For, the weights of the same bulk of air and water, are nearly as 1-222: 1000, therefore as 1.222: 1000 :: 333 feet: 27600 feet, or 54 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.

Corol. 4. From this proposition and the last it follows that the height is always the same, of a 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 different 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 the atmosphere above any places, D and d their densities, and H and the heights of the uniform columns, of the same densities and weights; Then H x D =

W,

and h x d = w

37.

PROP. XVII.

The density of the atmosphere, at different heights above the earth, decreases in such sort, that when the heights increase in arithmetical progression the densities decrease in geometrical progression.

Let the perpendicular line AP, erected on the earth, be conceived to be divided into a great number of very small 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, D, &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 matter from

that place upwards to the top of the atmosphere; therefore the quantity of matter in each stratum, is also as the whole quantity from that place upwards. Now, if from the whole weight at any place as B, the weight or quantity in the stratum B 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 always 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 its nth part; then

38. SCHOLIUM. 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 weights of air, at those altitudes.

So that, if D denote the density at the altitude A,

then A being as the log. of D, and a as the log. of d, the dif. of alt. A—a,

And if A = 0, or D the density at the surface of the earth; then any alt. above

Or, in general, the log. of d 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 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.

39. 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 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

D

where h will be of one constant

as log. ; assume h so that a = h× log. 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 foot, or one 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 density d at one foot above it; consequently 1= h × log. ; which, by the nature of logarithms,

27600
27599

is nearly h×

43429448
27600

h 63551

nearly; and hence h63551 feet; which

.

D

gives, for any altitude in general, this theorem, viz. a=63551 × log. '

M

M

or 63551 X log. feet, or 10592 x log. fathoms; where M is the column

m

m

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 M and m may be taken in any measure, either feet, or inches, &c.

40. Note, that this formula is adapted to the mean temperature of the air 55o. 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.

41. 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 6000 of the whole volume.

42. 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 535; therefore the corresponding change of temperature is 24° 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.

43. 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. 4363 feet, or 727 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. 2448 feet, or 408 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 syphons, pumps, barometers, &c; of which it may not be improper here to give a brief description.

OF THE SIPHON.

44. 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 syphon 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 And if the shorter leg be immersed into a vessel of water, and the syphon 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 syphon. Or, it may be set a running without filling the syphon 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 syphon by the pressure of the air on the water in the vessel.

end.

OF THE PUMP.

45. THERE are three sorts of pumps; the sucking, the lifting, and the forcing pump. By the former, water can be raised only to about 34 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 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 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.

46. When the pump is first to be worked, and the water is below the plug I; raise the end C of the handle, and the piston descending, compresses the air in HI, which by its spring shuts fast the valve K, and pushes up the valve H, 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.

OF THE AIR PUMP.

47. 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 upwards. 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.

48. 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,

cr: 1: the density after I stroke of the piston,

с