288. Notè. 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.

PROP. LVIII.

289. 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. 12282018 cubic inches.

Example 2. How many cubic inches of gunpowder are there in 1 lb. weight? Ans. 30 cubic inches nearly,

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

Ans. 3813 cubic feet.

PROP. LIX.

290. 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 its 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 the breadth and thickness each 12 feet; being the dimensions of one of the stones in the walls of Balbec?

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

Example 2. What is the weight of 1 pint, ale measure, of gunpowder?

Ans. 19 oz. nearly. Example 3. What is the weight of a block of dry oak, which measures 10 feet in length, 3 feet broad, and 23 feet deep? Ans. 433518 lb.

OF HYDRAULICS.

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

PROP. LX.

292. 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.

B

where AB and CD denote, not the diameters

at A and B, but the areas, or sections, there.

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 AB is equal to the column through CD, in the same time; or AB X length of its column= CD x 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.

293. 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 velocity at the top be

That at the bottom

And that at the sides

Dividing their sum by 3 gives

3) 210 sum;

70 the mean velocity,

which is to be multiplied by the section, to give the quantity discharged in a minute.

PROP. LXI.

294. The velocity with which a fluid runs out by a hole in the bottom or side of a vessel, kept always full, 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.

Now, by prop. 53, 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 B. 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.

295. 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.

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

297. 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 orifice, if it is of the same area.

Therefore if h denote the height of the fluid,

a the area of the orifice, and

g= 16 feet, or 193 inches;

then 2a ✔gh will be the quantity of water discharged in a second of time, or nearly 8/h cubic feet, when a and h are taken in feet.

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

298. SCHOLIUM.-When the orifice is in the side of the vessel, 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 hole, 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.

299. 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 whence it happens, that the real velocity through the orifice, is somewhat less than that of a single body only, urged with the same pressure of the superincumbent column of the fluid. And experiments on the quantity of water discharged through apertures, show that the velocity must be diminished, by those causes, rather more than the fourth part, when the orifice is small, or such as to make the mean velocity equal to that of a body falling through the height of the fluid above the orifice. Or else, that the orifice is not quite full of particles that spout out with the whole velocity, assigned in the proposition.

300. Experiments have also been made on the extent to which the spout of water ranges on a horizontal plane, and compared with the theory, by calculating it as a projectile discharged with the velocity acquired by descending through the height of the fluid. For, when the aperture is in the side of the vessel, the fluid spouts out horizontally with a uniform velocity, which, combined with the perpendicular velocity from the action of gravity, causes the jet to form the curve of a parabola. Then the distances to which the jet will spout on the horizontal plane BG, will be as the roots of the rectangles of the segments AC. CB, AD. DB, AE. EB. For the spaces BF, BG, are as the times and horizontal velocities; but the velocities are as AC, and the time of the fall, which is the same as the time of moving, or as CB; therefore the distance BF is as √AC. CB; and the distance BG as AD. DB.

I

H

K

E

And hence, if two holes are made equidistant from the top and bottom, they will project the water to the same distance; for if AC = EB, then the rectangle AC. CB is equal the rectangle AE. EB; which makes EF then the same for both. Or, if on the diameter AB a semicircle be described; then, because the squares of the ordinates CH, DI, EK are equal to the rectangles AC. CB, &c; therefore the distances BF, BG are as the ordinates CH, DI. And hence also it follows, that the projection from the middle point D will be farthest, for DI is the greatest ordinate,

These are the proportions of the distances: but for the absolute distances, it will be thus. The velocity through any hole, C, is such as will carry the water horizontally through a space equal to 2 AC in the time of falling through AC: but, after quitting the hole, it describes a parabola, and comes to F in the time a body will fall through CB; and to find this distance, since the times are as the roots of the spaces, therefore√AC. √CB :: 2AC : 2/AC. CB= 2CH = BF, the space ranged on the horizontal plane. And the greatest range BG = 2DI, or 2AD, or equal to AB.

And as these ranges answer very exactly to the experiments, this confirms the theory as to the velocity assigned.

PROP. LXII.

301. If a notch or slit EH, in form of a parallelogram, be cut in the side of a vessel, full of water, AD; the quantity of water flowing through it, will be of the quantity flowing through an equal orifice, placed at the whole depth EG, or at the base GH, in the same time; it being supposed that the vessel is always kept full.

E F

K

For the velocity at GH is to the velocity at IL, as √EG to EI, that is, as GH or IL to IK, the ordinate of a A parabola EKH, whose axis is EG. Therefore the sum of the velocities at all the points I, is to as many times the velocity at G, as the sum of all the ordinates IK to the sum of all the IL's, namely, as the area of the parabola EGH is to the area EGHF; that is, the quantity running through the notch EH, is to the quantity running through an equal horizontal area placed at GH, as EGHKE to EGHF, or as 2 to 3; the area of a parabola being of its circumscribing parallelogram. 302. Corol. 1. The mean velocity of the water in the notch, is equal to 3 of that at GH.

B

D

303. Corol, 2. The quantity flowing through the hole IGHL, is to that which would flow through an equal orifice placed as low as GH, as the parabolic frustum IGHK, is to the rectangle IGHL. As appears from the demonstration.

OF PNEUMATICS.

304. PNEUMATICS is the science which treats of the properties of air, or elastic fluids.