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14. PROP. VII. If M be the Magnitude of a body, S its Specific Gravity, and W its Weight, W=MS.

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It being found that the weight of a piece of Iron of any size the weight of a bulk of Water of the same size : 7-8 1, and the weight of a bulk of Silver: weight of an equal bulk of Water :: 105: 1, and so for other substances,-Tables have been formed in which the numbers 1, 7·8, 10-5, &c. are placed opposite the words "Water" "Iron" "Silver" &c.; and by means of these tables, as we shall immediately shew, the weight of any bulk of any of the substances so registered can be determined, if only the weight be known of some particular magnitude (a cubic inch suppose) of some one of them.

The numbers given in these Tables are called, by some writers, the "Specific Gravities" of the substances they are attached to, but the enunciation of Prop. vII. Art. 14, will not permit of their being called so here.

"The Tables of Specific Gravities" give

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Before proceeding to point out the use of the Tables, it will be necessary to shew that "the weights of any two substances are in the ratio of the numbers given by the Tables as corresponding to the substances,"

For w, w', w" being the respective weights of equal bulks of Water, Iron, and Silver, since, as explained above,

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Suppose then it be required to find the weight of a cubic foot of Iron, the weight of 10 cubic inches of Silver being 61 ounces nearly.

Weight of a cubic foot (or 12 x 12 x 12 cubic inches) of Silver,

= 12 x 12 x 12 x

61
10

ounces;

.. Weight

Let M = number of cubic inches in the body.

S = Specific Gravity of the body; i. e. let S be the number of grains that one cubic inch of it weighs.

W = number of grains which the whole body weighs.

grains grains c. inches

Then since W : S :: M:

W = M × S. Q. E. D.

c. inch

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15. PROP. VIII. When a body of uniform density floats on a fluid, the part immersed : the whole body the specific gravity of the body the specific gravity of the fluid.

Let a solid of uniform density float on a fluid with M cubic inches of it above the horizontal plane of the surface, and

N cubic inches below that surface.

=

M

N

Let S S. G. (Specific Gravity) of the solid; S'S. G. of the fluid.

Then (M + N) × S = weight of the solid. Prop. VII. Nx S'...............fluid displaced.

=

But because the body floats these two weights are equal; by Prop. vi.

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The part of it immersed the whole of the body :: S. G. of the body: S. G. of the fluid. Q. E. D. 16.

PROP. IX. When a body is immersed in a fluid, the weight lost whole weight of the body: the specific gravity of the fluid the specific gravity of the body.

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Let M be the number of cubic inches contained in a body of uniform density which is wholly immersed in a fluid; S the S. G. of the body, S" the S. G. of the fluid.

M

Then the pressure downwards of the solid is its weight, and if the solid be removed, and the space it filled be occupied by an equal bulk of the fluid, equilibrium will still remain. And if the fluid so added become solid, the equilibrium will continue to exist, and the pressure upwards of the surrounding fluid will remain the same as before.

Now under this supposition the pressure downwards of the fluid added is its weight. And since the pressure upwards of the surrounding fluid supports this weight, it must be exactly equal and opposite to it.

The pressure downwards therefore of the original solid, (i. e. its weight MS), must have been diminished by a pressure upwards arising from the

surrounding fluid exactly equalling in magnitude the weight of the fluid displaced, which is by Prop. VII equal to MS'.

.. Weight lost by the body: the whole weight of the body: MS: MS'

:: S: S'

:: S. G. of body S. G. of fluid. Q. E. D.

17. [It appears from the proof of the last Proposition, that the pressure of a fluid on a body wholly immersed in it acts vertically upwards, and is equal to the weight of the fluid which the body displaces.

If this pressure be less than the weight of the body, that is, if the S. G. of the fluid be less than that of the solid, the pressure downwards arising from the weight of the solid will be greater than the pressure of the surrounding fluid upwards, and the body will therefore sink to the bottom of the vessel.

But if the pressure of the fluid upwards be greater than the weight of the body immersed, that is, if the S. G. of the fluid be greater than that of the solid, the pressure upwards will be greater than the pressure downwards, and the body will therefore rise until the conditions of Proposition VIII are fulfilled,—when it will float.]

18. PROP. X. (1) To describe the Hydrostatic Balance; and (2) to shew how the Specific Gravity of a body may be found by it,-first, when its Specific Gravity is greater than that of the fluid in which it is weighed, and second, when it is less.

(1) The Hydrostatic Balance is the common balance with a hook attached

to the under part of one of its scales, so that bodies may be weighed either by putting them A into the scale, or by suspending them from it and letting them

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be immersed in a fluid, as here represented.

(2) FIRST, let the S. G. (Specific Gravity) of the body be greater than that of the fluid.

Since the S. G. of the solid is greater than that of the fluid, the body will sink in the fluid. Prop. IX. Cor.

Let S S. G. of the solid, S'S. G. of the fluid.

=

W = weight required to balance the body when placed in the scale.

W' = weight required when the body is immersed. .. W-W' = diminution of the body's weight in consequence of immersion.

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Whence, W and W' being known, S may be determined if S be given.

NEXT, let the S. G. of the body be less than the S. G. of the fluid.

When the body, in this case, is forced under the surface of the fluid, the pressure downwards on

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