20. Find the number of gallons contained in a spherical boiler whose depth is 4 feet, and radius of brim 6 feet.

Ans. 1620.

21. Find the solidity of a pyramid whose height is 30 feet, the base being an equilateral triangle whose side is IO feet. Ans. 433 feet.

22. The diameter of a well is 6 feet, and the depth 18 yards; the sinking of which cost 6d. per foot for the first 6 yards, Is. per foot for the next 6 yards, and 1s. 6d. per foot for the last 6 yards. Find the cost of sinking.

Ans. £76. 7s. 5d.

23. Find the content of a tapering tree whose length is 11 feet, and quarter girt 6 inches. Ans. 23 feet.

24. Find the curve surface of a cone whose height is 14 feet, and diameter of base 8 feet. Ans. 183'04 feet.

25. Find the number of gallons in a bucket whose top and bottom diameters are 15 and 10 inches, and depth 13 inches. Ans. 5.832.

26. The diameter of a globe is 15 inches; find the solidity of a segment whose height is 7 inches."

Ans.. 795.6.

27. Find the number of gallons in a trough whose dimensions are 4 ft. 6 in., 3 ft. 9 in., 2 feet. Ans. 210.

28. How many gallons are contained in a cylinder whose diameter is 3 feet, and depth 2 feet?

Ans. 88.

29. Find the curve surface of a cone whose diameter is 15 inches, and slant side 24 inches. Ans. 5655

30. Find the solidity of the frustum of a cone whose top and bottom diameters are 6 and 8 inches, and depth 91 inches. Ans. 41191

31. The length and breadth of the thick end of a wedge are 20 and 6 feet, the length of the thin end 22 feet, and the perpendicular from the thin end upon the thick end 16 feet; find the solidity. Ans. 3629 yards.

32. Find the solidity of a spherical segment whose height is 2 ft. 6 in., and radius 3 ft. Ans. 43 541 feet.

33. Find the curve surface of a cone whose altitude is 12 inches, and diameter 10 inches. Ans. 204.

34. The diameter of a globe is 6 inches; find the solidity and curve surface of a segment whose height is 2 inches. Ans. 29, 37.

35. Find the weight of a cast-iron pipe whose length is 9 feet, the bore 3 inches, and the thickness of the metal I inch. Ans. 3 cwt. 33 lbs.

36. Find the surface of a sphere whose diameter is 33 inches. Ans. 23 feet.

37. Find the curve surface of the frustum of a cone whose top and bottom diameters are 8 ft. 6 in. and 19 ft. ro in., and height 23 ft. 9 in. Ans. 1087. I. 5 feet.

38. Find the weight of a cannon ball whose diameter is 5 inches. Ans. 21322 lbs.

39.

Find the weight of a spherical shell whose diameter is 13 inches, the thickness of the metal being 11 inches.

Ans. 136 lbs.

40. The length and breadth of the thick end of the frustum of a wedge are 32 and 10 feet, of the thin end 19 and I feet, and the perpendicular from the thin end upon the thick end 24 feet; find the solidity. Ans. 133 yards.

41. Find the content of a regular triangular pyramid whose edge is 24 feet. Ans. 1629 158 feet.

42. The three edges of a parallelopipedon are as 7, 9, 11, and it contains 1901592 cubic feet; find the edges. Ans. 98, 126, 154.

43. Find the solidity of an anchor-ring whose thickness is 7 inches, and diameter 12 feet. Ans. 10972 feet.

44. Find the cost of gilding a ball 22 inches in diameter, at 4s. 92d. per foot. £2. 10s. 7 d.

Ι

45. If I lb. of sugar measure 25 cubic inches, find the cost of a sugar-loaf whose diameter is 10 inches, and slant side 24 inches, at 5d. per lb. Ans. IIS. 3d.

46. How many barrels of beer are there in a cylindrical vat whose depth is 15 inches and diameter 15 feet? Ans. 38.

47. Find the solidity of a prolate spheroid whose transverse and conjugate diameters are 64 and 43 feet. Ans. 76.819 feet.

48. Find the cost of painting the curve surface of a spherical segment whose height is 5 feet, and radius of base 8 feet, at 2d. per foot. Ans. £2. 18s. 3d.

49. Find the solidity of an oblate spheroid whose transverse and conjugate diameters are 15 and 11 feet. Ans. 12963 feet.

50. The lower part of a stack of wheat is the frustum of a cone whose top and bottom diameters are 15 and 12 feet, and height 9 feet; the upper part is a perfect cone whose height is 8 feet. Find how many sheaves it contains, each sheave containing 1 feet. Ans. 14123.

51. The transverse and conjugate diameters of a prolate spheroid are 14 and 8 feet; find the solidity of a segment whose height is 5 feet. Ans. 136.831 feet.

52. The upper section of a gas-holder is 110 feet in diameter, and the lower section 112 feet; each section contains 9 sheets, 2 feet deep, and each sheet has 24 rivets. Calculate the quantity of gas between two rivets of the upper and lower sections; and also the whole quantity of gas contained in 7 sheets 13 rivets, and in 12 sheets 8 rivets; supposing sheet of the lower section to be taken. up by the coupling.

I inch of upper section............ 792 feet

and

APPENDIX.

1. A lady, wealthy, kind, and fair,
Your aid, dear sir, would gladly share,
In finding of a plot of ground,

Which three right lines exactly bound.
The space and ambit, that you're told,
Both the same figure just unfold;
The sides-more data to supply,
Your skill she'd not severely try-
Are in arithmetic train,

And a right angle two contain.
Now sure this fair you may relieve,
And shew what science can achieve.
Mathematical Companion, 1812.

Let x-y, x, x+y be the sides; then

These equations reduce to

x-y=6,

x-4y=0;

.. y = 2, x = 8.

Hence the sides are 6, 8, 10.