Cor. 3. The water spouts out with the same velocity, whether it be downwards, or sideways, or upwards. And therefore, if it be upwards, it ascends nearly to the height above the hole. Cor. 4. The velocities and likewise the quantities of the spouting was ter, at different depths, will be as the square roots of the depths. Fig. 6. Fig. 7. B SCHOLIUM.–From hence are derived the rules for the construction of fountains or jets. Let ABC, fig. 7, be a reservoir of water, CDE a pipe coming out from it, to bring water to the fountain which spouts up at E, to the height £F, near to the level of the reservoir AB. In order to have a fountain in perfection, the pipe CD must be wide, and covered with a 1bin plate at Ė with a hole in it, not above the fifth or sixth part of the diameter of the pipe CD. And this pipe must be curve, having no angles. If the reservoir be 50 feet high, the diameter of the hole at E may be ar inch, and the diameter of the pipe 6 inches. In general the diameter of the hole E, ought to be as the square root of the height of the reservoir When the water runs through a great length of pipe, the jet will not rise so high. A jet never rises to the fall beight of ibe reservoir ; in a 5-feet jet it wants an inch, and it falls short by lengths which are as the sqnares of the heights; and smaller jets lose more. No jet will rise 300 feet high. A small fountain, (fig. 8.) is easily made by taking a strong bottle A, and filling it half full of water ; cement a tube BI very close in it, going near the bottom of the bottle. Then blow in at the top B, to compress the air within ; and the water will spout ont at B. If a fountain be placed in the sunshine and made to play, it will shiew all the colours of the rainbow, if a black cloth be placed beyond it. A jet goes higher if it is not exactly perpendicular; for then the opper part of the jet falls to one side without resisting the column below. The resistance of the air will also destroy a deal of its motion, and binder it from rising to the height of the reservoir. Also the friction of the tube or pipe of conduct has a great share in retarding the motion. If there be an upright vessel, as AF (fig. 9.), full of water, and several holes be made in the side as B, C, D, then the distances the water will spout upon the horizontal plane EL, will be as the square roots of the rectangles of the segments ABE, ACE, and ADE. For the spaces will be as the velocities and times. Brit (Cor. 4.) the velocity of the water flowing out of B, will be as ✓ AB, and the time of its moving (wbich is the same as the time of its fall) will be as ✓ BE: therefore the distance EH is as VAB X BE; and the space EL as VACE. And bence, if two boles are made equidistant from top and bottom, they will project the water to the same distance; for, if AB = DE, then ABE=ADE, wbich makes EH the same for both; and hence also it follows, that the projection from the middle point C will be furthest, for ACE is the greatest rectangle. These are the proportions of the distances; but for the olule distances it will be thus:--the velocity throngh any hole B, will . carry it through AB in the time of falling through AB; then to find huw far it will move in the time of falling through BE. Since these times are as the square roots of the heights, it will be v {AB : AB :: BE V BE : EH = AB. = V 2ABE; and so the space EL= GAB V2ACE. It is plain, these curves are parabolas. For the horizontal inotion being uniform, EH will be as the time ; that is, as ✓ BE or BE will be as EH”, which is the property of a parabola. Fig. 10. Fig. 9. E Б F Ен I If there be a broad vessel ABDC (fig. 10.) full of water, and the top AB fits exactly into it; and if the small pipe FE of a great lengtł be soldered close into the top, and if water be poured into the top of the pipe F, till it be full; it will raise a great weight laid upon the top, with ihe little quantity of water contained in the pipe; which weight will be nearly equal to a column of the Ouid, whose base is the top AB, and bciglit that of the pipe EF. For the pressure of the water against the , np AB, is equal to the weight of that column of water, by Prop. 3 and Cor, and Prop. 4. Cor. 2, page 722. But here the tube must not be too small. For in capillary tubes the attraction of the glass will take off its gravity. If a very small tube be immersed with one end in a vessel of water, the water will rise in the tube above the surface of the water; and the higher, the smaller the tube is. But, in quicksilver, it descends in the tube below the external surface, from the repulsion of the glass. To explain the operation of a syphon, (fig. 11.) which is a crooked pipe CDE, to draw liquors off. Set the syphon with the ends C, E, upwards, and fill it with water at the end E till it run out at C; to prevent it, clap the finger at C, and fill the other end to the top, and stop that with the finger. Then, keeping both ends stopt, invert the shorter end C into a vessel of water AB, and take off the fingers, and the water will ran out at E, till it be as low as C in the vessel ; provided the end E be always lower than C. Since E is always below C, the height of the coZumn of water DE is greater than that of CD, and therefore DE must outweigh CD and descend, and CD will follow after, being forced up by the pressure of the air, which acts upon the surface of the water in the vessel AB. The surface of the carth falls below the horizontal level only an inch in 620 yards; and in other distances the descents are as the squares of the distances. And, to find the nature of the curve DCG, (fig. 12 ) forming the jet IDG : Let AK be the height or top of the reservoir HF, and suppose the stream to ascend without any friction or resistance. By the laws of falling bodies the velocity in any place B, will be as ✓ AB. Put the semidiameter of the bole at D=d, and AD=h. Then since the same water passes through the sections at D and B; therefore (Prop. I.) the 1 velocity will be reciprocally as the section; w bence vh: :: V AB dd ; therefore and dd v h= BCV AB, whence BC da AB X BC*= HD; which is a paraboliform figure, whose asymplote is AK, for the nature of the cataractic curve DCG. And, if the fluid was to descend through a hole, as IC, it would form itself into the same figure GCD in descending. Fig. 11. D Fig. 12. F PROP.3.-The resistance any body meets with in moving through 1 fluid is as the square of the velocity. For, if any body moves with twice the velocity of another body equal to it, it will strike against twice as much of the fluid, and with twice the velocity, and therefore bas four times the resistance ; for that will be as the maiter and velocity. And, if it moves with thrice the velocity, it strikes against thrice as much of the fluid in the same time, with thrice the velocity, and therefore has nine times the resistance. And so on for all other velocities. Cor. If a stream of water, whose diameter is given, strike against an obstacle at rest; the force against it will be as the square of the velocity of the stream. For the reason is the same; since with twice or thrice the relocity, twice or thrice as much of the fluid impinges upou it, in the same time. PROP. 4.--The force of a stream of water against any plane obstacle at rest, is equal to the weight of a column of water, whose base is the section of the stream ; and height the space descended through by a falling body, to acquire that velocity. For let there be a reservoir whose height is that space fallen throngh: then the water (by Cor. 2. Prop. 2), flowing out at the bottom of the reservatory, has the same motion as the stream; but this is generated by the weight of that column of water, which is the force producing it. And that same motion is destroyed by the obstacle; therefore the force against it is the very same: for there is required as much force lo destroy as to generate any motion, Cor. The force of a stream of water flowing ont at a bole in the bot. tom of a reservatory, is equal to the weight of a column of the fluid of the same beight and wilosc base is the hole. PNEUMATICS. Prop. 1.-The air is a heavy body, and gravitates on all parts of the surface of the earth. That the air is a fluid is very plain, as it yields to any the least force that is impressed upon it, without making any sensible resistance. But, if it be moved briskly, by some very thin and light body, as a fan, or by a pair of bellows, we become very sensible of its motion against our hands or face, and likewise by its impelling or blowing away any light bodies, that lie in the way of its motion. Therefore the air being capable of moving other bodies by its impulse, must itself be a boily; and must therefore be heavy, like all other bodies, in proportion to the matter it con tains; and will consequently press upon all bodies placed under it. And, being a fluid, it will dilate and spread itself all over upon the earth; and like other fluids will gravitate upon, and press every where upon its surface. The gravity and pressure of the air is also evident from experiments. For, (see fig. 3, p. 723) if water, &c. be put into the tube ABF, and the air be drawn out of the end F, by an air-pump, the water will ascend in the nd F, and descend in the end A, by reason of the pressure at A, which was taken off or diminished at F. There are numberless experiments of this sort. And, though these properties and effects are certain, yet the air is a fluid so very fine and subtle, as to be perfectly transparent, and quite invisible to the eye. Cor. 1. The air, like other fluids, will, by its weight and fuidity, insinuate itself into all the cavities and corners within the carth ; and there press with so much greater force as the places are deeper. Cor. 2. Hence the atmosphere, or the whole body of air surrounding the earth, gravitates upon the surfaces of all other bodies, whether solid or fluid, and that with a force proportional to its weight or quantity of matter. For this property it must have in common with all other fluids. Cor. 3. Hence the pressure, at any depth of water, or other fluid, will be equal to the pressure of the fluid, together with the pressure of the atmosphere. Cor. 4. Likewise all bodies, near the surface of the earth, lose so much of their weight,) as the same bulk of so much air weighs. And, consequently, they are something lighter than they would be in a vacuum. But, being so very small, it is commonly neglected; though, in strictness, the true or absolute weight is the weight in vacuo. PROP. 2.—The air is an elastic fluid, or such a one as is capa. ble of being condensed or expanded. And it observes this law, that its density is proportional to the force that compresses it. These properties of the air are proved by experiments, of wbich there are innumerable. If you take a syringe, and thrust the handle inwards, you will feel the included air act strongly against your hand ; and the more you thrust the further the piston goes in, but the more it resists; and, taking away your hand, the handle returns back to where it was at Grst. This proves its elasticity, and also that air may be driven into a less space and condensed. Again, fill a strong bottle, fig. 8, half-full of water, and cement a pipe Bí closc in it,'going near the bottom: then inject air into the bottle through the pipe BI. Then the water will spout out at B, and form a jet: which proves, that the air is first condensed, and then by its spring Trives out the water, till it becomes of the same density as åt first, and - ben the spouting ccascs. If a vessel of glass AB, fig. 13, be filled with water in the vessel CD, and then drawn up with the bottom upwards: if any air is left in the top at A, the higher you pull it up, the more it expands; and, the further the glass is thrust down into the vessel CD, tho more the air is condensed. Take a crooked glass tube ABD, fig. 14, open at the end A, and close at D; pour in mercury to the heighi BC, but no bigher, and then the air in DC is in the same state as the external air. Thien pour in more mercury at A, and observe where it rises to in both legs, as to G and H. Then you may always see that the higher the mercury is in the leg BH, the less the space GD is, into which the air is driven. And, if the heiglit of the mercury FH be such as to equal thic pressure of the atmosphere, then DG will be half DC; if it be twice the pressure of the atmosphere, DG will be {DC, &c. So that the density is always as the weight or compression. And here the part CD is supposed to be cylindrical. Cor. 1. The space that any quantity of air takes up, is reciprocally as the force that compresses it. Cor. 2.--All the air near the earth is in a state of compression by the weight of the incumbent atmosphere. Cor 3.-The air is denser near the earth, or at the foot of a mountain, than at the top of and in bigla places; and, the bigher from the carth, the more rare it is. Cor. 4.—The spring or elasticity of the air is equal to the weight of the atmosphere above it, and produces the same eficets. For they always balance and sustain cach other. Cor. 5. Hence, if the density of the air be increased, its spring or elasticity will likewise be increased in the same proportion. Cor. 6.– From the gravity and pressure of the atmosphere upon the surfaces of fluids, the fluids are made to rise in pipes or vessels, when the pressure within is taken off. PROP. 3.— The expansion and elasticity of the air is increased by heat, and decreased by cold; or, heat expands, and cold con. denses, the air. This is also a matter of experience; for, tie a bladder very close with some air in it, and lay it before the fire, and it will visibly distend the bladder, and burst it, if the heat is continued, and increased high enough. If a glass vessel AB, (fig. 13,) with water in it, be turned upside down, with a little air in the top A, and be placed in a vessel of water, and lung over the fire, and any weight laid upon it to keep it down; as the water warms, the air in the top À will by degrees expand, till it fills the glass, and, by its elastic force, drive all the water out of the glass; and a good part of the air will follow, by continuing the vessel there. Many more experiments may be produced, proving the same thing. Prop. 4.-The air will press upon the surfaces of all fluids, wita any force, without passing through them, or entering into them. If this were not so, no machine, whose use or actions depends upo |