The Shulbasutras of Baudhayana, Apastamba, Katyayana and Manava: With Text, English Translation and Commentary c8.t20: Chapters 5 and 6: The Method of One Cord (Ekarajjuvidhi) and Two Cords (Dvirajjuvidhi) and Their Use in the Construction of Certain Altars

The Śulbasūtras of Baudhāyana, Āpastamba, Kātyāyana and Mānava: With Text, English Translation and Commentary c8.t20

Language: English

Chapters 5 and 6: The Method of One Cord (Ekarajjuvidhi) and Two Cords (Dvirajjuvidhi) and Their Use in the Construction of Certain Altars

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THE METHOD OF ONE CORD (*EKARAJJUVIDHI*) AND TWO CORDS (*DVIRAJJUVIDHI*) AND THEIR USE IN THE CONSTRUCTION OF CERTAIN ALTARS

**5.1-5.2.** *Construction of mahāvedi by one cord (ekarajjuvidhi)*. Āpastamba has described here a method for the construction of the *mahāvedi* (or *saumiki vedi*) with one cord (*ekarajjuvidhi*)a . It is an isosceles trapezium, having face 24 *prakramas*, base 30 *prakramas* and height 36 *prakramas*, .

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Let XS be the original cord, *a*, measuring 36 *prakramas*; and SY extra cord, a/2, measuring 18 *prakramas*.

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U and V are marks on the cord such that,

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VY = a/3 = 12 *prakramas*,

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YU = 5/12 a = 15 *prakramas*, and

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SU = a/2 - 5/12 a = a/12

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a Bag, 111—19.

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∴ XU = 13/12 a

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Now a² + (5/12 a)² = (13/12 a)² i.e. XY² + YU² = XU²

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Fig. 5. (a) cord, (b) right-angled triangle made by cord XY, (c) *mahāvedi*.

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The cord makes a right-angled triangle XYB. The right-angled triangle has been used for the construction of the isosceles trapezium ABCD where Δs XYU, *EFC*, *EBF*, *ELF* and *EKF* are equal and VY = AE = ED, and YU = BF = FC Here ABCD is the *mahāvedi* (Fig. 5(c)).

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**5.3-5.5.** *Method of two cords (dvirajjuvidhi)*. These rules deal with the construction of an isosceles trapezium with the help of two pieces of cords marked for the construction of right-angled triangles. Here three sets of two-cord relations are given :

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*First set* : From relation 3² + 4² = 5², the other relations obtained are:
(i) (3 + 3.3)² + (4 + 3.4)² = (5 + 3.5)² i.e. 12² + 16² = 20².
(ii) (3 + 4.3)² + (4 + 4.4)² = (5 + 4.5)² i.e. 15² + 20² = 25².

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*Second set* : The relations are :
(i) 5² + 12² = 13²
(ii) (5 + 2.5)² + (12 + 2.12)² = (13 + 2.13)² or 15² + 36² = 39².

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*Third set* : The relations used are :
(i) 8² + 15² = 17²
(ii) 12² + 35² = 37²

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**5.7** : *Area of mahāvedi*. The *mahāvedi* covers an area of 972 sq. *padas*. The rule gives hints how to calculate the area by proper transformation. The method is described here. The isosceles trapezium ABCD denoting *mahāvedi* has face, 24, base 30, and height 36 *padas*. For measuring the area, the portion *DCL* is cut off and placed in the northern side after inverting it and its new position becomes *AKB* (Fig. 6 ).

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Fig. 6

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Now the area *ABCD* = area *KBLD* = *EF* × *BL* = 36 × 27 = 972 square *padas*. [*BL* = ½ (*AD* + *BC*)]

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It is to be remembered in this connection that the area of the rectangle was already known to Āpastamba (*Āśl*. 3.6 and 3.7). The units of *mahāvedi* are given in *prakramas* and sometimes in *padas*. *padas* are sometimes calculated in terms of *prakramas* (vide *Bśl*. 4.12 - 4.14), though both of them are relative units (vide also *Āśl*.6.2).

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5.8-5.9.

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*Sautrāmanikī vedi*. The *sautrāmanikī vedi* is in the form of an isosceles trapezium having face 24/(√3) or 8√3, base 30/(√3) or 10√3, and altitude 36/(√3) or 12√3, and has an area 324 *sq. padas*. For detail, *vide Bśl*. 3.12.

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6.1.

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*Aśvamedha vedi*. Āpastamba has given hints to the construction of similar isosceles trapezium of area 1944 sq. *padas* for the *aśvamedha vedi*. Its area equals 36√2 × ½(24√2 + 30√2) or 1944 sq. *padas*. This is double of the size of the *mahāvedi*. Hence, for an isosceles trapezium of *n* times the size of the *mahāvedi*, *n* being an integer or a fraction, only the unit of measurement of the latter should be replaced by √n times the side of the *mahāvedi*.

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6.2.

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The unit of *prakrama* has been discussed (*Bśl*. 4.12-4.14) and *Āśl*. 5.7). Four officiating priests are usually required for the sacrifice, namely, *adhvaryu*, *hotṛ*, *brāhmaṇa* and the *āgnīdhriya*. The *adhvaryu*a should sweep the ground of altars three times and trace out the drawing with the help of the wooden sword.

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6.3-6.4.

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*Nirudapaśubandha vedi by one cord*. The construction of the *nirudapaśubandha vedi* is done according to *ekarajju* measure taught in *Āśl*. 5.1 and 5.2. Here the relation, $a² + (5/12a)² = (13/12a)², for a = 188,$ $i.e. (188)² + (78⅓)² = (203⅔)²$ has been used for the construction of isosceles trapezium, whose face in 86 *aṅg*., base 104 *aṅg*. and altitude 188 *aṅg*.

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The construction of another isosceles trapezium having face 3 *aratnis*, base 4 *aratnis*, and altitude 6 *aratnis*, has been obtained by the method of one-cord (*ekarajjuvidhi*). Here the following relation is used :

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a *Śrautakośa*, I, 213—14.

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$a² + (5/12 a)² = (13/12 a)², for a = 6,$ $or, 6² + (2½)² = (6½)²$

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**6.7-6.8.** *Paitṛki vedi and uttara vedi.* For the construction of the *paitṛki* and *uttara vedi*, the method of one-cord (*ekarajjuvidhi*) has been used. According to Āpastamba, the *paitṛki vedi* is a square of 5 *aratnis*, whereas the *uttara vedi* is a square of 10 *padas*. Hence the relation