१८

अथापरः १

Now the other type (of dronacit in the form of a circle).

अथापरः १

पुरुषस्य षोडशीभिर्विंशशतं सारत्निप्रादेशः सप्तविधः सम्पद्यते २

120 (square) bricks, each \(\frac{1}{16}\) of a (square) puruṣa (ṣoḍaśī, side\(=\frac{1}{4}\) pu. or 30 aṅg.) give the area of the seven-fold (fire-altar of \(7\frac{1}{2}\) sq. purușa) with (two) aratnis and (one) prādeśa.

पुरुषस्य षोडशीभिर्विंशशतं सारत्निप्रादेशः सप्तविधः सम्पद्यते २

तासामेकामपोद्धृत्य शेषाः परिमण्डलं करोति ३

One of them is taken away, and (the area equivalent to) the remaining (119 bricks) is transformed into a circle.

तासामेकामपोद्धृत्य शेषाः परिमण्डलं करोति ३

तत्पूर्वेण रथचक्रचिता व्याख्यातम् ४

This (kind of transformation into circle) has been explained in the case of the fire-altar in the form of a chariot wheel of the first type.

तत्पूर्वेण रथचक्रचिता व्याख्यातम् ४

षोडशीं पुरस्ताद्विशय उपधाय तया सह मण्डलं परिलिखेत् ५

The ṣoḍaśi (brick which is taken away) is placed in the middle of the east side (of the square equivalent to the area of 119 bricks) and with it the circle (of the same area as that of the square) is to be drawn.

षोडशीं पुरस्ताद्विशय उपधाय तया सह मण्डलं परिलिखेत् ५

यदवस्तादपच्छिन्नं तत्पुरस्तादुपदध्यात् ६

The western part (of the ṣoḍaśī brick) cut off (by the circle) is placed on its eastern side.

THE CONSTRUCTION OF FIRE-ALTARS IN THE FORM OF A CIRCULAR TROUGH, AND OF SAMŪHYA AND PARICĀYYA FIRE-ALTARS

CIRCULAR TROUGH

18.1-18.6. The construction of the fire-altar. For purposes of measurements, the ṣoḍaśī square bricks of area \(\frac{1}{16}\) sq. pu. or 900 sq. aṅg. (that is, 30 × 30 sq. aṅg.) are employed. Clearly, 120 ṣoḍaśi bricks equal \(\frac{120}{16}\) or \(7\frac{1}{2}\) 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- saptaviṁśatyaṅgulasahita-śatatrayāṅgulapramāṇena samacaturaśram kṛtvā….|

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 ṣoḍāśi 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 sodasi brick IFGJ cut by the circle is transferred to the top of the remain- ing portion, as KEHL. Now, KIJL, whose area remains the same as that of a ṣoḍaśi brick, represents the handle of the circular trough (also called oṣṭha).

Fig.66. Construction of the circular droṇacit.
The value of OM, the radius of the circle can be calculated from the formula
OM = \(\frac{a}{2}(2 + \sqrt{2})\), where a = AB = 327 aṅg. 9 ti.

Putting \(\sqrt{2}\) = 1.414, OM works out to 186 aṅg. 7 tila.

Alternatively, if AO = r,

2r^2 = a^2 = 107100 sq. aṅg.

r = 231 aṅg. 13 ti.

MN = \(\frac{1}{3}\left(r - \frac{a}{2} \right) = \frac{1}{3}\left(231 aṅg. 13 ti — 163 aṅg. 21 ti\right)\)

\(\frac{1}{3} \left(67 aṅg. 26 ti.\right)22 aṅg. 20 ti. \)

OM= 163 aṅg. 21 ti. + 22 aṅg. 20 ti = 186 aṅg. 7 ti.

The commentator D. gives the value of OM as 184 aṅg. 22 ti and that of MN as 21 aṅg. 1 ti. (tatra mandalakarane dvavimŝatitilasahita-caturasitiśatāngulo viṣkambhār- dhaḥ tena mandalakarane visaya upahitaṣoḍaśimadhye tilasahitaikavimśatyangulapramāņe maṇḍalam pravartate| ) The values given by the commentator are obviously wrong.

यदवस्तादपच्छिन्नं तत्पुरस्तादुपदध्यात् ६

प्रधीनां सप्तधा विभागः ७

The (four) circular segments (obtained by drawing the maximum possible square within the above circle and after placing 6 bricks of side equal to 1/12 of the side of the inscribed square on the base of the segment, as in the case of the chariot wheel) are each divided into 7 parts.

प्रधीनां सप्तधा विभागः ७

प्रधिमध्यमाः प्रक्रमव्यासा भवन्ति ८

Bricks in the middle of the segments are each 1 prakrama (30 añgulas) wide.

प्रधिमध्यमाः प्रक्रमव्यासा भवन्ति ८

चतुरश्राणामर्ध्याभिः सङ्ख्यां पूरयेत् ९

The number (of 200 bricks) is to be completed by bricks half of the square bricks (made with 1/12 of the side of the inscribed square).

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

\(PQ^2 = 2r^2 = \frac{2πr^2}{π} = \frac{2 x 107100}{π}\)

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.

Square bricks are made with side equal to \(\frac{1}{12}\) PQ, that is, 21 aṅg. 26 ti (= \(\frac{1}{12}\) of 261 aṅg. 6 ti), which is also the value given by the commentator (……tasya karaṇyā dvādaśena ṣaḍviṁśati tilasahitenaikaviṁśatyaṅgulena 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 one-twelfth type diagonally cut.

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

च-तुरश्राणामर्ध्याभिः सङ्ख्यां पूरयेत् ९

अपरस्मिन्प्रस्तारे प्रधिमध्यमामोष्ठ उपधाय यदवस्तात्तद्द्वेधा विभजेत् १०

In the other layer, the brick in the middle of the (eastern) segment is placed in the lip (-shaped handle) and the space below it is divided into 2 equal parts.

अपरस्मिन्प्रस्तारे प्रधिमध्यमामोष्ठ उपधाय यदवस्तात्तद्द्वेधा विभजेत् १०

स एष नवकरणो द्रोणचि-त्परिमण्डलः ११

This is the fire-altar in the form of a circular trough involving (the use of) nine types (of bricks).

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.

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. \(B_{1}\), \(B_{2}\), \(B_{3}\), \(B_{4}\) and \(B_{5}\) types are used in each segment; \(B_{6}\) and \(B_{7}\) in the handle; \(B_{8}\) is used in the centre of the segment in one layer; and \(B_{9}s\) are used to complete the number 200. The following particulars of the bricks are based on the commentary (Fig. 68).

\(B_{1}\) — square brick, ABCD, 1⁄2 of the side of the inscribed square : side = 21 aṅg. 26 ti.

\(B_{2}\) — 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.

\(B_{3}\) — 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.

\(B_{4}\) — four-sided brick, HDJI, in the pradhi ; HD = \(20\frac{1}{2}\) aṅg. ; JD = 50 aṅg. + 10 ti ; JI = 31 aṅg. + 25 ti. ; IH (curved) = 51 aṅg. + 18 ti. ; śara = 1 aṅg. +27 ti.

\(B_{5}\) — four-sided brick, IJKL, in the centre of the pradhi, pradhimadhyamā ; 17 = 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.

\(B_{6}\) — 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.

\(B_{7}\) — 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.

\(B_{8}\) — rectangular brick, QRST, used in the empty space in the pradhi in the 2nd layer, after 1 \(B_{5}\) is pushed into the ostha, is clearly half of a sodasi brick : QR= ST 30 aṅg. ; RS = QT = 15 aṅg.

\(B_{8}\) — triangular brick, UVW, half of \(B_{1}\) diagonally cut; UV = VW 21 aṅg. + 26 ti; UW = (21 aṅg. + 26 ti) \(\sqrt{2}\) The commentator incorrectly gives the value of UV = VW as 30 aṅg. 15 ti.

स एष नवकरणो द्रोणचि-त्परिमण्डलः ११

समूह्य परिचाय्यौ पूर्वेण रथचक्रचिता व्याख्यातौ १२

The construction of the samuhya and the paricayya (fire-altars) is explained in the same way as that of the fire-altar in the form of the chariot wheel as already discussed.

समूह्य परिचाय्यौ पूर्वेण रथचक्रचिता व्याख्यातौ १२

समूह्यस्य दिक्षु चात्वालान् खानयित्वा तेभ्यः पुरीषं समूह्योपदध्यात् १३

Pits are dug out in the (four) cardinal directions of (the space to be occupied by) the samuhya, and the earth is collected from them and placed on the samuhya (instead of the bricks).

समूह्यस्य दिक्षु चात्वालान् खानयित्वा तेभ्यः पुरीषं समूह्योपदध्यात् १३

परिचाय्य इष्टकानां देशभेदः १४

The positioning of bricks in the paricāyya is different (from that of the chariot

परिचाय्य इष्टकानां देशभेदः १४

तं सर्वाभिः प्रदक्षिणं परिचिनुयात् १५

wheel); these are placed all around in (concentric) circles turning towards right.

THE SAMŪHYA AND THE PARICĀYYA FIRE-ALTAR.

18.12-18.15. These sutras 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.

The paricāyya is, however, covered with bricks in concentric circles. According to Dvārakānātha, 6 equally spaced concentric circles (agnikṣetre samāntarālāni ṣaṇmaṇḍalāni) are drawn. The central circle, the nabhi, 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 (pratimaṇḍalam karaṇabhedaḥ | evam 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.

तं सर्वाभिः प्रदक्षिणं परिचिनुयात् १५