CHAPTER 1 DRAWING OF EAST-WEST LINE, CONSTRUCTION OF SQUARES AND FIXING THE PLACES OF THE AHAVANĪYA, GĀRHAPATYA, DAKSINĀGNI AND UTKARA ALTARS The Katyayana-sulbasutra, 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 ahavaniya, gārhapatya, dakṣiṇāgni and utkara altars. DRAWING OF East-West anD THE NORTH-SOUTH LINES 1.2. Let O be the pole, and a circle EPW be drawn with a cord of length equal to OP. NX E W Fig. 1. Let E and W be the eastern and western shadow points on the equinoctial day of the pole fixed at 0 (Fig. 1). Apte 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 prăci 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 ■ Apte, 1-16 COMMENTARY 265 point S is obtained. Then NS, gives the north-south line. In the Sulbasūtras the east- west line has always been drawn first presumably because of the importance attached to this direction. CONSTRUCTION of a Square or a Rectangle 1.3 This gives a general method of construction of a square or a rectangle. In a given cord marks are given for śroņi, amsa 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. 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. First cord. Let AB, the given measure be a, BC, the added length a, and D, the nirañchana mark (Fig. 2(a)) so that BC a BD 4 4 5a By definition, AD the diagonal 4 3a and DC =

the breadth 4 Clearly, AD2 DC2

• (52) ’ - (34) ’ - .2 a2 This is the expression for a right triangle ADC, of which AD is the diagonal (akṣṇayā), CD the breadth (tiryañmānī) and AC, the given measure for prāci. A A 00 B A D D Fig. 2 (a) First cord. (b) Second cord. 266 KĀTYAYANA-ŚULBASŪTRA Second cord. Let AB, the given measure be a, BC, the added length, D, be the nirañchana mark (Fig. 2(b)) so that 1 a α 13a 5a BD AD and CD 6 2 12 12 12 5a 2 13 +()=(冊。) 12 12 This satisfies the square relation: a2 + triangle. a and 2 2 of the right The term tiryaṁmāni literally means ’transverse’ or ‘oblique’ measure. In the Sulbasūtra, however, it has been used to signify the ‘breadth’ or ‘shorter side’ of a rectangle. Mahīdhara says: nirañchanenākarṣaṇe kṛte śronyamsa parichedikā jā rajjuḥ sā tiryańmāni. 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 prăci to obtain the square. For rectangle, poles are fixed at a distance equal to half the value prescribed in the text. The sakaṭamukha 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 prāgvamsa 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, Katyāyana followed the direction of older śāstras. RELATIVE POSITIONS OF THE Gārhapatya, Āhavaniya, Dakṣiṇāgni and Utkara LL A L G 1.9-1.10. Let A and G be the positions of āhavaniya and garhapatya fire-altars. As explained by Mahidhara, 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 ahavaniyamadhyāt paścimabhāge samacaturasramuktavidhinā kāryam (Mahidhara). At the southern śroņi point D of this square (daksiņaśronyam) 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 garhapatya point G towards east: paścimārdhe garhapatyasya madhyāt pūrvabhāge. Then U, the northern amsa point of Fig. 3. the square is the utkara. COMMENTARY CHATPER 2 267 UNITS OF MEASURES, PAITṚKĪ VEDI, MEASURES FOR DIAGONAL, THEOREM OF SQUARE, COMBINATION OF SQUARES. 2.1. Units of measures. Units like iṣā, akṣa, yuga and śamyā have been expressed here in angulas. This has been discussed under Bśl. 1.3. 2.3. Paitṛki vedi. The paitṛki vedi has been dealt with by Baudhayana (Bśl. 3.11). 2.3-2.6. Measures for the diagonal of a rectangle. Kâtyāyana has considered a parti- cular measure for a square as karaņi, so that its diagonal becomes tatkaraņi or dvikaraņi, (tat kṣetram dvaiguṇyādi kriyati ‘nayā sā tatkaraṇi—Mahidhara), for square on the diagonal is twice the original square. Likewise, the diagonal of a rectangle having sides as prāśvamānī and tiryaṁmāni is known as akṣṇayā. With the help of these technical terms Katyāyana has expressed the measures for the diagonal of a rectangle in the following two cases: and 12 + 32 22 + 62 (VTO)2 (√40)2 Other measures juga and famyā have been defined in Kśl. 2.1. 2.7 Theorem of square. Kātyāyana here enunciates the general theorem of square on the diagonal of a rectangle in the same language as did Baudhāyana (Bśl. 1.12) and Apastamba (Aśl. 1.4). b At the end of the enunciation, he remarks, iti kṣetrajñānam. The term kṣetra has been translated as ‘area’ by Thibauta and ‘figure’ by Datta. In the śulbasūtra, the area is technically expressed by bhūmi (Bśl. 1.6 and 1.9) and not kṣetra. Hence iti kṣetrajñānam means ’this is the knowledge of plane figures’. 2.8-2.9. Combination of two equal squares. Dvikaraṇi has been defined here as √2 a, where a is the measure. This is actually a method of combination of two equal squares each of side a into one of √/2a. Tritiyākaraṇi has been defined as navabhāgastraya of tṛkaraṇi : √3a, i.e. if tṛkarani then tṛtiyākaraṇi =

3. √/3a 9 1 a, √3 where a is the measure. Katyāyana’s rule is essentially the same as that of Baudhāyana (Bśl. 2.12). a Thibaut (2), 233-34. b Datta (2), 108. 268 KÄTYĀYANA-ŠULBASŪTRA 2.10-2.12. Construction of an isosceles trapezium. After explaining the meaning of dvikaraņi, tṛkaraṇī, and tṛtiyākaraṇi of a given measure Katyāyana gives the method of construction of the sautrāmaṇiki vedi, which is an isosceles trapezium having 24 face base

√/3 30 √3 36 and altitude

prakramas (vide Bŝl. 3.12). √3 2.13. Combination of two squares. Kātyāyana prescribes the same method of Baudhâ- yana for the combination of two different squares into a square (Bśl. 2.1.). CHAPTER 3 DIFFERENCE OF TWO SQUARES, TRANSFORMATION OF A RECTANGLE INTO A SQUARE AND A SQUARE INTO A RECTANGLE, AREAS OF FIGURES, PROBLEM OF CIRCLING A SQUARE AND QUADRATURE OF THE CIRCLE. 3.1-.34. The rule 3.1 deals with the construction of a square equal to the diffe- rence of two squares, 3.2-3.3 the transformation of a rectangle into a square and 3.4 transformation of a square into a rectangle. These rules have been given by Baud- hāyana (Bśl. 2.2, 2.5 and 3.4 respectively). The transformation of a rectangle into a square, when it is very large, is specially discussed by Katyāyana, as has also been done under Bśl 2.5. 3.5-3.10. These concern the areas of squares and rectangles and are essentially the same as those of Apastamba (Asl. 3.4-3.10) 3.11-3.12. For circling a square and the quadrature of the circle,Katyāyana gives the same rules as those by Baudhāyana (Bśl. 2.9-2.11) and Āpastamba (Āśl. 3.2, 3.3). CHAPTER 4 VACIT, CONSTRUCTION OF DRONACIT, TRIANGLE, RHOMBUS, TRANS- FORMATION OF TRIANGLE AND RHOMBUS INTO A SQUARE 4.1-4.2. Construction of dronacit. Kătyāyana discusses here the methods of drawing different squares required for the construction of droṇacit. First a square of area 7 sq. pu. is constructed and divided into 100 small squares by drawing ten parallel lines horizontally and ten vertically. Then small squares from one side are separated out and changed into a small square by the method of combination of squares (samāsa-vidhi), discussed in Kśl. 2.8 and 2.9. The remaining 90 small squares are likewise transformed into a single square. The former square is joined to the latter like a stalk. In the case of a circular droṇacit, the two squares mentioned above are COMMENTARY 269 to be transformed into two circles and joined together (Mahidhara). Katyāyana’s rule is basically different from that of Baudhayana (Bśl. 17.1.-18) and Āpastamba (Āśl. 13.4-5) and appears mathematically more sound. 4.3-4.4. Construction of triangle and rhombus. These two rules are the same as those of Baudhayana (Bśl. 2.7-2.8). 4.5. Transformation of an isosceles triangle into a square. For transformation of an isosceles triangle into a square, the isosceles triangle ECG is divided by the prāci line EF (Fig. 4). Now tr. ECF is transferred to the other side so that tr. EGH is now its new position. Thus tr. ECG is transformed into the rectangle EFGH. This rectangle is changed into a square by the sūtra Kśl. 3.2. It has also been discussed by Āpastamba (Ãśl. 12.4-12.8). C E H F Fig. 4. Transformation of a triangle into a square. 4.6. Transformation of a rhombus into a square. For transfor- ming a rhombus ENFS into a square, EF and NS, the east- west and north-south lines are joined (Fig. 5). The isosceles triangle NFS is clearly sub-divided into two triangles NOF and SOF. These are now transferred and after invertion placed in their positions as AEN and BES. Thus the rhombus ENFS is transformed into the rectangle ANSB. This rectangle is transformed into a square by the sutra Kśl. 3.2. A B N Ο F S tion of a rhombus E 4.7. Transformation of a triangle into a square. Katyāyana has Fig. 5 Transforma- hinted for the first time at a method of transforming into a square a triangle other than the isosceles. The rule is, how into a rectangle. ever, incomplete. According to commentator Mahidhara, ekakarṇa means tulyakarṇa, i.e. a figure having equal angles and dvikarṇa nānāvidhakarṇa, i.e. a figure of unequal angles. Whether by nānāvidhakarṇa Mahidhara meant an irregular pentagon cannot be definitely said. Šulbakāras were well acquainted with the method of converting an isosceles triangle into a square. Possibly they had also the knowledge of trans- formation of a pentagon of equal angles into a square by joining the angular points, dividing it into several isosceles triangles, and then joining them up into squares by the rule taught before. Kātyāyana has advised to break up pañcakarṇas of dvikarņa variety into a square. But there is no such method known to the fulbakāras by which an irregular pentagon can be broken up into squares. Kātyāyana’s pentagon ABCDE is of the type of hamsamukhi brick (Fig. 6), in which BF FC = HF EH and AB BF 2 A B D H F Fig. 6. = DC Description of this type of pentagonal hamsamukhi bricks is given by 270 KATYAYANA-SULBASŪTRA Baudhāyana (Bśl. 10.10). A pentagon of this type can be broken up into two or three squares, which again can be combined into a single square. CHAPTERS 5 AND 6 ENLARGED UNIT, CONSTRUCTION OF A SQUARE EQUAL TO N TIMES A GIVEN SQUARE AND EKĀDAŠINĪ FIRE-ALTAR. The whole of chapter 5 and the first five sūtras of chapter 6 are devoted to the discussion of enlarged unit required for measuring the areas of fire-altars from 81 sq. purușas to 101 sq. purușas. Most of these results agree with those of Baudha- yana and have been discussed under B§l. 5.1-5.6. In sūtra 6.2, Katyāyana says that ifs be the maximum enlarged unit in a prakrama for 101 fold fire-altar, then s2 equals 14 sq. prakramas. In the next sūtras he has pointed out that at each succes- sive constuction the value of the prakrama is to be increased by one seventh of the increased area, i.e. $2 1 + p 7 or 752 (1) 7+p, where p is the increased area. For the construction of 101 fold fire-altar the total increment from the 7 fold one is 94. Putting a = 94. 752 7 +94

then s2 101 7 101, 3 14 7 7 (2) But the formula (1) does not appear to be correct (vide Bśl. 5.1-5.6); it should be $2 1 + р 71 + since the enlargement in area starts from the normal 7 fold fire- altar, i.e. 71 sq. purușas. It may be that Kātyāyana simplified the rule for construc- tion of 101 fold fire-altar. Datta has suggested another rational of the formula (2) as follows. In the falcon-shaped fire-altar (second plan), the problem of proportionate enlargement is equivalent to the solution of the following quadratic equation: 1 4s2 + 2s 25 ( 1 + 1 ) + ( 3 + 1/10) s :) 1 7

• P2 2 1 or, 752 + 7 2
• P2 2 a Datta (2), 166-168.COMMENTARY 271 2 841 or, 75+
• 7p, 4 16 1 or, s 28 ( √841 + 112 p 1 When p 94, or, or, S

1 28 1 ( 11369 1

1. or, or, sa 1 784 ( ( 113702 11369 > $2 784 ( 79 11156 + 106 :) 133 when V11369 = 106 + approx. 212 19159 $2 14+ 83104 3 3 14 + 14 + approx. 245 13 13 19159 It is nearly equal to Kātyāyana’s value, s2 = 14 3 7 6.7. Construction of a square equal to n times the given square. This method undoubtedly hints at the construction of a square which is equivalent to ʼn times a given square. Let n number of equal squares each of side a are to be combined. Šulbakāras used isosceles triangles for different constructions. Here also Katyayana possibly consi- dered an isosceles triangle ABC, in which (n – 1) a, BC AB+ AC == Since BD = (n + 1) a, DC, AB AC, n 1 n + 1 BD a, and AB = a, 2 2 According to this rule, the altitude AD will produce the sum of n equal squares. Now AD2 AB2 BD2 n 2 a 2 =(+)’-(0) 2 a 2 na2 Fig. 7. 6.8-6.13. Construction of a fire-altar with enlarged areas was usually carried out by fixing the distance between the two poles (yupas) of ekādaśini. This distance is known as prakrama. The length of prakrama varies in the case of the enlarged fire- altar. There are various opinions on this point by ancient masters. This has been discussed under Bśl. 4.12-4.14.