[Pramāṇa](https://indike.org/pramana/) › [Studies](https://indike.org/pramana/ins) › [Science, technology and education](https://indike.org/pramana/ins.science) › [The Shulbasutras of Baudhayana, Apastamba, Katyayana and Manava: With Text, English Translation and Commentary](https://indike.org/pramana/ins.science.sen-bag-1983) › c8.t4 # 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 ### [c8.t4.1](https://indike.org/pramana/ins.science.sen-bag-1983.c8.t4.1) AREAS OF PRĀGVAṀŚA, MAHĀVEDI, SADAS, ETC., THEIR RELATIVE DISTANCES; CONSTRUCTION OF EKĀDAŚĪ AND AŚVAMEDHA VEDI, AND THE VALUE OF π ### [c8.t4.2](https://indike.org/pramana/ins.science.sen-bag-1983.c8.t4.2) \*\*4.1-4.11.\*\* The areas of different chambers and \*vedis\* as given by Baudhāyana are tabulated below : ### [c8.t4.3](https://indike.org/pramana/ins.science.sen-bag-1983.c8.t4.3) | \*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. | ### [c8.t4.4](https://indike.org/pramana/ins.science.sen-bag-1983.c8.t4.4) 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.\* ### [c8.t4.5](https://indike.org/pramana/ins.science.sen-bag-1983.c8.t4.5) 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. ### [c8.t4.6](https://indike.org/pramana/ins.science.sen-bag-1983.c8.t4.6) 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). ### [c8.t4.7](https://indike.org/pramana/ins.science.sen-bag-1983.c8.t4.7) 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. ### [c8.t4.8](https://indike.org/pramana/ins.science.sen-bag-1983.c8.t4.8) Use of Ekādaśinī in the Construction of Mahāvedi and Aśvamedha Vedi ### [c8.t4.9](https://indike.org/pramana/ins.science.sen-bag-1983.c8.t4.9) \*\*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 $(10 × 104 + 11 × 15 + 4)/24 = 50 ang. 18 tilas (approx.).$ ### [c8.t4.10](https://indike.org/pramana/ins.science.sen-bag-1983.c8.t4.10) 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. ### [c8.t4.11](https://indike.org/pramana/ins.science.sen-bag-1983.c8.t4.11) Fig. 30. ### [c8.t4.12](https://indike.org/pramana/ins.science.sen-bag-1983.c8.t4.12) 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. ### [c8.t4.13](https://indike.org/pramana/ins.science.sen-bag-1983.c8.t4.13) 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.). ### [c8.t4.15](https://indike.org/pramana/ins.science.sen-bag-1983.c8.t4.15) \*\*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\*, ### [c8.t4.16](https://indike.org/pramana/ins.science.sen-bag-1983.c8.t4.16) (i) π = c/d = 3. ### [c8.t4.17](https://indike.org/pramana/ins.science.sen-bag-1983.c8.t4.17) 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 : ### [c8.t4.18](https://indike.org/pramana/ins.science.sen-bag-1983.c8.t4.18) (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. ### [c8.t4.19](https://indike.org/pramana/ins.science.sen-bag-1983.c8.t4.19) Dvārakānāthaa has, however, improved upon results (ii) and (iii) by introducing certain corrections as follows : ### [c8.t4.20](https://indike.org/pramana/ins.science.sen-bag-1983.c8.t4.20) 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. ## Cites (6) - [Baudhāyanaśulbasūtra](https://indike.org/pramana/in.vedanga.kalpa.sulba.baudhayana.7.9) — Baudhāyana, Sanskrit - [Āpastambaśulbasūtra](https://indike.org/pramana/in.vedanga.kalpa.sulba.apastamba.7.2) — Āpastamba, Sanskrit - [Kātyāyanaśulbasūtra](https://indike.org/pramana/in.vedanga.kalpa.sulba.katyayana.6.8) — Kātyāyana (Vājasaneyi sūtrakāra), Sanskrit - [Śatapathabrāhmaṇa](https://indike.org/pramana/in.veda.brahmana.satapatha.9.2.1.4) — Yājñavalkya & Śāṇḍilya (Vedic teacher), Sanskrit - [Kṛṣṇayajurvedasaṃhitā](https://indike.org/pramana/in.veda.samhita.krsnayajur*taittiriya) — Sanskrit - [Mānavaśulbasūtra](https://indike.org/pramana/in.vedanga.kalpa.sulba.manava) — Sanskrit --- Source: [https://indike.org/pramana/ins.science.sen-bag-1983.c8.t4](https://indike.org/pramana/ins.science.sen-bag-1983.c8.t4) · Pramāṇa