30 that the atmosphere will balance a column of water 136 times as high as one of quicksilver; consequently And hence a common sucking pump will not raise water higher than about 33 or 34 feet. And a siphon will not run, if the perpendicular height of the top of it be more than about 33 or 34 feet. 368. 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. For, the weights of the same bulk of air and water, are nearly. as 1-222 to 1000; therefore as 1-222: 1000 :: 333 feet: 27600 feet, or 5 miles nearly. And so high the atmosphere would be, if it were all of uniform density, like water. But, instead of that, from its expansive and elastic quality, it becomes continually more and more rare, the farther above the earth, in a certain proportion, which will be treated of below, as also the method of measuring heights by the barometer, which depends on it. 369. Corol. 4. From this proposition and the last it follows, that the height is always the same, of an uniform atmosphere above any place, which shall be all of the uniform density with the air there, and of equal weight or pressure with the real height of the atmosphere above that place, whether it be at the same place, at different times, or at any dirffeent places or heights above the earth; and that height is always about 5 miles, or 27600 feet, as above found. For, as the density varies in exact proportion to the weight of the column, therefore it requires a column of the same height in all cases, to make the respective weights or pressures. Thus, if w and w be the weights of atmosphere above any places, D and d their densities, and H and h the heights of the uniform columns, of the same densities and weights; Then H X D = w, and hdw; therefore or x is equal to or h. The tem perature being the same, W D PROPOSITION PROPOSITION LXXVI. 370. The Density of the Atmosphere, at Different Heights above the Earth, Decreases in such Sort, that when the Heights Increase in Arithmetical Progression, the Densi ties Decrease in Geometrical Progression. LET the indefinite perpendicular line AP, erected on the earth, be conceived to be divided into a great number of very small equal parts, A, B, C, D, &c. forming so many thin strata of air in the atmosphere, all of different density, gradually decreasing from the greatest at A: then the density of the several strata, A, B, C, D, &c. will be in geometrical progression decreasing. For as the strata A, B, C, &c. are all of equal thickness, the quantity of matter in each of them, is as the density there; but the density in any one, being as the compressing force, is as the weight or quantity of all the matter from that place upward to the top of the atmosphere; therefore the quantity of matter in each stratum, is also as the whole quantity from that place upward. Now, if from the whole weight at any place as B, the weight or quantity in the stratum в be subtracted, the remainder is the weight at the next stratum c; that is, from each weight subtracting a part which is proportional to itself, leaves the next weight; or, which is the same thing, from each density subtracting a part which is proportional to itself, leaves the next density. But when any quantities are continually diminished by parts which are proportional to themselves, the remainders form a series of continued proportionals: consequently these densities are in geometrical progression. n-1 part, or the Thus, if the first density be D, and from each be taken its nth part; there will then remain its part, putting m for n-1; and therefore the series of denm m2 m3 m4 sities will be D,D, D, D, D, &c. the common ratio 12 723 of the series being that of n to m. SCHOLIUM. 371. Because the terms of an arithmetical series, are proportional to the logarithms of the terms of a geometrical series therefore different altitudes above the earth's sur face, face, are as the logarithms of the densities, or of the weights of air, at those altitudes. So that, if D denote the density at the altitude ▲, and d the density at the altitude a; D then a being as the log. of D, and a as the log. of d, the dif. of alt. A will be as the log. D-log. d. or log d And if A O, or D the density at the surface of the earth; then any altitude above the surface a, is as the log. of d D is Or, in general, the log. of place 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 deter mining the heights of mountains and other eminences, by the barometer, 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 be the 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 as log. ; assume h so that ah X log. where will be of one constant value for all altitudes; and to determine that value, let a case be taken in which we know the altitude a corresponding 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 VOL. II. = D Ģ g denote denote the density D at the lower place, and 27599 the less 27600 density dat 1 foot above it; consequently 1 = h × log. 27599 which, by the nature of logarithms, is nearly=hX -43429448 27600 h nearly; and hence h=63551 feet; which gives, 63551 D M M m for any altitude in general, this theorem, viz. a = 63551 × log. or == 63551 X log feet, or 10592 x log. m 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 tem peratures at the top and bottom of the altitude a, that altitude will vary by its 455th part; which must be added, when that medium exceeds 55°, otherwise subtracted. 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 go of the whole volume. 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 31°. So that the formula is, a = 10000 × log. fathoms, M 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 50° ? 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 397 fathoms. 378. Exam, 3, 378. Exam. 3. At what altitude is the density of the atmosphere only the 4th part of what it is at the earth's surface? 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 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 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 into the siphon by the pressure of the air on the water in the yessel. OF |