277. But the formula may be rendered much more convenient for use, by reducing the factor 10592 to 10000, by changing the temperature proportionally from 55°; thus, as the diff. 592 is the 18th part of the whole factor 10592; and as 18 is the 24th part of 435; therefore the corresponding change of temperature is 240, which reduces the 55° to 31°. So that the formula is, a = 10000 X M log. fathoms, when the temperature is 31 degrees; and for every degree m above that, the result is to be increased by so many times its 435th part. we have 55000. 278. Taking, instead of the logarithms, the first term of the logarithmic series B b for the altitude in feet: B and b, being the heights of B+b' the barometrical columns observed at the bottom and top of the hill. This formula is for the mean temperature 55°, and is easily remembered because the effective figures of the co-efficient are also 55. The reductions for any other temperature are the same as in the logarithmic rule. Ex. 1. To find the height of a hill when the pressure of the atmosphere is equal to 29.68 inches of mercury at the bottom, and 25-28 at the top; the mean temperature being 50°? Ans. 4352-4 feet, or 725-4 fathoms. Ex. 2. To find the height of a hill when the atmosphere weighs 29.45 inches of mercury at the bottom, and 26.82 at the top, the mean temperature being 33°? Ans. 406.28 fathoms. Ex. 3. At what altitude is the density of the atmosphere only the 4th part of what it is at the earth's surface? As. 6020 fathoms. By the weight and pressure of the atmosphere, the effect and operations of pneumatic engines may be accounted for, and explained; such as siphons, pumps, barometers, &c.; of which it will be proper here to give a brief description. OF THE SIPHON. 279. A Siphon, or Syphon, is any bent tube, having its two legs either of equal or of unequal length. If it be filled with water, and then inverted, with the two open ends downward, and held level in that position; the water will remain suspended in it, if the two legs be equal. For the atmosphere will press equally on the surface of the water in each end, and support them, if they are not more than 34 feet high; and the legs being equal, the water in them is an exact counterpoise by their equal weights; so that the one has no power to move more than the other; and they are both supported by the atmosphere. But if now the siphon be a little inclined to one side, so that the orifice of one end be lower than that of the other; or if the legs be of unequal length, which is the same thing; then the equilibrium is destroyed, and the water will all descend out by the lower end, and rise up in the higher. For, the air pressing equally, but the two ends weighing unequally, a motion must commence where the power is greatest, and so continue till all the water has run out by the lower end. And if the shorter leg be immersed into a vessel of water, and the siphon be set running as above, it will continue to run till all the water be exhausted from the vessel, or at least as low as that end of the siphon. Or, it may be set running without filling the siphon as above, by only inverting it, with its shorter leg into the vessel of water; then, with the mouth applied to the lower orifice A, suck out the air; and the water will presently follow, being forced up into the siphon by the pressure of the air on the water in the vessel. If a siphon be fixed in a vessel of water capable of rotation upon a vertical axis, and the orifice be lateral instead of at the bottom of the pipe, the reaction may be advantageously employed as a motive force. This is the principle of Mr. Busby's Hydraulic Orrery. OF THE PUMP. 280. There are three sorts of pumps: the Sucking, the Lifting, and the Forc ing Pump. By the first, water can be raised only to about 33 feet, viz. by the pressure of the atmosphere; Lut by the others, to any height; but then they require more apparatus and power. The annexed figure represents a common sucking pump. AB is the barrel of the pump, being a hollow cylinder, made of metal, and smooth within, or of wood for very common purposes. CD is the handle, moveable about the pin E, by moving the end C up and down. DT an iron rod turning about a pin D, which connects it to the end of the handle. This rod is fixed to the piston, bucket, or sucker, TG, by which this is moved up and down within the barrel, which it must fit very tight and close, that no air or water may pass between the piston and the sides of the barrel; and for this purpose it is commonly armed with leather. The piston is made hollow, or it has a perforation through it, the orifice of which is covered by a valve H opening upwards. I is a plug firmly fixed in the lower part of the barrel, also perforated, and covered by a valve K opening upwards. 281. When the pump is first to be worked, and the water is below the plug I; raise the end C of the handle, then the piston descending, compresses the air in HI, which by its spring shuts fast the valve K, and pushes up the valve HI, and so enters into the barrel above the piston. Then putting the end C of the handle down again, raises the piston or sucker, which lifts up with it the column of air above it, the external atmosphere by its pressure keeping the valve H shut the air in the barrel being thus exhausted, or rarefied, is no longer a counterpoise to that which presses on the surface of the water in the well; this is forced up the pipe, and through the valve K, into the barrel of the pump. Then pushing the piston down again into this water, now in the barrel, its weight shuts the lower valve K, and its resistance forces up the valve of the piston, and enters the upper part of the barrel, above the piston. Then, the bucket being raised, lifts up with it the water which had passed above its valve, and it runs out by the cock L; and taking off the weight below it, the pressure of the external atmosphere on the water in the well again forces it up through the pipe and lower valve close to the piston, all the way as it ascends, thus keeping the barrel always full of water. And thus, by repeating the strokes of the piston, a continued discharge is made at the cock L. 282. There is a farther limitation of the operation, than that which relates to the 33 feet. If the elastic force of the air within the tube joined to the weight of water in the tube equal the pressure of the atmosphere, the water cannot rise in the pump. To prevent this, the product of the stroke of the piston into 33 must always exceed the square of half the greatest altitude of the piston above the surface of the water in the well. Otherwise diminish the diameter of the sucking-pipe proportionally. OF THE AIR-PUMP. 283. NEARLY on the same principles as the water-pump, is the invention of the air pump, by which the air is drawn out of any vessel, like as water is drawn out by the former. A brass barrel is bored and polished truly cylindrical, and exactly fitted with a turned piston, so that no air can pass by the sides of it, and furnished with a proper valve opening upward. Then, by lifting up the piston, the air in the close vessel below it follows the piston, and fills the barrel; and being thus diffused through a larger space than before, when it occupied the vessel or receiver only, but not the barrel, it is made rarer than it was before, in proportion as the capacity of the barrel and receiver together exceeds the receiver alone. Another stroke of the piston exhausts another barrel of this now rarer air, which again rarefies it in the same proportion as before. And so on, for any number of strokes of the piston, still exhausting in the same geometrical progression, of which the ratio is that which the capacity of the receiver and barrel together exceeds the receiver, till this is exhausted to any proposed degree, or as far as the nature of the machine is capable of performing; which happens when the elasticity of the included air is so far diminished, by rarefying, that it is too feeble to push up the valve of the piston, and escape. 284. From the nature of this exhausting, in geometrical progression, we may easily find how much the air in the receiver is rarefied by any number of strokes of the piston; or what number of such strokes is necessary, to exhaust the receiver to any given degree. Thus, if the capacity of the receiver and barrel together, be to that of the receiver alone, as c to r, and 1 denote the natural density of the air at first; then r c:r: 1 : the density after 1 stroke of the piston, C 4" So if the barrel be equal to of the receiver; then cr 5" 5: 4; and =0.8" is=d the density after n turns. And if ʼn be 20, then 0·800 — *0115 n is the density of the included air after 20 strokes of the piston; which being the 86% part of 1, or the first density, it follows that the air is 86 times rarified by the 20 strokes. 285. Or, if it were required to find the number of strokes necessary to rarefy the air any number of times; because is the proposed density d; there fore, taking the logarithms, n × log. = log. d, and n = log. d the 1. r - 1. c' number of strokes required. So if r be of c, and it be required to rarefy the log. 100 air 100 times; then d = or '01; and hence n = = 20 nearly. 1. 51. 4 So that in 203 strokes the air will be rarefied 100 times. OF THE DIVING BELL AND CONDENSING 286. On the same principles, too, depend the operations and effect of the Condensing Engine, by which air may be condensed to any degree, instead of rarefied as in the air-pump. And, like as the air-pump rarefies the air, by extracting always one barrel of air after another; so, by this other machine, the air is condensed by throwing in or adding always one barrel of air after another; which it is evident may be done by only turning the valves of the piston and barrel, that is, making them to open the contrary way, and working the piston in the same manner; so that, as they both open upward or outward in the airpump, or rarefier, they will both open downward or inward in the condenser. 287. And on the same principles, namely, of the compression and elasticity of the air, depends the use of the Diving Bell, which is a large vessel, in which a person descends to the bottom of the sea, the open end of the vessel being downward; only in this case the air is not condensed by forcing more of it into the same space, as in the condensing engine; but by compressing the same quantity of air into a less space in the bell, by increasing always the force which compresses it. 288. If a vessel of any sort be inverted into water, and pushed or let down to any depth; then by the pressure of the water some of it will ascend into the vessel, but not so high as the water without, and will compress the air into less space, according to the difference between the heights of the internal and external water; and the density and elastic force of the air will be increased in the same proportion, as its space in the vessel is diminished. A G B So, if the tube CE be inverted, and pushed down into water, till the external water exceed the internal, by the height AB, and the air of the tube be reduced to the space CD; then that air is pressed both by a column of water of the height AB, and by the whole atmosphere which presses on the upper surface of the water; consequently the space CD is to the whole space CE, as the weight of the atmosphere is to the weights both of the atmosphere and the column of water AB. So that, if AB be about 34 feet, which is equal to the force of the atmosphere, then CD will be equal to CE; but if AB be double of that, or 68 feet, then CD will be CE; and so on. And hence, by knowing the depth AF, to which the vessel is sunk, we can easily find the point D, to which the water will rise within it at any time. For let the weight of the atmosphere at that time be equal to that of 34 feet of water; also, let the depth AF be 20 feet, and the length of the tube CE 4 feet: then, putting the height of the internal water DE = x, it is 34 AB: 34: CE: CD, that is 34+ AF - DE: 34: CE: CE- DE, or · 54 -x: 34:4:4 Ꮖ ; hence, multiplying extremes and means, 216 F 58x + x2 = 136, and the root is x2 very nearly =1414 of a foot, or 17 inches nearly; being the height DE to which the water will rise within the tube. 289. But if the vessel be not equally wide throughout, but of any other shape, as of a bell-like form, such as is used in diving; then the altitudes will not observe the proportion above, but the spaces or bulks only will accord with that proportion, namely 34 + AB: 34: capacity CKL capacity CHI, if it be common or fresh-water; and 33 + AB: 33: capacity CKL capacity CHI, if it be sea-water. From which proportion the height DE may be found, when the nature and shape of the vessel or bell CKL are known. OF THE BAROMETER. 290. THE Barometer is an instrument for measuring the pressure of the atmosphere and elasticity of the air, at any time. It is commonly made of a glass tube, of near 3 feet long, close at one end, and filled with mercury. When the tube is full, by stopping the open end with the finger, then inverting the tube, and immersing that end with the finger into a bason of quicksilver, on removing the finger from the orifice, the fluid in the tube will descend into the bason, till what remains in the tube be of the same weight with a column of the |