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History of Hindu Mathematics: A Source Book v2.c17.s4: 5. Fundamental Operations

History of Hindu Mathematics: A Source Book v2.c17.s4

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5. Fundamental Operations

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**Number of Operations.** The number of fundamental operations in algebra is recognised by all Hindu algebraists to be six, * of the same species (*jâti*) amongst unknowns; of different species they mean their separate statement."¹

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¹ *BBi*, p. 7.

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Nârâyaṇa:

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"Of the colours or letters of alphabets (representing the unknowns) addition should be made of those which are of the same species; and similarly subtraction. In the addition and subtraction of those of different species the result will be their putting down severally."²

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² *NBi*, I, R. 19.

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**Multiplication.** Brahmagupta says:

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"The product of two like unknowns is a square; the product of three or more like unknowns is a power of that designation. The multiplication of unknowns of unlike species is the same as the mutual product of symbols; it is called *bhâvita* (product or factum)."³

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³ *BrSpSi*, xviii. 42.

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Bhâskara II writes:

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"A known quantity multiplied by an unknown becomes unknown; the product of two, three or more unknowns of like species is its square, cube, etc.; and the product of those of unlike species is their *bhâvita*. Fractions, etc., are (considered) as in the case of knowns; and the rest (*i.e.*, remaining operations) will be the same as explained in arithmetic. The multiplicand is put down separately in as many places as there are terms in the multiplier and is then severally multiplied by those terms; (the products are then) added together according to the methods indicated before. Here, in the squaring and multiplication of unknowns, should be followed the method of multiplication by component parts, as explained in arithmetic."⁴

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⁴ *BBi*, p. 8.

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The same rules are given by Nârâyaṇa.⁵ The following illustration amongst others, is given by Bhâskara II:

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⁵ *NBi*, I, R. 21-2.

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“Tell at once, O learned, (the result) of multiplying five *yâvat-tâvat* minus one known quantity by three *yâvat-tâvat* plus two knowns.

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“Statement: Multiplicand *yâ* 5 *rû* 1̇; multiplier *yâ* 3 *rû* 2; on multiplication the product becomes *yâ va* 15 *yâ* 7 *rû* 2̇.”¹

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¹ *BBi*, p. 8.

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The detailed working of this illustration is shown by the commentator Kṛṣṇa (*c.* 1580) thus:

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“ya 3 & ya 5 & ru 1 / ru 2 & ya 5 & ru 1 ya va 15 ya 3 / ya 10 ru 2 / ya va 15 ya 7 ru 2,,

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**Division.** Bhâskara II states:

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“By whatever unknowns and knowns, the divisor is multiplied (severally) and subtracted from the dividend successively so that no residue is left, they constitute the quotients at the successive stages.”²

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Nârâyaṇa describes the method of division in nearly the same terms.³ As an example of division, Bhâskara II proposes to divide 18x²+24xy-12xz-12x+8y²-8yz-8y+2z²+4z+2 by -3x-2y+z+1. He simply states the quotient without indicating the different steps in the process. Kṛṣṇa supplies the details of the process which are substantially the same as followed at present.

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³ *NBi*, I, R. 23.

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**Squaring.** Only one rule for the squaring of an algebraic expression is found in treatises on algebra. It is the same as that stated before in the treatises on arithmetic, *viz.*, $(a+b)²=a²+b²+2ab;$ or, in its general form, (a + b + c + d + )² = a² + b² + c² + d² + + 2 ab.

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² *BBi*, p. 9.

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**Square-root.** For finding the square-root of an algebraic expression Bhâskara II gives the following rule:

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"Find the square-root of the unknown quantities which are squares; then deduct from the remaining terms twice the products of those roots two and two; if there be known terms, proceed with the remainder in the same way after taking the square-root of the knowns."

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Nârâyaṇa says:

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"First find the root of the square terms (of the given expression); then the product of two and two of them (roots) multiplied by two should be subtracted from the remaining terms; (the result thus obtained) is said to be the square-root here (in algebra)."

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Jñânarâja writes:

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"Take the square- *kṛ*, to do; hence literally, making equal) from the conditions of the proposed problem. Such preliminary work may require the application of one or more fundamental operations of algebra or arithmetic. The operations preliminary to the formation of a simple equation have been described by Pṛthûdakasvâmî (860) thus:

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"In this case, in the problem proposed by the questioner, *yâvat-tâvat* is put for the value of the unknown quantity; then performing multiplication, division, etc., as required in the problem the two sides shall be carefully made equal. The equation being formed in this way, then the rule (for its solution) follows."

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Bhâskara II's descriptions are fuller: He says :

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"Let *yâvat-tâvat* be assumed as the value of the accordance with the statement of the problem, the operations such as addition, subtraction, multiplication, division, rule of three, double rule of three, summation, plane figures, excavations, etc. And thus the two sides must be made equal. If the equality of the two sides is not explicitly stated, then one side should be multiplied, divided, increased or decreased by one's own intelligence (according to the problem) and thus the two sides must be made equal."¹

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¹ *NBi*, II, R. 3 (*Gloss*).

Plan of Writing Equations

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After an equation is formed, writing it down for further operations is technically called *nyâsa* (putting down, statement) of the equation. In the Bakhshâlî treatise the two sides of an equation are put down one after the other in the same line without any sign of equality being interposed.² Thus the equations $√(x+5)=s, √(x-7)=t$ appear as³

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² Datta, *Bakh. Math.*, (*BCMS*, XXI), p. 28.

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³ Folio 59, recto.

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The equation $x+2x+3× 3x+12× 4x=300$ is stated as⁴

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⁴ Folio 23, verso.

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This plan of writing an equation was subsequently abandoned by the Hindus for a new one in which the two sides are written one below the other without any sign of equality. Further, in this new plan, the terms of similar denominations are usually written one below the other and even the terms of absent denominations on either side are expressly indicated by putting zeros as their coefficients. Reference to the new plan is found as early as the algebra of Brahmagupta (628).¹ Pṛthûdakasvâmî (860) represented the equation² $10x - 8 = x² + 1$ as follows: $ya va & 0 & ya & 10 & ru & 8 / ya va & 1 & ya & 0 & ru & 1$ which means, writing x for *yâ* $x²· 0 + x· 10 - 8 / x² + x· 0 + 1$ $or 0x² + 10x - 8 = x² + 0· x + 1.$

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¹ *BrSpSi*, xvii. 43 (*vide infra*, p. 33). Compare also *BBi*, p. 127.

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² *BrSpSi*, xviii. 49 (*Com*).

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If there be several unknowns, those of the same kind are written in the same column with zero coefficients, if necessary. Thus the equation $197x - 1644y - z = 6302$ is represented by Pṛthûdakasvâmî thus:³ $ya & 197 & ka & 1644 & n & 1 & ru & 0 / ya & 0 & ka & 0 & n & 0 & ru & 6302$ which means, putting y for *kâ* and z for *nî*, $197x - 1644y - z + 0 = 0x + 0y + 0z + 6302.$

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³ *BrSpSi*, xviii. 54 (*Com*).

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The following two instances are from the *Bîjagaṇita* of Bhâskara II (1150):⁴ $(1) & ya & 5 & ka & 8 & n & 7 & ru & 90 / & ya & 7 & ka & 9 & n & 6 & ru & 62$ means (writing *x* for *yâ*, *y* for *kâ* and *z* for *nî*) $5x + 8y + 7z + 90 = 7x + 9y + 6z + 62.$ (2) $yâ gha 8 yâ va 4 kâ va yâ. bhâ 10$ $yâ gha 4 yâ va 0 kâ va yâ. bhâ 12$ means 8x³ + 4x² + 10y²x = 4x³ + 0x² + 12y²x, or 8x³ + 4x² + 10y²x = 4x³ + 12y²x.

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⁴ *BBi*, pp. 78, 101.

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In the above plan it will be noticed that the terms are ordered according to descending powers of the unknowns. Numerical coefficients are placed after the unknowns; if the coefficient be unity it is noted particularly and if the coefficient be fractional it is kept distinct from the unknown, that is, its denominator is so written as not to come under the unknown;¹ the minus sign is put over the numerical coefficient rather than on the unknown; and the absolute term is invariably put last on either side. As has been observed by Professor Smith,² this plan "in one respect was the best that has ever been suggested." For "it shows at a glance the similar terms one above the other, and permits of easy transposition."

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¹ For instance, see *BBi*, pp. 47 ff.

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The use of the old plan of writing equations is sometimes met with in later works also. For instance, in the MS. of Pṛthûdakasvâmî's commentary on the *Brâhma-sphuṭa-siddhânta*, an incomplete copy of which is preserved in the library of the Asiatic Society of Bengal (No. I B6) we find a statement of equations thus: "first side *yâvargaḥ* 1 *yâvakaḥ* 200 *rû* 0; second side *yâvargaḥ* 0 *yâvakaḥ* 0 *rû* 1500."³ That is to say, $x² + 200x + 0 = 0.x² + 0x + 1500.$

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² Smith, *History*, II, pp. 425, 426.

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³ *BrSpSi*, xii. 15 (*Com*).

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**Preparation of Equations.** The operation to be performed on an equation next to its statement (*nyâsa*) is technically known as *samaśodhana* (from *sama*, meaning equal or complete, and *śodhana*, clearance; hence literally meaning, equi-clearance or complete clearance¹) or simply *śodhana*. The nature of this clearance varies according to the kind of the equation. In the case of an equation in one unknown only, whether linear, quadratic or of higher powers, one side of it is cleared of the unknowns of all denominations and the other side of it of the absolute terms, so that the equation is ultimately reduced to one of the form $ax² + bx = c,$ where *a*, *b*, *c* may be positive or negative; some of them may be even zero. Thus Brahmagupta observes:

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¹ Colebrooke's rendering of the term as “equal subtraction”, though not literally inaccurate, is technically so; at least it is not happy.

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“From which the square of the unknown and the unknown are cleared, the known quantities are cleared (from the side) *below* that.”²

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² *BrSpSi*, xviii. 43.

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Pṛthûdakasvâmî explains:

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“This rule³ has been introduced for that case in which the two sides of the equation having been formed in accordance with the statement of the problem, there are present the square and other powers of the unknown together with the (simple) unknown. The absolute terms should be cleared off from the side opposite to that from which are cleared the square (and other powers) of the unknown and the (simple) unknown. When perfect clearance (*samaśodhana*) has

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³ The reference is to Brahmagupta's rule for the solution of a quadratic equation.

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been thus made..."¹

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¹ *BrSpSi*, xviii. 44 (*Com*).

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Śrîpati says:

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"From one (side) the square of the unknown and the unknown should be cleared by removing the known quantities; the known quantities (should be cleared) from the side opposite to that."²

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Similarly Bhâskara II:

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"Then the unknown on one side of it (the equation) should be subtracted from the unknown on the other side; so also the square and other powers of the unknown; the known quantities on the other side should be subtracted from the known quantities of another (*i.e.*, the former) side."³

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³ *BBi*, p. 44.

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Here we give a few illustrations. With reference to the equations from the commentary of Pṛthûdakasvâmî, stated on page 31, the author says:

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"Perfect clearance (*samaśodhana*) being made in accordance with the rule, (the equation) will be
$ya va 1 ya 10$
$ru 9″$
*i.e.*, x² - 10x = -9.

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² *SiŚe*, xvi. 17.

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The following illustration is from the *Bîjagaṇita* of Bhâskara II:⁴

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⁴ *BBi*, p. 63.

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"Thus the two sides are
$ya va 4 ya 34 ru 72$
$ya va 0 ya0 ru 90$
On complete clearance (*samaśodhana*), the residues of the two sides are

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*yâ va* 4 *yâ* 34 *rû* 0
*yâ va* 0 *yâ* 0 *rû* 18"

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*i.e.*, 4x² - 34x = 18.

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**Classification of Equations.** The earliest Hindu classification of equations seems to have been according to their degrees, such as simple (technically called *yâvat-tâvat*), quadratic (*varga*), cubic (*ghana*) and biquadratic (*varga-varga*). Reference to it is found in a canonical work of *circa* 300 B. C.¹ But in the absence of further corroborative evidence, we cannot be sure of it. Brahmagupta (628) has classified equations as: (1) equations in one unknown (*eka-varṇa-samîkaraṇa*), (2) equations in several unknowns (*aneka-varṇa-samîkaraṇa*), and (3) equations involving products of unknowns (*bhâvita*). The first class is again divided into two subclasses, *viz.*, (*i*) linear equations, and (*ii*) quadratic equations (*avyakta-varga-samîkaraṇa*). Here then we have the beginning of our present method of classifying equations according to their degrees. The method of classification adopted by Pṛthûdakasvâmî (860) is slightly different. His four classes are: (1) linear equations with one unknown, (2) linear equations with more unknowns, (3) equations with one, two or more unknowns in their second and higher powers, and (4) equations involving products of unknowns. As the method of solution of an equation of the third class is based upon the principle of the elimination of the middle term, that class is called by the name *madhyamâharaṇa* (from *madhyama*, "middle", *âharaṇa* "elimination", hence meaning "elimination of the middle term"). For the other classes, the old names given by Brahmagupta have been retained. This method of classification has been followed by subsequent writers.

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¹ *Sthânâṅga-sûtra*, Sûtra 747. For further particulars see Datta, *Jaina Math.*, (BCMS, XXI), pp. 119ff.

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Bhâskara II distinguishes two types in the third class, *viz.*, (*i*) equations in one unknown in its second and higher powers and (*ii*) equations having two or more unknowns in their second and higher powers. According to Kṛṣṇa (1580) equations are primarily of two classes: (1) equations in one unknown and (2) equations in two or more unknowns. The class (1), again, comprises two subclasses: (*i*) simple equations and (*ii*) quadratic and higher equations. The class (2) has three subclasses: (*i*) simultaneous linear equations, (*ii*) equations involving the second and higher powers of unknowns, and (*iii*) equations involving products of unknowns. He then observes that these five classes can be reduced to four by including the second subclasses of classes (1) and (2) into one class as *madhyamâharaṇa*.

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