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The Shulbasutras of Baudhayana, Apastamba, Katyayana and Manava: With Text, English Translation and Commentary c8.t5: Chapters 5, 6 and 7: Enlargement of Fire-Altar from 7½ Sq. Puruṣas to 101½ Sq. Puruṣas, Height of Altars and Their Peculiarities

The Śulbasūtras of Baudhāyana, Āpastamba, Kātyāyana and Mānava: With Text, English Translation and Commentary c8.t5

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English

Chapters 5, 6 and 7: Enlargement of Fire-Altar from 7½ Sq. Puruṣas to 101½ Sq. Puruṣas, Height of Altars and Their Peculiarities

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Enlarged Fire-altar and Units

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**5.1-5.6.** The *fire-altar* of the first construction has an area of 7½ sq. *puruṣas*. This has been discussed in subsequent chapters. The second construction has 8½sq. *puruṣas*, and according to Baudhāyana, it is increased by one square *puruṣa* successively from 7½ sq. *puruṣas* to 101½ sq. *puruṣas*. There is a mention of this type of increment upto 101½ sq. *puruṣas* in the *Śatapatha Brāhmaṇa*a. When the *fire-altar* is increased from the original 7½ sq. *puruṣas* by p square *puruṣas*, Baudhāyana says, the increment in each *puruṣa* equals the side of a square of 1 + 2p/15 sq. *puruṣas*. Hence the enlarged unit equals √(1 + 2p/15) and the area of the enlarged fire-altar 7½(1 + 2p/15) that is, 7½ + p. sq. *puruṣas* (*Bśl.* 5.6).

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a Datta (2), 149.

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With this length as unit of measure, the altar is constructed in the same manner as in original falcon-shaped altar. For obtaining the portion 2p/15, Baudhāyana perhaps divided the extra area into 15 equal parts by drawing 15 parallel lines or by drawing 3 parallel lines in one side and 5 parallel lines on the other after p square areas were turned into a square. Then two of these rectangular pieces were combined together with the help of *samāsavidi*.

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Both Āpastamba (*Āśl.* 8.6) and Kātyāyana (*Kśl.* 5.5) have given the same value of Baudhāyana, which equals the side of a square of 2/15(7½ + p) sq. *puruṣas* (where p = 1 sq. *puruṣa*). Kātyāyana has given the same value of enlarged sq. unit in two other forms as follows :

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(i) 1 + p(5/5.5 - ⅓·5/5.5) = 1 + 2p/15 (*Kśl.* 5.7);
and (ii) 1 + p(7/7.7 - 1 × (1 1/7)/120) = 1 + 2p/15 (*Kśl.* 5.7);

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where p = 1.

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This method of Kātyāyana is known as the method of increment by *puruṣa* (*puruṣābhyāsa*—*Kśl.* 5.4).

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Moreover, Āpastamba has devised a method of joining complete *puruṣas* in the body, wings and tail. If the increased area of p *puruṣas* be an exact mutiple or sub-multiple of the original *agni*, i.e. p = q × 7½, then the new unit comes out to be √(1 + q). This is known as increment on the whole posts (*sarvābhyāsa*, *Āśl.* 21.7).

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In the enlargement of the falcon-shaped fire-altar (first plan) let s be the enlarged unit in *puruṣa* and p the total increment in area. Then in the form of a quadratic equation this can be written as $body + 2 wings + tail = 7½ + p$ $or 4s² + 2 s(s + s/5) + s(s + s/10) = 7½ + p$

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a *saptavidhameva prathamaṃ vidadhītāthathaikottaramaikaśatavidhādekaśatavidhaṃ* (*Śat. Br*, X. 2.3. 17-18).

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$or 4s² + 12/5 s² + 11/10 s² = 7½ + p$ $or 15/2 s² = 7½ + p$ $or s² = 2/15 (7½ + p) = 1 + 2p/15$ $or s = √(1 + 2p/15)$ when p = 94, the maximum enlargement, $s = 13 8/15 = 14 approx.$

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The principle of enlargement of *agni* by increasing the length of the unit of measure was known in the time of the *Śatapatha Brāhmaṇa*.a It has described the construction of the *agni* of 101½ sq. *puruṣa*, where 14 or 14¾ times enlarged unit in *puruṣa* is taken up for its construction.b

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HEIGHT AND CHARACTERISTICS OF DIFFERENT LAYERS OF ALTAR AND BRICKS

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**5.7-7.3.** In the usual five-layer construction, the height of the altar is 32 *aṅg.*, height of each layer being 3⅖ *aṅg.* (*Bśl.* 5.7 and 7.2). According to Karavindasvāmī, it is 6 *aṅg.* Actually two layers are constructed. Construction of other layers is not required, since the 3rd and the 5th layer are the replica of the 1st and the 4th that of the 2nd. In the placement of bricks in different layers clefts between two layers are avoided. An altar of daily fire like the *gārhapatya* has 21 bricks in each layer, while a *kāmya* fire-altar is constructed with 1000 bricks, each layer containing 200 bricks. A fire-altar of 7½ sq. *puruṣas* is the normal construction. *Pañcacoḍā* and *nākasat* bricks have half the thickness of ordinary bricks ; hence one *pañcacoḍā* and one *nāksat* taken together are considered as one brick (*Bśl.* 5.17 and 7.2). They are usually on the fifth layer placed at the top. The spoiled or broken bricks or bricks of black colour are not used in the construction (*Bśl.* 7.1).

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Usually all the fire-altars from 1½ sq. *puruṣas* (one fold) to 101½ sq. *puruṣas* (101 fold) are not constructed. In some cases they are constructed without wings and tails.

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In the construction of fire-altars from 8½ to 101½ sq. *puruṣas*, how the units in sq. *puruṣas* are proportionately increased has been discussed. After all these are not normal constructions and are done in the construction for second and third time. In the second and third construction the fire-altar is constructed with more bricks and increased height. As regards its height and the number of bricks to be used in its construction, the *Taittirīya Saṃhitā* writes: “He should pile (the fire) of a thousand (bricks) when first piling (it); this world is commensurate with a thousand; verily he conquers this world. He should pile (it) of two thousands, when piling a second time, the atmosphere is commensurate with two thousands; verily he conquers the atmosphere. He should pile (it) of three thousands, verily he conquers yonder world. Knee-deep should he pile (it) when piling for the first time, verily with the *gāyatrī* he mounts this world; naval-deep should he pile it when piling for the second time, verily with the *triṣṭubh*, he mounts the atmosphere; neck-deep should he pile (it) when piling for the third time, verily with the *jagatī*, he mounts the yonder world''.a (Keith's translation):

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a *Śat. Br.* X.2.3. 11-14. Eggeling’s trn. is incorrect.

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b Datta (2), 159-60.

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The fire-altar (Chapter 6) has the characteristics of an animal. The head of the animal lies in the east and its back is compared with the east-west line. The east-west line is the central line of the altar and measurements are given with reference to this central east-west line. The bricks marked for south and north are placed accordingly on the southern and northern side of the altar. Diagrams of circle, a bull, a woman without ears, line, *darbha* grass are depicted on the bricks for their use for specific purposes. The bricks with different markings and symbols are also mentioned by Mānava in Chapter 7 of his *Mānava-śulbasūtra*.

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CONSTRUCTION OF GĀRHAPATYA FIRE-ALTAR (SQUARE).

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**7.4-7.7.** Baudhāyana says the *gārhapatya* fire-altar has the form of a square according to one tradition and of a circle according to another. The area is always one sq. *vyāyāma*. It is constructed with 21 bricks in each layer. For the placing of bricks in the square *gārhapatya*, the following two methods are adopted.

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*First Method.*

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One side of the square is divided into 3 parts by drawing 3 parallel lines and the other into 7 parts by drawing 7 parallel lines. These lines divide the square area into 21 rectangles and the bricks are accordingly made. This is for the first layer. For the second layer, the bricks are arranged differently in the square.

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*Second Method.*

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Three kinds of square bricks are made with 6th, 4th and 3rd part of a *vyāyāma*. The first layer is prepared with 9 bricks of the first kind and 12 bricks of the second, and the second layer with 6 bricks of the third kind and 16 bricks of the first.

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How the ancient altar builders determined the size of the bricks of different kinds and the number of bricks of each kind required for the construction of each layer is not known. The solution was obtained possibly on the following line.b

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*Empirical Method.*

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Suppose the sides of the three new types of bricks be *p*, *q*, and *r* th part of a *vyāyāma*, where *p*, *q*, *r* are rational integers. The minimum number of bricks must be three, since in each layer there are two types of square bricks and no two layers have identical arrangement of bricks.

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a *a sahasraṃ cinvīta prathamaṃ cinvānaḥ, sahasrasammito vā ayam loka imameva lokamabhi jayati, dvisahasraṃ cinvīta dvitīyaṃ cinvāno, dvisahasraṃ vā antarikṣamantarikṣamevābhi jayati; trisahasraṃ cinvīta trtīyaṃ cinvānastṛsahasro va asau loke'mumeva lokamabhi jayati | jānudaghnaṃ cinvīta prathamaṃ cinvāno... nābhidaghnaṃ cinvīta dvitīyaṃ... grīvādaghnaṃ cinvīta tṛtīyaṃ (Tait. S:V.* 6.8.2-3.

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b Datta (2), 180-183.

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Let the first layer have m number of bricks of the first kind and n bricks of the second kind, so that

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$. m + n = 21 / and m/(p²) + n/(q²) = 1 .. (1)$

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Similarly we can have another set of equations, if the second layer consists of s bricks of the third kind and t bricks of the first kind, as follows :

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$. s + t = 21 / s/(r²) + t/(p²) = 1 .. (2)$

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Baudhāyana's solutions are :
(i) x = 9, y = 12, p = 6, q = 4
(ii) x = 5, y = 16, p = 3, q = 6

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These two sets of values obviously satisfy equations (1) and (2) respectively.

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Solutions of equations (1) and (2) lead to the problem of solution of the simultaneous indeterminate equation of the type:

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$. x/(p²) + y/(q²) = 1 / x + y = 21 | ... (3)$

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Solving three, we write :

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$x = (p²(21 - q²))/(p² - q²), y = (q²(p² - 21))/(p² - q²)$

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The numbers x and y are considered positive by the *śulbakāras*. Hence if p > q, p² > 21 > q² since, s² > 21 > 4², p 5, q 4 if p < q, q 5, p 4

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Putting q = 1, 2, 3, 4, the values of p can be determined by trial, considering the value of x to be integral.

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Construction of Gārhapatya Fire-altar (Circular)

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**7.8.** For construction of a circular *gārhapatya* a square ABCD as large as possible is inscribed within a circle (Fig. 31) Āpastamba has directed that with half a *vyāyāma* as radius this circle is to be drawn; the end points of two diameters standing at right-angles to each other when joined give the largest possible square (*Āśl.* 7.5.).

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The square *ABCD* is then divided into 9 equal parts by 3 parallel lines from one side and 3 from the other side, and each segment is divided into 3 equal parts. Hence the whole circle is divided into 21 parts and the bricks are accordingly made for the first layer.

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Fig. 31

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Fig. 32

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In the second layer the square is so adjusted that the corners *A*, *B*, *C*, and *D* are placed in the middle points *E*, *F*, *G* and *H* of the segments.

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Baudhāyana and Āpastamba have considered the problem of quadrature of the circle whose area falls short of the circle.

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CONSTRUCTION OF DHIṢṆYA, ĀGNIDHRĪYA, MĀRJĀLĪYA AND CHANDA FIRE-ALTARS.

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**7.9-7.17.** The *dhiṣṇya* has one layer only. It may be a square or a circle. Generally it is covered with sand on which the fire is placed.

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The *āgnidhrīya* may also be a square or a circle. When it is in the form of a square, it is divided into nine small squares, and the central brick is replaced by a piece of stone. The same method is also given by Mānava (*Mśl.* 6.10). When it is circular, a round stone of 8 *aṅgulas* diameter is placed in the centre of the circle and the remaining area divided into eight equal parts (Fig. 32).

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The *mārjālīya* is divided into 3 equal parts by lines parallel to the north-south line. The eastern and western parts together are divided into 5 parts. No detail is available in the *Baudhāyana Śulba* as to how the eastern or western part is to be divided. The commentator Dvārakānātha gives the idea of dividing the eastern slab into two and western slab into three parts. Mānava has, however, suggested different constructions (*Mśl.* 6.11). The *mārjālīya*, when circular, is divided into six equal parts.

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The *chandaścit* is the fire-altar of *mantras*, not of bricks. The shape of the fire-altar is drawn on the ground and the whole ceremony performed. Instead of placing bricks, the sacrificer only touches the places on which the bricks are to be placed and mutters the same *mantras* at the appropriate time.

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