The Shulbasutras of Baudhayana, Apastamba, Katyayana and Manava: With Text, English Translation and Commentary c8.t15: Chapter 18: The Construction of Fire-Altars in the Form of a Circular Trough, and of Samūhya and Paricāyya Fire-Altars

The Śulbasūtras of Baudhāyana, Āpastamba, Kātyāyana and Mānava: With Text, English Translation and Commentary c8.t15

Language: English

Chapter 18: The Construction of Fire-Altars in the Form of a Circular Trough, and of Samūhya and Paricāyya Fire-Altars

18.1-18.6. *The construction of the fire-altar.* For purposes of measurements, the *ṣoḍaśī* square bricks of area 1/16 sq. *pu.* or 900 sq. *aṅg.* (that is, 30 × 30 sq. *aṅg.*) are employed. Clearly, 120 *ṣoḍaśī* bricks equal 120/16 or 7½ sq. *pu.*, the area of the fire-altar to be constructed. This area equals 108000 sq. *aṅg.* After deducting the area of 1 *ṣoḍaśī* brick of 900 sq. *aṅg.*, the balance of 107, 100 sq. *aṅg.* is converted into a square of which the side obtained by taking the square root works out to 327 *aṅg.* 9 *tila*. Dvārakānātha gives this value as follows : *tasya mūlamānīya navatilasahita-saptaviṃśatyaṅgulasahita-śatatrayaṅgulapramāṇena samacaturaśraṃ kṛtvā ... |*

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The next step is to convert the above-mentioned square into a circle by the method given in *sūtra Bśl.* 2.9. and already applied in the case of the fire-altar in the form of the chariot wheel. Let ABCD be the square of side equal to 327 *aṅg.* 9 *tila*. (Fig. 66 ). EFGH is the *ṣoḍaśī* brick placed at the middle of the eastern side AD of the square. With O as centre and OM as radius a circle equal in area of the square ABCD is drawn, cutting EFGH by the curved line (*dhanurvakrā*) IJ. The part of the *ṣoḍaśī* brick *IFGJ* cut by the circle is transferred to the top of the remaining portion, as *KEHL*. Now, *KIJL*, whose area remains the same as that of a *ṣoḍaśī* brick, represents the handle of the circular trough (also called *oṣṭha*).

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Fig.66. Construction of the circular *droṇacit*.

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The value of *OM*, the radius of the circle can be calculated from the formula *OM* = a/8 (2 + √2), where a = AB = 327 aṅg. 9 ti. Putting √2 = 1.414, *OM* works out to 186 *aṅg.* 7 *tila*. Alternatively, if AO = r, 2r² = a² = 107100 sq. aṅg. r = 231 aṅg. 13 ti.

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$MN = ⅓ (r - a/2) = ⅓ (231 aṅg. 13 ti - 163 aṅg. 21 ti)$ $= ⅓ (67 aṅg. 26 ti.) = 22 aṅg. 20 ti.$ $OM = 163 aṅg. 21 ti. + 22 aṅg. 20 ti = 186 aṅg. 7 ti.$

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The commentator D. gives the value of *OM* as 184 *aṅg.* 22 *ti* and that of *MN* as 21 *aṅg.* 1 *ti*. (*tatra maṇḍalakaraṇe dvāviṃśatitilasahita-caturaśītiśatāṅgulo viṣkambhārdhaḥ | tena maṇḍalakaraṇe viṣaya upahitaṣoḍaśīmadhye tilasahitaikaviṃśatyaṅgulapramāṇe maṇḍalaṃ pravartate |*) The values given by the commentator are obviously wrong.

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18.7-18.9. *Arrangement of bricks in the first layer.* A square *PQRS* as large as possible is inscribed within the circle of the fire-altar as constructed above. The side *PQ* is given by

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$PQ² = 2r² = (2 π r²)/(π) = (2 × 107100)/(π)$ $or PQ = 261 aṅg. 6 ti (for π = 3.14)$ The commentator's value is 261 *aṅg.* 4 *ti*, which closely agrees with our value.

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Square bricks are made with side equal to 1/12 PQ, that is, 21 *aṅg.* 26 *ti* (=1/12 of 261 *aṅg.* 6 *ti*), which is also the value given by the commentator (*......tasya karaṇyā dvādaśena ṣaḍviṃśati tilasahitenaikaviṃśatyaṅgulena pramāṇeneṣṭakāḥ kuryāt*). The inscribed square PQRS is filled with 144 such bricks. 6 such bricks are placed in each of the four segments in the middle of, and touching, the square; 24 such bricks are placed in the segments. The remaining space in each segment is divided into 7 parts as shown in Fig. 67 (a), making their number 28. The brick in the centre of the segment in between the row of 6 bricks and the circumference of the circle is 30 *aṅg.* broad and is called *pradhi madhyamā*. With 2 bricks in the handle, as cut by the circle already described, the number of bricks in the first layer totals 198. The deficit is met by replacing 2 corner bricks within the square by 4 half bricks of the onetwelfth type diagonally cut.

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Fig. 67. Arrangement of brirks in the circular *droṇacit*,—(a) first layer; (b) second layer.

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**18.10-18.11.** *Arrangement of bricks in the second layer.* In the second layer, the inscribed square is turned such that the corners point towards intermediate directions Fig. 67(b). The square is filled with 144 one-twelfth bricks as before and 24 similar bricks are placed in the segments. The remaining space in each segment is also divided into 7 parts as in the first layer, making their number 28. To avoid overlapping of edges, the *pradhi madhyamā* brick is placed in the handle, and the space below is equally divided. The total number aggregates to 198. The deficit is met by replacing 2 one-twelfth bricks in the square by 4 half bricks.

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*Types of bricks.* The *sūtras* do not specifically describe the types of bricks. From the descriptions, it is, however, clear that nine types of bricks are used. B₁, B₂, B₃, B₄ and B₅ types are used in each segment; B₆ and B₇ in the handle; B₈ is used in the centre of the segment in one layer; and B₉s are used to complete the number 200. The following particulars of the bricks are based on the commentary (Fig. 68).

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B₁ — square brick, ABCD, 1/12 of the side of the inscribed square : side = 21 *aṅg.* 26 *ti.*

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B₂ — corner brick, EFG, in the *pradhi* : EF = 33 *aṅg.* — 12 *ti* ; EG = 26 *aṅg* ; FG (curved) = 41 *aṅg* + 25 *ti* ; *śara* = 1 *aṅg.* + 6 *ti.*

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Fig. 68. Different types of bricks used in the circular *droṇacit*.

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B₃ — four-sided brick GECH in the *pradhi* ; GE = 26 *aṅg.* ; EC = 33 *aṅg.* — 12 *ti* ; CH = 42 *aṅg.* + 8 *ti* ; HG (curved) = 36 *aṅg.* + 16 *ti* ; *śara* = 30 *ti.*

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B₄ — four-sided brick, HDJI, in the *pradhi* : HD = 20½ *aṅg.* ; JD = 50 *aṅg.* + 10 *ti* ; JI = 31 *aṅg.* + 25 *ti.* ; IH (curved) = 51 *aṅg.* + 18 *ti* ; *śara* = 1 *aṅg.* + 27 *ti.*

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B₅ — four-sided brick, IJKL, in the centre of the *pradhi*, *pradhimadhyamā* ; IJ = 31 *aṅg.* + 25 *ti* ; JK = 30 *aṅg.* (commentator wrongly puts it as 32 *aṅg.*) ; KL = 31 *aṅg.* 25 *ti* ; LI (curved) = not given ; *śara* = 42 *ti* = 1 *aṅg.*+8 *ti.*

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B₆ — four-sided brick, INML, used in the western part of the handle, here called *oṣṭha* ; IN = ML = 9 *aṅg.* — 1 *ti* ; MN = 30 *aṅg.* ; LI (curved) = not given ; *śara* = 42 *ti.*

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B₇ — four-sided brick, OMNP, used in the eastern part of the *oṣṭha* ; OM = PN = 21 *aṅg.* + 1 *ti* ; MN = 30 *aṅg.* ; OP (curved) = not given ; *śara* = 42 *ti.*

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B₈ — rectangular brick, QRST, used in the empty space in the *pradhi* in the 2nd layer, after 1 B₅ is pushed into the *oṣṭha*, is clearly half of a *ṣoḍaśi* brick : QR = ST = 30 *aṅg.* ; RS = QT = 15 *aṅg.*

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B₉ — triangular brick, UVW, half of B₁ diagonally cut ; UV = VW = 21 *aṅg.* + 26 *ti* ; UW = (21 *aṅg.* + 26 *ti*) √2. The commentator incorrectly gives the value of UV = VW as 30 *aṅg.* 15 *ti.*

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THE SAMŪHYA AND THE PARICĀYYA FIRE-ALTAR.

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**18.12-18.15.** These *sūtras* deal with *samūhya* and *paricāyya* fire-altars. These altars are circular in shape and constructed in the same manner as the chariot wheel without spokes. The *samūhya* is not covered with bricks, but by loose earth dug out from pits, as stated in the *sūtra*.

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The *paricāyya* is, however, covered with bricks in concentric circles. According to Dvārakānātha, 6 equally spaced concentric circles (*agnikṣetre samāntarālānī ṣaṇmaṇḍalāni*) are drawn. The central circle, the *nābhi*, is divided into 8 equal parts. Beginning from the *nābhi*, the second and the third annular spaces are each divided into 16 equal parts ; the fourth annular space is divided into 32 equal parts ; and the fifth and the sixth annular spaces are each divided into 64 equal parts. In this way, 200 bricks are completed for the first layer. There are six different types of bricks, for the dimensions of radial divisions differ from one annular ring to the other (*pratimaṇḍalaṃ karaṇabhedaḥ | evaṃ ṣaṭ karaṇāni*). In the other layer, concentric circles are drawn in the middle of each annular ring and the inner circle (of the first layer) is removed. Here also we have 6 concentric circles but of different radii. The division of the *nābhi* and the succeeding annular rings is the same as before; but the radial lines are to be drawn such that these do not coincide with those of the first layer. All these are done to avoid the overlapping of edges. Here also 6 different types of bricks are used.

Linked texts (1)

Āpastambaśulbasūtra — Āpastamba, Sanskrit