History of Hindu Mathematics: A Source Book v2.c17.s2.t1: Symbols for Powers and Roots
History of Hindu Mathematics: A Source Book v2.c17.s2.t1
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Symbols for Powers and Roots
v2.c17.s2.t1.1
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
v2.c17.s2.t1.2
¹ Folio 59, recto; compare also folio 67, verso.
v2.c17.s2.t1.3
| 11 | *yu* | 5 | *mû* | 4 |
| 1 | | 1 | | 1 |
v2.c17.s2.t1.4
means √(11+5)=4
v2.c17.s2.t1.5
and
v2.c17.s2.t1.6
| 11 | 7+ | | *mû* | 2 |
| 1 | 1 | | | 1 |
v2.c17.s2.t1.7
means √(11-7)=2.
v2.c17.s2.t1.8
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
v2.c17.s2.t1.9
² *BBi*, p. 15.
v2.c17.s2.t1.10
*ka* 75 *ka* 54 means √9+√450+√75+√54.
v2.c17.s2.t1.11
**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
v2.c17.s2.t1.12
¹ *BMs*, Folio 22, verso.
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$0 & 2 & 3 & 4 & dṛśya & 200 / 1 & 1 & 1 & 1 & & 1 means x+2x+3x+4x=200.$
v2.c17.s2.t1.14
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:
v2.c17.s2.t1.15
² *Triś*, p. 29.
v2.c17.s2.t1.16
| *âdiḥ* 20 | *u* 0 | *gacchaḥ* 7 | *gaṇitaṁ* 245 |
v2.c17.s2.t1.17
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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$0 & 5 & yu & mû & 0 & 2*sa & 0 & 7+ & mû & 0 / 1 & 1 & & & 1 & & 1 & 1 & & 1$
v2.c17.s2.t1.20
which means $√(x+5)=s and √(x-7)=t,$ different unknowns have to be assumed at different vacant places.
v2.c17.s2.t1.21
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
v2.c17.s2.t1.22
¹ Folio 27, verso.
v2.c17.s2.t1.23
| 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 |
v2.c17.s2.t1.24
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.
v2.c17.s2.t1.25
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.
v2.c17.s2.t1.28
"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
v2.c17.s2.t1.32
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-
v2.c17.s2.t1.34
(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
v2.c17.s2.t1.35
The same list with a few additional names of colours appears in the algebra of Nârâyaṇa.² This writer has further added,
v2.c17.s2.t1.36
“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).”
v2.c17.s2.t1.37
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).”
v2.c17.s2.t1.38
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.
v2.c17.s2.t1.39
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.
Linked texts (5)
- Brāhmasphuṭasiddhānta — Brahmagupta, Sanskrit
- Bījagaṇita — Bhāskara II, Sanskrit
- Siddhāntaśekhara — Śrīpati, Sanskrit
- Bakhshālī Manuscript — Sanskrit
- Triśatikā (Pāṭīgaṇitasāra) — Śrīdhara (mathematician), Sanskrit