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The Shulbasutras of Baudhayana, Apastamba, Katyayana and Manava: With Text, English Translation and Commentary c8.t31: Chapters 5 and 6: Enlarged Unit, Construction of a Square Equal to N Times a Given Square and Ekādaśinī Fire-Altar

The Śulbasūtras of Baudhāyana, Āpastamba, Kātyāyana and Mānava: With Text, English Translation and Commentary c8.t31

Language
English

Chapters 5 and 6: Enlarged Unit, Construction of a Square Equal to N Times a Given Square and Ekādaśinī Fire-Altar

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The whole of chapter 5 and the first five *sūtras* of chapter 6 are devoted to the discussion of enlarged unit required for measuring the areas of fire-altars from 8½ sq. *puruṣas* to 101½ sq. *puruṣas*. Most of these results agree with those of Baudhāyana and have been discussed under *Bśl.* 5.1-5.6. In *sūtra* 6.2, Kātyāyana says that if *s* be the maximum enlarged unit in a *prakrama* for 101 fold fire-altar, then *s*² equals 14⁴⁄₇ sq. *prakramas*. In the next *sūtras* he has pointed out that at each successive constuction the value of the *prakrama* is to be increased by one seventh of the increased area,

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i.e. s² = 1 + p/7 ...... (1)

or 7*s*² = 7 + *p*, where *p* is the increased area.

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For the construction of 101 fold fire-altar the total increment from the 7 fold one is 94. Putting *a* = 94.

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7*s*² = 7 + 94 = 101,

then s² = 101/7 = 14 3/7 ...... (2)

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But the formula (1) does not appear to be correct (vide *Bśl.* 5.1-5.6); it should be s² = 1 + p/(7½), since the enlargement in area starts from the normal 7 fold fire-altar, i.e. 7½ sq. *puruṣas*. It may be that Kātyāyana simplified the rule for construction of 101 fold fire-altar.

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Dattaa has suggested another rational of the formula (2) as follows.

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In the falcon-shaped fire-altar (second plan), the problem of proportionate enlargement is equivalent to the solution of the following quadratic equation:

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a Datta (2), 166-168.

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4s² + 2s(s + ⅕) + s(s + 1/10) = 7 ½ + p,

or, 7s² + ½ s = 7 ½ + p,

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$or, (7s+¼)²=841/16+7p,$ $or, s=1/28(√(841+112p)-1)$ When p=94, $s=1/28(√11369-1) or, s²=1/784(11370-2√11369),$ $or, s²=1/784(11156+79/106), when √11369=106+133/212 approx.$ $or, s²=14+19159/83104$ $=14+3/(1324519159)=14+3/13 approx.$

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$It is nearly equal to Kātyāyana's value, s²=14 3/7$

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**6.7.** *Construction of a square equal to n times the given square.* This method undoubtedly hints at the construction of a square which is equivalent to n times a given square. Let n number of equal squares each of side a are to be combined. *Śulbakāras* used isosceles triangles for different constructions. Here also Kātyāyana possibly considered an isosceles triangle ABC, in which {cl} & BC=(n-1)\ a, / & AB+AC=(n+1)\ a, / Since & BD=DC, AB=AC, / & BD=\cfrac{n-1}\ a, and AB=\cfrac{n+1}a, According to this rule, the altitude AD will produce the sum of n equal squares. Now AD²[t] &=AB²-BD² / &=((n+1)/2a)²-((n-1)/2a)² / &=na²

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Fig. 7.

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**6.8-6.13.** *Construction of a fire-altar with enlarged areas was usually carried out by fixing the distance between the two poles (yūpas) of ekādaśinī.* This distance is known as *prakrama*. The length of *prakrama* varies in the case of the enlarged fire-altar. There are various opinions on this point by ancient masters. This has been discussed under *Bśl.* 4.12-4.14.

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The *Mānava-śulbasūtra*, in 16 chapters, is a mere compilation, and its value as a technical text appears to be of dubious nature. In most places it is corrupt, and the same topics are discussed in different places. The materials on *gārhapatya*, for instance are dispersed in chapters 1, 6, 9, 13, on *caturaśraśyenacit* in chapters 4, 5, 6 and 13, on *dhiṣṇyas* in chapters 6, 9 and 13 etc. It is full of paraphernalia about worship hardly of any consequence to altar construction. The arrangement is unsystematic. The text therefore bears no comparison with the three other *śulba* texts already commented upon.

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DETERMINATION OF EAST-WEST LINE, CONSTRUCTION OF *DĀRŚIKĪ VEDI*, SIZE AND RELATIVE PLACES OF *GĀRHAPATYA*, *ĀHAVANĪYA DAKṢIṆĀGNI*, *UTKARA* AND GENERAL RULE FOR DRAWING A SQUARE.

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**1.2-1.3.** *East-west line.* *Sūtras* 1.2 and 1.3 direct the fixation of the east-west line of the altar according to cardinal points. The rule is incomplete.

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**1.4-1.6.** *Dārśikī vedi.* The *dārśikī vedi* is in the shape of an isosceles trapezium having face 48 *aṅg.*, base 64 *aṅg.*, and altitude 96 *aṅg.* (*Fśl.* 3.6-3.7, *Aśl.* 4.5-4.6). The given verse is not very clear in describing the method of construction required for the purpose. Here the *prācī* is of 4 *aratnis* (96 *aṅg*) and the cord of 6 *aratnis* (144 *aṅg*) out of which a right triangle of sides 40, 96 and 104 is formed. By using this right triangle, the isosceles trapezium required for the *dārśikī vedi* is constructed. How the sides of the isosceles trapezium are cut off has been described by Baudhāyana and Āpastamba.

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**1.7-1.10.** *Gārhapatya, āhavanīya, dakṣiṇāgni and utkara.* Mānava describes *āhavanīya* as a square of one sq. *aratni*, *gārhapatya* and *dakṣiṇāgni* as circle and semi-circle of the same area. His incomplete method of circling a square appears to follow that of Baudhāyana, Āpastamba and Kātyāyana (*Bśl.* 2.9, *Aśl.* 3.2 and *Kśl.* 3.11). The method of finding the relative positions of these fires and of *dakṣiṇāgni* also differs from that given by other *śulbakāras*.

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**1.11-1.12.** *Construction of a square.* The rule gives a general method of construction of a square. If *a* be the original length of the cord, and *a*, the increased length of the cord, the *nirañchana* mark is given at a point dividing the total length 2*a* into two parts, 5/4 a and ¾ a. This satisfies the relation,

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$a² + (¾ a)² = (5/4 a)².$

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By using this relation which satisfies the condition of a right-angled triangle, the required circle is drawn. This method resembles that of Baudhāyana (*Bśl.* 1.4-1.5).

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UNITS OF CHARIOT, CONSTRUCTION OF *PAŚUBANDHA*, *PĀŚUKI*, *MĀRUTI*, *VARUṆA* AND *PAITṚKĪ VEDIS

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**2.1-2.3.** *Units, paśubandha vedi.* 1 īṣā = 188 aṅg., 1 akṣa = 104 aṅg., and 1 yuga = 86 *aṅg.* (vide *Bśl.* 1.3). The method of construction of *paśubandha* altar is not very clear.

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Fig. 1.

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**2.4.** *Pāśuki vedi.* The method is incomplete and may be reconstructed as follows. The *pāśuki vedi* is an isosceles trapezium having face 3 *aratnis*, base 4 *aratnis*, and altitude 6 *aratnis*. A cord AC (9 *aratnis* long) is used for its construction. Marks are given on it at *B*, *N*, *S*, *M* for obtaining *prācī*, *nirañchana*, *śroṇī* and *aṃsa* points, such that AB equals 6 *aratnis*, AN 6½ *aratnis*, BN, NS, SM each ½ aratni and CN 2½ *aratnis*. This satisfies the relation AB² + CN² = AN² or, AC² + CN² = AN², when the ends A and C are fixed on the east-west line, i.e. 6² + (2½)² = (6½)² holds. This is used for the construction of the isosceles trapezium DEFG (Fig. 1), the form of the *pāśuki vedi*.

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**2.5.** *Māruti and varuṇa vedi.* The *māruti vedi* is also an isosceles trapezium having face 3 *aratnis*, base 4 *aratnis* and altitude 6 *aratnis*. A cord of length 12 *aratnis* is taken and the *nirañchana* mark fixed at 7½ *aratnis* (½ + 2 + 2 + 1½ + 1½ = 7½) from one end; the remaining cord measures 4½ *aratnis* (½ + 1½ + 1½ + 1 = 4½). This satisfies the relation 6² + (4½)² = (7½)², which appears to have been used for the construction of the altar. A similar tedious technique is applied for the construction of the *varuṇa vedi*,a which is an isosceles trapezium having face 1½ *aratnis*, base 2 *aratnis*, and altitude 6 *aratnis*.

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**2.6-2.7.** *Paitṛkī vedi.* This appears to be a rhombus in shape, and pointing towards the cardinal directions. Van Gelder quoted Śivadāsa who prescribed a cord of 8 *aratnis* with marks at 4 *aratnis* and 5⅔ *aratnis*. This is obviously wrong, as these markings do not lead to the relationship for a right triangle. Simply a cord of 10 *aratnis* long is taken up and two ends are tied to the *prācī* points *E* and *W*, where EW equals 5⅓ *aratnis*. The cord is then stretched by the middle point on either side, fixing the points *G* and *H*. Hence EHWG is the required *paitṛkī vedi*.

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a Majumdsr (2).

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POSITIONS OF *PRĀGVAṂŚA*, *SADAS*, AND *HAVIRDHĀNA*, RELATIVE TO *MAHĀVEDI

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**3.1-3.4.** The distances for finding the positions of *prāgvaṃśa*, *sadas*, and *havirdhāna* relative to *mahāvedi* are given here. These values differ from those of Baudhāyana (*Bśl.* 4.1-4.11), although the dimensions of the *mahāvedi* remain the same in both the texts.

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**3.5.** The relation is : 3² + 1² = 10. This has been used by Kātyāyana in *Kśl.* 2.4.

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**3.6-3.9.** The *sada* is given as a rectangle, 27 × 9. Out of a rectangular area of breadth 10 *aṅgulas* in the eastern side of the *mahāvedi*, the rectangle of breadth 2½ *aṅg.* from east is for *śikhaṇḍini vedi* (*vide* *Bśl.* 4.12), and the next rectangle of 7½ *aṅg.* is known as *devyavedi*. The description of *kaukilī vedi* is not clear. According to Gelder, this represents an isosceles trapezium having *prācī* equal to 12 *prakramas*, base 10 *prakramas* and face 8 *prakramas*.

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