Ex. IV. Required the quantity of plastering in a room, the length being 14 feet 5 inches, breadth 13 feet 2 inches, and height 9 feet 3 inches to the under side of the cornice, which girts 8 inches, and projects 5 inches from the wall on the upper part next the ceiling-deducting only for a door 7 feet by 4? Ans. 53 yards 5 feet 3 inches of rendering, of ceiling, 18 5 6 39 011 of cornice. VIIL-PAINTERS WORK. PAINTERS' work is computed in square yards. Every part is measured where the colour lies; and the measuring line is forced into all the mouldings and corners. Windows are done at so much a piece. And it is usual to allow double measure for carved mouldings, &c. EXAMPLES. Ex. 1.—How many yards of painting contains the room which is 65 feet 6 inches in compass, and 12 feet 4 inches high? Ans. 893 yards. Ex. II. The length of a room being 20 feet, its breadth 14 feet 6 inches, and height 10 feet 4 inches; how many yards of painting are in it, deducting a fire-place of 4 feet by 4 feet 4 inches, and two windows each 6 feet by 3 feet 2 inches? Ans. 73 yards. Ex. III.-What costs the painting of a room at 6d. per yard; its length being 24 feet 6 inches, its breadth 16 feet 3 inches, and height 12 feet 9 inches; also the door is 7 feet by 3 feet 6 inches, and the window-shutters to two windows each 7 feet 9 inches by 3 feet 6 inches, but the breaks of the windows themselves are 8 feet 6 inches high, and 1 foot 3 inches deep-deducting the fire-place of 5 feet by 5 feet 6 inches? Ans. £3, 3s. 101d. IX-GLAZIERS WORK. GLAZIERS take their dimensions either in feet, inches, and parts, or feet, tenths, and hundredths. And they compute their work in square feet. In taking the length and breadth of a window, the cross bars between the squares are included. Also, windows of round or oval forms are measured as square, measuring them to their greatest length and breadth, on account of the waste in cutting the glass, EXAMPLES. Ex. 1.-How many square feet contains the window which is 4.25 feet long, and 2.75 feet broad? Ans. 113. Ex. 11.-What will the glazing a triangular skylight come to, at 10d. a foot; the base being 12 feet 6 inches and the perpendicular height 6 feet 9 inches? Ans. 1. 15s. 1ąd. Ex. III. There is a house with three tier of windows, three windows in each tier, their common breadth 3 feet 11 inches; now, The height of the first tier is 7 feet 10 inches, 6 8 5 4 of the third Required the expense of glazing, at 14d. per foot? Ans. £13, 11s. 104d. Ex. iv.-Required the expense of glazing the windows of a house at 13d. a foot; there being three stories, and three windows in each story; The height of the lower tier is 7 feet 9 inches, 6 6 of the middle of the upper 5 ................... and of an oval window over the door 1 ......... ......... 31 10/ the common breadth of all the windows being 3 feet 9 inches ? X.-PAVERS WORK. PAVERS' work is done by the square yard. And the content is found by multiplying the length by the breadth. EXAMPLES. Ans. £12, 5s. 6d. Ex. I. -What cost the paving a footpath at 3s. 4d. a yard; the length being 35 feet 4 inches, and breadth 8 feet 3 inches? Ans. £5, 7s. 114d. Ex. 11.-What cost the paving a court, at 3s. 2d. per yard; the length being 27 feet 10 inches, and the breadth 14 feet 9 inches ? Ans. £7, 4s. 54d. Ex. 111.—What will be the expense of paving a rectangular court-yard, whose length is 63 feet, and breadth 45 feet; in which there is laid a footpath of 5 feet 3 inches broad, running the whole length, with broad stones, at 3s. a yard— the rest being paved with pebbles, at 2s. 6d. a yard ? Ans. £40, 5s. 10 d. XI.-PLUMBERS' WORK. PLUMBERS' work is rated at so much a pound, or else by the hundred weight, of 112 pounds. Sheet lead used in roofing, guttering, &c., is from 7 to 12 lb. to the square foot. And a pipe of an inch bore is commonly 13 to 14 lb. to the yard in length. EXAMPLES. Ex. 1.-How much weighs the lead which is 39 feet 6 inches long, and 3 feet 3 inches broad, at 81⁄2 lb. to the square foot ? Ans. 1091 lb. Ex. II.-What cost the covering and guttering a roof with lead, at 18s. the cwt.; the length of the roof being 43 feet, and breadth or girt over it 32 feetthe guttering 57 feet long, and 2 feet wide the former 9-831 lb., and the latter 7.373 lb. to the square foot? Ans. £115, 9s. 11⁄2d. • XII TIMBER MEASURING PROBLEM I To find the area or superficial content of a bourd or plank. Multiply the length by the mean breadth. Note. When the board is tapering, add the breadths at the two ends toge ther, and take half the sum for the mean breadth. BY THE SLIDING RULE. Set 12 on B on the breadth in inches on A; then against the length in feet on B is the content on A, in feet and fractional parts. EXAMPLES. Ex. 1.-What is the value of a plank, at låd. per foot, whose length is 12 feet 6 inches, and mean breadth 11 inches? Ans. 1s. 5d. Ex. 11.—Required the content of a board, whose length is 11 feet 2 inches, and breadth 1 foot 10 inches. Ans. 20 feet, 5 inches, 8". Ex. III.—What is the value of a plank, which is 12 feet 9 inches long, and 1 foot 3 inches broad, at 24d. a foot? Ans. 3s. 33d. Ex. Iv.—Required the value of 5 oaken planks at 3d. per foot, each of them being 17 feet long, and their several breadths as follows; namely, two of 13 inches in the middle, one of 141⁄2 inches in the middle, and the two remaining ones, each 18 inches at the broader end, and 114 at the narrower. Ans. £1, 5s. 94d. PROBLEM II. To find the solid content of squared or four-sided timber. Multiply the mean breadth by the mean thickness, and the product again by the length, and the last product will give the content. That is, as the length in feet on C, is to 12 on D when the quarter girt is in inches, or to 10 on D when it is in tenths of feet; so is the quarter girt on D, to the content on C. Note 1.—If the tree taper regularly from the one end to the other; either take the mean breadth and thickness in the middle, or take the dimensions at the two ends, and half their sum will be the mean dimensions. Note 2.-If the piece do not taper regularly, but is unequally thick in some parts and small in others, take several different dimensions, add them all together, and divide their sum by the number of them, for the mean dimensions. Note 3.-The quarter girt is a geometrical mean proportional between the mean breadth and thickness, that is the square root of their product. Sometimes unskilful measurers use the arithmetical mean instead of it, that is, half their sum; but this is always attended with error, and the more so as the breadth and depth differ the more from each other. EXAMPLES. Ex. 1.—The length of a piece of timber is 18 feet 6 inches, the breadths at the greater and less end 1 foot 6 inches and 1 foot 3 inches, and the thickness at the greater and less end 1 foot 3 inches and 1 foot: required the solid content? Ans. 28 feet 7 inches. Ex. II.—What is the content of the piece of timber whose length is 24 feet, and the mean breadth and thickness each 1.04 feet? Ans. 26 feet. Ex. I.—Required the content of a piece of timber, whose length is 20-38 feet, and its ends unequal squares, the side of the greater being 19§, and the side of the less 97? Ans. 29-7562 feet. Ex. IV. Required the content of the piece of timber whose length is 27:36 feet, at the greater end the breadth is 1.78, and thickness 1·23; and at the less end the breadth is 104, and thickness 0.91? Ans. 41-278 feet. PROBLEM III. To find the solidity of round or unsquared timber. Multiply the square of the quarter girt, or of of the mean circumference, by the length, for the content. BY THE SLIDING RULE. As the length upon C: 12 or 10 upon D :: quarter girt, in 12ths or 10ths on D: content on C. Note 1.-When the tree is tapering, take the mean dimensions as in the former problems, either by girting it in the middle, for the mean girt, or at the two ends, and take half the sum of the two. But when the tree is very irregular, divide it into several lengths, and find the content of each part separately. Note 2. This rule, which is commonly used, gives the answer about less than the true quantity in the tree, or nearly what the quantity would be after the tree is hewed square in the usual way; so that it seems intended to make an allowance for the squaring of the tree. EXAMPLES. Ex. I.-A piece of round timber being 9 feet 6 inches long, and its mean quarter girt 42 inches; what is the content? Ans. 116 feet. Ex. II.-The length of a tree is 24 feet, its girt at the thicker end 14 feet, and at the smaller end 2 feet; required the content? Ans. 96 feet. Ex. III.-What is the content of a tree, whose mean girt is 3.15 feet, and length 14 feet 6 inches ? Ans. 8.9922 feet. Ex. IV. Required the content of a tree, whose length is 174 feet, which girts in five different places as follows; namely, in the first place 9.43 feet, in the second 7.92, in the third 6-15, in the fourth 474, and in the fifth 3-16? Ans. 42.519525 PRACTICAL QUESTIONS IN MENSURATION. 1. A plank is 14 feet 3 inches long, and I would have just a square yard slit off it; at what distance from the edge must the line be struck? Ans. 7 inches. 2. A wooden trough cost 3s. 2d. painting within, at 6d. per yard; the length of it is 102 inches, and the depth 21 inches; what is the width ? Ans. 274 inches. 3. The paving of a triangular court, at 18d. per foot, came to £100; the longest of the three sides is 88 feet; required the sum of the other two equal sides? Ans. 106.85 feet. 4. What is the side of that equilateral triangle whose paving at 8d. a foot, as the palisading the three sides did at area cost as much a guinea a yard? Ans. 72.746 feet. 5. Let a, b, c be the sides of a triangle respectively opposite to the angles A, B, C; then will the area of the triangle ABC be = a2 sin. B sin. C cosec. A. 6. Let a, b, c be the three sides of a triangle; put h=b+c and k=b—c; then will the area of the triangle be=√(h2—a2) (a2—k2). 7. Let the three sides be va, √b, √c; then prove that the area of the triangle is 4√2(ab+be+ca)-(a2+b2+c2). 8. A beam is 8 inches deep and 34 inches broad; what is the depth of another twice as large, which is 44 inches broad? Ans. 12.5263 inches. 9. Supposing the expense of paving a semicircular plot, at 2s. 4ď. per foot, come to £10; what is its diameter ? Ans. 14-7737 feet. 10. Two sides of an obtuse angled triangle are 20 and 40 poles; required the third side, that the triangle may contain just an acre of land? Ans. 58-876, or 23.099. 11. A circular fish-pond is to be made in a garden that shall enclose just half an acre; what must be the length of the chord that strikes the circle? Ans. 27 yards. 12. Having a rectangular marble slab, 58 inches by 27, I would have a square foot cut off parallel to the shorter edge; I would then have the like quantity divided from the remainder, parallel to the longer side; and this alternately |