History of Hindu Mathematics: A Source Book v2.c17.s1: 1. General Features

History of Hindu Mathematics: A Source Book v2.c17.s1

Language: English

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.

Linked texts (9)

Śatapathabrāhmaṇa — Yājñavalkya & Śāṇḍilya (Vedic teacher), Sanskrit
Uttarajjhayaṇa — Ardhamāgadhī Prakrit
Ṭhāṇaṃga — Sudharman, Ardhamāgadhī Prakrit
Aṇuogaddārāiṃ — Āryarakṣita, Ardhamāgadhī Prakrit
Brāhmasphuṭasiddhānta — Brahmagupta, Sanskrit
Āpastambaśulbasūtra — Āpastamba, Sanskrit
Brāhmaṇa — Sanskrit
Bakhshālī Manuscript — Sanskrit
Siddhāntaśiromaṇi — Bhāskara II, Sanskrit