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ENCYCLOPÆDIA PERTHENSIS.

PNEUMATIC S.

(Concluded from Volume Seventeenth.)

NOTHER eafy method is this: Let an apparatus abcdef, pl. CCLXXXI. fig. 48.) be made, rafting of a horizontal tube ae of even bore, a dge of a large diameter, and a swan-neck tube Let the ball and part of the tube geb be filled with mercury, fo that the tube may be in the fame horizontal plane with the furface de of the mercary in the ball. Then feal up the end a, and conDet fwith an air-pump. When the air is abftracted from the furface de, the air in a b will expand into a larger bulk ac, and the mercury in the pump-gage will rife to fome diftance below the barometric height. This distance, without farther calculation, will be the measure of the elafticity of the air preffing on the furface de, and therefore of the air in a c.

ceiver of an

e

The most exact method is to fufpend in the reair-pump a glass veffel, having a very narow mouth over a ciftern of mercury, and then abtract the air till the gage rifes to fome determired height. The difference between this and the barometric height determines the elafticity of the air in the receiver and in the fufpended veffel. Now lower down the veffel by the flipwire till its mouth is immerfed into the mercury, and admit the air into the receiver; it will prefs the mercury into the little veffel. Lower it ftill farther down, till the mercury within it is level with that without; then stop its mouth, take it out and weigh the mercury, and let its weight be Subtract this weight from the weight of the mercury, which would completely fill the whale veffel; then the natural bulk of the air will b, while its bulk, when the elafticity e in the rarefied receiver, was the bulk or capacity w of the veffel. Its denfity therefore correfponding to this elaAicity e, was And thus may the tation between the dentity and elafticity in all

ci⚫ be obtained.

Voz. XVIII. PART I

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Various experiments for this purpose have been made. Those made by M. de Luc, General Roy, Mr Trembley, and Sir George Shuckbourgh, are by far the moft accurate; but they are all confined to very moderate rarefactions. The general refult has been, that the elasticity of rarefied air is very nearly proportional to its density. No regular deviation from this law has been obferved, there being as many obfervations on one fide as on the other; but it is certainly worthy the attention of philofophers to determine it with precifion in the cafes of extreme rarefaction, where the irregularities are most remarkable. The great fource of error is a certain adhesive fluggishness of the mercury when the impelling forces are very small ; and other fluids can hardly be used; becaufe they either smear the infide of the tube and diminish its capacity, or they are converted into vapour, which alters the law of elafticity.

Upon the whole, we may affume the Boylean law, viz. that the elafticity of the air is propor tional to its denfity. The law deviates not in any fenfible degree from the truth in thofe cafes which are of the greatest practical importance, that is, when the denfity does not fo much exceed or fall fhort of that of ordinary air.

With refpect to the action of the particles on each other, the investigation is extremely eafy. We have feen that a force 8 times greater than the preffure of the atmosphere will comprefs common air into the 8th part of its common bulk, and give it 8 times its common denfity: and in this cafe the particles are at half their former diftance, and the number which are now acting on the furface of the pifton employed to comprefs them is quadruple of the number which act on it when it

is of the common density. Therefore, when this eightfold compreffing force is diftributed over a fourfold number of particles, the portion of it

which acts on each is double, In like manner, A when

x

when a compreffing force 27 is employed, the air is compreffed into one 27th of its former bulk, the particles are at of their former diftance, and the force is diftributed among 9 times the num. ber of particles; the force on each is therefore 3. In short, let be the distance of the particles, the number of them in any given vessel, and therefore the denfity will be as 3, and the number preffing by their elafticity on its whole internal furface will be as x2. Experiment fhows, that the compreffing force is as 3, which being diftributed over the number as x2, will give the force on each as x. Now this force is in immediate equilibrium with the elafticity of the particle immediately contiguous to the compreffing furface. This elafticity is therefore as x and it follows from the nature of perfect fluidity, that the particle adjoining to the compreffing furface preffes with an equal force on its adjoining particles on every fide. Hence the corpufcular repulfions exerted by the adjoining particles are inverfely as their diftances from each other; or the adjoining particles tend to recede from each other with forces inverfely proportional to their distances.

Sir ISAAC NEWTON was the first who reasoned in this manner from the phenomena. Indeed he was the first who had the patience to reflect on the phenomena with any precifion. His difcoveries in gravitation naturally gave his thoughts this turn, and he very early hinted his fufpicions that all the characteristic phenomena of tangible matter were produced by forces which were exerted by the particles at small and infenfible diftances: And he confiders the phenomena of air as affording an excellent example of this inveftigation, and deduces from them the law which we have now demonftrated; and fays that air confifts of particles which avoid the adjoining particles with forces inverfely proportional to their diftances, from each other. From this he deduces (in the 2d book of his Principles) feveral beautiful propofitions, determining the mechanical conftitution of the atmosphere. But he limits this action to the adjoining particles: and this is a remark of immenfe confequence, though not attended to by the numerous experimenters who adopt the

law.

The particles are supposed to act at a distance; this diftance is variable; and the forces diminish as the diftance increases. A very ordinary airpump will rarefy the air 125 times. The diftance of the particles is now 5 times greater than before; and yet they ftil repel each other: for the air of this denfity will ftill fupport the mercury in a fyphon-gage at the height of o'24 of an inch; and a better pump will allow this air to expand twice as much, and still leave it elastic. Thus, whatever is the distance of the particles of common air, they can act five times farther off. The question then is. Whether, in the ftate of common air, they really do act five times farther than the distance of the adjoining particles? While the particle a acts on the particle b with the force 5, does it alfo act on the particle c with the force 2'5, on the particle d with the force 1'667, on the particle e, with the force 1.25, on the particle, with the force 1, on the particle g, with the force

o'8333, &c.? Sir Ifaac Newton fhows plainly, th this is not the cafe; for if it were, the fenfib phenomena of condenfation would be totally d ferent from what we obferve. The force nece fary for a quadruple condensation would be the force must be 27 times greater. Two fpher times greater, and for a nonuple condenfatic filled with condensed air must repel each othe and two fpheres containing air that is rarer tha the furrounding air must attract each other, & All this will appear very clearly, by applying t air the reafoning which Sir Ifaac Newton has em ployed in deducing the fenfible law of mutu tendency of two fpheres, which confift of pai ticles attracting each other with forces propo tional to the fquare of the diftance inverfely. If we could fuppofe that the particles of air re pelled each other with invariable forces at all di tances within fome fmall and infenfible limit, thi would produce a compreffibility and elafticity fi milar to what we obferve. But this law of cor pufcular force is unlike every thing we obferve i nature, and to the laft degree improbable. W muft therefore continue the limitation of this mu tual repulfion of the particles of air, and be con tented for the prefent with having established it as an experimental fact, that the adjoining particles of air are kept afunder by forces inverfely proportional to their diftances; or perhaps it is better to abide by the fenfible law, that the denfity of air is proportional to the compreffive force. This law is abundantly sufficient for explaining all the subordinate phenomena, and for giving us a complete knowledge of the mechanical conftitution of our atmosphere.

SecT. VI. Of the HEIGHT of the ATMOSPHERE.

THE preceding view of the compreffibility of air muft give us a very different notion of the height of the atmosphere from what we deduced from our experiments. When the air is of the temperature 32° of Fahrenheit's thermometer, and the mercury in the barometer ftands at 30 inches, it will defcend one 10th of an inch if we take it to a place 87 feet higher. Therefore, if the air were equally denfe and heavy throughout, the atmosphere would be 30 X 10 X 87 feet, or 5 miles and ico yards. But it must be much higher; becaufe every ftratum as we afcend must be fucceffively rarer as it is lefs compreffed by incumbent weight. (See ATMOSPHERE, § 6.) Not knowing to what degree air expanded when the compreffion was diminifhed, we could not tell the fucceffive diminution of denfity and confequent augmentation of bulk and height; we could only fay, that feveral atmospheric appearances indicated a much greater height. Clouds have been feen much higher; but the phenomenon of the TWILIGHT is the most convincing proof of this. There is no doubt that the vifibility of the sky or air is owing to its want of perfect transparency, each particle (whether of matter purely aerial or heterogeneous) reflecting a little light.

Let (fig. 49.) be the laft particle of illuminated air, which can be feen in the horizon by a spectator at A. This must be illuminated by a ray SD b, touching the earth's furface at some point D. Now it is a known fact, that the degree of illumination

llumination called twilight is perceived when the fun is 18 below the horizon of the fpectator, that, when the angle EbS or ACD is 18 degrees; therefore &C is the fecant of 9° (it is lefs, viz. about 8, on account of refraction.) We know the earth's radius to be about 3970 miles: hence we conclude B to be about 45 miles; nay, a very fenfible illumination is perceptible much farther from the fan's place than this, perhaps twice as far, and the air is fufficiently denfe for reflect. ing a fenfible light at the height of nearly 200

miles.

We have feen that air is prodigiously expanfible. None of our experiments have diftinctly shown us any bint. But it does not follow that it is expanSible without end. It is much more probable that there is a certain distance of the parts in which they would arrange themfelves if they were not heavy. But at the very fummit of the atmosphere they will be a very fmall matter nearer to each other, on account of their gravitation to the earth. Till we know precifely the law of this mutual repulion, we cannot fay what is the height of the atmofphere. But if the air be an elastic fluid whofe denfity is always proportionable to the Compreffing force, we can tell what is its denfity any height above the furface of the earth; and we can compare the denfity fo calculated with the denfity difcovered by obfervation: for this tis measured by the height at which it fupparts mercury in the barometer. This is the direct mafure of the preffure of the external air; and we know the law of gravitation, we can tell What would be the preffure of air having the calcalated denfity in all its parts.

Suppofe a prifmatic or cylindric column of air reaching to the top of the atmosphere. Let this be divided into an indefinite number of ftrata of very fmall and equal depths or thicknefs; and let us fuppofe that a particle of air is of the fame weight at all diftances from the centre of the earth. The abfolute weight of any of these ftrata will on thefe conditions be proportional to the ber of particles, or the gravity of air contained in it; and fince the depth of each itratum is the Game, this quantity of air will be as the denfity of the ftratum; but denfity of any ftratum is as the compreffing force, i. e. as the preffure of the ftrata above it; i. e. as their weight; i.e. as their quantity of matter-therefore the quantity of air in each ftratum is proportional to the quantity of air above it; but the quantity in each ftratum is the difference between the column incumbent on its bottom and on its top: thefe differences are therefore proportional to the quantities of which they are the differences. But when there isa feries of quantites which are proportional to their own differences, both the quantities and their differences are in continual or geometrical progreffion: for let a, b, c, be three fuch quantities that

b: ca-b: b-c, then, by altern.
b: a b c :c-b and by compof.

b:

a=c : b

and a: bb : c therefore the denfities of thefe ftrata decreafe in a geometrical progreffion; that is, when the elevaions above the centre or furface of the earth in

creafe, or their depths under the top of the atmosphere decreafe, in an arithmetical progreffion, the denfities decrease in a geometrical progreffion.

Let ARQ (fig. 50.) reprefent the section of the earth by a plane through its centre O, and let m OAM be a vertical line, and AE perpendicular to OA will be a horizontal line through A, a point on the earth's furface. Let AE be taken to reprefent the density of the air at A; and let DH, parallel to AE, be taken to AE as the denfity at D is to the denfity at A: it is evident, that if a logiftick or logarithmic curve EHN be drawn, having AN for its axis, and paffing through the points E and H, the denfity of the air at any other point C, in this vertical line will be repre fented by CG, the ordinate to the curve in that point: for it is the property of this curve, that if portions AB, AC, AD, of its axis be taken in arithmetical progreffion, the ordinates AE, BF, CG, DH, will be in geometrical progreffion. It is another fundamental property of this curve, that if EK or HS touch the curve in E or H, the fubtangent AK or DS is a conftant quantity. A 3d fundamental property is, that the infinitely extended area MAEN is equal to the rectangle KAEL of the ordinate and fubtangent; and in like manner, the area MDHN is equal to SD × DH, or to KAX DH; confequently the area lying beyond any ordinate proportional to that ordinate.

Thefe geometrical properties of this curve are all analogous to the chief circumftances in the conftitution of the atmosphere, on the fuppofition of equal gravity. The area MCGN reprefents the whole quantity of aerial matter which is above C: for CG is the denfity at C, and CD is the thickness of the ftratum between C and D; and therefore CGD will be as the quantity of matter or air in it; and in like manner of all the others, and of their fums, or the whole area MCGN; and as each ordinate is proportional to the area above it, fo each denfity, and the quantity of air in each stratum, is proportional to the quantity of air above it: and as the whole area MAEN is equal to the rectangle KAEL, fo the whole air of variable denfity above A might be contained in a column KA, if, inftead of being compressed by its own weight, it were without weight, and compreffed by an external force equal to the preffure of the air at the furface of the earth. In this cafe, it would be of the uniform denfity AE, which it has at the furface of the earth, making what we have repeatedly called the homogeneous atmosphere.

Hence we derive this important circumftance, that the height of the homogeneous atmosphere is the fubtangent of that curve whofe ordinates are as the densities of the air at different heights, on the fuppofition of equal gravity. This curve may with propriety be called the ATMOSPHERICAL LOGARITHMIC; and as the different logarithmics are all characterised by their fubfequents, it is of importance to determine this one. It may be done by comparing the denfities of mercury and air. For a column of air of uniform denfity, reaching to the top of the homogeneous atmosphere, is in equilibrio with the mercury in the barometer. Now it is found, by the best experiments, that when mercury and air are of the temperature 32° 0. Fahrenheit's

A 2

4

PNEUMATICS.

Fahrenheit's thermometer, and the barometer ftands at 30 inches, the mercury is nearly 10440 times denfer than air. Therefore the height of the homogeneous atmosphere is 10440 times 30 inches, or 26100 feet, or 8700 yards, or 4350 fathoms, or 5 miles wanting 100 yards.

Or it may be found by obfervations on the barometer. It is found, that when the mercury and air are of the above temperature, and the barometer on the fea-hore ftands at 30 inches, if we carry it to a place 884 feet higher, it will fall to 29 inches. Now, in all logarithmic curves having equal ordinates, the portions of the axes intercepted between the corresponding pairs of ordinates are proportional to the fubtangents. And the fubtangents of the curve belonging to our common tables is 0'4342945, and the difference of the loga. rithms of 30 and 29 (which is the portion of the axis intercepted between the ordinates 30 and 29), or o'0147233, is to o°4342945 as 883 is to 26058 feet, or 8686 yards, or 4343 fathoms, or 5 miles, wanting 114 yards. This determination is 14 yards lefs than the other, and it is uncertain which It is extremely difficult to is the most exact. measure the refpective denfities of mercury and air; and in measuring the elevation which produces a fall of one inch in the barometer, an error of one 20th of an inch would produce all the difference. We prefer the laft, as depending on fewer circumftances. But all this investigation proceeds on the fuppofition of equal gravity, whereas we know that the weight of a particle of air decreases as the fquare of its diftance from the centre of the earth increases. In order, therefore, that a fuperior ftratum may produce an equal preffure at the furface of the earth, it must be denfer, because a particle of it gravitates lefs. The denfity, therefore, at equal elevations, muft be greater than on the fuppofition of equal gravity, and the law of diminution of density must be different.

Make OD: OA≈OA: Od; OC: OA=OA: Oc; OB: OA=OA: Ob, &c.; fo that Od, Or, Ob, OA, may be reciprocals to OD, OC, OB, OA; and through the points A, b, c, d, draw the perpendiculars AE, bf, cg, db, making them proportional to the denfities in A, B, C, D; and let us fuppofe CD to be exceeding ly fmall, fo that the density may be fuppofed uniform through the whole ftratum. Thus we have OD XOd=OA2, =OCX Oc

and Oc: Od=OD: OC:

and Oc: Oc-Od=OD: OD-OC,

or Oc: cd OD: DC;

and cd: CD-Oc: OD;

vitation of each particle. Therefore, cd X cg is a the preffure on C arifing from the weight of th ftratum DC; but edXcg is evidently the elemen of the curvilineal area AmnE, formed by the curv Therefore the fum of all the elements, fuch as Efghn and the ordinates AE, bf, cg, ab, &c. mn cdhg, that is, the area emng below cg, will be a the whole preffure on C, arifing from the gravita tion of all the air above it; but, by the nature of air, this whole preffure is as the denfity which it is of fuch a nature that the area lying below Or produces, that is, as cg. Therefore the curve Eg dinate. This is the property of the logarithmic beyond any ordinate cg is proportional to that orcurve, and Egn is a logarithmic curve.

The

But farther, this curve is the fame with EGN. For let B continually approach to A, and ultimately coincide with it. It is evident that the ultimate ratio of BA to Ab, and of BF to bf, is that of equality; and if EFK, Efk, be drawn, they will contain equal angles with the ordinate AE, and will cut off equal fubtangents AK, Ak. curves EGN, Egn are therefore the fame, but in oppofite pofitions. Laftly, if OA, Ob, Oc, Od, &c. be taken in arithmetical progreffion decreaf ing, their reciprocals OA, OB, OC, OD, &c. will be in harmonical progreffion increafing, as is well known; but, from the nature of the logarithmic curve, when OA, Ob, Oc, Od, &c. are in arithmetical progreffion, the ordinates AE, bf, cg, dh, &c. are in geometrical progreffion. Therefore when OA, OB, OC, OD, &c. are in harmonical progreffion, the denfities of the air at A, B, C, D, &c. are in geometrical progreffion; and thus may the denfity of the air at all elevations be difcovered. Thus to find the denfity of the air at K, the top of the homogeneous atmofphere, make OK: ÒA=OA: OL, and draw the ordinate LT, LT is the denfity at K.

Dr HALLEY was the firft who observed the re-
lation between the denfity of the air and the ordi-
nates of the logarithmic curve, or common loga-
rithms. This he did on the fuppofition of equal
gravity; and his difcovery is acknowledged by Sir
Ifaac Newton in Princip. ii. prop. 22. Schol. His
differtation on the fubject is in N° 185. of the Phil.
Tranf. Newton extended the fame relation to the
true ftate of the cafe, where gravity is as the fquare
of the diftance inverfely; and fhowed, that when
the distance from the earth's centre are in harmo-
nic progreffion, the denfities are in geometric pro-
greffion. He fhows indeed, in general, what pro-
greffion of the distance, on any fuppofition of gra
vity, will produce a geometrical progreffion of the
denfities, fo as to obtain a fet of lines OA, Ob,

or, becaufe OC and OD are ultimately in the ra- Oc, Od, &c. which will be logarithms of the den-
tio of equality, we have

ed: CD-Oc: OC=OA2: OC2,

and cd=CD X

AO2

OA' anded Xcg=CDx gx2

OC2;

but CDXcgXOA is the preffure at C arifing from

OC2

the abfolute weight of the ftratum CD. For this
weight is as the bulk, as the denfity, and as the
gravitation of each particle jointly. Now CD ex-
OA the ar-
prefes the bulk, cg the denfity, and

OC2

fities. The fubject was afterwards treated in a
more familiar manner by Cotes in his Hydroft. Lea.
and in his Harmonia Menfurarum; alfo by Dr
Brooke Taylor, Meth. Increment; Wolf in his A.
crometria; Herman in his Phoronomia; &c. &c.
and lately by Horfley. Phil. Tranf. tom. ixiv.

From thefe principles an important corollary is
deducible, viz. that the air has a finite density at
an infinite diftance from the centre of the earth,
namely, fuch as will be reprefented by the ordi-
nate OP drawn through the centre. It may
infinity
objected to this conclufion, that it would infer an

be

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