The Mānava-śulbasūtra, in 16 chapters, is a mere compilation, and its value as a technical text appears to be of dubious nature. In most places it is corrupt, and the same topics are discussed in different places. The materials on gārhapatya, for instance are dispersed in chapters 1, 6, 9, 13, on caturaśraśyenacit in chapters 4, 5, 6 and 13, on dhisnyas in chapters 6, 9 and 13 etc. It is full of paraphernelia about worship hardly of any consequence to altar construction. The arrangement is unsystematic. The text therefore bears no comparison with the three other sulba texts already commented upon. CHAPTER 1 DETERMINATION OF EAST-WEST LINE, CONSTRUCTION OF DĀRŠIKĪ VEDI, SIZE AND RELATIVE PLACES OF GĀRHAPATYA, ĀHAVANÎ YA DAKSIŅĀGNI, UTKARA AND GENERAL RULE FOR DRAWING 1.2-1.3. A SQUARE. East-west line. Sūtras 1.2 and 1.3 direct the fixation of the east-west line of the altar according to cardinal points. The rule is incomplete. 1.4-1.6. Dārśiki vedi. The dārŝiki vedi is in the shape of an isosceles trapezium having face 48 añg., base 64 arg., and altitude 96 ang. (Fśl. 3.6-3.7, Aśl. 4.5-4.6). The given verse is not very clear in describing the method of construction required for the purpose. Here the prăci is of 4 aratnis (96 ang) and the cord of 6 aratnis (144 ang) out of which a right triangle of sides 40, 96 and 104 is formed. By using this right triangle, the isosceles trapezium required for the dārsiki vedi is constructed. How the sides of the isosceles trapezium are cut off has been described by Baudhāyana and Āpastamba. 1.7-1.10. Gārhapatya, āhavaniya, dakṣiṇāgni and utkara. Mānava describes āhavaniya as a square of one sq. aratni, garhapatya and dakṣiņāgni as circle and semi-circle of the same area. His incomplete method of circling a square appears to follow that of Baudhāyana, Āpastamba and Katyāyana (Bśl. 2.9, A§l. 3.2 and Kśl. 3.11). The method of finding the relative positions of these fires and of dakṣiṇāgni also differs from that given by other sulbakāras. 1.11-1.12. Construction of a square. The rule gives a general method of construction of a square. If a be the original length of the cord, and a, the increased length of the cord, the nirañchana mark is given at a point dividing the total length 2a into two parts, 5 4 3 a and a. This satisfies the relation, 3 4 2 2 a2 + ( a ) = ( • ) 2 a COMMENTARY 273 By using this relation which satisfies the condition of a right-angled triangle, the required circle is drawn. This method resembles that of Baudhāyana (Bśl. 1.4- 1.5). CHAPTER 2 UNITS OF CHARIOT, CONSTRUCTION OF PAŚUBANDHA, PĀŠUKI, MĀRUTI, VARUŅA AND PAITṚKĪ VEDIS 86 2.1-2.3. Units, paśubandha vedi. 1 iṣā — 188 añg., 1 akṣa = 104 añg., and 1 yuga ang. (vide Bśl. 1.3). The method of construction of pasubandha altar is not very clear. 2.4. Pāśuki vedi. The method is incomplete and may be reconstructed as follows. The pāśuki vedi is an isosceles trape- zium having face 3 aratnis, base 4 aratnis, and altitude 6 aratnis. A cord AC (9 aratnis long) is used for its construction. Marks are given on it at B, N, S, M for obtaining prācī, nirañchana, śroņi and amsa points, such that AB equals 6 aratnis, AN 61 aratnis, BN, NS, SM eacharatni and CN 21 aratnis. This satisfies the relation AB2 + CN2 AN2 or, AC2 + CN2 AN2, when the ends A and C are fixed on the east- west line, i.e. 62 + (21)2 (61)2 holds. This is used for the construction of the isosceles trapezium DEFG (Fig. 1),the form of the pasuki vedi.

DAG FN Fig. 1. 2.5. Māruti and varuņa vedi. The māruti vedi is also an isosceles trapezium having face 3 aratnis, base 4 aratnis and altitude 6 aratnis. A cord of length 12 aratnis is taken and the nirañchana mark fixed at 71⁄2 aratnis (§ + 2 + 2 + 1} + 1} = 7}) from one end; the remaining cord measures 41 aratnis (} + 1} + 1} + 1 = 41). This satisfies the relation 62 + (41)2 (71)2, which appears to have been used for the construction of the altar. A similar tedious technique is applied for the construction of the varuna vedi,a which is an isosceles trapezium having face 1 aratnis, base 2 aratnis, and altitude 6 aratnis.

2.6-2.7. Paitṛki vedi. This appears to be a rhombus in shape, and pointing towards the cardinal directions. Van Gelder quoted Śivadāsa who prescribed a cord of 8 aratnis with marks at 4 aratnis and 53 aratnis. This is obviously wrong, as these mark- ings do not lead to the relationship for a right triangle. Simply a cord of 10 aratnis long is taken up and two ends are tied to the prācī points E and W, where EW equals 51⁄2 aratnis. The cord is then stretched by the middle point on either side, fixing the points G and H. Hence EHWG is the required paitṛki vedi. a Majumdsr (2). 274 MĀNAVA-ŚULBASŪTRA CHAPTER 3 POSITIONS OF PRĀGVAMŚA, SADAS, AND HAVIRDHĀNĀ, RELATIVE TO MAHĀVEDI 3.1-3.4. The distances for finding the positions of prāgvaņśa, sadas, and havirdhāna relative to mahāvedi are given here. These values differ from those of Baudhayana (Bśl. 4.1-4.11), although the dimensions of the mahavedi remain the same in both the texts. 3.5. The relation is: 32+ 12 10. This has been used by Kātyāyana in Kśl. 2.4. 3.6-3.9. The sada is given as a rectangle, 27 x 9. Out of a rectangular area of breadth 10 angulas in the eastern side of the mahāvedi, the rectangle of breadth 21 ang, from east is for sikhaṇḍini vedi (vide Bśl. 4.12), and the next rectangle of 71⁄2 ang. is known as devyavedi. The description of kaukili vedi is not clear. According to Gelder, this represents an isosceles trapezium having prăci equal to 12 prakramas, base10 prakramas and face 8 prakramas. CHAPTERS 4 AND 5 UNITS OF MEASURES AND WEIGHTS, BRICKS 4.1-4.6. The six rules provide a table of units of measure. 4.7-4.8. Sizes of bricks and different layers are generally stated. Chapter 5 describes a method of measuring areas in a square syena (caturasra- fyenacit). Two bamboo rods are taken, one measuring 120 ang. (one puruşa) in length, the other 144 ang. In the second bamboo rod marks are given at a distance of 120 ang., 132 añg. and 144 ang. from one end. Two middle marks are given in these two rods at a distance of 60 ang. from the same ends. Then a pañcāngi cord is formed in the following way (vide Mśl 13.15). A cord AB of length 2 purușas (240 añg) is taken and three marks are given, one at the middle of the cord C and one each at the middle of the two halves, i.e. at D and E. (Fig. 2). The cord is fixed by two poles at its eastern end A and the western end B; poles are likewise fixed at C, D, and E. The two bamboo rods are then stretched towards south from D and E respectively so as to meet at F, 120 ang from the end of each. The first bamboo rod is held over CF so as to obtain G at 120 ang from C. Now the second bamboo rod is stretched from pole A towards south and the first rod from G towards east so as to meet at H, 120 ang from the end of each rod. H is the south-eastern corner of the COMMENTARY 275 ātmā. Likewise, the north-eastern corner I and the two western corners J and K of the square body are fixed. The area of the body is thus 2402 sq. ang or 4 sq. pu. H A C M Q K Fig. 2. Construction of a caturaŝraśyenacit. To construct the northern wing poles are fixed at L and M, the middle points of the two halves of the southern side HK. By stretching the two bamboo rods from L and M towards south the point N where they meet at a distance of 120 ang from the end is determined. The second bamboo rod is now held over GN and the point O 144 ang from G is obtained. Then by stretching the second rod from L and the first rod from 0 the point P is fixed and likewise the point Q, so that the rectangle LPQM constitutes the southern wing. Likewise the northern wing is formed. The area of each wing measures 120 × 144 sq. ang or 13 sq. pu and that of two wings 2 sq. pu. To obtain the tail, one has to proceed in the same manner as for the wing; but the mark with the second bamboo rod stretched from B westwards should be given at a distance of 132 ang. In other words, a rectangle 120 × 132 sq. ang or 116 sq. pu is to be formed. For the head a square 60 × 60 sq. ang (according to Śivadāsa, a rectangle of 60 × 75 sq. ang) is to be formed at the middle of the eastern side. In this construction, the body, the two wings and the tail (4 + 23 + 17%) account for 7 sq. pu, with the head needing an additional area and thus deviating from the traditional area of a fire-altar of this type. We have seen that Baudhāyana did not provide his square syenacit with a head and strictly restricted himself to the area of 7 sq. pu. 276 MĀNAVA-ŚULBASŪTRA CHAPTER 6 GĀRHAPATYA, ĀGNIDHRĪYA, BRĀHMAṆĀCCHAMŚA, MĀRJĀLĪ YA AND CATURAŚRAŚYÈNACIT 6.1-6.9. Gārhapatyaciti. Six kinds of bricks are used for the construction of gārhapatya fire-altar. Their measurements are: 12 ang × 24 ang.; 24 ang × 24 añg.; 12 ang. × 12 ang.; 12 ang × 6 ang.; 12 ang. × 18 ang.; and 30 ang. × 30 ang. The height of these bricks is 6 ang.; the heights of nākasad and pañcacodă are half of these, but whether the heights of ṛtavyā and vaiśvadevi bricks are also half is not clear. Månava gives no idea as to how to arrange the bricks in the first and second layers (compare with MŚl. 13.6-13.13). The sūtra 6.7 is so vague and inadequate that no attempt has been made to reconstruct the arrangement of bricks. 6.10. Āgnidhriya, hotriya, brāhmaṇācchaṛśa and mārjālīya. A similar description of āgnidhriya square of side 36 ang. divided into 9 equal parts with a stone being placed at the centre is met with in the Baudhayana-sulba. The descriptions of hotriya, mārjaliya and brāhmaṇācchamsa are different in different places (vide Mśl. 13.23-13.29). 6.11-6.15. Caturafrasyenacit. The placement of bricks in two layers of caturaśraśyenacit is hinted at. Rectangular (18 ang×12 ang) and square bricks (12 ang × 12 ang., also 30 × 30 sq. ang) are used for this purpose. There is no mention of the total number of bricks required for each layer. What can be ascertained from the rules is that the first layer contains 98 adhyardhā (18 × 12) bricks (40 in the eastern and western sides of the ātmā, 48 in the eastern and western sides of the two wings and 10 in the head) and the second layer 72 adhyardhās (40 in the southern and northern sides of the ātmā, 22 on either side of the tail and 10 in the head). The remaining space is to be filled by square bricks. Van Gelder suggested 80 square bricks of size. 30 × 30 sq ang. and 128 square bricks of size 12 × 12 sq. ang. making the total for the first layer 306.a For the second layer the total number of bricks was likewise shown to be 269. Several other alternatives are possible, but that would be a futile exercise. The sūtra 6.14 lays down how to perform worships of three and six days. CHAPTERS 7 AND 8 These two chapters describe the construction of suparnaciti, not found in earlier sulba literature. In this structure various bricks such as viśvajyoti, ṛtavyā, svayamātṛ, apasyā, prāṇabhṛt, vaiśvadevi, vāyavyā, chanda, virāja, vikarṇī have been used. The description is mostly of a general nature. a Van Gelder, 294 I COMMENTARY 277 CHAPTER 9 AREAS OF GĀRHAPATYA, DHIṢNYAS AND PLACING OF BRICKS IN DIFFERENT YAJÑAS The garhapatya is a square citi of 9216 sq. angulas. The square gārhapatya has side 96 ang. There are eight dhiṣṇyas, namely, āgnidhriya, māṛjāliya, and six others within the sadas, viz. hotṛi, maitrāvaruņa (or prasāstri), brāhmaṇācchamsin, potri, nestr and acchāvaka. Each dhiṣṇa has an area of 1296 sq. añgulas; a square dhiṣṇa is of side 36 ang. The fire-altar is 111600 sq. angulas in area. Now, 111600 sq. ang. 78 sq. pu. An area of 1 sq. pu. for the head is added to original 71⁄2 sq. pu. agni. Here the break-up is as follows (for measures vide Mśl. 11.2-11.8). Atman 400 sq. padas -20 padas x 20 padas

2 pu. × 2 pu. 4 sq. pu. Each wing

120 sq. padas = 10 padas x 12 padas 1 pu. × pu. Tail 12 10 = 110 sq. padas 10 padas x 11 padas 6 sq. pu. 5 × 11 11 1 pu. × pu. 10 10 sq. pu. Head 25 sq. padas 5 padas × 5 padas 1 1 1 2 pu. x pu. www.*** 2 4 sq. pu. 11 3 Total area 4 + 2. + + 7

5 10 4 sq. pu. 4 The placement of bricks has been described in a general way. CHAPTER 10 THE SULBAVID, ŚAŃKU, ROPE, MEASUREMENT OF VOLUME, PROPERTIES OF RIGHT-ANGLED TRIANGLE The qualifications of fulbabid and the nature of saňku and rope are described. The sulbabid is one who is versed in geometry (measurement of areas), calculations, and altar construction and who takes up as a profession the measurement of areas. 278 MĀNAVA-SULBASŪTRA The ground for the construction of altars should be plane, the sanku or poles must be straight and the cord smooth. For the volume measure (Msl. 10.9) length, breadth and height are multiplied. In a right-angled triangle, a2+ b2=c2 where a == length, b breadth and c = hypotenuse (Mśl. 10.10). CHAPTER 11 UNITS OF MEASUREMENT, CIRCLING A SQUARE, AREAS OF PLANE FIGURES, VALUE OF T, QUADRATURE OF THE CIRCLE, USE OF PANCANGI CORD, MEASURES FOR DIAGONAL OF A RECTANGLE 11.1-11.8. Units of measure. The units of human measure become short or long depending on the stature of the sacrificer. However, the table runs as follows : 1 yava=6 mastard seeds; 1 mastard seed=6 cords of hair; 1 aratni=2 prādeŝas; 1 prakrama=1 aratni or 2 prādeśas (in citi measure); 1 purușa=120 angulas=5 aratnis or 10 padas; 1 yuga==86 angulas; 1 akṣa=104 añgulas; the ratha measures are according to the prescription of the text. I 11.9-11.10. Circling a square. The method of circling a square described by Mānava in Mśl. 1.8a is repeated. Here the word, viskambha meaning ‘diagonal’ has been used; it should be viskambhardha meaning ‘radius’. This rule has been explained. by Baudhayana (Bśl. 2.9).

2a2; 11.11-11.12. Areas of figures. If d be diagonal of a quare of side a, then d2 that is, the square on the diagonal produces twice the area of the original square; similarly D2 2d2 4a2, where D is the diagonal of the square drawn on the diagonal of the original square of side a; and so on. The area of a rectangle with breadth 2 pu. and length 8 pu is 16 sq. pu. (Mśl. 11.18). 11.13. Value of π. Value of π. If c be the circumference, d the diameter of a circle, d C = 5 + 3d = 3·2 d C or 3.2. d Baudhayana has also given a similar approximate value of # as 3 (Bl. 4.15). 11.14-11.16. Quadrature of the circle. Possibly these are not problems of quadrature of the circle. Ordinary squares are drawn without any mathematical significance. COMMENTARY 279 11.17. Properties of right triangle. The relations 32+42-52, and (3n)2 + (4n) 2 (5n)2, where n is any quantity, hold good for any right-angled triangle. 11.19-11.28. Measurement of śroņi and amsa points by pañcāñgi cord. How a cord with five marks (pañcāñgi) is used to measure the western (śroņi) and eastern corners (amsa) of citis other than kanka and alaja has been explained in a general way. CHAPTER 12 DIAGONAL OF A RIGHT TRIANGLE This chapter deals with the method of calculating the diagonal of a right triangle when its other sides are known. Sometimes out of three sides any two are known, when the third can be calculated. The length is known as measure or pramāṇa or pārśvamāni, breadth veseșa or tiryaṁmāni, and diagonal akṣṇayā. a a 5 (i) If length

a, breadth a, 2 12 12 13 diagonal produces a square equal to a 12 ) 2 a a a 13 (ii) If length 2 و diagonal + a, 24 24 5 2 then breadth produces a square equal to a 24 By applying this, two fold producer (√/2a), three fold producer (√3a), twenty- one fold producer (√21 a) used for the aśvamedha vedi, and 101 fold producer are obtained. This also justifies that 12 + (V10)? = 11, 2 CHAPTER 13 CONSTRUCTION OF SAUMIKĪ VEDI, GÄRHAPATYA (BOTH SQUARE AND CIRCULAR), CATURAŚRAŚYENA OF ANOTHER TYPE, ÄGNIDH- RĪYA, HOTRÏYA, BRĀHMANĀCCHAMŚA AND MĀRJĀLĪYA 13.1-13.5 Saumiki vedi. Here the construction of sautrāmani, saumiki and paśubandha fire-altars has been hinted at. The saumiki fire-altar is in the form of an isosceles 280 MĀNAVA-SULBASŪTRA trapezium having face 8 √3, base 12 √3, and altitude 12 √3. This is meaningful (Ãśl. 5.8-5.9),but the description regarding the other two is not very clear. 13.6-13.13. Construction of garhapatya vedi. The garhapatya vedi has two forms, e.g. square and circular, each covering an area of either one square vyāvāma or one square purușa. The fire-altar always contains 21 bricks in each layer. Mānava has given almost correct solution to each case. For square gārhapatya of one square vyāyāma, he advises like Baudhāyana, the making of 21 bricks each of length} of a vyāyāma and breadth of a vyāyāma (Mśl. 13.7). In the second layer directions of length and breadth are interchanged. For square gārhapatya of one square purușa, he advises the whole area to be divided into 18 rectangular parts each of length of a purușa and breadth of a purușa (Mśl. 13.8.-13.9). Clearly, each brick measures 40 ang × 20 ang. Three corner bricks are replaced by those of size, 20 ang × 20 ang. thereby making the number of bricks 21. In the next layer the length and breadth are interchanged. In the circular gārhapatya of one sq. purușa, a circle is drawn with a radius half a purușa. The altar is covered with 21 bricks by four types of bricks, the sizes of which are not given. This may be done by laying 12 square bricks of type I, 4 triangular bricks of type II of which one side is curved, and 4 bricks of type III, of which one side is curved and the other straight. One brick of type III is halved (type IV) to make the number 21. For the other layer the direction is only changed. 13.14-13.22. Construction of caturasrasyenacit. This gives an incomplete description of another type of caturaśrasyenacit with pañcami (24 ang. × 24 ang.), adhyardhā (36 ang × 24 ang) pāda (12 ang. × 12 ang.), and ardhapāda (6 ang. × 12 ang.) brick. Admitting that the text is defective and the measures of bricks are uncertain, Gel- der has given a tentative plan, but we shall make no such attempt. Like the previous type it has a square body, 2 rectangular wings, a rectangular tail, and a square head. There is no mention that the citi is constructed with 200 bricks. 13.23-13.29. Construction of agnidhriya, hotriya, brāmaṇācchaṛśa. Compare with Mśl. chapters 6 and 9. Dhiṣṇyās are squares of size 36 ang. × 36 añg., but the description is different at different places. CHAPTER 14 VAKRAPAKṢA ŚYENA, KANKA AND ALAJA 14.1-14.6. Parts of the body in syena, alaja and kanka. Measured with a square brick of size one-fourth of a purușa, the different parts of a syena, alaja and kaňka fire-altar comprise areas shown in Table 1.COMMENTARY Table 1. Areas of different parts of ŝyena, alaja, and kańka fire-altar. wings head atman tail feet Total Syena 75 4. 26 15 120 alaja 75 2 26 17 120 kanka 75 7 26 8 4 120 281 The area of each fire-altar is given correctly as 120 × or 71⁄2 sq. pu. IT 14.7-14.20. Layout of vakrapakṣaśyena. For measuring vakrapakṣałyena of 71⁄2 sq. pu. a cord with 12 parts has been used; each part is equal to 30 añgulas. The alternative of 123 parts does not agree with remaining directions. A rough sketch of both parts and brick structures are given by Gelder, which do not agree with the textual description. Four types of bricks are used for this purpose. They are square (40 ang. × 40 ang.), triangular (30 ang. × 30 ang., 30 √2 ang.), triangular half (30 ang., 15 √2 añg, 15 √2 añg.) and five-cornered bricks. There is no mention that the layer is to be covered with 200 bricks. CHAPTERS 15 AND 16 PRAUGACIT, UBHAYATA PRAUGA, SAMŪHYA, DRONA, RATHACAKRACIT The descriptions are mostly inadequate for drawing the actual diagrams of ubhajata prauga, samūhya, droṇa and rathacakra fire-altars These can, however, be understood by reference to Baudhāyana and Apastamba. For praugacit a rectangle of 15 sq. pu. is to be drawn, and half of this area is required for the purpose. In the droṇacit of 1000 bricks, each layer is constructed with 200 bricks. Of two chariot wheels (rathacakracit) of different sizes, one has an area of 71 sq. pu. and the other three times as large.