58. How many shot are in an incomplete oblong pile, the length and breadth of the base being 50 and 20, and the length and breadth at the top 38 and 8? Ans. 8190 balls 59. Required the weight of lead in a pipe 600 yards long, the diameter of the bore being 14 inches, and the thickness of the metal inch. Ans. 10448.2744 lbs.

60. Required the content of a frustum of a cone, of which the greatest diameter is 60 inches, the diagonal between the farthest extremities of the diameters 66, and the slant side 30 inches. Ans. 293.61 imp. gallons.

61. If a heavy sphere, of which the diameter is 4 inches, is dropt into a conical glass full of water, of which the diameter is 5 inches, and the altitude 6 inches, How much water will run over? Ans. 26.27215 cubic inches.

62. Suppose it is found that a ship, ging, &c. displaces 50,000 cubical feet weight of the vessel ?

with its ordnance, rig of water, What is the Ans. 1395-0893 tons.

63. If a solid inch of metal weighs 8 ounces avoirdupois, What is its specific gravity?

Ans. 13824.

64. If a man weighs 192 lbs., and the specific gravity of his body be 1200, How much cork must be tied to him to make him swim? Ans. 10 lbs.

65. If a cube of solid fir, 12 inches each way, sinks 6 inches in water, What is its specific gravity?

Ans. 500. 66. Four solid inches of copper is to be made into a hollow cube. How thick must the metal be that it may swim in one inch depth of water? Ans. 01863 inches

67. If two solid feet of feathers weigh 4 lbs., What will the same quantity weigh when compressed into the bulk of half solid foot, supposing a solid foot of air to weigh 13 oz.? Ans. 4 lbs. 1.8 oz.

68. If a man standing at the side of a river hears his voice reflected from the opposite bank in 3 seconds of time, What is the breadth of the river? Ans. 1713 feet.

69. I saw the flash of a gun fired from a ship at sea, and 38 seconds afterwards I heard the report. How far was the ship distant from me? Ans. 7 miles

70. Observing a battery of cannon, I counted 17 seconds on my watch between the times of seeing the flash and of hearing the report. How far was I distant from the battery? Ans. 377 miles.

71. The frustum of a cone is 5.7 inches in height, the dia

R

eter at the top 3.7 inches, and that at the bottom 4.23 inches. equired the difference between the contents of the hoofs into hich it is divided by a plane passing through the opposite tremities of its diameters. Ans. 7.0532 cubic inches.

72. Required the contents of the hoofs into which a cone of hich the height is 6 inches, the top diameter 3, and the bot ́m diameter 4 inches, is divided by a plane passing from the ige of the top to the centre of the base.

ns. The less hoof 15-2628, the greater 42.8568 cubic inches. 73. Suppose a cubic inch of common glass to weigh 1.4921 1. avoirdupois, one of sea-water 59542 oz., and one of brandy 368 oz. How much force will be required to buoy up in the a an imperial gallon of brandy in a bottle, of which the weight f the glass in air is 3.84 lbs. ? Ans. 20-669 oz.

74. How far will a body descend from a state of rest in 20 econds? Ans. 6433 feet. 75. If a body is projected perpendicularly in free space with velocity of 10,000 feet per second, To what height would it scend, and in what time would it again reach the earth?

Ans. 2948 miles, and in 621113 seconds.

76. Suppose that at the moment a body is projected up AB vith the velocity acquired by falling down it, another body begins to fall down it, In what point will they meet, AB being 1029 feet? Ans. 772 feet from the bottom.

77. Suppose that a body is projected downwards with a velocity of 64 feet per second, and in 2 seconds after another body is projected down with a velocity of 258 feet, In what time will it overtake the other? Ans. 14 second.

78. A person from a window 20 feet high observes in a mirror placed 12 feet from the foundation of the house the top of a spire 100 feet high. Required the distance of the observer from the spire. Ans. 72 feet.

79. Melville's Monument in St Andrew's Square, Edinburgh, is 136 feet 4 inches high, and the statue on the top 14 feet high. At what distance from the base of the monument does the statue subtend the greatest angle?

Ans. 143.1622 feet.

80. Two trees, 100 feet asunder, are placed, the one at the distance of 100 feet, and the other 50 feet from a wall. What is the shortest distance that a person must pass over in running from one tree to touch the wall, and then to the other tree? Ans. 171.334 feet. 81. I took two stations A and B at the distance of 150 feet

from each other, and in the same straight line with an inaccessible spire; then from A, the station nearest the spire, in a line per. pendicular to the line AB, I measured AC 160 feet, and set up a pole at the extremity C; and from B, the other station in a line also perpendicular to AB, I measured the distance BD 275.5 feet, when I observed that the spire and the pole at C were in the same straight line with the point D. Required the distance of the spire from the station A.

Ans. 207-79 feet. 82. What is the weight of a sphere of oak 6 feet in diameter, its specific gravity being 925? Ans. 2-91895 tons.

83. To what depth would a cube of beech 2 feet 6 inches in the side sink in water? Ans. 2-13 inches.

84. A horse's tether of 40 yards in length is fixed in the cir cumference of a circular field whose diameter is 350 yards How much will it allow him to graze? And, supposing that the end of the tether is removed to the circumference of the secondary circle, and in a line with the centre of the field, What additional space would he be enabled to graze?

Ans. First 2391-2695 square yards; and afterwards 3061-1712 square yards.

85. The axes of a punch-bowl in the form of the segment of an oblong spheroid are to each other as 3 to 4, the depth is of the longer axis, and the diameter of its top is 20 inches. What number of rounds may a company of 30 persons drink out of it, using a conical glass of which the top diameter is 1 inches, and the depth 2 inches? Ans. 38 01499 rounds.

86. A certain island is 73 miles in circumference, and if 2 men set out from the same point in the same direction, the one travelling at the rate of 5 and the other at the rate of 3 miles an hour, In what time will they be together again?

Ans. 361⁄2 hours. 87. Required the solidity of the greatest cone which can be cut out of an oblong spheroid of which the axes are 40 and 60 inches.

Ans. 22340-26 feet when the axis of the cone is in the minor axis of the spheroid, and 14893-51 when the axis of the cone is in the major axis.

88. Suppose a cone 20 feet high, and the diameter of the base 6 feet, is cut through the axis 5 feet from the bottom, at an angle of 60 degrees. Required the solidity of the sections. Ans. Solidity of the upper 82.296 feet. Solidity of the under 106.2 feet.

APPENDIX.

SECTION I.

GENERAL PRINCIPLES OF GEOMETRY.

He demonstrations of many of the rules given in Trigonoetry and Mensuration were judged too long to be inserted in e text; they are, therefore, added here, and to them are prexed the general principles of geometry upon which they epend.

A straight line may be drawn between two points, by laying ruler or another straight line by these points, and tracing a ne along the side of it.

But the only original method of producing a straight line , by stretching a hair or thread through the two points; nd as the thread assumes invariably the same position as often 3 it is stretched through the same points, and a less portion of lies between the points when it is stretched, than when it es loosely between them, it follows,

First, That a straight line between two points has only one

osition.

Secondly, That both sides of a straight line are exactly like.*

Thirdly, That a part of a straight line is in every respect imilar to another part of it, or to another straight line of the ame length.

Fourthly, That the straight line is the shortest distance rom one point to another.

From these properties of a straight line it is inferred,

1st, That two straight lines will coincide when they are aplied to one another, in what way soever the application is made.

2d, That one straight line cannot cut another in more points than one.

* If a hair stretched between the points Aand B coincide with the trace AB, and if then the part of it at A be brought

-B

o B, and that at B to A, so that the upper side of it may now be the ower one, the stretched hair will again coincide with the trace AB.

3d, And consequently that two straight lines can neither have a common segment nor enclose a space.

4th, That a straight line is less than a curve, or than the sum of any number of straight lines joined together, which terminate at the same points with it.

5th, That straight lines which have the same position, in respect of the same straight line, must either coincide or be parallel to one another.*

PROPOSITION I. Two triangles ABC, GHK are equal in every respect, when an angle BAC and the two sides AB, AC, which contain it in one of them, are respectively equal to an angle HGK, and the sides GH, GI containing it in the other.

For, if the triangle ABC lie on GHK, so that A be on G and AB on GH, then AC will lie along GK, for the angle A= G, and B will be on H, and C on K; therefore BC will coincide with HK, the triangle ABC with GHK, the angle B with H, and C with K. They are all therefore equal.

B

CH

PROP. II. If a side AB, and the two adjacent angles at A and B of one triangle ABC, be equal to a side DE, and the adjacent angles at D and E of another, the triangles are in all respects equal.

D

Да

For, if the triangle ABC be laid on DEF, A on D, and AB on DE, then B will be on E, AC on DF and BC on EF, because the angles at A and B are equal to those at D and E; therefore, the angle C shall be on F, CE F and the triangle ABC will coincide altogether with DEF, and be equal to it.

B

A

E

IC/H
B

F

D G

* If the straight lines AB and CD intersect in E, the angle CEB shows their relative situations; and these situations would remain though they should intersect in any other point of CD, as at D; in which case AB would become FG, and EC would coincide with DE. Of course, if the angle EDG be equal to CEB, the lines AB and FG would have the same direction, and if they have the same direction, the angle EDG would be equal to CEB; and for the same reason the angle HEB would be equal to EFD. These things seem to follow immediately from the definitions of straight line and of an angle, and, if admitted as principles, they would render several parts of geometry easy, which are at present difficult.

at

a