule is quite unnecessary.

v2.c17.s26.13

**Brahmagupta's Rule.** Brahmagupta gives the following rule for the solution of a quadratic indeterminate equation involving a factum :

v2.c17.s26.14

“With the exception of an optional unknown, assume arbitrary values for the rest of the unknowns, the product of which forms the factum. The sum of the products of these (assumed values) and the (respective) coefficients of the unknowns will be absolute quantities. The continued products of the assumed values and of the coefficient of the factum will be the coefficient of the optionally (left out) unknown. Thus the solution is effected without forming an equation of the factum. Why then was it done so?”¹

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¹ *BrSpSi*, xviii. 62-3, *vide supra*, p. 297.

v2.c17.s26.16

The reference in the latter portion of this rule is to the method of the unknown writer. The principle underlying Brahmagupta's method is to reduce, like the Greek Diophantus (*c.* 275),¹ the given indeterminate equation to a simple determinate one by assuming arbitrary values for all the unknowns except one. So it is undoubtedly inferior to the earlier method. Brahmagupta gives the following illustrative example:

v2.c17.s26.17

"On subtracting from the product of signs and degrees of the sun, three and four times (respectively) those quantities, ninety is obtained. Determining the sun within a year (one can pass as a proficient) mathematician."²

v2.c17.s26.18

² *BrSpSi*, xviii. 61.

v2.c17.s26.19

If x denotes the signs and y the degrees of the sun, then the equation is xy - 3x - 4y = 90. Thus this problem, as that of Bhâskara II (*infra*), appears to have some relation with that of the Bakhshâlî work. Pṛthûdakasvâmî solves it in two ways. *Firstly*, he assumes the arbitrary number to be 17, then x &= 1/1 ( (90.¹ + 3.4)/17 + 4 ) = 10, / y &= 1/1 (17 + 3) = 20. *Secondly*, he assumes arbitrarily y = 20. On substituting this value in the above equation, it reduces to 20x - 3x = 170; whence x = 10.

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¹ Heath, *Diophantus*, pp. 192-4, 262.

Mahâvîra's Rule

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Mahâvîra (850) has not treated equations of this type. There are, however, two problems in his *Gaṇita-sâra-saṃgraha* which involve similar equations. One of them is to find the increase or decrease of two numbers (a, b) so that the product of the resulting numbers will be equal to another optionally given number (d). Thus we are to solve $(a ± x)(b ± y) = d,$ $or xy ± (bx + ay) = d - ab.$

v2.c17.s26.t1.2

The rule given for solving this is :

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“The difference between the product of the given numbers and the optional number is put down at two places. It is divided (at one place) by one of the given numbers increased by unity and (at the other) by the optional number increased by the other given number. These will give in the reverse order the values of the quantities to be added or subtracted.”¹

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¹ GSS, vi. 284.

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That is to say, $.x &= (d ~ ab)/(d + b) / y &= (d ~ ab)/(a + 1), .x &= (d ~ ab)/(b + 1) / y &= (d ~ ab)/(d + a).$

v2.c17.s26.t1.6

Thus the solutions given by Mahâvîra are much cramped. The other problem considered by him is to separate the capital, interest and time when their sum is given : If x be the capital invested and y the period of time in months, then the interest will be mxy, where m is the rate of interest per month. Then the problem is to solve $mxy + x + y = p.$

v2.c17.s26.t1.7

Mahâvîra solves this equation by assuming arbitrary values for y.²

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² GSS, vi. 35.

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**Śrîpati’s Rule.** Śrîpati (1039) gives the following rule :

v2.c17.s26.t1.10

“Remove the factums from one side, the (simple) unknowns and the absolute numbers from the other. The product of the coefficients of the unknowns being added to the product of the absolute quantity and the coefficient of the factum, (the sum) is divided by an optional number. The quotient and the divisor should be added arbitrarily to the greater or smaller of the coefficients of the unknowns. These divided by the coefficient of the factum will be the values of the unknowns in the reverse order.”¹

v2.c17.s26.t1.11

$i.e., . x &= 1/a(m + c) / y &= 1/a((ad + bc)/m + b) , . x &= 1/a((ad + bc)/m + c) / y &= 1/a(m + b) ,$ where m is arbitrary.

v2.c17.s26.t1.12

**Bhâskara II’s Rule.** Bhâskara II (1150) has given two rules for the solution of a quadratic indeterminate equation containing the product of the unknowns. His first method is the same as that of Brahmagupta:

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“Leaving one unknown quantity optionally chosen, the values of the other should be assumed arbitrarily according to convenience. The factum will thus be reduced and the required solution can then be obtained by the first method of analysis.”²

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Bhâskara’s aim was to obtain * two unknowns can be obtained with much difficulty.”¹ So he describes a second method “by which they can be obtained with little difficulty.”

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¹ “Evamekasmin vyakte râśau kalpite sati bahûnâyâsenabhinnau râśî jñâyete”—*BBi*, p. 124.

v2.c17.s26.t1.16

“Transposing the factum from one side chosen at pleasure, and the (simple) unknowns and the absolute number from the other side (of the equation), and then dividing both the sides by the coefficient of the factum, the product of the coefficients of the unknowns together with the absolute number is divided by an optional number. The optional number and that quotient should be increased or diminished by the coefficients of the unknowns at pleasure. They (results thus obtained) should be known as the values of the two unknowns reciprocally.”²

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² *BBi*, pp. 124f.

v2.c17.s26.t1.18

This rule has been elucidated by the author thus:

v2.c17.s26.t1.19

“From one of the two equal sides the factum being removed, and from the other the unknowns and the absolute number; then dividing the two sides by the coefficients of the factum, the product of the coefficients of the unknowns on the other side added to the absolute number, is divided by an optional number. The optional number and the quotient being arbitrarily added to the coefficients of the unknowns, should be known as the values of the unknowns in the reciprocal order. That is, the one to which the coefficient of the *kâlaka* (the second unknown) is added, will be the value of the *yâvat-tâvat* (the first unknown) and the one to which the coefficient of the *yâvat-tâvat* is added, will be the value of the *kâlaka*. But if, after that has been done, owing to the magnitude, the statements (of the problem) are not fulfilled, then from the optional number and the quotient, the coefficients of the unknowns should be subtracted, and (the remainders) will be the values of the unknowns in the reciprocal order."

v2.c17.s26.t1.20

Thus Bhâskara's solutions are .x & = c/a ± m' / y & = b/a ± n'} .x & = c/a ± n' / y & = b/a ± m'} where m' is any arbitrary number and n' = 1/(m')(bc/(a²) + d/a).

v2.c17.s26.t1.21

The *rationale* of these solutions is as follows : axy = bx + cy + d, or xy - b/ax - c/ay = d/a, or (x - c/a)(y - b/a) = d/a + bc/(a²) = m'n', say. Then, either .x - c/a & = ± m' / y - b/a & = ± n'} or .x - c/a & = ± n' / y - b/a & = ± m'}; whence the solutions.

v2.c17.s26.t1.22

**Bhâskara's Proofs.** The same *rationale* of the above solutions has been given also by Bhâskara II with the help of the following illustrative example. He observes that the proof "is twofold in every case : one geometrical (*kṣetragata*), the other algebraic (*râśigata*)."¹

v2.c17.s26.t1.23

¹ *BBi*, p. 125.

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*Example.* "The sum of two numbers multiplied by four and three, added by two is equal to the product of those numbers. Tell me, if thou knowest, those two numbers.”¹

v2.c17.s26.t1.25

¹ *BBi*, pp. 123, 125.

v2.c17.s26.t1.26

*Solution.* “Having performed the operations as stated, the sides are $xy=4x+3y+2.$ The product of the coefficients of the unknowns plus the absolute term is 14. Dividing this by an optional number (say) unity, the optional number and the quotient are 1, 14. To these being arbitrarily added 4, 3, the coefficients of the unknowns, the values of (x, y) are (4, 18) or (17, 5). (Dividing) by (the optional number) 2, (other values will be) (5, 11) or (10, 6).”²

v2.c17.s26.t1.27

² *BBi*, p. 125.

v2.c17.s26.t1.28

*Geometrical Proof.* “The second side of the equation is equal to the factum. But the factum is the area of an oblong quadrilateral of which the base and upright are the unknown quantities. Within this figure (Fig. 15) exist four x’s, three y’s and the absolute number 2. From this figure on taking off four x’s and y minus four multiplied by its own coefficient, (*i.e.*, 3), it becomes this (Fig. 16).

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Fig. 15

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Fig. 16

v2.c17.s26.t1.31

The other side of the equation being so treated there results 14. This must be the area of the figure remaining at the corner (see Fig. 16) within the rectangle representing the factum, and is Bhâskara II further observes : "Thus the proof of the solution of the factum has been shown to be of two kinds. What has been said before—the product of the coefficients of the unknowns together with the absolute number is equal to the area of another rectangle inside the rectangle representing the factum and lying at a corner—is sometimes otherwise. For, when the coefficients of the unknowns are negative, the factum-rectangle will be inside the other rectangle at one corner ; and when the coefficients of the unknowns are greater than the base and upright of the factum-rectangle, and are positive, the other will be outside the factum rectangle and at a corner, as (Figs. 17, 18).

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When it is so, the coefficients of the unknowns lessened by the optional number and the quotient, will be the values of x and y."¹

v2.c17.s26.t1.34

¹ *BBi*, p. 127.

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