face, are as the logarithms of the densities, or of the weights of air, at those altitudes. So that, if o denote the density at the altitude a, the density at the altitude a; then a being as the log. of p, and a as the log. of d, D the dif. of alt. 4-a will be as the log. p-log, d. or log.. A And if a=0, or p the density at the surface of the earth ; then any altitude above the surface a, is as the log. of D Or, in general, the log. of is as the altitude of the one place d above the other, whether the lower place be at the surface of the earth, or any where else, And from this property is derived the method of determin ing the heights of mountains and other eminences, by the ba rometer, which is an instrument that measures the pressure or density of the air at any place. For, by taking with this instrument, the pressure or density, at the foot of a hill for instance, and again at the top of it, the difference of the logarithms of these two pressures, or the logarithm of their quotient, will be as the difference of altitude, or as the height of the hill; supposing the temperatures of the air to the be same at both places, and the gravity of air not altered by the different distances from the earth's centre, ་ 372. But as this formula expresses only the relations be-. tween different altitudes with respect to their densities, recourse must be had to some experiment, to obtain the real altitude which corresponds to any given density, or the density which corresponds to a given altitude. And there are various experiments by which this may be done. The first, and most natural, is that which results from the known specific gravity of air, with respect to the whole pressure of the atmosphere on the surface of the earth. Now, as the altitude a is always D D as log.; assume h so that a=hXlog. where h will be of d' one constant value for all altitudes; and to determine that value, let a case be taken in which we know the altitude a core responding to a known density d; as for instance, take a=1 foot, or 1 inch, or some such small altitude; then, because the density D may be measured by the pressure of the atmosphere, or the uniform column of 27600 feet, when the temperature is 55°; therefore 27600 feet will denote the density p at the VOL. II lower 30 lower place, and 27599 the less density d at 1 foot above it; hence h=63551 feet; which gives, for any altitude in gene DM ral, this theorem, viz. a 63551 X log. or 63551X log. M m M m d' feet, or 10592 Xlog. fathoms; where м is the column of mercury which is equal to the pressure or weight of the atmosphere at the bottom, and m that at the top of the altitude a; and where м and m may be taken in any measure, either feet or inches, &c. 373. Note, that this formula is adapted to the mean temperature of the air 55°. But, for every degree of temperature different from this, in the medium between the temperatures at the top and bottom of the altitude a, that altitude will vary by its 435th part; which must be added, when that medium exceeds 55°, otherwise subtracted. 1 374: Note, also, that a column of 30 inches of mercury varies its length by about the part of an inch for every degree of heat, or rather of the whole volume. 9600 32.0 375. 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 24°, which reduces the 55° to 31o. So that the formula is, a = 10000 × log. fathoms, M when the temperature is 31 degrees; and for every degree above that, the result is to be increased by so many times its 435th part. 376. Exam. 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 50o ? Ans. 4378 feet, or 730 fathoms. 377. Exam. 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. 2385 feet, or 3971 fathoms. ! 378. Exam. 3. At what altitude is the density of the atmosphere only the 4th part of what it is at the earth's sur face? Ans. 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 may not be improper here to give a brief description. OF THE SIPHON. 379. THE 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 sup port 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 a running as above, it will continue to run till all the water be exhausted out of the vessel, or at least as low as that end of the siphon. Or, it may be set a 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 the air out; and the water will presently follow, being forced up in the siphon by the pressure of the air on the water in the vessel. OF " an iron rod turning about a pin p, which connects it to the end of the handle. This rod is fixed to the piston, bucket, or sucker, FG, 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 н 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. 381. When the pump is first to be worked, and the water is below the plug 1; raise the end c of the handle, then the piston descending, compresses the air in HI, which by its spring shuts fast the valve x, and pushes up the valve H, and so enters into the barrel above the piston. Then put ting the end 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, f barrel its weight shuts the lower valve к, 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 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.. OF THE AIR-PUMP. 382. 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 cápacity 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 pis ton and escape. 383. 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 |