Thus, if the first density be D, and from each be taken its nth part; then

39. 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,

will be as the log. D — log. d or log,

D

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

the surface a, is as the log. of

D

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.

D

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 resul's 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 as log. ; assume h so that a = h× log., 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 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

d

27600

above it ; consequently 1= h× log. 27599; which, by the nature of logarithms,

is nearly X

•43429448
27600

h

nearly; and hence h=65551 feet; which

63551

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

M

ገቢ

M

or 63551 X log.feet, or 10592 × log. fathoms; where M 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 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 55o, 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 whole volume.

of the

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 above that, the result is to be increased by so many times its 435th part.

m

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 mean temperature being 50"?

EXAM. 2. To find the height of a 29-45 inches of mercury at the bottom, and perature being 33° ?

the bottom, and 25-28 at the top;

Ans. 4363 feet, or 727 fathoms hill when the atmosphere weighs 26-82 at the top, the mean temAns. 2443 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 end. 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.

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 perfo ration 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 coninned 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 cylindri cal, 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 unus 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 I denote the natural density of the air at first; then,

So, if the barrel be equal to of the receiver; then cr:: 5 : 4; and 5" = 0·8" is = d the density after n turns. And if n be 20, then 0·8a — •0115 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 rarefied by the 20 strokes.

49. 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;

log, d 1. r I. c'

the

therefore taking the logarithms n log. = log. d, and n = number of strokes required. So, if r be 3 of c, and it be required to rarefy the

air 100 times; then d = fŋ or '01; and hence n =

log. 100 1. 5

So that in 20 strokes the air will be rarefied 100 times.

=203 nearly.

OF THE DIVING BELL, AND CONDENSING MACHINE.

50. Os 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 upwards, or outwards, in the air-pump, or rarefier, they will both open downwards, or inwards, in the condenser.

51. 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 downwards; 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.

52. If a vessel of any sort be inverted into water, and pushed or let down to any depth in it; 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 interval 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.

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 CF, as the weight of the atmosphere, is to the weights Loth of the atmosphere and the column of