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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

62

rain water weighed 624 pounds avoirdupois ; consequently, (0725)

0 03616898lb. 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 beavy 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.

Bodies.

Sp. gra. Wt. Ib. av.

Wt. oz. tr.

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Fine gold
Standard gold
Coast gold ...
Quicksilver...
Lead ...
Fine silver
Standard silver
Cast silver ...
Copper
Plate brass ...
Cast brass
Steel
Bar iron
Block tin
Cast iron
Loadstone
Blue slate
Veined marble
Common glass
Flint stone
Portland stone ...
Free stone ...
Brick
Alabaster

19.640 19:520 18.888 13 762 11:313 11.092 10-629 10°528 8.769 8.350 8104 7.850 7.764 7.238 7:135 5:106 3.500 2702 2.600 2.582 2:570 2.352 2.000 1 888

0°7103587 0-7060185 0.6898703 0.4976574 (-4091696 0.4011501 0:3844400 0-3807870 0•3171658 0.2942593 0.2929832 0.2839265 0.2808159 0.2617901 Oʻ2580647 O‘1846788 0·1264914 0-0977286 0-0940393 0.0933883 0·0929543 0·0915788 00723379 0-0683061

10:359273 9.969625 9.911707 7.384411 5.084010 5.850035 5 556769 5503967 4747121 4.404273 4.27 2409 4.142127 4:031361 3.861519 3-806568 2-724083 1.867272 1.429411 1:360841 1.351419 1.345139 1.231038 1.046801 0.988456

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TABLE CONTINUBD.

Bodies.

Sy gra. Wt. lb. av.

Wt. oz. tr.

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Ivory

1.832

0·0662606 0.958489 Brimstone

1.800 0.0651049 0.949494 Clay....

1.712 0.0619213 0-904498 Lignum vitæ

1.327 0·01479862 0.699936 Coal....

1.255 0-0453921 0.661959 Pitch

1.150 00415943 0.606576 Mahogany wood

1.063 0.0384475 0-560691 Dry box wood

1.030 0.037 2530 0·543874 Milk

1.030 } Sea water

0.0372530 0°543272 Rain water.

1.000 0.0361690 0-577-458 Red wine

0.993 0·0359158 0-523766 Bees' wax

0.995 0·0359881 0:124820 Linseed oil

0.939 0·0337095 0:491591 Proof spirits, or brandy

0.927 0·0335503 0.484268 Dry oak

0.915 0'0330946 0°489003 Olive oil

0.913 0.0330222 0 $81569 Beech

0 854 0·0308883 0-450449 Dry elm

0.800 Dry ash

00989352 0-421966 Dry wainscot.

0747 0.0270182 0.39 1011 Dry yellow fir

0*657 0·0237630 0-346539 Cedar

0613) 0·0221715 0.323332 Dry white deal

0.569 0.0205801 0.300123 Cork

0.240 0.0186805 0.176390 Air

0.0012 0.0000434 0-000633 Note. — 7000 grains make 1 lb. avoirdapois, and 5760 grains make 1 lb troy; therefore, as 1 lb. avoirdupois : 1 lb. troy :: 7000 : 5760, or as 700 :

576 576, consequently, 1 lb. avoirdupois, multiplied by gives 1 lb. troy, and

700 700 1 lb. troy, multiplied by

576

gives i lb. avoirdupois. For example, 18 ounces

700 is a pound troy; then X12=14:58f the number of troy ounces in one

576 ponnd avoirdupois, and 14.584 multiplied into any number ander Wt. Ib. av. in the Table, will give its opposite number under Wt. oz, tr. ; on the contrary,

567 if -be multiplied by any puniber under Wt. oz. tr., it will produce il

12 X 700 opposite or borizontal number under Wt. Ib. 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. Ex. What is the solidity of a block of marble, weighing 8 tons 1 cwt, in cubic feet?

Now 8 tons 14 cwt.=19488 lb.

19448 Theo 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 weiglit 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 12096 X '0330946=4009122816 lb. the weight.

OF THB

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 Heraedron, 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.

Surfaces and Solidities of the five regular Solids.

No. of
sides.

Names.

Surfaces.

Solidities.

4 6 8 12 20

Tetraedron
Hexaedron
Octaedron
Dodecaedron
Icosaedron

1.73205
6-00000
3.46410
20-64573
8:66025

0:11785 1.00000 0:47140 7.66312 2:18169

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

Thus 1•73205 * 9=15458845 superficies.
And 0:11785 X 27= 3:18195 solidity.

Answer { superficies = 24

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

} i Er. 3. Required the superficies and solidity of the octaedron, wbose linear side is 2. Answer { superficies = 13-8564 solidity

}

3.7712 5 Er. 4. What is the superficies and solidity of the dodecaedron, whose linear side is 2 ? Answer { superficies = 82:58292 solidity =

} Er. 5. What is the superficies and solidity of an icosaedron, whose linear side is 2 ? Answer { superficies = 3464)

} 3 17.45352

OF MEASURING

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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, ĞH, IK, LM, and No, are respectively 5ft., 5ft. 6in., 6ft., 7ft., gft., and 8ft., and the distance of AC, CE, EG, or CI, is 3 feet, required the area of the curve:

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AB=5
NO=8

2)13

6 6 half the sum of the outside ordinates
CD= 5 6
EF= 6 0
GH= 7 0

IK= 90
LM=10 0

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.

132 superficies. 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 i foot from each other.

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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 some

hing less than the truth; but if bounded a concave curve or curves, the area will be something greater than the truth; and if the extremi

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