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331. Note. The several sorts of wood are supposed to be dry. Also, as a cubic foot of water weighs just 1000 ounces avoirdupois, the numbers in this table express, not only the specific gravities of the several bodies, but also the weight of a cubic foot of each, in avoirdupois ounces; and therefore, by proportion, the weight of any other quantity, or the quantity of any other weight, may be known, as in the next two propositions.

PROPOSITION LXVII.

332. To find the Magnitude of any Body, from its Weight.

As the tabular specific gravity of the body, Is to its weight in avoirdupois ounces, So is one cubic foot, or 1728 cubic inches, To its content in feet, or inches, respectively. Example 1. Required the content of an irregular block of common stone, which weighs 1 cwt, or 112 lb?

Ans. 12281 cubic inches. Example 2. How many cubic inches of gunpowder are there in 1 lb. weight? Ans. 294 cubic inches nearly. Example 3.

Example 3. How many cubic feet are there in a ton weight of dry oak? Ans. 38+ cubic feet.

PROPOSITION LXVIIІ.

333. To find the Weight of a Body from its Magnitude.

As one cubic foot, or 1728 cubic inches,
Is to the content of the body,
So is the tabular specific gravity,
To the weight of the body.

Example 1. Required the weight of a block of marble, whose length is 63 feet, and breadth and thickness each 12 feet; being the dimensions of one of the stones in the walls of Balbeck?

Ans. 683 ton, which is nearly equal to the burden of an East-India ship.

:

Example 2. What is the weight of 1 pint, ale measure, of Ans. 19 oz. nearly.

gunpowder?

Example 3. What is the weight of a block of dry oak, which measures 10 feet in length, 3 feet broad, and 2 feet

deep or thick?

Ans. 4335

lb.

OF HYDRAULICS.

334. HYDRAULICS is the science which treats of the motion of fluids, and the forces with which they act upon bodies.

PROPOSITION LXIX.

335. If a Fluid Run through a Canal or River, or Pipe of various Widths, always filling it; the Velocity of the Fluid in different Parts of it AB, CD, will be reciprocally as the Transverse Sections in those Parts.

THAT is, veloc. at a: veloc. at C:CDAB; where AB and CD denote, not the diameters at A and B, but the areas or

sections there.

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For, as the channel is always equally full, the quantity of water running through AB is equal to the quantity running through CD, in the same time; that is, the column through

AR

AB is equal to the column through CD, in the same time; or AB X length of its column umn = CD × length of its column; therefore AB : CD :: length of column through CD: length of column through AB. But the uniform velocity of the water, is as the space run over, or length of the columns; therefore AB : CD :: velocity through CD : velocity through AB.

336. Corol. Hence, by observing the velocity at any place AB, the quantity of water discharged in a second, or any other time, will be found, namely, by multiplying the section AB by the velocity there.

But if the channel be not a close pipe or tunnel, kept always full, but an open canal or river; then the velocity in all parts of the section will not be the same, because the velocity towards the bottom and sides will be diminished by the friction against the bed or channel; and therefore a medium among the three ought to be taken. So, if the velo.

city at the top be

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100 feet per minute,

60

50

3) 210 sum;

dividing their sum by 3, gives 70 for the mean velocity, which is to be multiplied by the section, to give the quan tity discharged in a minute.

PROPOSITION LXX.

337. The Velocity with which a Fluid Runs out by a Hole in the Bottom or Side of a Vessel, is Equal to that which is Generated by Gravity through the Height of the Water above the Hole; that is, the Velocity of a Heavy Body acquired by Falling freely through the Height AB.

DIVIDE the altitude AB into a great number of very small parts, each being 1, their number a, or a = the altitude AB.

B

Now, by prop. 61, the pressure of the fluid against the hole B, by which the motion is generated, is equal to the weight of the column of fluid above it, that is the column whose height is AB or a, and base the area of the hole в. Therefore the pressure on the hole, or small part of the fluid 1, is to its weight, or the natural force of gravity, as a to 1. But, by art. 28, the velocities generated in the same body in any time, are as those forces; and because gravity generates the velocity 2 in descending through the small space 1, therefore 1:a::2:2a, the velocity generated by the pressure of the column of fluid in the same time. But 2a is also, by corol. 1 prop. 6, the velocity generated by gravity in descending through a or AB. That is, the velocity of the issuing water, is equal to that which is acquired by a body in falling through the height AB.

The same otherwise.

Because the momenta, or quantities of motion, generated in two given bodies, by the same force, acting during the same or an equal time, are equal. And as the force in this case, is the weight of the superincumbent column of the fluid over the hole. Let the one body to be moved, be that column itself, expressed by ab, where a denotes the altitude AB, and b the area of the hole; and the other body is the column of the fluid that runs out uniformly in one second suppose, with the middle or medium velocity of that interval of time, which is thu, if v be the whole velocity required. Then the mass thu, with the velocity u, gives the quantity of motion thu xv, or hv2, generated in one second, in the spouting water: also 2g, or 32 feet, is the velocity generated in the mass ab, during the same interval of one second; consequently ah x 2g, or 2ahg, is the motion generated in the column ab in the same time of one second. But as these two momenta must be equal, this gives hv2 = 2ahg : hence then v2 = 4ag, and v = 2√ag, for the value of the velocity sought; which therefore is exactly the same as the velocity generated by the gravity in falling through the space a, or the whole height of the fluid.

For example, if the fluid were air, of the whole height of the atmosphere, supposed uniform, which is about 54 miles, or 27720 feet = a. Thén 2√ag = 2√27720 × 16+= 1335 feet = v the velocity, that is, the velocity with which common air would rush into a vacuum.

338. Corol. 1. The velocity, and quantity run out, at different depths, are as the square roots of the depths. For the velocity acquired in falling through AB, is as ✓ AB.

339. Corol. 2. The fluid spouts out with the same velocity, whether it be downward or upward, or sideways; because the pressure of fluids is the same in all directions, at the same depth. And therefore, if an adjutage be turned upward, the jet will ascend to the height of the surface of the water in the vessel. And this is confirmed by experience,

4

by which it is found that jets really ascend nearly to the

height of the reservoir, abating a small quantity only, for the friction against the sides, and some resistance from the air and from the oblique motion of the fluid in the hole.

340. Corol. 3. The quantity run out in any time, is equal to a column or prism, whose base is the area of the hole, and its length the space described in that time by the velocity acquired by falling through the altitude of the fluid. And the quantity is the same, whatever be the figure of the ori fice, if it is of the same area.

Therefore, if a denote the altitude of the fluid, and h the area of the orifice,

also g = 16 feet, or 193 inches;

then 2h ag will be the quantity of water discharged in a second of time; or nearly 8th a cubic feet, when a and b arë taken in feet.

So, for example, if the height a be 25 inches, and the orifice b=1 square inch; then 2h ag = 2√25 × 193 = 139 cubic inches, which is the quantity that would be discharged per second.

SCHOLIUM.

341. When the orifice is in the side of the vessel, then the velocity is different in the different parts of the hole, being less in the upper parts of it than in the lower. However, when the hole is but small, the difference is inconsiderable, and the altitude may be estimated from the centre of the whole, to obtain the mean velocity. But when the orifice is pretty large, then the mean velocity is to be more accurately computed by other principles, given in the next proposition.

342. It is not to be expected that experiments, as to the quantity of water run out, will exactly agree with this theory, both on account of the resistance of the air, the resistance of the water against the sides of the orifice, and the oblique motion of the particles of the water in entering it. For, it is not merely the particles situated immediately in the column over the hole, which enter it and issue forth, as if that column only were in motion; but also particles from all the surrounding parts of the fluid, which is in a commotion quite around; and the particles thus entering the hole in all directions, strike against each other, and impede one another's motion: from which it happens, that it is the particles in the centre of the hole only that issue out with the whole velocity due to the entire height of the fluid, while the other particles towards the sides of the orifices pass out with decreased velocities; and hence the medium velocity through the orifice, is somewhat less than that of a single body only, urged with the same pressure of the superincumbent column

of

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