पृष्ठम्:Ganita Sara Sangraha - Sanskrit.djvu/४६२

264 GANITASĀRASANGRAHA. measure of the depth also is known, in refation 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. 27. 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? 20 The problems contemplated in this stanza ale (a) to find out the full latitude of an inverted pyramid or cone and (b) to find out the dimensions of the cross section thercof 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 pyrand 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, I taken as the height of the whole pyramid axh ab and the measure of a side of the cross section of the pyramid at any given height 1epresented by h₁a (H-h₂) 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.