The Shulbasutras of Baudhayana, Apastamba, Katyayana and Manava: With Text, English Translation and Commentary c8.t13: Chapter 16: The Construction of a Fire-Altar in the Form of a Chariot Wheel (Rathacakraciti)

The Śulbasūtras of Baudhāyana, Āpastamba, Kātyāyana and Mānava: With Text, English Translation and Commentary c8.t13

Language: English

Chapter 16: The Construction of a Fire-Altar in the Form of a Chariot Wheel (Rathacakraciti)

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**16.1-16.2.** Fire-altars in the form of a chariot-wheel are of two types : (a) a square piece with four circular segments attached one on each side so as to give the whole structure a circular shape, and (b) a circular wheel provided with spokes. Both types are used for sacrificial purposes and are described in this chapter.

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THE CHARIOT WHEEL WITH CIRCULAR SEGMENTS.

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**16.3.** *The construction and brick types.* The method of constructing a circle equivalent to that of a square area, — in this case 7½ sq. pu., has been given in rule 2.9 and has already been discussed. Fig. 59 represents the required circle of area 7½ sq. pu., within which is drawn the largest possible square ABCD. The space bounded by each side of the square and the arc of the circle is called *pradhi* (segment) ; there are four such segments.

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Fig. 59

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Let the side of the square AB or AD be a and the radius of the circle AO be r.

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$r = √((15×120×120)/(2 π)) ang.$ $= 185.45 ang. or 185 ang. 15 ti.$

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Again, a² = 2r² = (2×15×120×120)/(2 π) $or a = √((15×120×120)/(π))$ $= 262.27 or 262 ang. 9 ti.$

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In the above calculations, π has been taken to be 3.14. D. gives the value of r as 185 ang. 14 ti. (*madhye śaṅkuṁ nihatya pañcāśītiśatāṅgulena caturdaśatilayuktena parimaṇdala bhramayet*). His value of a is 262 ang. 7 ti (*tasya madhye viṣkambhārdhadvikaraṇyā (r √2) saptatilādhikayā dviṣaṣṭiśatadvayāṅgulayā samacaturaśraṁ kuryāt*). It appears that in giving these values Dvārakanātha used the more approximate correct value of π.

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The square bricks are then made with each side equal to the twelfth part of AD, which, according to D's value of the side of the inscribed square, is 21 ang. 29 ti (that is, 22 ang - 5 ti).

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**16.4-16.5.** *Placement of bricks.* 144 bricks of the type mentioned above can be placed within the square ABCD. In the segments, 6 such bricks are placed adjoining the middle side of the square and the remaining space is divided into 8 parts (Fig. 60 (a), (b)). Thus each segment contains 14 bricks and four segments account for 56 bricks, making the total number 200. Regarding the placement of bricks in each segment, D. explains as follows : *pradhimūlamadhye ṣaṭ caturaśrā upadhāya tasya pradheḥ śeṣamaṣṭadhā vibhajet | upahitānāṁ ṣaṇṇāṁ pārśvayordve dve | mukhe catasra iti |*

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D. also gives the measurements of these eight bricks as follows (only four will do) :
(1) The corner brick *abg* : ab = 33 ang. - 7 ti ; bg = 26 ang + 3 ti ; ag = 42 ang. - 3 ti ;
(2) The 4 sided brick *bchg* : bc = 33 ang - 7 ti ; ch = 42 ang + 11 ti ; hg = 36 ang + 26 ti ; bg = already given.
(3) The 4 sided brick *deih* : de = 33 ang. - 7 ti ; ei = 30 ang - 16 ti ; ih = 34 ang. ; hd = 20½ ang.
(4) The 4 sided brick *efji* : ef = 33 ang. - 7 ti ; fj = 32½ ang. ; ji = 33 ang. - 4 ti ; ei - already given.

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Fig. 60 (a) *Rathacakraciti* with 4 segments— 1st layer (after Bürk, 368).
(b) *Rathacakraciti* with 4 segments—2nd layer (after Bürk, 368).

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*ag*, *hg*, *ih* and *ji* are curved. D. neglected the curvature of *ag*, but considered the rest by giving the values of the *śara*, the distance between the centre of the arc and the respective chord in each case. For *hg*, *ih* and *ji*, these values are 31 *ti*; 26 *ti*., and 25 *ti*. respectively. Calculation of arcs and their *śaras* were obviously considered important mathematical exercises in medieval India when these commentaries were prepared.

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In the second layer the inscribed square is turned such that its corner lies in the centre of each segment of the first layer in order to avoid the overlapping of the edges of the bricks between layers.

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THE CHARIOT WHEEL WITH SPOKES.

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**16.6-16.11.** *The measurement and the construction of the wheel.* For purposes of measurement of a wheel of this type, square bricks each of area 1/30 sq. pu. are used. The total area of the fire-altar, 7½ sq. pu. will therefore involve (15 × 30)/2 or 225 bricks. The side of each brick a is given by :

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$a = √((120 × 120)/30) = √480 ang. = 21 ang. 31 tila.$

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This agrees with the value given by D. — . . . . . *labdhamekaviṁśatiraṅgulaya ekatriṁśattilāḥ | teneṣṭakāḥ samacaturaśrāḥ kārayet |*

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Fig. 61. Construction of the chariot wheel (*rathacakra*) fire-altar.

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Now, 225 is a squared number 15². A chariot wheel with spokes consists of the central circular part, the nave, the spokes and the outer circular rim, the felly. The spokes connecting the nave and the felly alternate with empty spaces. In the procedure suggested, an area equivalent to the empty spaces between spokes is first added and then removed. This area is here taken to be equivalent to that of 64 square bricks of the type described above, probably because 225 plus 64 make 289, a squared number 17². Hence the direction of arranging 289 bricks in the form of a square so that each side contains 17 bricks. This is done in two stages; at first 256 bricks are arranged in a square, each side containing 16 bricks, and then 33 bricks, — 17 plus 16, are placed along two adjoining sides. In this way the square *ABCD* is formed (Fig. 61).

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In the central region of the above square, another square *EFGH* of 16 bricks is formed. As D. explains, it is constructed out of the central square of 25 bricks by putting 4 poles, one each at the centre of the corner bricks, at *E*, *F*, *G* and *H*, and then joining them. The square *EFGH* forms the nave (*nābhi*).

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In the same manner, another square *IJKL* is formed of 144 bricks, centrally placed within *ABCD*. This is again done out of a square of 169 bricks, each side containing 13 bricks. A pole is placed at the centre of each of the four corner bricks, *I*, *J*, *K* and *L*, and these poles are joined. The area between the squares *IJKL* and *EFGH* is equivalent to the area of 128 (144-16) bricks, half of which, that is, 64, is used for making the spokes (*arā*) and the remaining half (64 bricks) for the empty spaces (*vedi*).

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The space between the squares *ABCD* and *IJKL*, equivalent to 145 (289-144) bricks constitutes the felly (*nemi*).

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Then three squares *ABCD*, *EFGH* and *IJKL* are turned into circles of radii *OM*, *OP* and *ON* respectively, according to the method given in rule 2.9. The space between the felly and the nave, that is between the circles of radii *ON* and *OP* is radially divided into 32 equal parts and half of them, in alternate order, is removed. In this way, an area equivalent to that of 64 bricks is rejected, and the remaining area of the wheel with the nave, spokes and the felly is exactly equal to that of 225 bricks or 7½ sq. *pu.*

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**16.12-16.14.** *Placement of bricks in the first layer.* The felly (*nemi*) is divided into 64 equal parts. We have seen that the annular space comprising the spokes and the empty spaces was divided into 32 equal parts by radial lines. These are projected into the felly, dividing it at first into 32 equal parts, and each part is then equally divided by radial lines confined between the outer and the inner circumference. Now a concentric circle passing through the middle of these two circumferences of the felly is drawn, dividing its space into 128 parts. Note that the area of each part lying outside the middle circumference is different from that lying within. Accordingly, two types of bricks are required to cover the felly, as D. points out—*bāhyāścatuḥṣaṣṭi | tāsāmekaṁ karaṇam | antataścatuḥṣaṣṭi | tāsāmekaṁ karaṇam |*

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Fig. 62. Arrangement of bricks in the 1st layer (after Thibaut).

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Each spoke is divided into 4 parts, thereby involving the use of 64 bricks of four different types. Finally the nave is divided into 8 equal parts to make the total number of bricks in the first layer 200. Thus seven different types of bricks are required to cover the first layer—*evaṁ sapta karaṇāni | eṣa prathamaḥ prastāraḥ | evaṁ śatadvayasampattiḥ |*—D.

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Fig. 63. Arrangement of bricks in the 2nd layer (after Thibaut).

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**16.15-16-20.** *Placement of bricks in the second layer.* The general principle of avoiding overlapping of cleavages of bricks in the two layers is to be kept in view. For this reason, a circle shorter by one-fourth (¾ OP) is drawn within the nave; its distance from the edge is therefore a quarter. In the felly, a concentric circle is drawn so that its distance from the inner edge is one-fourth the breadth of the felly (*nemimabhyantarato nemipramāṇacaturthabhāge'tīte parikṛṣet*—D.)

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The annular space of the felly between this new circle and the outermost one is divided into 64 equal parts and filled with bricks of one type. Each of the 16 spokes extending from the newly drawn inner circle in the nave upto the newly drawn inner circle of the felly is divided into 5 parts and 5 different types of bricks are used. The spokes will thus involve the use of 80 bricks. In between spokes, in the region of the felly, there are 16 spaces, each of which is divided into two equal parts, and 32 bricks of another type are used to fill them up (*arāṇāmantarāleṣu vedīpradeśasamīpe nemyāṁ dve dve iṣṭake / .... tāsāmekaṁ karaṇaṁ / tā dvātriṁśat*—D.). Likewise, there are 16 spaces in the nave between the spokes and outside the inner circle; 16 bricks of another size are used to cover them. The remaining central part of the nave is radially divided into 8 parts and 8 bricks of still another type are employed to cover them. In this way, the total number of 200 (64 + 80 + 32 + 16 + 8) bricks of 9 different types are used. With the 7 types used for the first layer, and 9 for the second 16 different types of bricks are used to construct the fire-altar in the shape of a chariot wheel with spokes.