The Shulbasutras of Baudhayana, Apastamba, Katyayana and Manava: With Text, English Translation and Commentary c8.t28: Chapters 18, 19 and 20: Construction of a Fire-Altar in the Form of a Falcon with Curved Wings and Extended Tail—Second Type

The Śulbasūtras of Baudhāyana, Āpastamba, Kātyāyana and Mānava: With Text, English Translation and Commentary c8.t28

Language: English

Chapters 18, 19 and 20: Construction of a Fire-Altar in the Form of a Falcon with Curved Wings and Extended Tail—Second Type

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**18.1-18.2.** These *sūtras* are repetitions of 15.1 and 15.2.

**18.3.** *Areas of different parts of the falcon.* One-sixteenth or *ṣoḍaśī* square bricks are used here for purposes of measurements. The same method was used by Baudhāyana for measuring the circular *droṇacit* (*BŚl.* 18.2). As already explained, the area is 1/16 sq. *pu.* or 900 sq. *aṅg.*, that is, a square of side ¼ *pu.* or 30 *aṅg.* Clearly, 120 *ṣoḍaśī* bricks make 7½ sq. *pu.* On this basis, measurements of the different parts of the fire-altar are as follows :

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| Head | — | 3 | *ṣoḍaśī* | = 3/16 sq. *pu.* | or | 2,700 | sq. *aṅg.* |
| Body | — | 40 | ,, | = 5/2 ,, | or | 36,000 | ,, |
| Wings | — | 62 | ,, | = 31/8 ,, | or | 55,800 | ,, |
| Tail | — | 15 | ,, | = 15/16 ,, | or | 13,500 | ,, |
| | | 120 | ,, | 7½ ,, | | 108,000 | ,, |

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Note that the areas of the head and the body are the same as those of the first type of the falcon. Shapes of the different parts of the falcon are now described.

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**18.4.** *The body.* It agrees with the body of the first type of falcon not only in area but also in shape (vide 15.7, 16.1). It is only differently described. One starts with the same rectangle of breadth 1½ *pu.* or 180 *aṅg.* and length 2 *pu.* or 240 *aṅg.* The four corners are cut off, each by an isosceles right triangle of side 60 *aṅg.* (Fig. 15(b)). The area of each such triangle is 1800 sq. *aṅg.* equivalent of 2 *ṣoḍaśī* bricks, so that a total of 8 bricks are taken out from 48 bricks that the rectangle holds, leaving an area equivalent of 40 bricks. Regarding the cutting of the corners, Kapardi advises the construction of 4 squares of side ½ *pu.* and cutting them off diagonally—*yathā śroṇyaṁseṣu catvāri caturaśrāṇi ardhapuruṣapramāṇāni kṛtvā akṣṇayānyārdhāni nirasyet.*

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**18.5.** *The head.* This is done in the same way as the head in the first type (Fig. 15(d)). A square of side 60 *aṅg.* contains 4 *ṣoḍaśīs* of which 1 is removed by the cutting off of the two eastern corners as described in the rule.

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**18.6-18.8.** *The wings.* 1/16 *sq. pu.* equals a rectangular area, 120 *aṅg.* long by 7½ *aṅg.* broad. If this area is added to the rectangle, 240 *aṅg.* × 120 *aṅg.* along the common side 120 *aṅg.*, we get the rectangle *ABCD* so that AB = CD = 120 *aṅg.* and AD = BC = 247½ *aṅg.* (Fig. 19(a)). About the addition of 7½ *aṅg.* to the side 240 *aṅg.*, Kapardi says—*puruṣaṣoḍaśabhāgāścārdhonāṣṭāṅgulamātraṃ dakṣiṇe pakṣe cāyāmaḥ* /

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At the end, the rectangular strip *EDFC* of breadth 30 *aṅg.* is made and divided into 4 squares, 30 × 30 *aṅg.* each. Each square is diagonally cut and the outer half removed. *ABCD* whose area is 2 1/16 *sq. pu.* is equivalent of 33 *ṣoḍaśī* bricks. An area equivalent of 2 *ṣoḍaśīs* is discarded by diagonally cutting the 4 squares, leaving an area equal to that of 31 *ṣoḍaśī* bricks.

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In the middle of the rectangle less the portion where four squares were drawn up, the east-west line *GHI* is drawn perpendicular to the sides *BF*, *AE*. *BF* is 217½ *aṅg.* and *BI* 108¾ *aṅg.* The point *H* on the east-west line is obtained by stretching a cord or a rod of 1 *pu.* such that *BH* equals 1 *p.u.* (Sundararāja explains—*pakṣasya sārdhasaptadaśadviśatāṅgulasya madhye lekhāṃ kṛtvā pakṣāpyasyāparānte puruṣamātraṃ veṇuṃ niyamya tasyāṃ lekhāyāṃ nipātayet / sā yatra nipatati lekhāyāṃ tatra nitodam kuryāt / *)

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The point *G* is obtained by making *GH* equal to 1 *pu.* *GA*, *GE*, *HB* and *HF* are joined. Each of these sides equals 1 *pu.* Thus *ABHFEGA* together with the 4 diagonally intersected half squares EK₁L₁, L₁K₂L₂, L₂K₃L₃ and L₃CF at the south end represents the shape of the southern wing. The northern wing is obtained in the same manner.

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Fig. 19. Falcon-shaped fire-altar—(a) wing, (b) tail.

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**19.1.** *The tail* . As per measurements given, *ABCD* is the shape of the tail, where AD = 60 *aṅg.*, BC = 240 *aṅg.*, and AE (= DF = BE = CF) = 90 *aṅg.* The measurement of each of the two sides *AB* and *DC* is given in the text as *aṣṭādaśakaraṇī*. It means a side that produces a square equal to the area of 18 squares. Obviously, these 18 squares are 18 *ṣoḍaśīs*, that is, 18/16 *sq pu.* or 900 × 18 sq. *aṅg.* The required side is therefore ¼√18 pu. or 30√18 ang. The point is explained by Karavinda as follows: *aṣṭādaśa karotītyaṣṭādaśakaraṇī | ṣoḍaśīnāṃ prākṛtatvāttāsām aṣṭādaśānāṃ karaṇī | te khalu triṣoḍaśīpramāṇanavatyangulasamacaturaśrasyākṣṇayābhūte |* Thus, Karavinda puts it as the diagonal of a square of side 90 *aṅg.* (Fig. 19(b)), which again equals 30√18 ang. That the area of the tail is equivalent to that of 15 *ṣoḍaśīs* is clear from the figure.

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**19.2-19.8.** *Types of bricks.* 6 types of bricks have been used in covering the alternate layers of the fire-altar. These are—

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B₁ — the four-sided one-sixteenth (*ṣoḍaśī*) brick, of which ab = ⅛ pu. or 15 *aṅg.*, bc = ¼ pu. or 30 *aṅg.*, cd = ⅜ pu. or 45 *aṅg.* and da = ¼√2 pu. or 30√2 ang. The area is 30 × 15 + ½ × 30 × 30 = 900 sq. ang.

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B₂ — the half-brick (*ardheṣṭakā*), *e.g.*, a half *ṣoḍaśī*, diagonally cut; ab = bc = ¼ pu. or 30 *aṅg.*, ac = ¼√2 pu.

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B₃ — the quarter brick (*pādeṣṭakā*), *e.g.* ¼ *ṣoḍaśī*, diagonally cut; bc = ¼ pu.; ab = ac = ⅛√2 pu. or 15√2 ang.

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Fig. 20. Six types of bricks for the falcon-shaped fire-altar.

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B₄ — the brick, suitable for use in either wing (*pakṣeṣṭakā*), is a parallelogram of sides ¼ and 1/7 pu. The shape is so given that one diagonal *ac* is longer than the other *bd*, whose dimensions are given by Sundararāja as 40 *aṅg.* 12 *ti* and 27 *aṅg.* 20 *ti*. The purpose is to make it fit in the wing, so that the inclinations are similar to those of the wing at either side of the bending. Clearly *de* is 1/7 of 108¾ ang. or 15 *aṅg.* 18 3/14 ti (Sundararāja gives this value as 15 *aṅg.* 18 *ti*). The shapes of B₄, B₅ and B₆ are further explained in 19.8.

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B₅ — The brick suitable for use in the middle of the wing (*pakṣamadhyīyā*) at the bending. This is just 2 B₄s joined along the longer side.

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B₆ — The brick suitable for use at the end of the tail (*pakṣāgriyā*) broken in the form of four triangles. It consists of two parts, e.g. parallelogram *abcd* and the triangle *ecd*, and is a combination of B₄ and B₂ joined about the common side ¼ pu. The inclination is so adjusted that the parallelogram part fits in the parallelogram part and the triangular in the triangular part of the wing.

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**19.9-20.4.** *Placement of bricks in the first layer.* The placement of bricks is clearly explained in Fig. 21. The rules start with the placement of B₃ bricks, —4 at the tip of the head *ABC*, 5 west of the line *DE*, 11 east of the line *KF* joining the eastern points of juncture of the wings with the body, 11 west of *JG*, the western line of juncture, 5 each on the eastern and the western side of *IH*, the junction line between the tail and the body, and finally 15 at the end of the tail *LM*. Thus 56 B₃s are used (*tā evaitāḥ ṣaṭpañcāśatpādeṣṭakāḥ*—Kapardi).

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4 B₆s are placed at each end *OP* of the two wings, such that the triangular parts cover the triangular ends and the parallelogram parts part of the adjoining parallelogram of the wing. 4 B₆s are placed at either junction *FG*, *JK* of the wings with the body such that the triangular parts lie in the body. The total number of B₆s used is 16. North of *FG* and south of *JK* each, 4 B₁ bricks are placed in the body with their diagonally cut sides fitting exactly with the similar diagonal sides of the B₆s. The remaining space in either wing is covered by 4 B₅s at the bending *MN* and by 40 B₄s, —20 B₄s each on either side of the bending ; B₄s are turned eastwards. (*catvāriṃśatā catvāriṃśatā pakṣeṣṭakābhiḥ prāgāyatābhiḥ pakṣau pracchādayet*—Karavinda).

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The spaces of the fire-altar now left out are in the head between the rows of B₃ bricks, in the body between the B₃s at the eastern and western ends and in the middle enclosed on east and west sides by B₃s and on south and north sides by B₁s, and in the tail between B₅s at the juncture and the end. These spaces are to be

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Fig. 21. Arrangement of bricks in the first layer of the śyenacit (2nd. type) with curved wings and extended tail (north wing not shown) (after Būrk).

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covered by B₁s, such that at the inclined edges at the four corners of the body and the two sides of the tail, the diagonal sides (½√2 pu., *saviśeṣāḥ*, as Kapardi explains) face outwards; elsewhere 2 B₁s lie with their diagonals touching each other so as to form a rectangle 60 *aṅg.* × 30 *aṅg.*, as the geometry clearly indicates. The number of bricks and their types in the different parts of the fire-altar are given in Table 7.

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| Parts of the *citi* | Brick types | | | | | Total |
| | B₁ | B₃ | B₄ | B₅ | B₆ | |
| Head | 2 | 4 | | | | 6 |
| Body | 28 | 32 | | | | 60 |
| Wings, including part of body | | | 80 | 8 | 16 | 104 |
| Tail | 10 | 20 | | | | 30 |
| Total | 40 | 56 | 80 | 8 | 16 | 200 |

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**20.5-20.12.** *Placement of bricks in the second layer.* In the second layer, 2 B₁s are placed at the tip of the head with their diagonal sides facing outwards. West of them 2 B₁s are placed, partly covering the head and the body, but with their diagonal sides facing inwards. There are two ways of doing it, as shown in Fig. 22(a) and (b). The space between the two diagonals is filled by 2 B₂s on each side of the 2 B₁s at the juncture, 1 B₂ being placed with its diagonal side facing outwards. Thereafter, B₁ bricks are to be placed at the corners of the body. Būrk, in his Fig. 63a, placed 6 such bricks,—2 in the eastern and 4 in the western corners. All commentators have interpreted rule 20.7 as the placement of 4 B₁s, — one in each of the 4 corners (*śroṇyamseṣu catasraḥ ṣoḍaśyo vāhyaviśeṣāḥ pratyantaviśeṣāḥ*—Kapardi. *tāsām sandhiṣu.........ṣoḍaśyaḥ catasraḥ*—Karavinda.) The placing of 2 B₂s,—one on either side of the western corners, to maintain the symmetry with the eastern corners, has also been suggested. The rules then direct the placement of 4 B₂s at the end of each wing, 6 B₂s at both sides of the tail (3 on each) and B₄s in the wings turned towards north 20.8, 20.9). From the geometry of the wing, it is clear that the parallelogram spaces in either wing can be divided exactly into 56 (7 × 8) small parallelograms for containing 56 B₄s. Būrk's Fig. 63 shows 64 (8 × 8) which is incorrect. About 56 B₄s, Kapardi says : *ṣaṭpañcāśadviparyasya kāritāḥ*; about 7 rows south north, in the wings, Karavinda's statement is : *pakṣayorudicyaḥ pakṣeṣṭakāḥ sapta rītayaḥ* |

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a Bürk, 385

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Rule 20.10 directs that the remaining space in the body and the tail is to be covered by B₁s. Now, after covering the body with 4 B₁s and 4 B₂s and part of 2 B₁s at the juncture with the head (total area covered is equivalent of 7 *ṣoḍaśi* bricks), a space equivalent of 33 *ṣoḍaśi* bricks remains to be filled up; in the tail, the space yet to be covered is clearly that of 12 *ṣoḍaśis*. Hence, Karavinda's comment—*avaśiṣṭamātmani trayastriṃśat ṣoḍaśisthānaṃ pucchadvādaśa ṣoḍaśisthānam*. With 18 bricks placed in the head, body and tail, 120 in the two wings and 45 *ṣoḍaśi*-spaces, the total number comes to 183. Then the geometry of the remaining space in the body is such that only 30 B₁s can be accommodated and the remaining three *ṣoḍaśi* spaces (30 × 30 sq. *aṅg.*) can be covered by 4 B₂s and 4 B₃s (20.11). In that way, Kapardi arrived at 188 bricks, still short of 12 (*evaṃ dvādaśonaṃ śatadvayam /*) The arrangement of 188 bricks is shown in Fig. 22(a).

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The deficit of 12 bricks can be met by replacing 12 B₁s by B₂s and B₃s, as indicated in 20.11. The role of *aṇukās*, 30 × 30 sq. *aṅg.*, in parallelogram form is not properly understood, for the same objective can be achieved by half bricks (B₂).

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(a)

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(b)

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Fig. 22(a). Arrangement of bricks in the second layer, following Kapardi, and showing the deficit of 12 bricks. (North wing not shown).

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Fig. 22(b). Arrangement of bricks in the second layer, following Karavinda, showing the placement in the head, body and tail only.

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However, a complete scheme explaining how 200 bricks can be laid in the second layer has been given by Karavinda. There is no change in the arrangement of bricks in the two wings, totalling 120. Accordingly, the placement of bricks in the head, the body and the tail only is shown in Fig. 22(b). The head and its juncture with the body are covered, as in Fig. 22(a), by 4 B₁s and 2 B₂s; at the juncture the dispositions of 2 B₁s and 2 B₂s are reversed : *śirasi prāṅmukhe bāhyaviśeṣe dve ṣoḍaśyau upadadhyāt / tayoḥ paścāt prāṅmukhe bāhyaviśeṣe viśaye dve ardhe / tayoḥ*

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*paścāttadviśeṣaśliṣṭaviśeṣe dve ṣoḍaśyau viśaye ātmani* / The bricks at the *viśaya* are flanked on either side by 1 B₂ as before, forming a line *AB*.

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The body west of the line *AB* is divided into six rows in the east-west direction : *tataḥ paścāt ātmani prācyaṣṣaḍritayaḥ* / Beginning from the south, the first row contains 8 bricks,—1 B₁ at each of the eastern and the western corners and 6 B₂s in between. The second row has 2 B₁s at the western corner end, then 1 B₁, 6 B₂s, again 1 B₁ and 2 B₂s, of which the one at the eastern corner has already been mentioned. There are thus 12 bricks (*dvādaśeṣṭakā eṣā* /) In the third layer, starting from west of the juncture line *AB* upto the line of juncture *CD* in the tail, there are 8 B₁s, of which 4 are turned eastwards and 4 westwards : *tṛtīyasyāṃ rityāṃ śiro'pyayaṣoḍaśyāḥ paścādārabhya āpucchāpyayamāṣṭau ṣoḍaśya upadheyāḥ* / *tāsāṃ catasraḥ prācyaścatasraḥ praticyaḥ* / The arrangements in the 4th, 5th and 6th rows are the same as those in the 3rd, 2nd and 1st respectively.

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Table 8. *Bricks in different parts of the citi—second layer (after Karavinda).*

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| Parts of the *citi* | Brick type | | | Total |
| | B₁ | B₂ | B₄ | |
| Head, with part of juncture with body | 2 | 2 | | 4 |
| Body, with part of junctures with head and tail | 24 | 32 | | 56 |
| Wings | | 8 | 112 | 120 |
| Tail, with part of juncture with body | 10 | 10 | | 20 |
| Total | 36 | 52 | 112 | 200 |

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In the tail, 3 B₂s are placed on either side as before. In the second row, south-north, the remaining space is covered by 2 B₁s in the middle and 2 B₂s on either side, and in the last row at the end by 6 B₁s as in Fig. 22(a). The total number of bricks and their types, as per Karavinda's description, are given in Table 8.

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DRAWING OF EAST-WEST LINE, CONSTRUCTION OF SQUARES AND FIXING THE PLACES OF THE *ĀHAVANĪYA*, *GĀRHAPATYA*, *DAKṢIṆĀGNI* AND *UTKARA* ALTARS

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The *Kātyāyana-śulbasūtra*, in six chapters, is essentially a geometrical work containing the main principles of geometry and some problems involved in altar construction. Kātyāyana has made some reference to different *vedis* and *agnis* without any details of their construction with bricks and tried to explain geometrical results as such. In this chapter he has dealt with the method of drawing east-west and north-south lines, the construction of squares and the determination of the relative positions of *āhavanīya*, *gārhapatya*, *dakṣiṇāgni* and *utkara altars*.

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DRAWING OF EAST-WEST AND THE NORTH-SOUTH LINES

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**1.2.** Let *O* be the pole, and a circle *EPW* be drawn with a cord of length equal to *OP*.

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Fig. 1.

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Let *E* and *W* be the eastern and western shadow points on the equinoctial day of the pole fixed at *O* (Fig. 1). Aptea suggests that the actual east-west line was determined by the shadow of the pole on the equinox day and verified by the rising and setting points of the star Kṛttikā. Then *EW* is in east-west or *prācī* line. Two knots are given at the two ends of a cord which is double of the original cord and are fastened at the poles at *E* and *W*. The cord is then stretched towards north by its middle point and a mark *N* is given at it. This is the north point. Similarly south point S is obtained. Then NS, gives the north-south line. In the *Śulbasūtras* the east-west line has always been drawn first presumably because of the importance attached to this direction.

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a Apte, 1-16

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CONSTRUCTION OF A SQUARE OR A RECTANGLE

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**1.3** This gives a general method of construction of a square or a rectangle. In a given cord marks are given for *śroṇi*, *aṃsa* and *nirañchana* points. Two knots are fixed at the two ends of the cord; then fixing the two ends of the cord to the poles at the two ends of the east-west line, the cord is drawn by the *nirañchana* mark on either side of the line. By interchanging the knots at the two ends, the operation is repeated. Further details as to the length of the cord corresponding to a given distance between the two poles (the length of the altar) and where the *nirañchana* mark is to be given are discussed in the subsequent rules.

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**1.4-1.9.** Rules 1.4 and 1.5 give direction for determining the *nirañchana* points and are used for the construction of square and rectangles.

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*First cord.* Let AB, the given measure be a, BC, the added length a, and D, the *nirañchana* mark (Fig. 2(a)) so that $BD = BC/4 = a/4.$

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By definition, AD = the diagonal = 5a/4 $and DC = the breadth = 3a/4$

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Clearly, AD² - DC² = (5a/4)² - (3a/4)² = a²

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This is the expression for a right triangle ADC, of which AD is the diagonal (*akṣṇayā*), CD the breadth (*tiryaṅmānī*) and AC, the given measure for *prācī*.

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Fig. 2 (a) First cord. (b) Second cord.

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*Second cord.* Let AB, the given measure be a, BC, the added length, a/2, and D, be the *nirañchana* mark (Fig. 2(b)) so that $BD = ⅙ · a/2 = a/12, AD = 13a/12, and CD = 5a/12.$

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This satisfies the square relation: a² + (5a/12)² = (13/12a)² of the right triangle.

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The term *tiryaṅmāni* literally means 'transverse' or 'oblique' measure. In the *śulbasūtra*, however, it has been used to signify the 'breadth' or 'shorter side' of a rectangle. Mahīdhara says: *nirañchanenākarṣaṇe kṛte śroṇyaṃsa parichedikā yā rajjuḥ sā tiryaṅmāni.*

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After finding the perpendicular lines to the east-west line with the help of any of the above cords, poles are fixed upon the perpendicular lines at a distance equal to half the measure from the *prācī* to obtain the square. For rectangle, poles are fixed at a distance equal to half the value prescribed in the text.

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The *śakaṭamukha* means a figure resembling the fore-part of a cart and represents an isosceles triangle. It is also constructed out of square or rectangle (*Bśl.* 2.7-2.8). The *prāgvaṃśa* is a rectangle (*Bśl.* 4.1), the *śālā* a rectangle (Mahīdhara) and the *sada* also a rectangle (*Bśl.* 4.4). In the construction of these altars the north-south line is considered the reference line. For construction, decrease and increase of measures of altars, Kātyāyana followed the direction of older *śāstras*.

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RELATIVE POSITIONS OF THE *Gārhapatya*, *Āhavanīya*, *Dakṣiṇāgni* AND *Utkara

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Fig. 3.

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**1.9-1.10.** Let A and G be the positions of *āhavanīya* and *gārhapatya* fire-altars. As explained by Mahīdhara, the distance AG is to be reduced by one-third. With a cord equal to this reduced length, that is AB, a square EFGD is drawn in the eastern part (Fig. 3), that is, from point A westward,: *pūrvārdhe āhavanīyamadhyāt paścimabhāge samacaturasramuktavidhinā kāryam* (Mahīdhara). At the southern *śroṇī* point D of this square (*dakṣiṇaśroṇyam*) the fire is to be placed. To determine the place of *utkara*, a similar figure UJKL is drawn in the western part, that is, from the *gārhapatya* point G towards east: *paścimārdhe gārhapatyasya madhyāt pūrvabhāge*. Then U, the northern *aṃsa* point of the square is the *utkara*.

Linked texts (1)

Baudhāyanaśulbasūtra — Baudhāyana, Sanskrit