2. What is the content of the piece of timber, whose length is 243ft, and the mean breadth and thickness each 1.04ft? Ans. 26 ft.

3. Required the content of a piece of timber, whose length is 20-38ft, and its ends unequal squares, the side of the greater being 19ĝin, and the side of the less 97in. Ans. 29 7562ft. 4. Required the content of the piece of timber, whose length is 27.36ft; at the greater end the breadth is 1:78ft and thickness 1:23ft; and at the less end the breadth is 1'04ft, and thickness 0.91ft. Ans. 41.278ft.

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 10th, on D content on C.

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 taking half the sum of the two; or by girting it in several places, then adding all the girts together, and dividing the sum by the number of them, for the mean girt: but when the tree is very irregular, divide it into several lengths, and find the content of each part separately.

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.

On this subject, however, Hutton's Mensuration, pt. v. sect. 4, may be advantageously consulted.

EXAMPLES.

1. A piece of round timber being 9ft 6in long, and its mean quarter girt 42in; what is the content? Ans. 116 ft.

2. The length of a tree is 24ft, its girt at the thicker end 14ft, and at the smaller end 2ft; required the content. Ans. 96ft. 3. What is the content of a tree whose mean girt is 3·15ft, and length 14ft 6in ? Ans. 8-9922ft. 4. Required the content of a tree, whose length is 174ft, and which girts in five different places as follows, namely, in the first place 9:43ft, in the second 7.92ft, in the third 6.15ft, in the fourth 4.74ft, and in the fifth 3·16ft.

Ans. 42.519525.

PRACTICAL EXERCISES IN MENSURATION.

1. What difference is there between a floor 28ft long by 20 broad, and two others, each of half the dimensions: and what do all three come to at 45s per square of 100ft? Ans. dif. 280ft; amount 18 guineas.

2. An elm plank is 14ft 3in 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 in.

3. A ceiling contains 114yds 6ft of plastering, and the room is 28ft broad; what is the length of it? Ans. 369ft. 4. A common joist is 7in deep, and 24in thick; but I want a scantling just as big again, that shall be 3in thick; what will the other dimension be?

Ans. 113in.

5. A wooden trough, length 102in, and depth 21in, cost me 3s 2d painting, within, at 6d per yd: what was the width ? Ans. 27 in. 6. If my court-yard be 47ft 9in square, and I have laid a foot-path with Purbeck-stone, of 4ft wide, along one side of it; what will paving the rest with flints come to at 6d per square yd? Ans. 51 168 0žd.

7. A ladder, 36ft long, may be so placed, that it shall reach a window 30.7ft from the ground on one side of the street; and, by only turning it over, without moving the foot out of its place, it will do the same by a window 18.9ft high on the other side; what is the breadth of the street, and the angle of elevation of the second window from the first?

Ans. the street is 49:4414ft wide; and the elevation is 13° 25′ 24′′. 8. The paving of a triangular court, at 18d per ft, came to 1007; the longest of the three sides was 88ft; required the sum of the other two equal sides?

Ans. 106.85ft.

9. The perambulator, or surveying-wheel, is so contrived, as to turn twice in the length of a pole, or 164ft; required the diameter. Ans. 2.626ft.

10. In turning a one-horse chaise within a ring of a certain diameter, it was observed, that the outer wheel made two turns, while the inner made but one : the wheels were both 4ft high; and, supposing them fixed at the statutable distance of 5ft asunder on the axle-tree, what was the circumference of the track described by the outer wheel? Ans. 62-832ft. 11. What is the side of that equilateral triangle, whose area cost as much paving at 8d a ft, as the palisading the three sides did at a guinea a yd?

Ans. 72.746ft. 12. A roof, which is 24ft 8in by 14ft 6in, is to be covered with lead at 81b per square ft; find the price at 18s per cwt. Ans. 221 19s 104d.

13. Having a rectangular marble slab, 58in 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 repeated, till there shall not be the quantity of a foot left; what will be the dimensions of the remaining piece * ? Ans. 20 7in by 6'086.

14. Given two sides of an obtuse-angled triangle, which 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.

15. How many bricks will it take to build a wall, 10ft high, and 500ft long, of a brick and half thick, reckoning the brick 10 inches long, and four courses to the foot in height? Ans. 72000.

16. How many bricks will build a square pyramid of 100ft on each side at the base, and also 100ft perpendicular height, the dimensions of a brick being supposed 10in long, 5in broad, and 3in thick? Ans. 3840000.

17. If, from a right-angled triangle, whose base is 12, and perpendicular 16ft,

*This question may be solved neatly by an algebraical process, as may be seen in the Ladies' Diary for 1823. In applying the formulæ there found, the term to stop at is that whose ordinal number is the number of entire feet in the slab which in the present case is 10, since 58.27 = 107ft.

a line be drawn parallel to the perpendicular, cutting off a triangle whose area is 24ft; required the sides of this triangle. Ans. 6, 8, and 10.

18. If a round pillar, 7in across, have 4ft of stone in it; of what diameter is the column, of equal length, that contains 10 times as much? Ans. 22.136in. 19. A circular fish-pond is to be made in a garden, that shall take up half an acre; required the length of the cord that strikes the circle? Ans. 27 yds.

20. When a roof is of a true pitch, the rafters are å of the breadth of the building. Now supposing the eaves-boards to project 10in on a side, what will the new ripping a house cost, that measures 32ft 9in long, by 22ft 9in broad on the flat, at 15s per square? Ans. 87 15s 91d. 21. A cable, which is 3ft long and 9in in compass, weighs 221b; required the weight of a fathom of a cable which measures a foot round? Ans. 783lb.

22. A plumber has put 281b per square foot into a cistern, 74in and twice the thickness of the lead long, 26in broad, and 40 deep; he has also put three stays across it within, 16in deep, of the same strength, and reckons 22s per cwt for work and materials: a mason has in return paved him a workshop, 22ft 10in broad, with Purbeck-stone, at 7d per ft; and upon the balance finds there is 3s 6d due to the plumber: what was the length of the workshop, supposing sheet lead of an inch thick to weigh 5.8991b per ft? Ans. 32.2825ft.

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23. The girt or outside circumference of a vessel is 44in, the hoop is lin thick, and the height of the vessel is 24in; required its content in imperial gallons. Ans. 9 7892 gallons.

24. If 20ft of iron railing weigh half a ton, when the bars are an inch and quarter square; what will 50ft come to at 3d per lb, the bars being but of an inch square? Ans. 201 Os 2d.

25. It is required to find the thickness of the lead in a pipe, of an inch and quarter bore, which weighs 14lb per yd in length; the cubic foot of lead weighing 11325 ounces? Ans. ⚫20737in.

26. Supposing the expense of paving a semicircular plot, at 2s 4d per ft, come to 107; what is its diameter ? Ans. 14.7737ft.

27. What is the length of a chord which cuts off one-third of the area from a circle whose diameter is 289 ? Ans. 278 6716.

28. My plumber has set me up a cistern, and, his shopbook being burnt, he has no means of bringing in the charge, and I do not wish to take it down to have it weighed; but by measure he finds it contains 64 square feet, and that it is precisely of an inch in thickness. If lead was then wrought at 21 per fother of 193cwt., can we from these items make out the bill, allowing 63 oz for the weight of a cubic inch of lead? Ans. 4l 11s 2d.

29. What will the diameter of a globe be, when the solidity and superficial content are expressed by the same number? Ans. 6. 30. A sack, that would hold 3 bushels of corn, is 22 in broad when empty; what will another sack contain, which, being of the same length, has twice its breadth? Ans. 12 bushels. 31. A carpenter, is to put an oaken curb to a round well, at 8d per foot square; the breadth of the curb is to be 8in, and the diameter within 3ft: what will be the expense? Ans. 5s 9åd.

32. A gentleman has a garden 100ft long, and 80ft broad; and a gravel walk is to be made of an equal width half round it: determine both by construction and calculation the breadth of the walk, to take up just half the ground.

Ans. 25.968ft.

33. The top of a may-pole, being broken off by a blast of wind, struck the ground at 15ft from the foot of the pole; what was the height of the whole may-pole, supposing the length of the broken piece to be 39ft? Ans. 75ft.

34. Seven men bought a grinding-stone of 60in diameter, each paying part of the expense; what part of the diameter must each grind down for his share? Also, exhibit the solution by a geometrical construction.

Ans. the 1st 4:4508, 2d 4.8400, 3d 5-3535, 4th 6.0765, 5th 7.2079, 6th 9.3935, 7th 22.6778in.

35. A maltster has a kiln, that is 16ft 6in square; but he wants to pull it down, and build a new one, that may dry three times as much as the old one; what must be the length of its side?

Ans. 28ft 7in.
Ans. 64.

36. How many 3in cubes may be cut out of a 12in cube? 37. How long must the tether of a horse be, that will allow him to graze an acre of ground? Ans. 39 yds. 38. What will the cost of painting a conical spire come to at 8d per yd; the height being 118ft, and the circuit of the base 64ft? Ans. 147 Os 8 d. 39. The diameter of an old standard corn bushel is 184in, and its depth 8in; what must be the diameter of that bushel whose depth is 74in? 40. The ball on the top of St. Paul's church is 6ft diameter; it cost at 34d per square inch?

Ans. 19·1067in. what did gilding Ans. 2371 10s 1d.

41. What will a frustum of a marble cone come to at 12s per ft; the diameter of the greater end being 4ft, that of the less end 11ft, and the length of the slant side 8ft? Ans. 301 1s 101d.

42. Divide a cone into three equal parts by sections parallel to the base, and find the heights of the three parts, that of the whole cone being 20in.

Ans. the upper 13.867, the middle 3.605, the lower 2.528. 43. A gentleman has a bowling-green, 300ft long, and 200ft broad, which he wishes to raise 1ft higher, by means of the earth to be dug out of a ditch surrounding it to what depth must the ditch be dug, supposing its breadth to be every where 8ft? Ans. 78ft.

44. How high above the earth must a person be raised, that he may see of its surface and under what angle will the earth then appear?

Ans. to the height of the earth's diameter; angle 38° 56′ 32′′. 45. A cubic foot of brass is to be drawn into wire of inch in diameter; what will the length of the wire be, allowing no loss in the metal?

Ans. 97784 797 yds, or 55mls 984-797yds.

46. Of what diameter must the bore of a cannon be, which is cast for a ball of 24lb weight, so that the diameter of the bore may be of an inch more than that of the ball? Ans. 5.647in.

47. Supposing the diameter of an iron 91b ball to be 4in, it is required to find the diameter of the several balls weighing 1, 2, 3, 4, 6, 12, 18, 24, 32, 36, and 42lb, and the calibre of their guns, allowing of the calibre, or of the ball's diameter, for windage.

49

48. Supposing the windage of all mortars to be of the calibre, and the diameter of the hollow part of the shell to be 7 of the calibre of the mortar: it

is required to determine the diameter and weight of the shell, and the quantity or weight of powder requisite to fill it, for each of the several sorts of mortars, namely, the 13, 10, 8, 5'8, and 4.6in mortar.

49. If a heavy sphere, whose diameter is 4in, be let fall into a conical glass, full of water, whose diameter is 5, and altitude 6in; it is required to determine how much water will be displaced. Ans. 26-272 cubic in, or nearly pint.

50. The dimensions of a sphere and cone being the same as in the last question, and the cone only full of water; what part of the axis of the sphere is immersed in the water? Ans. 546 parts of an inch.

51. (1) If R and r be the radii of two spheres inscribed in a cone, so that the greater may touch the less, and that planes are drawn to touch the spheres at their intersections with the axis of the cone: it is required to prove that the volumes of the three cones thus cut off by the planes, and estimated from the vertex, are respectively expressed by

(2) It is likewise required to prove that if any number of spheres are inscribed in a cone to touch each other in succession, that they will be in geometrical progression.

52. If a person, in an air balloon, ascend vertically from London, to such height that he can just see Oxford in the horizon; required his height above the earth, supposing its circumference to be 25000 miles, and the distance between London and Oxford 49-5933 miles? Ans. nearly m, or 547 yds 1ft.

53. In a garrison there are three remarkable objects, A, B, C, the distances of which from one to another are known to be, AB = 213, AC = 424, and BC = 262 yds. I am desirous of knowing my position and distance at a place or station S, from whence I observed the angle ASB = 13° 30′, and the angle CSB : 29° 50′, both by geometry and trigonometry, the point S being on the same side of AC with B. Ans. AS 605.7122, BS = = 429 6814, CS: =524.2365. 54. Required the same as in the last question, when the point B is on the other side of AC, supposing AB = 9, AC 12, and BC = 6 furlongs; also the angle ASB = 33° 45′, and the angle BSC = 22° 30'.

=

Ans. AS 10·65, BS = 15.64, CS = 14.01. 55. It is required to determine the magnitude of a cube of standard gold, which shall be equal to £960000000; supposing a guinea to weigh 5dwts 9 grs. Ans. 23 549ft.

56. The ditch of a "fortification is 1000ft long, 9ft deep, 20ft broad at the bottom, and 22 at the top; how much water will fill the ditch?

Ans. 1177867gall nearly. 57. If the diameter of the earth be 7930 miles, and that of the moon 2160