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OF SPECIFIC GRAVITY.

Def. 1. The specific gravity of a body, is the relation that the weight of a magnitude of one kind of body, has to the weight of an equal magnitude of another kind.

2. In this comparison of the weight of bodies, it is convenient to consider one body as the standard or unit, to which the others are to be compared ; and as rain water is nearly alike in all places, it is the most convenient standard.

3. It has been found, by repeated experiments, that a cubic foot of

5

rain water weighed 624 pounds avoirdupois; consequently, (1728)

003616898lb. is the weight of one cubic inch of rain water.

4. The knowledge of the specific gravities of bodies, is of great use in computing the weights of such bodies, as are too heavy or too unshapely to have their weight discovered by other means

A TABLE..

Showing the specific gravity to rain water, of metals, and other bodies; and the weight of a cubic inch of each, in parts of a pound avoirdupois, and an ounce troy.

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Note.-7000 grains make 1 lb. avoirdupois, and 5760 grains make 1lb troy; therefore, as 1 lb. avoirdupois : 1 lb. troy :: 7000 : 5760, or as 700: 576, consequently, 1 lb. avoirdupois, multiplied by gives 1 lb. troy, and

576

700

0.913

0.0330222

0.481569

0 854

0.0308883

0:450449

0.800

0-0289352

0-421966

0.747

0.0270182

0.391011

0.657

0.0257630

0-346539

0.613)

0.0221715

0.323332

0.569

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ponnd avoirdupois, and 14.585 multiplied into any number under Wt. lb. av. in the Table, will give its opposite number under Wt. oz. tr.; on the contrary, if 567 -be multiplied by any number under Wt. oz. tr., it will produce ite 12X700 opposite or horizontal number under Wt. lb. av.

Prob. 1. The weight of a body being given, to find its solidity. Divide the given weight in pounds avoirdupois, by the tabular weight corresponding to the name of the same kind, and the quotient will be the solidity in cubic inches; and if the quotient is divided by 1728, you will have the number of cubic feet.

Er. What is the solidity of a block of marble, weighing 8 tons 14 cwt. in cubic feet?

Now 8 tons 14 cwt.=19488 lb.

19448

Then

0977286

+1728=115 ̊4 cubic feet the solidity.

Prob. 2. The linear dimension, or solidity, of a body being given, to find its weight.

Multiply the cubic inches contained in the body, by the tabular weight corresponding to the name of the same kind, and the product will give the weight in pounds avoirdupois.

Er. What is the weight of a piece of oak, of a rectangular form, whose length is 50in. the breadth 18in. and the depth 12in. ?

Now 56 x 18 x 12-12096 inches.

Then 12096x 0330946=4003122816 lb. the weight.

OF THE

FIVE REGULAR SOLIDS.

Definitions.

1. A regular solid, is a body that either may be incribed or circumscribed by a sphere, in such a manner as to be contained under equai and similar planes; alike posited, and equally distant from the centre of the sphere.

2. The Tetraedron, is contained under four equilateral triangles. 3. The Hexaedron, is contained under six equal squares.

4. The Octaedron, is contained under eight equilateral triangles. 5. The Dodecaedron, is contained under twelve equilateral and equiangular pentagons.

6. The Icosaedron, is contained under twenty equilateral triangles. Prob. 1. To find the superficies, and solidity, of any of the five regular bodies.

To find the superficies. Multiply the area (taken from the following table) by the square of the linear edge of the solid, for the superficies.

To find the solidity.

Multiply the tabular solidity by the cube of the linear edge, for the solid content.

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Er. If the linear edge or side of a tetraedron be 3, required its superficial and solid content.

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Er. 2. What is the surface and solidity of the hexaedron, whose side is 2?

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Answer { superficies = 24
{solidity

}

= 8

Er. 3. Required the superficies and solidity of the octaedron, whose linear side is 2.

Answer { superficies = 13-8564
solidity

= 3.7712

}

Er. 4. What is the superficies and solidity of the dodecaedron, whose linear side is 2?

Answer { superficies = 82-58292

solidity = 61.30496

}

Ex. 5. What is the superficies and solidity of an icosaedron, whose linear side is 2?

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IRREGULAR SURFACES AND SOLIDS.

Def. An irregular surface or solid, is such a surface or solid which have their bounds by lines or surfaces in any manner whatever, of no particular kind of form or shape, but merely accidental, according as they are to be found or given.

Prob. 1. To measure any irregular surface whatever, by means of equidistant ordinates.

Method 1. To the half sum of the two outside ordinates, add the sum of all the other remaining ordinates; multiply the whole sum by the distance between any two ordinates, and the product will be the superficial content.

Er. 1. Let fig. 1 be the curve proposed, whose equidistant ordinates, AB, CD, EF, GH, IK, LM, and NO, are respectively 5ft., 5ft. 6in., 6ft., 7ft., 9ft., and 8ft., and the distance of AC, CÉ, EG, or CI, is 3 feet, required the area of the curve.

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Ex. 2. Let ABCD, fig. 2, be a circle, whose diameter AC, or BD, is 10 feet, it is required to find the area by means of equidistant ordinates, marked 3ft. 4ft. 4'5ft. 4'9ft. and 5ft. being at the distance of 1 foot from each other.

0
5

2,5

2.5 half the sum of the outside ordinates.

3

4

45

4.9

189 area of one quarter.

4

75.6 feet, area of the whole.

If the diameter, which is 10 feet, be multiplied by 7854, the product, 78 54, will be the area. From hence it appears, that this mode of operation, by means of equidistant ordinates, is exceedingly near the truth in measuring irregular planes; for it will produce the area of a circle, which is one of the most oblique curves possible, as the ends raise quite perpendicular to the axis, from only 10 equidistant spaces within the 1-26th part of the truth; and would be still nearer when applied to measuring any plane surface, where it is bounded partly by concave and partly by convex curves: because, if wholly bound by a convex curve, or curves, the area will be something less than the truth; but if bounded by a concave curve or curves, the area will be something greater than the truth; and if the extremi

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