The Science of the Shulba: A Study in Early Hindu Geometry c21.s1: Terminology

The Science of the Śulba: A Study in Early Hindu Geometry c21.s1

Language: English

Terminology

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In the *Sulba*, as in later Hindu mathematics, the fraction is called *aṃśa*, *bhāga*, meaning "part," "portion." Once in the *Āpastamba Śulba* and *Kātyāyana Śulba* each, it is called *kalā*.¹ This term is interesting inasmuch as it was used as early as the *Ṛgveda* ² to denote particularly a sixteenth part. It is also employed symbolically for the number sixteen. The unit fraction is indicated by a compound of either of those terms with a cardinal number, *e.g.,* *pañcadaśa-bhāga* = 1/15 (*ĀpŚl*, x. 3; *KŚl*, v. 4); *tri-bhāga* = 1/3 (*KŚl*) or with an ordinal number, *e.g.,* *pañcama-bhāga* = 1/5 (*ĀpŚl*, ix. 7, x. 2; *KŚl*, v. 6). Oftentimes in the latter case, the word *bhāga* is omitted, so that we have only an ordinal to denote a fraction, *e.g.,* *pañcama* (or "fifth") = 1/5, *dvādaśa* ("twelfth") = 1/12, *trayodaśa* ("thirteenth") = 1/13, etc.³

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¹ *ĀpŚl*, iii. 10; *KŚl*, iii. 11. Compare also *Chāndogya Upaniṣad*, vi. 7. 1.

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² *ṚV*, viii. 47. 17.

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³ *BŚl*, i. 61, ii. 67; *ĀpŚl*, ix. 7, xii. 1, etc. It should be noted that *dvādaśa*, *ṣoḍaśa*, etc., are also used in the cardinal sense.

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In the *Mānava Śulba*,⁴ we find a few very strange and unusual instances where *dvi-guṇa*, *tri-guṇa* and *catur-guṇa* are employed to denote, respectively 1/2, 1/3, and 1/4. But there we also meet with such usual use as *dvi guṇa* = 2-times, *pañca-guṇa* = 5-times.⁵ The former are, indeed, highly ambiguous applications.

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⁴ *MāŚl*, v. 5.

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⁵ *Ibid*, ii. 5, 6.

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It is much noteworthy that the authors of the *Sulba* did not restrict themselves to the use of unit fractions only, as is known to have been the case with the early Egyptian, Babylonian, and Chinese mathematicians.¹ In the *Śulba*, the unit fraction has not, indeed, any special significance attached to it. We find in them the frequent use of the general fraction. Their mode of expressing it is exactly the same as that of later Hindu writers. Thus 3/8 is called *tri-aṣṭama* ("three-eighths"), 2/7 *dvi-saptama* ("two-sevenths").² Kātyāyana mentions 14 3/7 prakramas as *caturdaśa prakramān trīṁśca prakrama-saptabhāgān* (or "14 prakramas and three of the seventh parts of a prakrama").³ 3/4 is sometimes called *catur-bhāgona* ("less one-fourth"),⁴ that is, 1 - 1/4.

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¹ D. E. Smith, *History of Mathematics*, in two volumes, Boston, 1925; Vol. II, pp. 208 ff.

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² *ĀpŚl*, xix. 2, 6.

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³ *KŚl*, vi. 2.

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⁴ *ĀpŚl*, xv. 5, xix. 1.

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A peculiar mode of expressing certain fractions is sometimes found in the *Śulba*, e.g., *ardha-navama*, which literally means "containing a half for its ninth," is used to denote "eight and a half"; *ardha-daśama* ("containing a half for its tenth") " = 9½, and so on.⁵ Such a term evidently carries with it the concrete concept of the operation of measuring.

Linked texts (6)

Chāndogyopaniṣad — Sanskrit
Ṛgvedasaṃhitā — Sanskrit
Mānavaśulbasūtra — Sanskrit
Baudhāyanaśulbasūtra — Baudhāyana, Sanskrit
Āpastambaśulbasūtra — Āpastamba, Sanskrit
Kātyāyanaśulbasūtra — Kātyāyana (Vājasaneyi sūtrakāra), Sanskrit