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The Shulbasutras of Baudhayana, Apastamba, Katyayana and Manava: With Text, English Translation and Commentary c8.t4: Chapter 4: Areas of Prāgvaṃśa, Mahāvedi, Sadas, Etc., Their Relative Distances; Construction of Ekādaśī and Aśvamedha Vedi, and the Value of π

The Śulbasūtras of Baudhāyana, Āpastamba, Kātyāyana and Mānava: With Text, English Translation and Commentary c8.t4

Language
English

Chapter 4: Areas of Prāgvaṃśa, Mahāvedi, Sadas, Etc., Their Relative Distances; Construction of Ekādaśī and Aśvamedha Vedi, and the Value of π

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AREAS OF PRĀGVAṀŚA, MAHĀVEDI, SADAS, ETC., THEIR RELATIVE DISTANCES; CONSTRUCTION OF EKĀDAŚĪ AND AŚVAMEDHA VEDI, AND THE VALUE OF π

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**4.1-4.11.** The areas of different chambers and *vedis* as given by Baudhāyana are tabulated below :

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| *Name of altar* | *Geometrical shape* | *Measurement* |
| *āgnīdhriya* | square | side = 5 aratnis. |
| *cātvāla* | square | side = 36 aṅgulas. |
| *dhiṣṇas* | circle | diameter = 2 prādeśas. |
| *havirdhāna* | square | side = 10 or 12 prakramas. |
| *mahāvedi* | isosceles trapezium | face = 24 padas, base = 30 padas, altitude = 36 padas,; the units may be also in *prakramas*. |
| *mārjālīya* | square | side = 5 aratnis. |
| *prāgvaṁśa* | rectangle | length = 16 prakramas, breadth = 12 prakramas; or length = 12 prakramas, breadth = 10 prakramas. |
| *sadas* | rectangle | length = 27 aratnis, breadth = 10 prakramas; or length = 18 aratnis, breadth = 10 prakramas. |

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The *uparavā* is a square of side 12 *aṅg.* or a circle of radius 6 *aṅg.* drawn within a square of side 24 *aṅg.* Two *uparavās* are generally placed together at a distance of 12 *aṅgulas.*

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The rites and ceremonies in connection with the construction of the above and various other altars are commonly found in the *Taittirīya Saṁhitā* and *Śatapatha Brāhmaṇa*, but any clear mention of their special magnitudes is very rare.

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Baudhāyana has made categorical mention of spatial magnitudes besides the methods of construction here and there (*BŚl.* 7.9, *ĀŚl.* 7.2.). Baudhāyana has discussed the construction of the *mahāvedi* in the next rule. Āpastamba has specially treated the dimension, area and construction of *mahāvedi* (or *saumikī vedi*) in chapter 5 of his *Āpastamba-śulbasūtra*. The *mahāvedi* has much older tradition and its method of construction is mentioned in the *Śatapatha Brāhmaṇa* (IX. 2.1.4).

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As regards relative distance of other *vedis* within the *mahāvedi* it is known from the *Baudhāyana śulba* that the *sadas* is at a distance of 1 *prakrama* (or *pada*) from the western side of the *mahāvedi*, and the *havirdhāna* at a distance of 4 *prakramas* (or *padas*) from it, and in the eastern side of the *havirdhāna*, 9 prakramas (or *padas*) still remain. Hence the relative distance is 1 + 10 (*sadas*) + 4 + 12 (*havirdhāna*) + 9 (rest) = 36. According to Mānava (*Mśl.* 3.1-3.3), this is 1 + 9 (*sadas*) + 4 + 12 + 10 = 36. In this connection it is important to note that the altitude of the *mahāvedi* is 36.

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Use of Ekādaśinī in the Construction of Mahāvedi and Aśvamedha Vedi

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**4.12-4.14.** In *ekādaśinī vedi* there are 11 posts arranged in a row, each placed at a distance of 1 *akṣa* from its immediate next. The diameter of each hole in which the pole is placed is a *pada*, and a space of 4 *aṅg.* is left on both sides of the two end-poles. Hence the space covered equals 10 *akṣa* + 11 *padas* + 8 *aṅg.* that is, 1213 *aṅg.* For the construction of the eastern part (i.e. face) of the *mahāvedi* with 11 poles, the length of 1213 *aṅg.* is divided by 24 to calculate the value of each *prakrama*. The distance between the two poles is known as *prakrama*. The length of *prakrama* varies for enlarged altars. For this reason the length of *prakrama* in *mahāvedi* is different from that of *aśvamedha vedi*. Since the face of the *mahāvedi* is 24 *prakramas*, according to Dvārakānātha, one *prakrama* equals $(10 × 104 + 11 × 15 + 4)/24 = 50 ang. 18 tilas (approx.).$

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According to Kātyāyana (*Kśl.* 6.8-6.13), it is the 24th part of 11 *paravān*, 10 *akṣa* that is, 48 *aṅg.* 28 tilas (Mahīdhara). Kātyāyana has not considered any space left beyond the two-poles. Mahīdhara has suggested 1 *paravān* to be 12 *aṅg.*; the *pada* according to Baudhāyana is 15 *aṅg.*, while it is 12 *aṅg.* in other texts.

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Fig. 30.

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The east-west line (*prācī*) of the *mahāvedi* is likewise constructed with 11 poles. Only a rectangular block of half a *pada* is cut off from the eastern side of the altar and the end-pole is placed, so that half of the end-pole is to be considered inside the altar and half outside. For construction of east-west line, the extra 8 *aṅg.* that is left beyond the end-poles discussed above is not required in this case.

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For the construction of the *aśvamedha vedi*, by the use of 21 poles, the value of each *prakrama* is taken to be 24th part of 20 *akṣa* + 21 *padas* + 8 *aṅg.* which is equal to (20 × 104 + 21 × 15 + 8)/24 = 100 ang 4 tilas (approx.).

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**4.15.** If d be the diameter of each hole (required in the construction of altar), and c its circumference, then according to this *sūtra*,

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(i) π = c/d = 3.

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This is undoubtedly a rough value known to the *śulbakāras*. In the problem of circling a square and quadrature of the circle, Baudhāyana has given three other values, which are a little better when compared with the correct value of π (= 3.14159...) as will be evident from the following :

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(ii) π = 4/(r²) = 3.0883......
where r = 1 + ⅓ (√2 - 1);
(iii) π = 4 ( 1 - ⅛ + 1/8.29 - 1/8.29.6 + 1/8.29.6.8 ) = 3.0885 ;
and (iv) π = 4 ( 1 - 2/15 )² = 3.004.

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Dvārakānāthaa has, however, improved upon results (ii) and (iii) by introducing certain corrections as follows :

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π = 4/([ 1 + ⅓ (√2 - 1) ]²) × ( 118/117 )² = 3.141109......
and π = 4 ( 1 - ⅛ + 1/8.29 - 1/8.29.6 + 1/8.29.6.8. )²
× ( 1 + ½ · 3/133 )²
= 3.157991.

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