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 1 and h the heights of the uniform columns, of the same densities and

W

weights; then H X D = w, and h× d = w; therefore D or h the temperature being the same.

or H is equal to

w

d

285. PROP. With regard to the atmosphere, at different heights above the earth, this law obtains that when the heights increase in arithmetical progression, the densities 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 de. creasing.

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 в, 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 cach 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.

Thus, if the first density be D, and from each be taken

part, putting m for n 1; and therefore the series of den.

286. 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 surface, 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 A,

and d.

the density at the altitude a;

then a being as the log. of D, and a as the log. of d,

D

the dif. of alt. Aa will be as the log. Dlog. d, or log. d And if a = 0, or D the density at the surface of the earth;

D

then any altitude above the surface a, is as the log. of ď

Or, in general, the log. of is as the altitude of the one 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 determining the heights of mountains and other eminences, by the barometer, which (art. 302) 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 al tered by the different distances from the carth's centre.

287. But as this formula expresses only the relations between 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

The

are various experiments by which this may be done. 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 alti

D

D

tude a is always as log. assume h so that a=hX log. d'

where h 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 denote the density D at the lower place, and 27599 the less 27600

density d at 1 foot above it; consequently 1=hXlog.

27599

⚫43429448

which, by the nature of logarithms, is nearly h×· 27600

h

63551

nearly; and hence h=63551 feet; which gives, for any altitude in general, this theorem, viz. a = 63551 ×

D

M

M

;

m

log. or = 63551 X log. ď 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.

288. 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.

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.

289. 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.

290. Taking, instead of the logarithms, the first term of B- -b the logarithmic series, we have 55000. for the altitude B+b'

in feet в and b, being the heights of 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.

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.

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. 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 will be proper here to give a brief description.

OF THE SIPHON.

201. 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 po- A sition; 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 sup. ported 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 i VOL. II.

36

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.

292, THERE are three sorts of pumps; the Sucking, the Lifting, and the Forcing Pump. By the first, water can be raised only to about 33 feet, viz. by the pressure of the atmosphere; but 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.

DF an iron rod turning about a pin D, which connects it to the