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History of Hindu Mathematics: A Source Book v2.c17: Chapter III. Algebra

History of Hindu Mathematics: A Source Book v2.c17

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Contents

Chapter III. Algebra

1. General Features

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**Name for Algebra.** The Hindu name for the science of algebra is *bîjagaṇita*. *Bîja* means "element" or "analysis" and *gaṇita* "the science of calculation." Thus *bîjagaṇita* literally means "the science of calculation with elements" or "the science of analytical calculation." The epithet dates at least as far back as the time of Pṛthûdakasvâmî (860) who used it. Brahmagupta (628) calls algebra *kuṭṭaka-gaṇita*, or simply *kuṭṭaka*.¹ The term *kuṭṭaka*, meaning "pulveriser", refers to a branch of the science of algebra dealing particularly with the subject of indeterminate equations of the first degree. It is interesting to find that this subject was considered so important by the Hindus that the whole science of algebra was named after it in the beginning of the seventh century. Algebra is also called *avyakta-gaṇita* or "the science of calculation with unknowns" (*avyakta*=unknown) in contradistinction to the name *vyakta-gaṇita* or "the science of calculation with knowns" (*vyakta*=known) for arithmetic including geometry and mensuration.

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¹ See Bibhutibhusan Datta, "The scope and development of the Hindu Gaṇita," *IHQ*, V, 1929, pp. 479-512; particularly pp. 489f.

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**Algebra Defined.** Bhâskara II (1150) has defined algebra thus:

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"Analysis (*bîja*) is certainly the innate intellect assisted by the various symbols (*varṇa*), which, for the

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instruction of duller intellects, has been expounded by the ancient sages who enlighten mathematicians as the sun irradiates the lotus; that has now taken the name algebra (*bîjagaṇita*).”¹

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¹ *BBi*, p. 99.

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That algebraic analysis requires keen intelligence and sagacity has been observed by him on more than one occasion.

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“Neither does analysis consist in symbols, nor are there different kinds of analyses; sagacity alone is analysis, for wide is imagination.”²

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² *BBi*, p. 49; *SiŚi*, *Gola*, xiii. 5.

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“Analysis is certainly clear intelligence.”³

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³ *L*, p. 15; *SiŚi*, *Gola*, xiii. 3.

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“Or intelligence alone is analysis.”⁴

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⁴ *BBi*, p. 49.

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In answer to the question, “if (unknown quantities) are to be discovered by intelligence alone what then is the need of analysis ?” he says:

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“Because intelligence is certainly the real analysis; symbols are its helps. The innate intelligence which has been expressed for the duller intellects by the ancient sages, who enlighten mathematicians as the sun irradiates the lotus, with the help of various symbols, has now obtained the name of algebra.”⁵

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⁵ *BBi*, p. 100.

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Thus, according to Bhâskara II, algebra may be defined as the science which treats of numbers expressed by means of symbols, and in which there is scope and primary need for intelligent artifices and ingenious devices.

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**Distinction from Arithmetic.** What distinguishes algebra from arithmetic, according to the Hindus, will be found to some extent in their special names. Both deal with symbols. But in arithmetic the values of the symbols are *vyakta*, that is, known and definitely determinate, while in algebra they are *avyakta*, that is, unknown, indefinite. The relation between these two branches of *gaṇita* is considered by Bhâskara II to be this:

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“The science of calculation with unknowns is the source of the science of calculation with knowns.”¹

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He has put it more explicitly and clearly thus:

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“Algebra is similar to arithmetic in respect of rules (of fundamental operations) but appears as if it were indeterminate. It is not indeterminate to the intelligent; it is certainly not sixfold,² but manifold.”³

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The true distinction between arithmetic and algebra, besides that of symbols employed, lies, in the opinion of Bhâskara II, in the demonstration of the rules. He remarks:

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the other symbolical (*râśigata*).”¹ We do not know who was the first in India to use geometrical methods for demonstrating algebraical rules. Bhâskara II (1150) ascribes it to “ancient teachers.”²

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¹ *BBi*, p. 125.

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² *BBi*, p. 127.

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**Importance of Algebra.** The early Hindus regarded algebra as a science of great importance and utility. In the opening verses of his treatise³ on algebra Brahmagupta (628) observes:

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³ Forming chapter xviii of his *Brâhma-sphuṭa-siddhânta*.

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“Since questions can scarcely be known (*i.e.*, solved) without algebra, therefore, I shall speak of algebra with examples.

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“By knowing the pulveriser, zero, negative and positive quantities, unknowns, elimination of the middle term, equations with one unknown, factum and the Square-nature, one becomes the learned professor (*âcârya*) amongst the learned.”⁴

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⁴ *BrSpSi*, xviii. 1-2.

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Similarly Bhâskara II writes:

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“What the learned calculators (*sâmkhyâḥ*) describe as the originator of intelligence, being directed by a wise being (*satpuruṣa*) and which alone is the primal cause (*bîja*) of all knowns (*vyakta*), I venerate that Invisible God as well as that Science of Calculation with Unknowns... Since questions can scarcely be solved without the reasoning of algebra—not at all by those of dull perceptions—I shall speak, therefore; of the operations of analysis.”⁵

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⁵ In the first part of this passage every principal term has been used with a double significance. The term *sâmkhyâḥ* (literally, “expert calculators”) signifies the “Sâmkhya philosophers” in one sense, “mathematicians” in the other; *satpuruṣa* “the self-existent being of the Sâmkhya philosophy” or “a wise mathematician”; *vyakta* “manifested universe” or “the science of calculation with knowns.”

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Nârâyaṇa (1350) remarks:

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"I adore that Brahma, also that science of calculation with the unknown, which is the one invisible root-cause of the visible and multiple-qualitied universe, also of multitudes of rules of the science of calculation with the known."

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"As out of Him is derived this entire universe, visible and endless, so out of algebra follows the whole of arithmetic with its endless varieties (of rules). Therefore, I always make obeisance to Śiva and also to (*avyakta-*) *gaṇita* (algebra)."

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He adds:

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"People ask questions whose solutions are not to be found by arithmetic; but their solutions can generally be found by algebra. Since less intelligent men do not succeed in solving questions by the rules of arithmetic, I shall speak of the lucid and easily intelligible rules of algebra."

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**Scope of Algebra.** The science of algebra is broadly divided by the Hindus into two principal parts. Of these the most important one deals with analysis (*bîja*). The other part treats of the subjects which are essential for analysis. They are: the laws of signs, the arithmetic of zero (and infinity), operations with unknowns, surds, the pulveriser (or the indeterminate equation of the first degree), and the Square-nature (or the so-called Pellian equation). To these some writers add concurrence and dissimilar operations, while others include them in arithmetic. At the end of the first section of his treatise on algebra Bhâskara II is found to have observed as follows:

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¹ ^1 *NBi*, I, R. 1.

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² ^2 *NBi*, II, R. 1.

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³ ^3 *NBi*, I, R. 5-6.

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⁴ ^4 All writers, except Brahmagupta and Śrîpati, are of the latter opinion.

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"(The section of) this science of calculation which is essential for analysis has been briefly set forth. Next I shall propound analysis, which is the source of pleasure to the mathematician."¹

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¹ *BBi*, p. 43.

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Analysis is stated by all to be of four kinds, for equations are classified into four varieties (*vide infra*). Thus each class of equations has its own method of analysis.

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**Origin of Hindu Algebra.** The origin of Hindu algebra can be definitely traced back to the period of the *Sulba* (800-500 B.C.) and the *Brâhmaṇa* (*c.* 2000 B.C.). But it was then mostly geometrical.² The geometrical method of the transformation of a square into a rectangle having a given side, which is described in the important *Sulba* is obviously equivalent to the solution of a linear equation in one unknown, *viz.*, $ax = c².$ The quadratic equation has its counterpart in the construction of a figure (an altar) similar to a given one but differing in area from it by a specified amount. The usual method of solving that problem was to increase the unit of measure of the linear dimensions of the figure. One of the most important altars of the obligatory Vedic sacrifices was called the *Mahâvedi* (the Great Altar). It has been described to be of the form of an isosceles trapezium whose face is 24 units long, base 30 and altitude 36. If x be the enlarged unit of measure taken in increasing the size of the altar by m units of area, we must have $36x × ((24x + 30x))/2 = 36 × ((24 + 30))/2 + m,$ or 972x² = 972 + m.

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² Bibhutibhusan Datta, *The Science of the Śulba*, Calcutta, 1932.

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Therefore x = √(1 + m/972).

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If m be put equal to 972 (n—1), so that the area of the enlarged altar is n times its original area, we get x = √n, some particular cases of which are described in the *Sulba*. The particular cases, when n = 14 or 14 3/7, are found as early as the *Śatapatha Brâhmaṇa*¹ (*c.* 2000 B.C.).

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¹ *ŚBr*, X. 2. 3. 7ff.

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The most ancient and primitive form of the "Fire-altar for the sacrifices to achieve special objects" was the *Śyenacit* (or "the altar of the form of the falcon").

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Fig. 1.

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Its body (*ABCD*) consists of four squares of one square *puruṣa* each; each of its wings (*EFGH*, *E'F'G'H'*) is a rectangle of one *puruṣa* by one *puruṣa* and a *prâdeśa* (= 1/10 of a *puruṣa*). This Fire-altar was enlarged in two ways: *first*, in which all the constituent parts were affected in the same proportion; *second*, in which the breadth of the portions *LFGM* and *L'F'G'M'* of the wings were left unaffected. If *x* be the enlarged unit for enlargement in the first case we shall have to solve the quadratic equation

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$2x × 2x + 2x × (x + x/5) + x × (x + x/10) = 7½ + m,$

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where *m* denotes the increment of the Fire-altar in size.

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$Therefore x² = 1 + 2m/15.$

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In particular, when m = 94, we shall have

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$x² = 13 8/15 = 14 (approximately),$

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which occurs in the *Śatapatha Brāhmaṇa*.

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In the second case of enlargement the equation for *x* will be

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$2x × 2x + 2x × (x + ⅕) + x × (x + 1/10) = 7½ + m,$ $or 7x² + ½x = 7½ + m,$

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which is a complete quadratic equation.

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The problem of altar construction gave rise also to certain indeterminate equations of the second degree such as,

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(1) & x² + y² = z², / (2) & x² + a² = z²;

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and simultaneous indeterminate equations of the type

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ax + by + cz + dw &= p, / x + y + z + w &= q.

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Further particulars about these equations will be given later on.

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**Coefficient.** In Hindu algebra there is no systematic use of any special term for the coefficient. Ordinarily, the power of the unknown is mentioned when the reference is to the coefficient of that power. In explanation of similar use by Brahmagupta his commentator Pṛthûdakasvâmî writes “the number (*aṅka*) which is the coefficient of the square of the unknown is called the ‘square’ and the number which forms the coefficient of the (simple) unknown is called ‘the unknown quantity.’ ”¹ However, occasional use of a technical term is also met with. Brahmagupta once calls the coefficient *saṁkhyâ*² (number) and on several other occasions *guṇaka*, or *guṇakâra* (multiplier).³ Pṛthûdakasvâmî (860) calls it *aṅka*⁴ (number) or *prakṛti* (multiplier). These terms reappear in the works of Śrîpati (1039)⁵ and Bhâskara II (1150).⁶ The former also used *rûpa* for the same purpose.⁷

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¹ *BrSpSi*, xviii. 44 (*Com*).

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² *BrSpSi*, xviii. 63.

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³ *BrSpSi*, xviii. 64, 69-71.

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⁴ *BrSpSi*, xviii. 44 (*Com*).

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⁵ *SiŚe*, xiv. 33-5.

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⁶ *BBi*, pp. 33-4.

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⁷ *SiŚe*, xiv. 19.

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**Unknown Quantity.** The unknown quantity was called in the *Sthânâṅga-sûtra*⁸ (before 300 B.C.) *yâvat-tâvat* (as many as or so much as, meaning an arbitrary quantity). In the so-called Bakhshâlî treatise, it was called *yadṛcchâ*, *vâñchâ* or *kâmika* (any desired quantity).⁹ This term was originally connected with the Rule of False Position.¹⁰ Âryabhaṭa I (499)

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⁸ *Sûtra* 747; *cf.* Bibhutibhusan Datta, “The Jaina School of Mathematics,” *BCMS*, XXI, 1929, pp. 115-145; particularly pp. 122-3.

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⁹ *BMs*, Folios 22, verso; 23, recto & verso.

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¹⁰ Bibhutibhusan Datta, “The Bakshshâlî Mathematics,” *BCMS*, XXI, pp. 1-60; particularly pp. 26-8, 66.

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calls the unknown quantity *gulikâ* (shot). This term strongly leads one to suspect that the shot was probably then used to represent the unknown. From the beginning of the seventh century the Hindu algebraists are found to have more commonly employed the term *avyakta* (unknown).¹

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¹ *BrSpSi*, xviii. 2, 41; *SiŚe*, xiv. 1-2; *BBi*, pp. 7ff.

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**Power.** The oldest Hindu terms for the power of a quantity, known or unknown, are found in the *Uttarâdhyayana-sûtra* (c. 300 B.C. or earlier).² In it the second power is called *varga* (square), the third power *ghana* (cube), the fourth power *varga-varga* (square-square), the sixth power *ghana-varga* (cube-square), and the twelfth power *ghana-varga-varga* (cube-square-square), using the multiplicative instead of the additive principle. In this work we do not find any method for indicating odd powers higher than the third. In later times, the fifth power is called *varga-ghana-ghâta* (product of cube and square, *ghâta* = product), the seventh power *varga-varga-ghana-ghâta* (product of square-square and cube) and so on. Brahmagupta's system of expressing powers higher than the fourth is scientifically better. He calls the fifth power *pañca-gata* (literally, raised to the fifth), the sixth power *ṣaḍ-gata* (raised to the sixth); similarly the term for any power is coined by adding the suffix *gata* to the name of the number indicating that power.³ Bhâskara II has sometimes followed it consistently for the powers one and upwards.⁴ In the *Anuyogadvâra-sûtra*⁵, a work written before the commencement of the Christian Era, we find certain interesting terms for higher powers, integral as well as fractional, particularly successive squares (*varga*) and square-roots (*varga-mûla*). According to it the term *prathama-varga* (first square) of a quantity, say *a*, means a²; *dvitîyavarga* (second square) = (a²)² = a⁴; *tṛtîya-varga* (third square) = ((a²)²)² = a⁸; and so on. In general,

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² Chapter xxx, 10, 11.

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³ *BrSpSi*, xviii. 41, 42.

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⁴ *BBi*, p. 56.

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⁵ *Sûtra* 142.

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*n*th *varga* of a = a² × 2 × 2 × to n terms = a2n.

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Similarly, *prathama-varga-mûla* (first square-root) means √a; *dvitîya-varga-mûla* (second square-root) = √()(√a) = a1/4; and, in general,

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*n*th *varga-mûla* of a = a1/2n.

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Again we find the term *tṛtîya-varga-mûla-ghana* (cube of the third square-root) for (a1/2³)³ = a3/8.

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The term *varga* for "square" has an interesting origin in a purely concrete concept. The Sanskrit word *varga* literally means "rows," or "troops" (of similar things). Its application as a mathematical term originated in the graphical representation of a square, which was divided into as many *varga* or troops of small squares, as the side contained units of some measure.¹

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¹ G. Thibaut, *Śulba-sûtras*, p. 48. Compare also Bibhutibhusan Datta, "On the origin of the Hindu terms for root," *Amer. Math. Mon.*, XXXVIII, 1931, pp. 371-6.

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**Equation.** The equation is called by Brahmagupta (628) *sama-karaṇa*² or *samî-karaṇa*³ (making equal) or more simply *sama*⁴ (equation). Pṛthûdakasvâmî (860) employs also the term *sâmya*⁵ (equality or equation); and Śrîpati (1039) *sadṛśî-karaṇa*⁶ (making similar). Nârâyaṇa (1350) uses the terms *samî-karaṇa*, *sâmya* and *samatva* (equality).⁷ An equation has always two *pakṣa* (side). This term occurs in the works of Śrīdhara (*c.* 750), Padmanābha¹ and others.²

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² *BrSpSi*, xviii. 63.

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³ *BrSpSi*, xviii, subheading for the section on equations.

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⁴ *BrSpSi*, xviii. 43.

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⁵ *BrSpSi*, xii. 66 (*Com*).

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⁶ *SiŚe*, xiv. 1.

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⁷ *NBi*, II, R. 2-3.

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¹ The algebras of Śrīdhara and Padmanâbha are not available now. But the term occurs in quotations from them by Bhâskara II (*BBi*, pp. 61, 67).

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² *BrSpSi*, xviii. 43 (*Com*); *SiŚe*, xiv. 14, 20; *BBi*, pp. 43-4.

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**Absolute Term.** In the Bakhshālî treatise³ the absolute term is called *dṛśya* (visible). In later Hindu algebras it has been replaced by a closely allied term *rūpa*⁴ (appearance), though it continued to be employed in treatises on arithmetic.⁵ Thus the true significance of the Hindu name for the absolute term in an algebraic equation is obvious. It represents the visible or known portion of the equation while its other part is practically invisible or unknown.

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³ *BMs*, Folio 23, verso; Folio 70, recto and verso (c).

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⁴ *BrSpSi*, xviii. 43-4; *SiŚe*. xiv, 14, 19; etc.

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⁵ *Triś*, pp. 11, 12.

3. Symbols

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**Symbols of Operation.** There are no special symbols for the fundamental operations in the Bakhshālî work. Any particular operation intended is ordinarily indicated by placing the tachygraphic abbreviation, the initial syllable of a Sanskrit word of that import, after, occasionally before, the quantity affected. Thus the operation of addition is indicated by *yu* (an abbreviation from *yuta*, meaning added), subtraction by + which is very probably from *kṣa* (abbreviated from *kṣaya*, diminished), multiplication by *gu* (from *guṇa* or *guṇita*, multiplied) and division by *bhâ* (from *bhāga* or *bhājita*, divided). Of these again, the most systematically employed abbreviation is that for the operation of subtraction and next comes that of division. In the case of the other two operations the indicatory words are often written in full, or altogether omitted. In the latter case, the particular operations intended to be carried out are understood from the context. We take the following instances from the Bakhshâlî Manuscript:

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(i)¹ 0 & 5 / 1 & 1 yu means x/1 + 5/1, and 11 / 1 yu 5 / 1 means 11/1 + 5/1.

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¹ Folio 59, recto.

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(ii)² 3 & 3 & 3 & 3 & 3 & 3 & 3 & 10 gu / 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 / means 3 × 3 × 3 × 3 × 3 × 3 × 3 × 10.

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² Folio 47, recto.

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(iii)³ 0 & 1 & 3 & 2 & 5 & 3 & 7 & 4 & 9 / 1 & 1 & 2 & 1 & 2+ & 1 gu & 2+ & 1 gu & 2+ means x(1+3/2) + 2x(1+3/2) - 5x/2 + 3x(1+3/2) - 7x/2 + 4x(1+3/2) - 9x/2.

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³ Folio 25, verso. The beginning and end of this illustration are mutilated but the restoration is certain.

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(iv)⁴ 1 & 1 & 1 & 1 bhâ & 36 / 1+ & 1 & 1 & 1 & / 2 & 3 & 4+ & 6 & 1 / means 36/((1-½)(1+⅓)(1-¼)(1+⅙)).

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⁴ Folio 13, verso.

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(v)⁵ 40 bhâ & 160 & 13 / 1 & 1 & 1 / & & 2 means 160/40 × 13½.

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⁵ Folio 42, recto.

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In later Hindu mathematics the symbol for subtraction is a dot, occasionally a small circle, which is placed above the quantity, so that 7 or 7 (°) means -7; other operations are represented by simple juxtaposition.

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Bhâskara II (1150) says, "Those (known and unknown numbers) which are negative should be written with a dot (*bindu*) over them."¹ A similar remark occurs in the algebra of Nârâyaṇa (1350).² Their silence about symbols of other fundamental operations proves their non-existence.

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¹ *BBi*, p. 2.

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² *NBi*, I, R. 7.

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**Origin of Minus Sign.** The origin of · or ° as the minus sign seems to be connected with the Hindu symbol for the zero, 0. It is found to have been used as the sign of emptiness or omission in the early Bakhshâlî treatise as well as in the later treatises on arithmetic (*vide infra*).³ It is placed over the number affected in order to distinguish it from its use in a purely numerical significance when it is placed beside the number. The origin of the Bakhshâlî minus sign (+) has been the subject of much conjecture. Thibaut suggested its possible connection with the supposed Diophantine negative sign ⋔ (reversed ψ, tachygraphic abbreviation for λειψις meaning wanting). Kaye believes it. The Greek sign for minus, however, is not ⋔, but ↑. It is even doubtful if Diophantus did actually use it; or whether it is as old as the Bakhshâlî cross.⁴ Hoernle⁵ presumed the Bakhshâlî minus sign to be the abbreviation *ka* of the Sanskrit word *kanita*, or *nu* (or *nû*) of *nyûna*, both of which mean diminished and both of which abbreviations in the Brâhmî characters would be denoted by a cross. Hoernle was right, thinks Datta,⁶ so far as he sought for the origin of + in a tachygraphic abbreviation of some Sanskrit word. But as neither the word *kanita* nor *nyûna* is found to have been used in the Bakhshâlî work in connection with the subtractive operation, Datta finally rejects the theory of Hoernle and believes it to be the abbreviation *kṣa*, from *kṣaya* (decrease) which occurs several times, indeed, more than any other word indicative of subtraction. The sign for *kṣa*, whether in the Brâhmî characters or in Bakhshâlî characters, differs from the simple cross (+) only in having a little flourish at the lower end of the vertical line. The flourish seems to have been dropped subsequently for convenient simplification.

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³ p. 16.

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⁴ *Cf.* Smith, *History*, II, p. 396.

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⁵ *IA*, XVII, p. 34.

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⁶ Datta, *Bakh. Math.*, (BCMS, XXI), pp. 17-8.

Symbols for Powers and Roots

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The symbols for powers and roots are abbreviations of Sanskrit words of those imports and are placed after the number affected. Thus the square is represented by *va* (from *varga*), cube by *gha* (from *ghana*), the fourth power by *va-va* (from *varga-varga*), the fifth power by *va-gha-ghâ* (from *varga-ghana-ghâta*), the sixth power by *gha-va* (from *ghana-varga*), the seventh power by *va-va-gha-ghâ* (from *varga-varga-ghana-ghâta*) and so on. The product of two or more unknown quantities is indicated by writing *bhâ* (from *bhâvita*, product) after the unknowns with or without interposed dots; e.g., *yâva-kâgha-bhâ* or *yâvakâghabhâ* means (ya)² (ka)³. In the Bakhshâlî treatise the square-root of a quantity is indicated by writing after it *mû*, which is an abbreviation for *mûla* (root). For instance,¹

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¹ Folio 59, recto; compare also folio 67, verso.

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| 11 | *yu* | 5 | *mû* | 4 |
| 1 | | 1 | | 1 |

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means √(11+5)=4

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and

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| 11 | 7+ | | *mû* | 2 |
| 1 | 1 | | | 1 |

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means √(11-7)=2.

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In other treatises the symbol of the square-root is *ka* (from *karaṇî*, root or surd) which is usually placed before the quantity affected. For example,² *ka* 9 *ka* 450

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² *BBi*, p. 15.

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*ka* 75 *ka* 54 means √9+√450+√75+√54.

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**Symbols for Unknowns.** In the Bakhshâlî treatise there is no specific symbol for the unknown. Consequently its place in an equation is left vacant and to indicate it vividly the sign of emptiness is put there. For instance,¹

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¹ *BMs*, Folio 22, verso.

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$0 & 2 & 3 & 4 & dṛśya & 200 / 1 & 1 & 1 & 1 & & 1 means x+2x+3x+4x=200.$

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The use of the zero sign to mark a vacant place is found in the arithmetical treatises of later times when the Hindus had a well-developed system of symbols for the unknowns. Thus we find in the *Triśatikâ*² of Śrîdhara (*c.* 750) the following statement of an arithmetical progression whose first term (*âdiḥ*) is 20, number of terms (*gacchaḥ*) 7, sum (*gaṇitaṁ*) 245 and whose common difference (*uttaraḥ*) is unknown:

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² *Triś*, p. 29.

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| *âdiḥ* 20 | *u* 0 | *gacchaḥ* 7 | *gaṇitaṁ* 245 |

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This use of the zero sign in arithmetic was considered necessary as algebraic symbols could not be used there. Lack of an efficient symbolism is bound to give rise to a certain amount of ambiguity in the representation of an algebraic equation especially when it contains more than one known. For instance, in³

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³ *BMs*, Folio 59, recto.

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$0 & 5 & yu & mû & 0 & 2*sa & 0 & 7+ & mû & 0 / 1 & 1 & & & 1 & & 1 & 1 & & 1$

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which means $√(x+5)=s and √(x-7)=t,$ different unknowns have to be assumed at different vacant places.

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To avoid such ambiguity, in one instance which contains as many as five unknowns, the abbreviations of ordinal numbers, such as *pra* (from *prathama*, first), *dvi* (from *dvitîya*, second), *tṛ* (from *tṛtîya*, third), *ca* (from *caturtha*, fourth) and *paṁ* (from *pañcama*, fifth), have been used to represent the unknowns; *e.g.*,¹

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¹ Folio 27, verso.

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| 9 *pra* | 7 *dvi* | 10 *tṛ* | 8 *ca* | 11 *paṁ* | yutaṁ jâtaṁ pratyaika-(krameṇa) |
| 7 *dvi* | 10 *tṛ* | 8 *ca* | 11 *paṁ* | 9 *pra* | 16\|17\|18\|19\|20 |

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which means x₁ (= 9) + x₂ (= 7) = 16; x₂ (= 7) + x₃ (= 10) = 17; x₃ (= 10) + x₄ (= 8) = 18; x₄ (= 8) + x₅ (= 11) = 19; x₅ (= 11) + x₁ (= 9) = 20.

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Āryabhaṭa I (499) very probably used coloured shots to represent unknowns. Brahmagupta (628) mentions *varṇa* as the symbols of unknowns.² As he has not attempted in any way to explain this method of symbolism, it appears that the method was already very familiar. Now, the Sanskrit word *varṇa* means "colour" as well as "letters of the alphabet," so that, in later times, the unknowns are generally represented by letters of the alphabet or by means of various colours such as *kâlaka* (black), *nîlaka* (blue), etc. Again in the latter case, for simplification, only initial letters of the names are generally written. Thus Bhâskara II (1150) observes, "Here (in algebra) the initial letters of (the names of) knowns and unknowns should be written for implying them."³ It has been stated before that at one time the unknown quantity was called *yâvat-tâvat* (as many as, so much as). In later times this name, or its abbreviation *yâ*, is used for the unknown. According to the celebrated Sanskrit lexicographer Amarasiṃha (*c.* 400 A.D.), *yâvat-tâvat* denotes measure or quantity (*mâna*). He had probably in view the use of that term in Hindu algebra to denote "the measure of an unknown" (*avyakta mâna*). In the case of more unknowns, it is usual to denote the first by *yâvat-tâvat* and the remaining ones by alphabets or colours. Pṛthûdakasvâmî (860) says:

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² *BrSpSi*, xviii. 2, 42, 51, etc.

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³ *BBi*, p. 2; see also *NBi*, I, R. 7.

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"In an example in which there are two or more unknown quantities, colours such as *yâvat-tâvat*, etc., should be assumed for their values."¹

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He has, indeed, used the colours *kâlaka* (black), *nîlaka* (blue), *pîtaka* (yellow) and *harîtaka* (green).

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Śrîpati (1039) writes:

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"*Yâvat-tâvat* (so much as) and colours such as *kâlaka* (black), *nîlaka* (blue), etc., should be assumed for the unknowns."²

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Bhâskara II (1150) says:

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"*Yâvat-tâvat* (so much as), *kâlaka* (black), *nîlaka* (blue), *pîta* (yellow), *lohita* (red) and other colours have been taken by the venerable professors as notations for the measures of the unknowns, for the purpose of calculating with them."³

"In those examples where occur two, three or more unknown quantities, colours such as *yâvat-

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(variegated), *kapilaka* (tawny), *piṅgalaka* (reddish-brown), *dhûmraka* (smoke-coloured), *pâtalaka* (pink), *śavalaka* (spotted), *śyâmalaka* (blackish), *mecaka* (dark blue), etc. Or the letters of alphabets beginning with *ka*, should be taken as the measures of the unknowns in order to prevent confusion.”¹

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The same list with a few additional names of colours appears in the algebra of Nârâyaṇa.² This writer has further added,

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“Or the letters of alphabets (*varṇa*) such as *ka*, etc., or the series of flavours such as *madhura* (sweet), etc., or the names of dissimilar things with unlike initial letters, are assumed (to represent the unknowns).”

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Bhâskara II occasionally employs also the tachygraphic abbreviation of the names of the unknown quantities themselves in order by names, the equations may be formed by the intelligent (calculator).”

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It should be noted that *yâvat-tâvat* is not a *varṇa* (colour or letter of alphabet). So in its inclusion in the lists of *varṇa*, as found enumerated in the Hindu algebras—though apparently anomalous—we find the persistence of an ancient symbol which was in vogue long before the introduction of colours to represent unknowns. To avoid the anomaly Muralidhara Jha¹ has suggested the emendation *yâvakastâvat* (*yâvaka* and also; *yâvaka* = red) in the place of *yâvat-tâvat*, as found in the available manuscripts. He thinks that being misled by the old practice, the expression *yâvakastâvat* was confused by copyists with *yâvat-tâvat*. In support of this theory it may be pointed out that *yâvaka* is found to have been sometimes used by Pṛthûdakasvâmî to represent the unknown.² Bhâskara II has once used simply *yâvat*.³ Nârâyaṇa used it on several occasions. The origin of the use of names of colours to represent unknowns in algebra is very probably connected with the ancient use of differently coloured shots for the purpose.

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¹ See the Preface to his edition of Bhâskara’s *Bîjagaṇita*.

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² *BrSpSi*, xii. 15 (*Com*); xii. 18 (*Com*).

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³ *BBi*, p. 50.

4. Laws of Signs

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**Addition.** Brahmagupta (628) says:

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“The sum of two positive numbers is positive, of two negative numbers is negative; of a positive and a negative number is their difference.”⁴

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⁴ *BrSpSi*, xviii. 30.

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Mahâvîra (850):

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“In the addition of a positive and a negative number (the result) is (their) difference. The addition of two positive or two negative numbers (gives) as much positive or negative numbers respectively."¹

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Śrīpati (1039):

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"In the addition of two negative or two positive numbers the result is their sum; the addition of a positive and a negative number is their difference."² then they must be added together."¹ Mahâvîra:

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¹ *BrSpSi*, xviii. 31-2.

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"A positive number to be subtracted from another number becomes negative and a negative number to be subtracted becomes positive."²

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² *GSS*, i. 51.

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Śrîpati:

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"A positive (number) to be subtracted becomes negative, a negative becomes positive; (the subsequent operation is) addition as explained before."³

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³ *SiŚe*, xiv. 3.

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Bhâskara II:

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"A positive (number) while being subtracted becomes negative and a negative becomes positive; then addition as explained before."⁴

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⁴ *BBi*, p. 3.

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Nârâyaṇa:

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"Of the subtrahend affirmation becomes negation and negation affirmation; then addition as described before."⁵

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⁵ *NBi*, I, R. 9.

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**Multiplication.** Brahmagupta says:

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"The product of a positive and a negative (number) is negative; of two negatives is positive; positive multiplied by positive is positive."⁶

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⁶ *BrSpSi*, xviii. 33.

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Mahâvîra:

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"In the multiplication of two negative or two positive numbers the result is positive; but it is negative in the case of (the multiplication of) a positive and a negative number."⁷

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⁷ *GSS*, i. 50.

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Śrîpati:

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"On multiplying two negative or two positive

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numbers (the product is) positive; in the multiplication of positive and negative (the result is) negative.”¹

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¹ *SiŚe*, xiv. 4.

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Bhâskara II:

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“The product of two positive or two negative (numbers) is positive; the product of positive and negative is negative.”²

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² *BBi*, p. 3.

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The same rule is stated by Nârâyaṇa.³

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