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 BF 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. XII. 22. 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 C 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. Corol. 1. The mean velocity of the water in the notch, is equal to 3 of that at GH. B 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 demonstra tion. OF PNEUMATICS. 23. PNEUMATICS is the science which treats of the properties of air, or elastic fluids. PROP. XIII. 24. Air is a heavy fluid body; and it surrounds, and gravitates on, all parts of the surface of the earth. THESE properties of air are proved by experience. That it is a fluid, is evident from its easily yielding to any the least force impressed on it, without making a sensible resistance. But when it is moved briskly, by any means, as by a fan or pair of bellows; or when any body is moved very briskly through it; in these cases we become sensible of it as a body, by the resistance it makes in such motions, and likewise by its impelling or blowing away any light substances. So that, being capable of resisting, or moving other bodies by its impulse, it must itself be a body, and be heavy, like all other bodies, in proportion to the matter it contains; and therefore it will press on all bodies that are placed under it. E D Also, as it is a fluid, it will spread itself all over on the earth; and, like other fluids, it will gravitate and press every where on the earth's surface. 25. The gravity and pressure of the air is also evident from many experiments. Thus for instance, if water or quicksilver, be poured into the tube ACE, and the air be suffered to press on it, in both ends of the tube, the fluid will rest at the same height in both legs of the tube : but if the air be drawn out of one end as E, by any means, then the air pressing on the other end A, will press down the fluid in this leg at B, and raise it up in the other to D, as much higher than at B, as the pressure of the air is equal to. By which it appears, not only that the air does really press, but also what the quantity of that pressure is equal to. And this is the principle of the 26. The air is also an elastic fluid, being condensible and expansible. And the law it observes is this, that its density is proportional to the force which compresses it. THIS property of the air is proved by many experiments. Thus if the handle of a syringe be pushed inwards, it will condense the enclosed air into less space, thereby showing its condensibility. But the included air, thus condensed, will be felt to act strongly against the hand, resisting the force compressing it more and more; and, on withdrawing the hand, the handle is pushed back again to where it was at first. Which shows that the air is elastic. 27. Again fill a strong bottle half full of water, and then insert a pipe into it, putting its lower end down near to the bottom, and cementing it very close round the mouth of the bottle. Then, if air be strongly injected through the pipe, as by blowing with the mouth or otherwise, it will pass through the water from the lower end, ascending into the parts before occupied with air at B, and the whole mass of air become there condensed, because the water is not compressible into a less space. But, on removing the force which injected the air at A, the water will begin to rise from thence in a jet, being pushed up the pipe by the increased B elasticity of the air B, by which it presses on the surface of the water, and forces it through the pipe, till as much be expelled as there was air forced in; when the air at B will be reduced to the same density as at first, and, the balance being restored, the jet will cease. 28. Likewise, if into a jar of water AB, be inverted an empty glass tumbler CD, or such like, the mouth downwards; the water will enter it, and partly fill it, but not near so high as the water in the jar, compressing and condensing the air into a less space in the upper parts CD, and causing the glass to make a sensible resistance to the hand in pushing it down. Then, on removing the hand, the elasticity of the internal condensed air throws the glass up again. All these showing that the air is condensible and elastic. A K H C B f D B 29. Again, to show the rate or proportion of the elasticity to the condensation; take a long crooked glass tube, equally wide throughout, or at least in the part BD, open at A, but close at the other end B. Pour in a little quicksilver at A, just to cover the bottom to the bend at CD, and to stop the communication between the external air and the air in BD. Then pour in more quicksilver, and mark the corresponding heights at which it stands in the two legs: so, when it rises to H in the open leg AC, let it rise to E in the close one, reducing its included air from the natural bulk BD to the contracted space BE, by the pressure of the column He; and when the quicksilver stands at I and K, in the open leg, let it rise to F and G in the other, reducing the air to the respective spaces BF, BG, by the weights of the columns If, Kg. Then it is always found that the condensations and elasticities are as the compressing weights, or columns, of the quicksilver, and the atmosphere together. So, if the natural bulk of the air DB be compressed into the spaces BE, BF, BG, or reduced by the spaces DE, DF, DG, which are,, of DB, or as the numbers 1, 2, 3; then the atmosphere, together with the corresponding columns He, If, Kg, will also be found to be in the same proportion, or as the numbers 1, 2, 3. And then He = A, If = A, and Kg = 3A; where A is the weight of the atmosphere. Which shows, that the condensations are directly as the compressing forces. And the elasticities are in the same ratio, since the columns in AC are sustained by the elasticities in BD. From the foregoing principles may be deduced many useful remarks, as in the following corollaries, viz. M A E B C G F 30. Corol. 1. The space which any quantity of air is confined in, is reciprocally as the force that compresses it. So, the forces which confine a quantity of air in the cylindrical spaces AG, BG, CG, are reciprocally as the same, or reciprocally as the heights, AD, BD, CD. And therefore, if to the two perpendicular lines, DA, DH, as asymptotes, the hyperbola IKL be described, and the ordinates AI, BK, CL be drawn; then the forces which confine the air in the spaces AG, BG, CG, will be directly as the corresponding ordinates AI, BK, CL, since these are reciprocally as the abscisses AD, BD, CD, by the nature of the hyperbola. KKK Corol. 2. All the air near the earth is in a state of compression, by the weight of the incumbent atmosphere. Corol. 3. The air is denser near the earth, than in high places; or denser at the foot of a mountain than at the top of it. And the higher above the earth, the less dense it is. Corol. 4. The spring or elasticity of the air, is equal to the weight of the atmosphere above it; and they will produce the same effects; since they always sustain and balance each other. Corol. 5. If the density of the air be increased, preserving the same heat or temperature; its spring or elasticity will likewise be increased, and in the same proportion. Corol. 6. By the gravity and pressure of the atmosphere, on the surface of fluids, the fluids are made to rise in any pipes or vessels, when the spring or pressure within is decreased or taken off, PROP. XV. 31. Heat increases the elasticity of the air, and cold diminishes it. Or, heat expands, and cold condenses the air. This property is also proved by experience. 32. Thus, tie a bladder very close with some air in it; and lay it before the fire then as it warms, it will more and more distend the bladder, and at last burst it, if the heat be continued, and increased high enough. But if the bladder be removed from the fire, as it cools it will contract again, as before. And it was upon this principle, that the first air-balloons were made by Montgolfier: for, by heating the air within them, by a fire underneath, the hot air distends them to a size which occupies a space in the atmosphere, whose weight of common air exceeds that of a balloon. 33. Also, if a cup or glass, with a little air in it, be inverted into a vessel of water; and the whole be heated over the fire, or otherwise: the air in the top will expand till it fill the glass, and expel the water out of it; and part of the air itself will follow, by continuing or increasing the heat. Many other experiments, to the same effect, might be adduced, all proving the properties mentioned in the proposition. 34. SCHOLIUM.-So that, when the force of the elasticity of air is considered, regard must be had to its heat or temperature; the same quantity of air being more or less elastic, as its heat is more or less. And it has been found, by experiment, that the elasticity is increased by the 435th part, by each degree of heat, of which there are 180 between the freezing and boiling heat of water. 35. N.B. Water expands about the 2ʊʊʊ part, with each degree of heat (Sir Geo. Shuckburgh, Philos. Trans. 1777, p. 560, &c). Also, the water 836 when the barom. is at 29-27, mercury 11227) which are their mean heights in this 36. The weight or pressure of the atmosphere, on any base at the earth's surface, is equal to the weight of a column of quicksilver, of the same base, and the height of which is between 28 and 31 inches. THIS is proved by the barometer, an instrument which measures the pressure of the air, and which is described below. For, at some seasons, and in some places, the air sustains and balances a column of mercury, of about 28 inches; but at other times it balances a column of 29 or 30, or near 31 inches high; seldom in the extremes 28 or 31, but commonly about the means 29 or 30. A variation which depends partly on the different degrees of heat in the air near the surface of the earth, and partly on the commotions and changes in the atmosphere, from winds and other causes, by which it is accumulated in some places, and depressed in others, being thereby rendered denser and heavier, or rarer and lighter; which changes in its state are almost continually happening in any one place. But the medium state is commonly about 291⁄2 or 30 inches. Corol. 1. Hence the pressure of the atmosphere on every square inch at the earth's surface, at a medium, is very near 15 pounds avoirdupois. For, a cubic foot of mercury weighing 13600 ounces nearly, an inch of it will weigh 7.866 or almost eight ounces, or near half a pound, which is the weight of the atmosphere for every inch of the barometer on a base of a square inch; and therefore 30 inches, or the medium height, weighs very near 143 pounds, Corol. 2. Hence also, the weight or pressure of the atmosphere, is equal to that of a column of water from 32 to 35 feet high, or on a medium 33 or 34 feet high. For, water and quicksilver are in weight nearly as 1 to 13·6; so that the atmosphere will balance a column of water 13.6 times as high as one of quicksilver; consequently 13.6 times 28 inches 381 inches, or 313 feet, 394 inches, or 32% feet, 13.6 times 31 inches = 422 inches, or 35 feet. And hence a common sucking pump will not raise water higher than about 34 feet. And a syphon will not run, if the perpendicular height of the top of it be more than about 33 or 34 feet. Corol. 3. If the air were of the same uniform density at every height up to the top of the atmosphere, as at the surface of the earth; its height would be about 5 miles at a medium. |