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History of Hindu Mathematics: A Source Book v2.c17.s5: 7. Linear Equations in One Unknown

History of Hindu Mathematics: A Source Book v2.c17.s5

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7. Linear Equations in One Unknown

v2.c17.s5.1

**Early Solutions.** As already stated, the geometrical solution of a linear equation in one unknown is found in the *Śulba*, the earliest of which is not later than 800 B.C. There is a reference in the *Sthânâṅga-sûtra* (*c.* 300 B.C.) to a linear equation by its name (*yâvat-tâvat*) which is suggestive of the method of solution¹ followed at that time. We have, however, no further evidence about it. The earliest Hindu record of doubtless value of problems involving simple algebraic equations and of a method for their solution occurs in the Bakhshâlî treatise, which was written very probably about the beginning of the Christian Era. One problem is:²

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¹ Datta, *Jaina Math.*, (*BCMS*, XXI), p. 122.

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² *BMs*, Folio 23, recto.

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“The amount given to the first is not known. The second is given twice as much as the first; the third

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thrice as much as the second ; and the fourth four times as much as the third. The total amount distributed is 132. What is the amount of the first?”

v2.c17.s5.6

If x be the amount given to the first, then according to the probelm, $x + 2x + 6x + 24x = 132.$

v2.c17.s5.7

**Rule of False Position.** The solution of this equation is given as follows: “ ‘Putting any desired quantity in the vacant place’ ; any desired quantity is || 1 ||; ‘then construct the series’

v2.c17.s5.8

| 1 | 2 | 2 | 3 | 6 | 4 |
| :-: | :-: | :-: | :-: | :-: | :-: |
| 1 | 1 | 1 | 1 | 1 | 1 |

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‘multiplied’ || 1 | 2 | 6 | 24 |; ‘added’ 33. ‘Divide the visible quantity’ | 132 | / | 33 |; (which) on reduction becomes | 4 | / | 1 |. (This is) the amount given (to the first).”¹

v2.c17.s5.10

¹ *Ibid.*

v2.c17.s5.11

The solution of another set of problems in the Bakhshâlî treatise, leads ultimately to an equation of the type² $ax + b = p.$ The method given for its solution is to put any arbitrary value g for x, so that $ag + b = p', say.$ Then the correct value will be $x = (p - p')/a + g.$

v2.c17.s5.12

² *Vide infra*, pp. 48f.

v2.c17.s5.13

The above method of solution of a linear equation was known in the middle ages, amongst Arab and European algebraists, by the name of the Rule of False Position. It is noteworthy that the terms *yaddṛcchâ*, *vâñcchâ*, and *kâmika* of the Bakhshâlî treatise are equivalent to the term *yâvat-tâvat*. So the origin of this latter term seems to be connected with the Rule of False Position. It is interesting to find that the rule was once given so much importance in Hindu algebra that the section dealing with it was named after it.

Disappearance from Later Algebra

v2.c17.s5.t1.1

The Rule of False Position bespeaks of an early stage of development of the science of algebra when there was no symbol for the unknown. It naturally disappears with the introduction of a system of notations.¹ This will account for the complete disappearance of the Rule of False Position from the later Hindu treatises on algebra beginning with that of Âryabhaṭa I (499). There are found, however, very limited applications of it in the arithmetical treatises of Śrîdhara (*c.* 750), Mahâvîra (850) and Bhâskara II (1150). This can be accounted for easily. The problems which have been solved by those writers with the help of the Rule of False Position are really of algebraic nature, though incorporated into arithmetical treatises. But as the use of algebraic symbols and notations is not permissible in arithmetic, recourse had to be taken to that Rule. For instance, we take the following problem from the *Gaṇita-sâra-saṁgraha* of Mahâvîra :

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¹ Smith, *History*, II, p. 437.

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"The sum of ⅛, ¼ of ⅓, ⅕ of ½, ⅕ of ¾ of ⅙, of a certain number is equal to ½. What is that unknown (number) ?"²

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² *GSS*, iii. 108.

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Mahâvîra gives the following rule for finding out the unknown in a problem of this kind :

v2.c17.s5.t1.6

“Put down one for the value of the unknown ; then in accordance with the previous rule (find) the sum (of its parts) ; divide the known (number) by that (sum) ; the quotient will be (the value of) the unknown in compound fractions.”

v2.c17.s5.t1.7

**Operation with an Optional Number.** Bhâskara II describes a method called *Iṣṭa-karma* or “operation with an optional number.” This may be illustrated by the following example :

v2.c17.s5.t1.8

“What is that number which multiplied by five, diminished by its third part and (then) divided by ten, will become, together with its one-third, half and one-fourth parts, equal to seventy minus two ?”

v2.c17.s5.t1.9

*i.e.*, (5x-5x/3)/10 + x/3 + x/2 + x/4 = 70 - 2.

v2.c17.s5.t1.10

Bhâskara assumes x = 3 and then calculates $(5 × 3 - 5 × 3/3)/10 + 3/3 + 3/2 + ¾ = 17/4.$ He then states $x = 68 × 3 ÷ 17/4 = 48.$ He observes : “Similarly, in every example, by whatever the (required) number is multiplied or divided, by whatever fraction of the number it is found to have been increased or diminished, assuming an optional number, on it perform the same operations in accordance with the statement of the problem; by that, which results, divide the known number multiplied by the assumed number; the quotient will be the (required) number.”

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¹ *GSS*, iii. 107.

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² *L*, p. 10.

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³ *L*, p. 11.

v2.c17.s5.t1.14

**Solution of Linear Equations.** Āryabhaṭa I (499) says :

v2.c17.s5.t1.15

“The difference of the known “amounts” relating to the two persons should be divided by the difference of the coefficients of the unknown.¹ The quotient will be the value of the unknown, if their possessions be equal.”

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¹ The original is *gulikāntara* which literally means “the difference of the unknowns.” But what is implied is “the difference of the coefficients of the unknown.” As has been observed by Pṛthūdakasvâmî, according to the usual practice of Hindu algebra, “the coefficient of the square of the unknown is called the square (of the unknown) and the coefficient of the (simple) unknown is called the unknown.” *BrSpSi*, xviii. 44 (*Com*).

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*Ā*, ii. 30.

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This rule contemplates a problem of this kind : Two persons, who are equally rich, possess respectively a, b times a certain unknown amount together with c, d units of money in cash. What is that amount ?

v2.c17.s5.t1.19

If x be the unknown amount, then by the problem $ax + c = bx + d.$

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Therefore $x = (d - c)/(a - b).$

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Hence the rule.

v2.c17.s5.t1.22

Brahmagupta says :

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“In a (linear) equation in one unknown, the difference of the known terms taken in the reverse order, divided by the difference of the coefficients of the unknown (is the value of the unknown).”³

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³ *BrSpSi*, xviii. 43.

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Śrīpati writes :

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“First remove the unknown from any one of the sides (of the equation) leaving the known term ; the reverse (should be done) on the other side. The difference of the absolute terms taken in the reverse order

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divided by the difference of the coefficients of the unknown will be the value of the unknown.”¹

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¹ *SiŚe*, xiv. 15.

v2.c17.s5.t1.29

Bhâskara II states :

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“Subtract the unknown on one side from that on the other and the absolute term on the second from that on the first side. The residual absolute number should be divided by the residual coefficient of the unknown ; thus the value of the unknown becomes known.”²

v2.c17.s5.t1.31

² *BBi*, p. 44.

v2.c17.s5.t1.32

Nârâyaṇa writes :

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“From one side clear off the unknown and from the other the known quantities ; then divide the residual known by the residual coefficient of the unknown. Thus will certainly become known the value of the unknown.”³

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³ *NBi*, II, R. 5.

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For illustration we take a problem proposed by Brahmagupta :

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“Tell the number of elapsed days for the time when four times the twelfth part of the residual degrees increased by one, plus eight will be equal to the residual degrees plus one.”⁴

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⁴ *BrSpSi*, xviii. 46.

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It has been solved by Pṛthûdakasvâmî as follows :

v2.c17.s5.t1.39

“Here the residual degrees are (put as) *yâvat-tâvat*, *yâ* ; increased by one, *yâ* 1 *rû* 1; twelfth part of it, ya 1 & ru 1 / 12 &; four times this, ya 1 & ru 1 / 3 &; plus the absolute quantity eight, ya 1 & ru 25 / 3 &. This is equal to the residual degrees plus unity. The statement of both sides tripled is $ya 1 & ru 25 / ya 3 & ru 3$

v2.c17.s5.t1.40

The difference between the coefficients of the unknown is 2. By this the difference of the absolute terms, namely 22, being divided, is produced the residual of the degrees of the sun, 11. These residual degrees should be known to be irreducible. The elapsed days can be deduced then, (proceeding) as before.”

v2.c17.s5.t1.41

In other words, we have to solve the equation $T4/2(x + 1) + 8 = x + 1,$ which gives $x + 25 = 3x + 3,$ or $2x = 22.$ Therefore $x = II.$

v2.c17.s5.t1.42

The following problem and its solution are from the *Bîjagaṇita* of Bhâskara II :

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“One person has three hundred coins and six horses. Another has ten horses (each) of similar value and he has further a debt of hundred coins. But they are of equal worth. What is the price of a horse ?

v2.c17.s5.t1.44

“Here the statement for equi-clearance is : 6x + 300 = 10x - 100. Now, by the rule, ‘Subtract the unknown on one side from that on the other etc.,’ unknown on the first side being subtracted from the unknown on the other side, the remainder is 4x. The absolute term on the second side being subtracted from the absolute term on the first side, the remainder is 400. The residual known number 400 being divided by the coefficient of the residual unknown 4x, the quotient is recognised to be the value of x, (namely) 100.”[¹]

v2.c17.s5.t1.45

There are a few instances in the Bakhshâlî work where a method similar to that of later writers appears to have been followed for the solution of a linear equation. One such problem is: Two persons start with different initial velocities (a₁, a₂); travel on successive days distances increasing at different rates (b₁, b₂). But they cover the same distance after the same period of time. What is the period?

v2.c17.s5.t1.46

¹ *BBi*, pp. 44f.

v2.c17.s5.t1.47

Denoting the period by x, we get

v2.c17.s5.t1.48

$a₁ + (a₁ + b₁) + (a₁ + 2b₁) + ... to x terms / = a₂ + (a₂ + b₂) + (a₂ + 2b₂) + ... to x terms,$

v2.c17.s5.t1.49

or $a₁ + ((x - 1)/2)b₁ x = a₂ + ((x - 1)/2)b₂ x;$

v2.c17.s5.t1.50

whence $x = (2(a₂ - a₁))/(b₁ - b₂) + 1,$

v2.c17.s5.t1.51

which is the solution given in the Bakhshâlî work:

v2.c17.s5.t1.52

"Twice the difference of the initial terms divided by the difference of the common differences, is increased by unity. The result will be the number of days in which the distance moved will be the same."

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