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The Science of the Shulba: A Study in Early Hindu Geometry c22: Appendix: Some Technical Terms of the Sulba

The Science of the Śulba: A Study in Early Hindu Geometry c22

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Appendix: Some Technical Terms of the Sulba

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*Line of Symmetry of the Vedi.*—Every one of the altars of various shapes that have been described in the *Sulba*, has a line of symmetry. Some which are square, rectangular or circular (with or without spokes) have, indeed, more than one such line. But primary importance is always attached even in those cases only to one of them. That line of symmetry of an altar is technically called the *pṛṣṭhyā*. This term is derived from the word *pṛṣṭha* (or "back") and so means "the line marking the back or rather the back-bone of the altar." It has its origin in the comparison of the altar with an animal which occurs repeatedly in the *Saṁhitā* and *Brāhmaṇa*. For instance, the *Taittirīya Saṁhitā* observes, "The Fire-altar is an animal."¹

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¹ *TS*, v. 2. 10. 1.

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*Configuration of the Vedi.*—A sacrificial altar is built in such a configuration as to place its principal line of symmetry always along the west-to-east direction. Hence it is also called the *prācī*, or "the eastward line." This line, as has been already observed, is of primary importance in the geometry of the *Sulba*. For all constructions are described in the *Sulba* invariably with reference to it. The sides of an altar lying on either sides of its *prācī*, whether parallel to it or not, are called its *pārśvamānī*, from *pārśva* = "side" and *māna* = "measure," and hence meaning literally the "side measure;" those which are at right angles to the *prācī* are called the *tiryaṅmānī* or the transverse measure," from *tiryak* = "transverse," *māna* = "measure." The latter term is oftentimes called in short *tiryak*, *tiraścīna* or *tiraścī* ("transverse") which are sometimes further abbreviated into *tiraḥ*. These terms are very old and occur in the earliest literatures of the Hindus.¹ The transverse sides are again distinguished into *paścāttiraścī* ("the western transverse side") and *purastāttiraścī* ("the eastern transverse side").² The former is also called the *mukha* (="the face") and the latter the *pada* (="the base") of the altar.

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¹ For instance see *TS*, vi. 2.4.5; *MaiS*, iii. 8.4; *ŚBr*, vii. 1.1.18; etc.

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² *BŚl*, i. 72. 76; *TS*, vi. 2.4.5.

c22.7

*Line*.—The line is called in the *Sulba*, *lekhā* or *rekhā*, both the terms being identical, as, according to the rules of the Sanskrit Grammar, the alphabets *l* and *r* can be replaced mutually. A straight line is distinguished as *ṛjulekhā*, *ṛju* meaning "straight."³

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³ *BŚl*, ii. 32.

c22.9

*Rectilinear Figures*.—In the *Sulba*, we discern two different systems of nomenclature for the rectilinear geometrical figures.⁴ In one system the naming is according to the number of angles or corners in the figures, and the names are formed by the juxtaposition of the number names with *aśra* or *asra* which ordinarily means "corner," "angle," *e.g.*, *tryasra* ("triangle"), *caturasra* ("quadrangle"), etc. These names were introduced in the time of the *Śrauta-sūtra* (*c.* 2000-1500 B.C.). Still older names were compounds ending with *srakti* (= "angle," "corner"). Thus the name *catuḥsrakti*, which literally means, the "quadrangle," occurs in the *Vājasaneyī Saṃhitā*, *Taittirīya Saṃhitā*, *Śatapatha Brāhmaṇa*, *Āpastamba Śrauta*, *Baudhāyana Śulba* and other works. In the *Ṛgveda*, we find the term *navasrakti* referring to the "nine corners" of the heaven. In the *Kātyāyana Śulba Pariśiṣṭa*,¹ we have compound names for rectilinear figures ending with *karṇa*. The Sanskrit word *karṇa* means the "ear." Applied to geometrical figures, it implies the "angle"; hence *trikarṇa* = "triangle," *pañcakarṇa* = "pentagon." The word *karṇa* degenerated into *koṇa* in the *Prākṛta* languages. So in the Ardha Māgadhī work *Sūryaprajñapti*,² we get the names *trikoṇa* (= "trigonon"), *catuṣkoṇa* (= "tetragonon"), *pañcakoṇa* (= "pentagon") etc. These terms are, however, accepted in later Sanskrit literature.³ The term *aśra* or *asra* in a compound name sometimes denotes the "side". Thus Baudhāyana once described a square as *catuḥsrakti* ("four-cornered") and *sama-caturasra* ("equi-four-sided").⁴ So the terms *tryasra*, *caturasra*, etc., will also mean respectively "trilateral," "quadrilateral," etc.⁵ Thus we get a second system of naming retilinear geometrical figures according to the number of sides they possess.

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⁴ For fuller information on this point, see the author's article, "On the Hindu Names for the Rectilinear Geometrical Figures," in the *Journal of the Asiatic Society of Bengal*, N.S., XXVI, 1930, pp. 283-290.

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¹ *KŚlP*, iv. 7-8.

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² *Sūryaprajñapti*, Sūtra 19, 25.

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³ See for instance, the Pariśiṣṭas *of the Atharva-veda*, xxiii. 1 5 xxv. 1, 3, 6, 7, etc. ; *Arthaśāstra of Kauṭilya*, ii. 11. 29.

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⁴ *BŚl*, i. 79.

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⁵ *Cf.* *ĀpŚ*, xx. 12.

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An isosceles triangle is denoted by the term *praüga*. This word is probably derived from *pra* + *yuga*, meaning "the forepart of the shafts of a chariot." A rhombus is similarly called *ubhayataḥ praüga* ("praüga on both sides") inasmuch as it is divided into two praügas by a diagonal. Both these terms are as old as the *Taittirīya Saṃhitā*⁶ and continued to be used in the same sense in posterior works, the *Brāhmaṇa* and *Śrauta* including *Śulba*.

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⁶ *TS*, v. 4. 11.

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A square is generally called *sama-caturasra* (*sama* = "equal"). It is oftentimes, of course when there is no chance of ambiguity the context being clear, shortened into *caturasra*,¹ and occasionally even into simple *sama*.² Thibaut is responsible for the opinion that in the term *sama-caturasra*, the word *sama* refers to the equality of four sides and *caturasra* implies that the four angles of the figure are right angles.³ A more plausible interpretation would be that *sama* refers to the form or shape of the figure which is to be the same in every respect, *caturasra* implying a quadrangle or quadrilateral. It will then be consistent with the term *dīrgha-caturasra* for the rectangle,⁴ which signifies that the form of the *caturasra* is in this case *dīrgha* (or "longish"). The rectangle is sometimes called in short the *dīrgha*.⁵ The term for a quadrilateral of unequal sides is the *viṣama-caturasra* (literally, "inequilateral quadrilateral"). But in contradistinction from the *sama-caturasra* or the square, that term may denote also the rectangle.⁶

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¹ *BŚl*, i. 22. 50. 51; *ĀpŚl*, i. 5; ii. 4-5; etc.

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² *ĀpŚl*, i. 5.

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³ Thibaut, *Śulvasūtras*, p. 7.

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⁴ *BŚl*, i. 36. 38; *ĀpŚl*, ii. 7, iii. 1; etc.

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⁵ *ĀpŚl*, i. 4.

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⁶ See *KŚl*, iii. 4.

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When all the angles of a polygon are equal, it is said to be of *eka-karṇa* (literally, "one-angled") variety; and when not so, of *dvi-karṇa* (literally, "two-angled") variety, implying that in this variety the angles of the figure are of more than one size).⁷

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⁷ Vide supra p. 81, foot-note 2.

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The diagonal is called the *akṣṇa* or *akṣṇayā* ("that which goes across or transversely," that is, "the crossline'').¹ In relation to the instrument of measurement, it is sometimes designated as the *akṣṇayā-rajju* ("the diagonal cord")² and at other times the *akṣṇayā-veṇu* ("the diagonal bamboo-rod").³ The diagonal is also denoted by *karṇa*, meaning "the line going across the angle" or "the line going across from corner to corner."⁴

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¹ *BŚl*, i. 52; iii. 55, 65; *BŚr*, x. 19, xix. 1. Compare *ṚV*, viii. 7. 35; *ĀpŚl*, ii. 5. Rāma, the commentator of the *Kātyāyana Śulba*, is of opinion that this line is so called because it divides the figure into two *akṣi* or "eyes":

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² *BŚl*, i. 50, *ĀpŚl*, i. 325.

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³ *ĀpŚl*, ix. 3.

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⁴ *BŚr*, xix. 1; *KŚr*, xvii. 6. 3.

c22.32

*Circle*.—In the *Śulba*, the circle is designated the *maṇḍala* ("round"),⁵ *pari-maṇḍala* ("round on all sides");⁶ the circumference, *parināha*⁷ ("bounding line on all sides"), and the diameter, *viṣkambha*⁸ or *vyāsa* ("breadth"). The centre of the circle is called *madhya* ("middle"). But this term is also used in more general sense for the middlemost point of a square or rectangle,⁹ or of a line.¹⁰ The segment of the circle is denoted by the term *pradhi*.¹¹

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⁵ *BŚl*, i. 23, 24, 58, 59; *ĀpŚl*, iii. 2, 3.

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⁶ *ĀpŚl*, vii. 6. 13; *BŚl*, ii. 63. 70.

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⁷ *BŚl*, i. 113.

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⁸ *BŚl*, i. 23, 25, 26; *ĀpŚl*, iii; *ĀpŚl*, vii. 10.

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⁹ *BŚl*, i. 58; *ĀŚl*, iii. 2.

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¹⁰ *BŚl*, i. 56, 57, 73; *ĀpŚl*, ii. 1.

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¹¹ *BŚl*, ii. 71-2.

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It is perhaps noteworthy that the direction of rotation was used to be indicated in the Vedic Age by means of an

c22.41

"अक्षिवत् नयति चेक्षमक्ष्णया ⋯ चतुरस्रे कोणात् प्रतिकोणं नीता हि मध्यरज्जुश्चतुरस्रमक्षिद्वयरूपतां नयति। तथा अस्मिन्नपि तिर्यङ् मानीपार्श्वमान्यन्तौ नीताक्ष्णया भवति।" *KŚl*, ii. 7 (Com.)

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arc of a circle very likely with an arrow-head at one extremity. Thus we have the terms *dakṣiṇāvṛta lekhā* (‘the line turning rotationally towards the right’’) and *savyāvṛta lekhā* (‘the line turning rotationally towards the left’’).¹ Again a rotation is called *dakṣiṇā-prāk*, if it is towards the east by the south, and *dakṣiṇā-pratyak*, if it is towards the west by the south.²

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¹ *BŚl*, ii. 30-31.

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² *ĀpŚl*, viii. 9-10.

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*Area.*—In the early Hindu geometry a figure is generally denoted by the term *kṣetra*³ and its area by *bhūmi*.⁴ Occasionally, however, the term *kṣetra* is employed also in the sense of an area.⁴

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³ *BŚr*, xix. 7-9.

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⁴ *BŚr*, xxvi. 25; *BŚl*, i. 56, 57, 82; *ĀpŚl*, i. 5. *KŚl*, iii. 11.

c22.48

*Fundamental Operations.*—Addition is called *samāsa* (‘putting together’’) and the sum obtained *samasta* (‘whole,’’ ‘‘total’’).⁵ Subtraction is called *nirhāra* (‘deduction’’) and the remainder *śeṣa*.⁶ Division is *bhāga*, *vibhāga*. One term deserves special notice : it is *abhyāsa*. This word, formed from *abhi* and *āsa*, means radically ‘‘repetition,’’ ‘‘reduplication.’’ It then came to denote, in its various declinations, the operation of addition⁷ as well as of multiplication.⁸ Whence it seems that the early Hindus recognised multiplication to be a kind of addition.

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⁵ See *BŚl*, i. 59; *ĀpŚl*, ii. 4.

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⁶ *BŚl*, i, 58; *ĀpŚl*, ii. 6, 7.

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⁷ *BŚl*, ii. 4, 9, 11; *ĀpŚl*, i. 9, 2; ii. 1.

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⁸ *ĀpŚl*, v. 3.

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