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History of Hindu Mathematics: A Source Book v2.c17.s2: 3. Symbols

History of Hindu Mathematics: A Source Book v2.c17.s2

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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.

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