Chapter 8 - Calculations regarding Excavations (khata-vyavahara)

1. I bow in religious devotion with my head (bent downwards) to Jina Vardhamana, whose foot-stool is honoured by the crowns worn by all the chief gods, who is omniscient, ever-enduring, unthinkable, and infinite in form, and is (further) like the young (rising) sun in relation to the lotus-lakes representing the good and worthy people that are his devotees. 2. I shall now give out the (three) varieties of karmantika, aundraphala, and suksmaphala (in relation to excavations), which varieties are all derived from those various kinds of geometrical figures, mentioned before, as results obtained by multiplying them by (quantities measuring) depth. This seventh subject of treatment is the subject of excavations. A stanza regarding the conventional assumption (implied in this chapter) :- 3. The quantity of earth required to fill an excavation measuring one cubic hasta is 3,200 palas. From that (same cubie volume of excavation) 3,600 palas (of earth) may be taken out. The The rule for arriving at the cubical contents of excavations:- 4. Area multiplied by depth gives rise to the approximate measure of the cubical contents in a regular excavation. sums of (all the various) top dimensions with the corresponding bottom dimensions are halved; and then (these halved quantities of the same denomination are all added, and their sum is) divided by the number of the said (halved quantities). Such is the process of arriving at the average equivalent value. 2. The term Aundra in Aundraphala is rather strange Sanskrit and is perhaps related to the Hindi word aug meaning 'deep.' 3. The idea in this stanza evidently is that one cubic hasta of compressed earth weighs 3,600 palas, while 3,200 palas of earth are sufficient to fill loosely the 8 pace of 1 cubic hasta. 4. The latter half of this stanza evidently gives the process by which we may arrive at the dimensions of a regular excavation fairly equivalent to any given irregular excavation.

OF CULTURE MINISTRY OF CUL VERNMENT OF INDIA sarakara SSL CHAPTER VIII. CALCULATIONS REGARDING EXCAVATIONS. 259 Examples in illustration thereof. 5. In relation to (an equilateral) quadrilateral area (representing the section of a regular excavation), the sides and the depth are 8 hastas (each in measure). In respect of this regular excavation, what may be the value of the cubical contents here? 6. In relation to an (equilateral) triangular area (representing the section of a regular excavation), the sides are 32 hastas each, and in the depth there are found 36 hastas and 6 angulas. What is the calculation (of the contents) here? 7. In relation to a (regular) circular area representing (the section of) a regular excavation, the diameter is 108 hastas, and the depth (of the excavation) is 165 hastas. (Now), give out what the cubical contents are. 8. In relation to a longish quadrilateral area (forming the section) of a regular excavation, the breadth is 25 hastas, the side (measuring the length) is 60 hastas and the depth (of the excavation) is 108 hastas. Quickly give out (the cubical contents of this regular excavation). The rule for arriving at the accurate value of the cubical contents in the calculation relating to excavations, after knowing the result designated karmantika as well as the result designated aundra and with the aid of these results :- 9-11. The values of the base and the other sides of the figure representing the top sectional area are added respectively to the values of the base and the corresponding sides of the figure representing the bottom sectional area. The (several) sums (so arrived at) are divided by the number of the sectional areas taken into consideration (in the problem). The (resulting) quantities are 9-11. The figures dealt with in this rule are truncated pyramids with rectangular or triangular bases, or truncated cones all of which have to be conceived as turned upside down. The rule deals with three different kinds of measures of the cubical contents of excavations. Of these, two, viz., the Karmantika and Aundra measures give only the approximate values of the contents. The accurate measure is calculated with the help of these values. If K represents the Karmantika-phala and A represents the Aundra-phala A-K then the accurate measure is said to be equal to + K, i.e., K+ A.

multiplied with each other (as required by the rules bearing upon the finding out of areas when the values of the sides are known). The area (so arrived at), when multiplied by the depth, gives rise to the cubical measure designated the karmantika result. In the case of those same figures representing the top sectional area and the bottom sectional area, the value of the area of (each of) these figures is (separately) arrived at. The area values (so obtained) are added together and then divided by the number of (sectional) areas (taken into consideration). The quotient (so obtained) is multiplied by the value of the depth. This gives rise to (the cubical measure designated) the aundra result. If one-third of the difference between these two results is added to the karmantika result, it indeed becomes the accurate value (of the required cubical contents). Examples in illustration thereof. 12. There is a well whose (sectional) area happens to be an equilateral quadrilateral. The value (of each of the sides) of the top (sectional area) is 20 (hastas), and that (of each of the sides) of the bottom (sectional area) is only 16 (hastas). The depth is 9 (hastas). O you who know calculation, tell me quickly what the cubical measure here is. 13. There is a well whose (sectional) area happens to be an equilateral triangular figure. The value (of each of the sides) of the top (sectional area) is 20 (hastas), and that (of each of the sides) of the bottom (sectional area) is 16; the depth is 9 (hastas). What is the value of the karmantika cubical measure, of the If a and b be the measures of a side of the top and bottom surfaces respectively of a truncated pyramid with a square base, it can be easily shown that the accurate measure of the cubical contents is equal to h (a + b + ab), where h is the height of the truncated pyramid. The formula given in the rule for the accurate measure of the cubical contents may be verified to be the same as this with the help of the following values for the Karmantika and Aundra results given in the zule:K= * x h; A = x h. Similar verifications may be arrived at in the case of truncated pyramids having an equilateral triangle or a rectangle for the base, and also in the case of truncated cones.

261 aundra cubical measure, and of the accurate cubical measure here? 14. There is a well whose (sectional) area happens to be regularly circular. The (diameter of the) top (sectional area) is 20 dandas, and that of the bottom (sectional area) is only 16 dandas. The depth is 12 dandas. What may be the karmantika, the aundra, and the accurate cubical measures here ? 15. In relation to (an excavation whose sectional area happens to be) a longish quadrilateral figure (i.e., oblong), the length at the top is 60 (hastas), the breadth is 12 (hastas); at the bottom, these are (respectively) half (of what they measure at the top). The depth is 8 (hastas). What is the cubical measure here ? 16. (Here is another well of the same kind), the lengths (of whose sectional areas) at the top, at the middle, and at the bottom are (respectively) 90, 80, and 70 (hastas), and the breadths are (respectively) 32, 16, and 10 hastas. This is 7 (hastas) in depth. (Find out the required cubical measure.) 17. In relation to (an excavation whose sectional area happens to be) a regular circle, the diameter at the mouth is 60 (hastas), in the middle 30 (hastas), and at the bottom 15 (hastas). The depth is 16 hastas. What is the calculated result giving its cubical measure? 18. In relation to (an excavation whose sectional area happens to be) a triangle, each of the three sides measures 80 hastas at the top, 60 hastas in the middle, and 50 hastas at the bottom. The depth is 9 hastas. What is the calculated result giving its cubical contents? The rule for arriving at the value of the cubical contents of a ditch, as also for arriving at the value of the cubical contents of an excavation having in the middle (of it) a tapering projection (of solid earth) :- 19-20. The breadth (of the central mass) increased by the top-breadth of the surrounding ditch, and (then) multiplied by 19-204. These stanzas deal with the measurement of the cubic contents of a ditch dug round a central mass of earth of any shape. The central mass may be in section a square, a rectangle, an equilateral triangle, or a circle ;

three, gives rise to the value of the (required) perimeter in the case of triangular and circular excavations. In the case of a quadrilateral excavation, (this same value of the required perimeter results) by multiplying the quantity four (with the value of the breadth as before). In the case of excavations having central masses tapering upwards or downwards the operation (for Karmantikaphala) is (to add the value of) half the breadth of the excavation to (that of the breadth of) the central nass, and (for Aundraphala), to add (the value of) the breadth (of the excavation to the value of the breadth of the central mass); then (the procedure is) as (given) before. Examples in illustration thereof. 21. The already mentioned trilateral, quadrilateral, and circular (areas) have ditches thrown round them. The breadth. measures 80 dandas, and the ditches are as much as 4 (dandas) in breadth, and 3 (dandas) in depth. (Find ont the cubical contents.) and the excavation may be of the same width both at the bottom and the top, or may be of diminishing or increasing width. The rule enables us to find out the total length of the ditch in all these cases. I. When the width of the ditch is uniform, the length of ditch (d+ b) x 3 in the case of an equilateral triangular or circular ditch, where d is the measure of a side or of the diameter of the central mass and b is the width of the ditch: but this length (d + b) x 4 in the case of a square excavation with a central mass, square in section. II. When the ditch is tapering to a point at the bottom or the top, the length of the ditch for finding out the Karmantika-phala. d + x 4, according as the central mass (1) is in section trilateral or circular, or (2) square. Length of ditch for finding out Aundra-phala x 4 respectively. (a + b) x 3 and (d + b) These expressions have to be multiplied by half of the width of the ditch and by its depth for finding out the respective cubical phalas. The formulas given above in relation to triangular and circular excavations give only approximate results. With the aid of the total length of the ditch so obtained, the cubical contents are found out in the case of ditches with sloping sides by applying the rule given in stanzas 9 to 11 above.

263 22. The length of a longish quadrilateral is 120 (dandas) and the breadth is 40. The ditch around is as big as 4 dandas in breadth and 3 in depth. (Find out the cubical contents.) The rule for arriving at the value of the cubical contents of an excavation, when the depth of the excavation varies (at various points), and also for arriving, when the cubical contents of an excavation are known, at the depth of digging necessary in the case of another (known) area (so that the cubical contents may be the same): 23. The sum of the depths (measured in different places) is divided by the number of places; this gives rise to the (average) depth. This multiplied by the top area (of the excavation) gives rise to the (required) cubical contents of the excavation in the case where that area is trilateral, quadrilateral or circular. The cubical contents (of a given excavation), when divided by the (known) value of another area, gives rise to the depth (to which there should be digging, so that the resulting cubical contents may be the same). Examples in illustration thereof. 24. In an equilateral quadrilateral field, the ground covered by which has an extent measured by 4 hastas (in length and breadth), the excavations are (in depth) 1, 2, 3, and 4 hastas (in four different cases). What is the measure of the average depth (of the excavations) ? 25. There is a well with an equilateral quadrilateral section, the sides whereof are 18 hastas in measure; its depth is + hastas. With the water of this (well), another well measuring 9 hastas at each of the sides (of the section) is filled. What is the depth (of this other well) ? When the measures of the sides of the top (sectional area) and also of the bottom (sectional area) are known, and when the 22). For finding out the total length of the surrounding ditch when the central mass of earth is rectangular in section, the measures of the sides as increased by the width or half the width of the ditch are added together, socording as the Kai mantika or the Aundra result is required.

measure of the depth also is known, in relation to a certain given excavation, the rule for arriving at the value of the sides (of the resulting bottom section) at any optionally given depth, and also for arriving at the (resulting) depth (of the excavation) if the bottom is reduced to a mere point: 26: The product resulting from multiplying the (given) depth with (the given measure of a side at the) top, when divided by the difference between the measures of the top side and the bottom side gives rise here to the (required) depth (when the bottom is) made to end in a point. The depth measured (from the pointed bottom) upwards (to the position required) multiplied by (the measure of the side at) the top and (then) divided by the sum of the side measure, if any, at the pointed bottom and the (total) depth (from the top to the pointed bottom), gives rise to the side measure (of the excavation at the required depth). An example in illustration thereof. 271. There is a well with an equilateral quadrilateral section. The (side) measure at the top is 20 and at the bottom 14. The depth given in the beginning is 9. (This depth has to be) further (carried) down by 3. What will be the side value (of the bottom here)? What is the measure of the depth, (if the bottom is) made to end in a point? 26. The problems contemplated in this stanza are (a) to find out the full latitude of an inverted pyramid or cone and (b) to find out the dimensions of the cross section thereof at a desired level, when the altitude and the dimensions of the top and bottom surfaces of a truncated pyramid or cone are given. If, in a truncated pyramid with square base, a is the measure of a side of the base and b that of a side of the top surface and h the height, then according to the rule given here, H taken as the height of the whole pyramid and the reasure of a side of the cross section of the pyramid at any given height represented by a (H-h₁) axh a-b H These formulas are applicable in the case of a cone as well. In the rule the measure of the side of section forming the pointed part of the pyramid is required to be added to H, the denominator in the second formula, for the reason that in some cases the pyramid may not actually end in a point. Where, however, it does end in a point, the value of this side has to be zero as a matter of course.

265 The rule for arriving at the value of the cubical contents of a spherically bounded space :-- 28. The half of the cube of half the diameter, multiplied by nine, gives the approximate value of the cubical contents of a sphere. This (approximate value), multiplied by nine and divided by ten on neglecting the remainder, gives rise to the accurate value of the cubical measure. An example in illustration thereof. 29. In the case of a sphere measuring 16 in diameter, calculate and tell me what the approximate value of (its) cubical measure is, and also the accurate measure (thereof). The rule for arriving at the approximate value as well as the accurate value of the cubical contents of an excavation in the form of a triangular pyramid, (the height whereof is taken to be equal to the length of one of the sides of the equilateral triangle forming the base) :- 30. The cube of half the square of the side (of the basal equilateral triangle) is multiplied by ten; and the square root (of the resulting product is) divided by nine. This gives rise to the approximately calculated value (required). (This approximate) value, when multiplied by three and divided by the square root of 281. The volume of a sphere as given here is (1) approximately () and (2) accurately (2) X The correct formula for the cubi- 2*10 cal contents of a sphere is 3 and this becomes osmparable with the above value, if r is taken to be 10. Both the MSS. read 10 , making it appear that the accurate value is of the approximate 9 which makes the accurate valna value; but the text adopted is 10 of the approximate one. It is easy to see that this gives a more accurate result in regard to the measure of the cubical contents of a sphere than the other reading. 30. Algebraically represented the approximate value of the cubical contents of a triangular pyramid according to the rule comes to a? * 12 9 20; and the accurate value becomes equal to 18 a 12 V 2; where 34

ten, gives rise to the accurately calculated cubical contents of the pyramidal excavation. An example in illustration thereof. 314. Calculate and say what the approximate value and the accurate value of the cubical measure of a triangular pyramid are, the side of the (basal) triangle whereof is 6 in length. When the pipes leading into a well are (all) open, the rule for arriving at the value of the time taken to fill the well with water, when any number of optionally chosen pipes are together (allowed to fill the well). 32-33. (The number one representing) each of the pipes is divided by the time corresponding to each of them (separately); and (the resulting quotients represented as fractions) are reduced so as to have a common denominator; one divided by the sum of these (fractions with the common denominator) gives the fraction of the day (within which the well would become filled) by all the pipes (pouring in their water) together. Those (fractions with the common denominator) multiplied by this resulting fraction of the day give rise to the measures of the flow of water (separately through each of the various pipes) into that well. An example in illustration thereof. 3. There are 4 pipes (leading into a well). Among them, each fills the well (in order) in 1,,, of a day. In how much of a day will all of them (together fill the well, and each of them to what extent) ? In the Fourth Subject of Treatment named Rule of Three, an example (like this) has (already) been given merely as a hint ; the a gives the measure of the altitude of the pyramid as also of a side of the basal equilateral triangle. It may be easily seen that both these values are somewhat wide of the mark, and that the given approximate value is nearer the correct value than the Bo-called accurate value,

267 subject (of that example) is expanded here and is given out in detail. 35-36. There is at the foot of a hill a well of an equilaterally quadrilateral section measuring 9 hastas in each of the (three), dimensions. From the top of the hill there runs a water channel, the section whereof is (uniformly) an equilateral quadrilateral having 1 angula for the measure of a side. (As soon as the water flowing through that channel begins to fall into the well), the stream is broken off at the top; and (yet), with it (that well) becomes filled in with water. Tell me the height of the hill and also the measure of the water in the well. 37-88. There is at the foot of a hill a well of an equilaterally quadrilateral section measuring 9 hastas in each of the (three) dimensions. From the top of the hill, there runs a water channel, (the section whereof is throughout) a circle of 1 angula in diameter. As soon as the water (flowing through the channel) begins to fall into the well, the stream is broken off at the top. With the water filling the whole of the channel, that well becomes filled. O friend, calculate and tell me the height of the mountain and also the measure of the water.. 39-40. There is at the foot of a hill a well of an equilaterally quadrilateral section measuring 9 hastas in (each of the) three dimensions. From the top of the hill there runs a water ebannel, (the section whereof is throughout) triangular, each side measuring 1 angula. As soon as the water (flowing through that channel) begins to fall into the well, the stream is broken off at the top. With the water (filling the whole of the channel) that (well) becomes filled. O friend, calculate and tell (me) the height of the mountain and the measure of the water. 35 to 42. The reference here is to the example given in stanzas 15-16 of chapter V-vide also the footnote thereunder. The volume of the water is probably intended to be expressed in vahas. (Vide the table relating to this kind of volume measure in stanzas 36 to 38, chapter I.) It is stated in the Kanarese commentary that 1 cubic angula of water is equal to 1 karsa. Then according to the table given in stanza 41 of chapter I, 4 karsas make one pala; according to stanza 44 in the same chapter, 12 palas make one prastha; and stanzas 36 to 37 therein give the relation of the prastho to the vaha.

41-42. There is at the foot of a hill a well of an equilaterally quadrilateral section measuring 9 hastas in (each of the) three dimensions. (From the top of the hill) there runs a water channel, (the section whereof is uniformly) 1 angula broad at the bottom, 1 angula at (each of) the dug (side slopes), and 2 angulas in length (at the top). As soon as the water (flowing through that channel) begins to fall into the well, the stream is broken off at the top. With the water (filling the whole of the channel) that well becomes filled. What is the height of the hill and (what) the measure of the water P Thus ends the section on accurate measurements in the calculations relating to excavations. Calculations Relating to Piles (of Bricks). Hereafter, in (this) chapter treating of operations relating to excavations, we will expound calculations relating to (brick) piles. Here there is this convention (regarding the unit brick). 434. The (unit) brick is 1 hasta in length, half of that in breadth, and 4 angulas in thickness. With such (bricks all) operations are to be carried out. The rule for arriving at the cubical contents of a given excavation in a field and also at the number of bricks corresponding to the above cubical contents. 44. The area at the mouth (of the excavation) is multiplied by the depth; this (resulting product) is divided by the cubic measure of the (unit) brick. The quotient so obtained is to be understood as the (cubical) measure of a (brick) pile; that same (quotient) also happens to be the measure of the number of the bricks. Examples in illustration thereof. 45. There is a raised platform equilaterally quadrilateral (in section) having a side measure of 8 hastas and a height of 9 44. The oubical measure of the brick pile here is evidently in terms of the unit brick.

hastas. That (platform) is built up of bricks. O you who know calculation, say how many bricks there are (in it). 46. A raised platform, equilaterally triangular (in section), having 8 hastas (as its side measure), and 9 hastas as height, has been constructed with the aforesaid bricks. Calculate and say how many bricks there are in this (structure). 47. A raised platform, circular in section, having a diameter of 8 hastas and a height of 9 hastas is built up-with (the same aforesaid) bricks. O you who know calculation, say how many bricks there are in it. 48. In the case of (a raised platform having) an oblong (section), the length is 60 hastas, the breadth 25 hastas, and the height is 6 hastas. Give out in this case the measure of that brick pile. 49. A boundary wall is 7 hastas in thickness, 24 hastas in length and 20 hastas in height. How many are the bricks used in building it ? 50. The thickness of a boundary wall is 6 hastas at the top and 8 hastas at the bottom; its length is 24 hastas and height 20 hastas. How many are the bricks used in building it ? 51. (In the case of a raised sloping platform), the heights are (respectively) 12, 16 and 20 hastas (at three different points); 20 60-51. In finding out the cubical contents of the wall, the average breadth 116 + 24 3 1 12 calculated accor ding to the rule, given in the latter half of stanza 4 above, is used; so the Karman. tika value is taken into consideration here. This 51. sloping platform is bounded at its two ends by two vertical planes, the top and the surfaces side alone being sloping. The top forms an inclined

the measures of the breadth at the bottom are (respectively) 7, 6 and 5, (the same) at the top being 4, 3 and 2 hastas; the length is 24 hastas. (Find out the number of bricks in the pile). The rule for arriving, in relation to a given raised platform (part of) which has fallen down, at the number of bricks found (intact) in the unfallen (part) and also at the number of bricks found in the fallen (part):- 52. The difference between the top (breadth) and the bottom (breadth) is multiplied by the height of the fallen (portion) and divided by the whole height. (To the resulting quotient) the value of the top (breadth) is added. This gives rise to the measure of the basal breadth in relation to the upper (fallen portion) as well as to the top breadth in relation to the lower (intact portion). The remaining operation has been already described. An example in illustration thereof. 53. (In relation to a raised platform), the length is 12 hastas the breadth at the bottom is 5 hastas, (the breadth) at the top is 1 hasta, and the height all through is 10 hastas. (A measure of) 5 hastas (in height) of that (platform) gets broken down and falls. How many are those (unit) bricks there (in the broken and the unbroken parts of the platform)? When a (high) fort-wall is broken down obliquely, the rule for arriving at the number of bricks which remain intact and of the bricks that have fallen down :plane, the breadth of which is 2 hastas at the raised end and 4 hastas at the other end. Vide diagram in the margin of page 269. 52. The measure of the top-breadth of the standing par of the platform which is the same as the bottom-breadth of the fallen part of the platform-is (ab) d algebraically +b; where a is the bottom-breadth, b is the toph breadth, h the total height and d the height of the fallen part of the raised platform. This formula can be easily shown to be correct by applying the properties of similar triangles. The operation referred to in the rule as having been already described is what is given in stanza 4 above.

271 54. The bottom (breadth) and the top (breadth) are (each) doubled. To these are added (respectively) the top (breadth) and the bottom (breadth). The (resulting) quantities are (respectively) increased and decreased by the height (above the ground) of the unbroken (part of the wall); and (then the quantities so obtained) are multiplied by the length and also by the sixth part of the (total) height. (Thus) the number of bricks intact and the number of bricks fallen off may be obtained in order. Examples in illustration thereof. 553. This high fort-wall (of measurements already given, struck by a cyclonic wind) has been (obliquely) from the bottom, broken down along the diagonal section. In relation thereto, how many are the bricks intact and the bricks fallen down? 56. The same high fort-wall has been broken down by the cyclone obliquely after leaving over 1 hasta from the bottom. How many are the bricks that remain intact and how many the bricks that have fallen down P The rule for arriving at the growing number of layers (of bricks) in relation to the central height of a fort-wall, and (rate of the) diminution of layers (also) for arriving at the 54. If a be the breadth at the bottom, b the breadth at the top, h the total 12 10 B height and / the length of the wall, and d the height above the ground of the unbroken part of the wall then th (2 a + b + d), and , and the (26 +a-d) represent the number of bricks intact and the number of bricks fallen off. The figure in the margin shows the wall mentioned in stanza 56; and ABCD ndicate the plane along which the wall fractured when it broke.

(happening to be the diminution in breadth) on both the sides (of the wall in passing from below upwards) :- 573. The height (of the central section) divided by the height of the given brick gives rise to the (required) measure of the layers (of bricks). This (number) is diminished by one and (then) divided by the difference between the top (breadth) and the bottom (breadth). The resulting quotient gives (in itself) the value of the (rate of the) diminution (in breadth) measured in terms of the layers. Examples in illustration thereof. 58. The breadth of a high fort-wall is 7 hastas at the bottom Its height is 20 hastas. It is built so as to have 1 hasta (as its breadth) at the top. With the aid of bricks of 1 hasta in height, (find out) the (measure of the) growth of the (central) layers and of the (rate of) diminution (in the breadth). 591-60. In a regularly circular well, 4 hastas in diameter, a wall of 1 hastas in thickness is built all round by means of (the already mentioned typical) bricks. The depth of that (well) is 3 hastas. If you know, calculate and tell me, O friend, how many are the bricks used in the building. In relation to a structure built of bricks (around a place), the rule for arriving at the value of the cubical contents (of that structure), when the breadth at the bottom (of the structure) is given and also the breadth at the top:-- 61. Twice the (average) thickness of the structure has added to it the given length and the breadth (of the place). The sum (so obtained) is doubled, and the result is the (total) length (of the structure when it is) in (the form of) an oblong. This (resulting quantity), multiplied by the (given) height and the (already mentioned average) thickness, gives rise to the (required) cubical measure). 59-60. The bricks contemplated here is the unit brick mentioned in stanza 43 above. This problem does not illustrate the rule given above in stanza 57. but it has to be worked according to the rales given in stanzas 191-20 and 44+ of this chapter.

An example in illustration thereof. 273 62. In relation to the (place known) as vidyadhara-naghara, the breadth is 8, and the length is 12. The thickness of the surrounding wall is 5 at the bottom and 1 at the top. Its height is 10. (What is the cubic measure of this wall ?) Thus ends (the section on) the measurement of (brick) piles in the operations relating to excavations. Hereafter, we shall expound the operations relating to the work done with saws (in sawing wood). The definitions of terms in relation thereto are as follow:- 63. Two hastas less by six angulas is what is called a kisku. The number measuring the courses of cutting from the beginning to the end of a given (log of wood) has the name of marga (or way). 64-66. Then, in relation to collections (of logs) of wood of not less than two varieties, consisting of teak logs and other such logs hereafter to be mentioned, the number of angulas measuring the breadth, and those measuring the length, and the number of margas are (all three) multiplied together. The resulting product is divided by the square of the number of angulas found in a hasta. In operations relating to saw-work, this gives rise to a valuation (of the work as measured) in (what is known as) pattikas. In relation to logs (of wood) consisting of teak logs and other such logs, the number of hastas measuring the breadth and of those measuring the length are multiplied with each other, and (then) multiplied by the number of margas, and (thereafter) divided by the pattikas as above determined; this gives rise to the numerical measure of the work done by means of the saw. 63 to 67. Kisku = 14 hasta. Marga is the name given to any desired course or line of sawing in a log of wood. The extent of the cut surface in a log of wood measures ordinarily the work done in sawing it provided that the wood is of a definite hardness assumed to be of anit value. This extent of the cut surface is measured by means of a special unit area which is called a pattika and is 96 angulas in length and one kisku or 42 angul asin breadth. It is easy to see that a pattika is thus equal to seven square hastas. 35

67-67. In relation to (logs of wood obtained from) trees named saka, arjuna, amla-vetasa, sarala, asita, sarja and dunduka, and also (in relation to varieties of wood) named sriparni and plaksa, the marga is 1 in each case, the length is 96 angulas, and the breadth is 1 kisku (for arriving at the measure of a pattika). An example in illustration thereof. 68. In relation to a log of teak wood, the length is 16 hastas, the breadth is 3 hastas and the margas (or saw-courses) are 8 in number. How many are (the units of saw-work) done here ? Thus ends the section on saw-work in the (chapter on) operations relating to excavations. Thus ends the seventh subject of treatment known as Operations relating to Excavations in Sarasangraha, which is a work on arithmetic by Mahaviracarya.