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57. A circular pond occupies half an English acre ; required the perimeter of a square circumscribed about the pond.

Anf. 1009 links. 58. Three farmers, A, B, and C, had each an inclosure. B's inclosure contains 100 English acres. A's inclosure, and that of Bs, are together it times as large as that of C's; and B's and C's are togerhet 3 times as large as that of A's ;-required the extent of A's and C's

Anf. A had 60 acres and C 120. 59. Supposing A's inclosure, as in last question, to be in the forn of a rhombus, and one of its acute angles 30°, required the expence of inclosing it with a wall 6 feet high, it feet thick, at 5 guineas per rood, standard measure.

Anf. 423 391 guineas. 60. The pit wheel of a meal-mill contains 60 сogs, and makes 16 revolutions in a minute. It drives a trundle of 8 teeth. How many revolutions will the stone make per minute ?

Anf. 120 revolutions. 61. The advantage gained by water-falls of different heights is as the square root of the heights. Now, supposing a fall of 4 feet sufficient to overcome 10 cwt. 3 qr. 7 lb. of friction, what friction will the same quantity of water overcome from a fall of 16 feet?

Anf. 1 ton i cwt. 2 qrs. 14 lb. 62. A wright engaged to build a common corn-mill under the following restrictions. The stone must revolve 75 times per minute; the trundles to have 8 rungs, the driver 80 cogs, and the fall 16 feet high. Required the diameter of the waterwheel to produce the greatest effect possible,

Anf, 13 feet 6 inches 11 pts. 8". Note. The greatest effect is obtained when the float-boards

move with one-third the velocity of the impinging fluid. 63. A tapering round tree 10 feet long, whose diameter at the greater end is 3 feet, at the less 2 feet, being hurled down a regular declivity describes a segment of a circle. How far

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distant from the greater end is the centre of the segment ?

Anf. 30 feet. 64. A barrel is filled with pure spirits, and weighs, when full, 66 lb. How many gallons does it contain, allowing 6 lb. for the weight of the barrel?

Anf. 8.29 gallons. 65. A column of the atmosphere, whose base is a square inch, weighs 15 lb. ; and supposing the atmosphere to press equally in all directions,-required the pressure upon a middle. fized man, whose surface may be reckoned 16 square feet.

Anf. 34560 lb. 66. Suppose the atmosphere, in a mean state, balance mercury in the barometer 291 inches high, required the height to which water may be raised by means of a puinp, the state of the atmosphere being the same.

Anf. 34** feet. 67. Suppofc the earth's mean distance from the sun is 82 millions of miles, and goes round him in 365 days 5 hours 49 minutes, at what rate does it travel per hour ?

Anf. 58776 mile's. 68. Light passes from the sun to the earth in 8 minutes 15 seconds of tin.e, and the velocity of the earth in its orbit is 58776 nules per hour, required the proportion they bear to each other.

Anf. The velocity of light is to that of the earth as

993931 is to 979 6. 69. In what time would the earth fall to the sun at the rate of 58776 miles per hour ?

Anf. 58 days 3 hours 7 minutes 38 seconds. 70. The paving of a square inscribed in a semicircle, whose fide coincides with the diameter, and whose opposite angles are in the circumference, at od. per square foot, colt 331. 155. required the diameter of the circle. Anf. 22.3606 yards.

71. A triangle, whose three sides are 800, 640, and 360 feet, is inscribed in a circle ; it is required to find the diameter of the circle.

Anf. 820.211 feet.

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72. The

72. The three sides of a triangular pyramid are 312, 360 and 26, and altitude 100 feet; it is required to find the solidity of a cone circumscribed about the pyramid, and whose altitude is equal to that of the pyramid. Anf 3981978 cubic feet.

73. Required the dimensions of a cone, whose area of the base, curve superficies, and solid content, are in geometrical progression, and the area of the base equal to the rectangle of the base's diameter and axis. Anf. The diameter of the cone's base is 24.7036, and axis

19.4022. 74. The area of an equilateral triangle being 720, required the side.

Anf. 40.7776. 75. Suppose I have a circular inclosure of an acre of ground, how long ought a cord be, that, fastened in the circumference of the inclosure as centre, will strike an arch that wil. divide the said inclosure into two equal parts?

Anf. 45.47898 yards76. A reservoir is supplied from a pipe of 6 inches bore. How many pipes of 3 inches bore will be sufficient to discharge the same quantity?

Ans. 4 pipes.

ALGEBRA

APPENDIX.

ALGEBRA

Algebra is a general method of computation, by which many useful problems in geometry and arithmetic are solved, which, without its aid, would be impossible. The principles on which the rules are founded are similar to those in common arithmetic.

Certain symbols and characters are admitted into this fcience, to give it that extent and excellence which it pofleffes above all other methods of computation. Through all the steps of an algebraic operation, these symbols may be fo conducted as to be preferved distin@ly in view, with their relations and affections to each other, and at last to produce a canon, or general rule, by which not only the question proposed is folved, but every other question of the like conditions. Whereas, in the course of an arithmetical operation, the original numbers dilo appear.

NOTATION.

1.-Algebraic signs only affect those symbols to which they are prefixed.

2.-Quantity is that which is made up of parts, or is capable of being encreased by addition or diminished by subtraction. Hence a quantity may be introduced into an algebraic compu3 D

tation

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tation two different ways, either as a decrement, or as an increment, that is, as a negative quantity, or as a positive one.

SIGNS.

3. + (plus) signifies addition, or that the quantity to which it is prefixed is positive *.

(minus) signifies Subtraction, or that the quantity to which it is prefixed is negative. x fignifies Multiplication to

fignifies Division.
Equal to, or the sign of equality.

(the radical sign) denotes the square root of the quan Eity annexed.

4.-A quantity may be represented by any symbol or character. It is, however, a pretty general custom to use the first letters of the alphabet, a, b, c, &c. for known quantities, and the last letters, x, y, E, for unknown ones. In the following com pendium we will follow the general method.

To examplify these ligns, iet us suppose a=3, b=8, c=12, d=10, 1=4, m=6, p=1, and s=5.

5.-Then the sum of a and b is represented thus, a+b=nr. The difference of d and ?

d-p=9.

The

* When no sign is prefixed to a quantity, + is understood.

+ When no sign is placed between two quantities, X is understood.

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