Sen, Bag - Translation

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 We shall explain (the method of measuring areas by) the combination of the cords. Having put a pole on a level ground and described a circle round it by means of a cord (fastened to the pole), a pole is fixed on each of the two points where the end of the pole’s shadow touches (the two halves of the circle). This (line joining the two points) is the east-west line (prācī). Then after doubling (a given) cord, two loops (made at its two ends) are fixed at the two poles (of the prāci), and (the cord is stretched towards south by its middle point where) a pole is fixed; the same is repeated to the north. This (line joining the two poles) is the north-south line (udīcī). Two loops are fixed at the two ends of a cord. Marks are (to be given) at the fronis, the amsas, the nirañchana and the samāsabhangas. A pole is fixed at each end of the east-west line (of desired length); likewise (a pole is fixed at each of) the two froņis (west corners) and the two amsas (east corners). Having fixed the loops at the two poles (on the east-west line), the cord is to be stretched by the nirañchana mark towards the south-east corner. The same is done towards the north-east corner. After interchanging (the loops of the cord on the poles), the same is repeated. This is the method (of construction of squares and rectangles) in all cases. Having doubled (the length of) a given measure, a mark is made at one- fourth of the added length; this is the nirañchana mark. (The length upto the nirañchana is) the diagonal (akṣṇayā) and the remainder the breadth (tiryaňmānī). Or else, half the measure is added (to the measure) and a mark is made at the sixth part of the added length; this is the nirañchana. (The length up to the nirañchana is) the diagonal and the remainder the breadth. For (constructing) a square (samacaturaśra), a pole (is to be fixed) at half the measure. For (constructing) a rectangle (dirghacaturaśra), (the pole is to be fixed) at half (of the value) prescribed in the text. The same is for (the construction of) a triangle (śakaṭa-mukha). By these (methods) are explained the measurements of the sacrificial chamber (prāgvamśa), the altars, as also the other chambers (śālā). In these cases the north-south line (is used in the same manner) as in the east-west line. The same is for the sadas (tent). By the word aparimita is to be meant a measure greater than the given measure. 1.9 1.10 TRANSLATION 121 For decrease (nirhāsa) or increase (vṛddhi) as also for addition, the method according to the direction of the text is to be followed. We shall explain in what follows how to find the southern agni by (the method of) the third. (A cord of a length equal to) the distance between the garhapatya and the āhavaniya is increased by one-sixth or one-seventh (of its length) and the length so increased is divided into three equal parts; the cord is stretched towards the south by the mark given at one-third from the other (western) end; at the point (thus obtained) the fire (is to be placed). The opposite point in the north is the place for the utkara (pit). 1.11 Alternately, with a cord of length equal to the distance (between the āhavaniya and the gārhapatya fire) reduced by one-third, a square is drawn in the eastern half; the fire (dakṣiṇāgni) (is placed) at the śroņi (that is, at the south-western corner of the square). By reversing, the rubbish heap (utkara) (is placed) at the amsa (that is, at the north-east corner). 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2 (The fire-altar is) measured by the units of a chariot expressed in angulas; 188 angulas make one iṣā (pole); 104 angulas make one akṣa (axle of a cart); 86 angulas make one yuga (yoke); and 32 angulas make one famyā. For (the construction of) the paitṛki (vedi), a square of 2 purusas (in area) is constructed and a pole is fixed at the middle of each side; this is the solution. (The terms) karaṇī, tatkaraṇī, tiryanmānī, pārśvamānī and akṣṇayā denote cords (measuring the sides of the areas). The diagonal (of a right triangle) of which the breadth is pada and the length 3 padas is √/10 padas. Similarly, the diagonal (of a right triangle) of which the breadth is 2 padas and the length 6 padas is √40 padas. The juga and the samya measures as seen (in the case of the uttara vedi) have already been stated. The (area of the) square drawn on the diagonal of a rectangle is equal to the sum of (areas of) the squares drawn separately on its breadth and length; this is the property of plane figures (concerning rectangles). The diagonal of a square produces a square twice as large (that is, the diagonal equals √2 times the side of the square). The measure is to be increased by its third and this (third) again by its own fourth less the thirtyfourth part (of that fourth); this is the (the value of) diagonal of a square (whose side is the measure); this is approximate. 122 KATYAYANA-SULBASŪTRA 2.10 The diagonal (of a rectangle) of which the breadth is the side of a given square (pramāṇa) and the length the side of a square twice as large (dvikaraṇī) equals the side of a square thrice as large (trkaraṇī). 2.11 Thereby is explained the side of a square one-third the area of a given square (tṛtīyakaraṇī). It is the side of a square one-ninth the area of the square thrice as large (tṛkarani). (Alternatively,) the side of three-ninth the square of the original (pramāņa) equals the side of a square one-third the area of a given square (tṛtiyakarani). 2.12 The side of a square one-third the area of a given square (tṛtiyakaraṇī) in prakrama is used in (the construction of) the sautrāmaṇi (sacrificial altar). 2.13 (By the preceding rules) the combination of squares of equal size is explained. To find a combination of two squares of different measures, a (rectangular) part is cut off from the larger (square) with the side of the smaller; the diagonal of the cut-off (rectangular) part is the side of the combined square. 3.1 3.2 3.3 3.4 3.5 3.6 3 If it is desired to remove a square from another, a (rectangular) part is cut off from the larger (square) with the side of the smaller one (to be removed); two poles are fixed on (two ends of) the cut and the line (thus obtained by joining the poles) is placed across so as to touch the opposite side; by this contact (the side) is cut off. With the cut-off (part) the difference (of two squares) is obtained. (If it is desired) to transform a rectangle into a square (its breadth is taken as the side of a square and this square on the breadth is cut off from the rectangle), the remainder of the rectangle is divided into two (equal parts) and placed in its east and south side; (the empty space in the corner) is filled up by a (small extra) square piece. The removal of it (of the square piece from the square thus formed to get the required square) has been stated. If (the rectangle be) very long, it should be cut again and again (into squares) making equal (the length to its) breadth; these squares are combined into one square; (to this is) added the (remaining) portion (of the rectangle) after transforming it suitably. This is the method of addition (ekaḥ samāsaḥ). A square intended to be transformed into a rectangle is cut off by its diagonal (by drawing a line from its north-west corner to south-east corner); one portion (again is subdivided) into two equal parts and are added, one to its east and the other to the north; for a trapezium the side is cut off and added as it fits. By unit measure the square unit is to be understood. On specific direction, it may be otherwise. Square on a side of 2 units is 4; on 3 units it is 9 and on 4 units it is 16. TRANSLATION 123 3.7 3.8 3.9 The number of units (in the side of a square) measured by a cord, when multiplied by the same, will give the total number of squares. Half of the unit will produce its one-fourth, one-third its one-ninth, one- fourth its one-sixteenth. This is (the method of) deduction (nirhāsa) which has been explained before. This is according to (the injunction) of the śāstra. 3.10 Enlargement (vivṛddhi) and deduction (hrāsa) are made, depending on the unit of length in a cord (rajjupramāņa). 3.11 3.12 If it is desired to transform a square into a circle (a cord of length) half the diagonal (of the square) is stretched from the centre to the east (a part of it lying outside the eastern side of the square); with one-third (of the part lying outside) added to the remainder (of the half diagonal), the (required) circle is drawn. This gives the solution. If it is desired to transform a circle into a square, its diameter is divided into fifteen parts and two of them are removed. The remaining (thirteen parts) will be the side (of the square). 4.1 4.2 4.3 4.4 The altar in the shape of a trough, chariot wheel, falcon, triangle, rhombus and a kind of pot in the shape of a wheel are the (citis of the corresponding) fire-altars. For construction of trough (droṇacit) a square equal to seven-fold fire-altar together with its wings and tail (i.e., 71⁄2 sq. purușa) is drawn and the tenth part of the area (after being transformed into a square) is joined to the original square like a stalk. This is according to some (teacher). The original square (having an area of 7 sq. puruşa) is to be divided horizontally and vertically by drawing parallel lines from its tenth part; the (ten small squares) are then combined into a square (by the ekasamāsa method) and separated out. The remaining (squares are changed to a similar shape, i.e., a square). (Next) the previous part is to be joined to the latter (towards west like a stalk). The same (process) is adopted in the case of a circular (droṇacit). In (the case of) an isosceles triangle (prauga) a square whose area is double the area of (the seven-fold) fire-altar with its wings and tail (i.e. 2×71=15 sq. puruşa) (is constructed) and a pole is put in the middle of the eastern side. This pole is then joined (by means of a cord) with poles at south-west and north-west corners. This gives (the construction for) the fire-altar. In the case of rhombus (ubhayata prauga) a rectangle which has an area twice as much as in the seven-fold fire-altar together with its wings and tail, is con- structed and poles (are fixed) at the middle point of each of its sides. This will give the solution. 124 4.5 4.6 4.7 KATYAYANA-SULBASŪTRA (Having desired) to transform an isosceles triangle (prauga) into a square, the former is divided by the prācī line, one (of its parts) is placed on the · opposite side after inverting it, and (the rectangle so formed) is transformed into a square by the (known) method of addition. This is the solution. (Wishing to transform a rhombus into a square), it (the former) is bisected by its transverse middle line and recombined as before. By this (method) the transformation of a triangle (into a square) is explained. This also explains the method of (constructing) a pentagon. A pentagon of equal angles is to be cut off into isosceles triangle and that having two types of angles into square. 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5 How one hundred-and-one-fold fire-altar (from the original seven-fold fire- altar) is obtained gradually (by adding one square purușa) will be explained. Two times, three times of the original fire-altar (which is 7 sq. purușa) is to be constructed always by the method of addition. Upto twentyone-fold, the fire-altar is to be increased by the addition of one square purușa. For (the purpose of) adding one (sq.) purușa to the original falcon-shaped fire-altar, a square equal (in area) to (that of) the original fire-altar with its wings and tail (i.e., 71⁄2 sq. purușa) is to be constructed and to it is added one (sq.) puruşa (by the method of samāsa.) The original fire-altar is to be divided into fifteen equal parts. Two of these parts are to be transformed into a square (by the samāsa method. This will give the (new) unit (pramāņa) of the purușa. One-fifth (of a purușa) is the measure for the bṛhatī brick and one-tenth (of a purușa) is for the padamātrā. Or, an area of one square purușa is to be divided (into 25 parts) by (drawing) five lines both ways. Five of these small parts are to be transformed into a square, third part of which is cut off. The remaining (two parts) is added to one square purușa. This is another method (of determining the enlarged square unit). One-fifteenth of a purușa is 8 añgulas. 5 aratnis, 10 vitastis, 120 añgulas are each equal to one purușa. One pada measures 12 añgulas. Or, an area of one (sq.) puruşa is to be divided by seven (lines) drawn from both sides; seven (parts) of these are to be combined (into a square); from this combined sum, (a rectangle) of 14 añg. by 1 puruşa (to be transformed into a square) is to be subtracted (by the method of nirhāsa). The TRANSLATION 125 5.11 remainder (which is a square) is added to one (sq.) purușa. This is another method. By aratni and vitasti, no enlargement with (purușa) is to be understood; it is a number according to Śruti (i.e. enlargement of puruşa is to be made with its submultiple). 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6 As the (area) of the fire-altar is increased, so is (increased) the unit of the bricks. This we will explain. The side of a square of area 14 (square) prakramas will give the unit in prakrama for 101-fold fire-altar (construction above twentyone-fold). Or, in the second and following construction, there should be an increase (of one square) prakrama for every seven constructions. (At each successive construction), instead of the (original) prakrama the value of the prakrama should be enlarged by its one-seventh. Such (seventh part increase) is continued upto 101-fold fire-altar. There should be no increase in the antaḥpatya and the garhapatya agni. The yoni will remain (fixed), for, according to Śruti, (the number of) yoni never increases with the (number of the) birth. Accordingly, the increase of the yoni is prohibited. The transverse (side) is to measure one less than the number of squares (of equal area) intended to be combined into a square; the two sides (together) should be one more than that; an (isosceles) triangle is to be formed (with the sides and the transverse side as the base); the altitude will produce it (the required square). The increase of the altar depends on the distance between the poles (yupas). This we are going to explain. The twentyfourth part of the cord measuring 11 paravān 10 akṣa (1172 ang.) of the chariot measure forms the (required) unit in prakrama. 6.10 Having constructed an altar with this unit (i.e. 1172 añg.), a (rectangular) portion of 12 angulas from the east of the altar is to be cut off, and then having considered the east-west line are placed the poles in the holes (made for this purpose). 6.11 Or, the half (of the holes used for the two end yūpas) at both sides should be inside the altar (and the other half outside). This is according to Śruti. 6.12 According to another (group of teachers), the first and the last pole (yupa) should be placed inside the altar; this is natural. 6.13 It is an altar having a lock on the crown of the head (śikhaṇḍinī vedi).