Various attempts have been made to ascertain the height to which the atmos phere is extended all round the earth. These commenced soon after it was dis covered, by means of the Torricellian tube, that air is endued with weight and pres sure. And had not the air an elastic pow er, but were it every where of the same density, from the surface of the earth to the extreme limit of the atmosphere, like water, which is equally dense at all depths, it would be a very easy matter to deter mine its height, from its density, and the column of mercury which it would coun terbalance in the barometer tube ; for, it having been observed that the weight of the atmosphere is equivalent to a column of 30 inches, or 21 feet of quicksilver, and the density of the former to that of the latter as 1 to 11040: therefore the 'height of the uniform atmosphere would be 11040 times 21 feet, that is 27600 feet, or little more than five miles and a quar ter. But the air, by its elastic quality, expands and contracts ; and it being found, by repeated experiments in most nations of Europe, that the spaces it oc cupies, when compressed by different weights, are reciprocally proportional to those weights themselves ; or, that the more the air is pressed, so much the less space it takes up ; it follows, that the air in the upper regions of the atmosphere must grow continually more and more rare, as it ascends higher ; and indeed that, according to that law, it must neces sarily be extended to an indefinite height. Now, if we suppose the height of the whole divided into innumerable equal parts, the quantity of each part will be as its density ; and the weight of the whole incumbent atmosphere being also as its density ; it follows, that the weight of the incumbent air is every where as the quantity contained in the subjacent part ; which causes a difference between the weights of each two contiguous parts of air. But, by a theorem in arithmetic, when a magnitude is continually dimi nished by the like part of itself; and the remainders the same, there will be a se ries of continued quantities, decreasing in geometrical progression : therefore, if, according to the supposition, the altitude of the air, by the addition of new parts into which it is divided, do continually increase in arithmetical progression, its density will be diminished, or, which is the same thing, its gravity decreased, in continued geometrical proportion. And hence, again, it appears that, according to the hypothesis of the density being always proportional to the compressing force, the height of the atmosphere must necessarily be extended indefinitely. And, farther, as an arithmetical series adapted to a geometrical one is analogous to the logarithms of the said geometrical one ; it follows, therefore, that the altitudes are proportional to the logarithms of the den sities, or weights of air : and that any height taken from the earth's surface, which is the difference of two altitudes to the top of the atmosphere, is propor tional to the difference of the logarithms of the two densities there, or to the loga rithm of the ratio of those densities, or their corresponding compressing forces, as measured by the two heights of the barometer there.
It is now easy, from the foregoing pro perty, and two or three experiments, or barometrical observations, made at known altitudes, to deduce a general rule to de termine the absolute height answering to any density, or the density answering to any given altitude above the earth. And, accordingly, calculations were made upon this plan by many philosophers, particu larly by the French ; but it having bean found that the barometrical observations did not correspond with the altitudes, as measured in a geometrical manner, it was suspected that the upper parts of the at mospherical regions were not subject to the same laws with the lower ones, in re gard to the density and elasticity. .Ana indeed, when it is considered that the at mosphere is a heterogeneous mass of par ticles of all sorts of matter, some elastic, and others not, it is not improbable but this may be the case, at least in the re gions very high in the atmosphere, which it is likely may more copiously abound with the electrical fluid. Be this how
ever as it may, it has been discovered that the law above given holds very well, for all such altitudes as are within our reach, or as far as to the tops of the highest mountains on the earth, when a correction is made for the difference of the heat or temperature of the air only, as was fully evinced by M. De Luc, in a long series of observations, in which he determined the altitudes of hills, both by the barometer, and by geometrical measurement, from which he deduced a practical rule to al low for the difference of temperature. Similar rules have also been deduced, from accurate experiments, by Sir George Shuckburgh and General Roy, both con curring to spew, that such a rule for the altitudes and densities holds true for all heights that are accessible to us, when the elasticity of the air is corrected on account of its density : and the result of their experiments spewed, that the dif ference of the logarithms of the heights of the mercury in the barometer, at two stations, when multiplied by 10000, is equal to the altitude, in English fathoms, of the one place above the other ; that is, when the temperature of the air is about 31 or 32 degrees of Fahrenheit's ther mometer, and a certain quantity more or less, according as the actual temperature is different from that degree.
But it may be shown, that the same rule may be deduced, independent of such a train of experiments as those referred to, merely by the density of the air at the surface of the earth. Thus, let D denote the density of the air at one place, and d the density at the other ; both measured by the column of mercury in the barome trical tube ; then the difference of alti tude between the two places will be pro portional to the log. of D—the log. of d, or to the log. of— 1) . But as this formula expresses only the relation between dif ferent altitudes, and not the absolute quantity of them, assume some indeter minate, but constant, quantity h, which multiplying the expression log. , may be equal to the real difference of altitude a, that is, a = h X log. of — Then, to determine the value of the general quan tity h, let us take a case, in which we know the altitude a that corresponds to a known density d; as for instance, taking a = 1 foot, or 1 inch, or some such small altitude : then, because the density D may be measured by the pressure of the whole atmosphere, or the uniform column of 27600 feet, when the temperature is 55° ; therefore 27600 feet will denote the den sity D at the lower place, and 27599 the less density d at 1 foot above it ; conse 00 quently 1 h x log.
(127— which, by 27599' the nature of logarithms, is nearly -= h x .43429448 1 2760U or 63551 nearly ; and hence we find Is = 63551 feet ; which gives us this formula for any altitude a in general, viz. a = 63551 x log. or a = 63551 x log. feet, or 10592 x log. fathoms ; where M denotes the column of mercury in the tube at the lower place and in that at the upper. This formula is adapted to the mean temperature of the air 55°: but it has been found, by the experiments of Sir George Shuckburgh and general Roy, that for every degree of the thermometer, different from 55°, the altitude a will vary by its 435th part ; hence, if we would change the factor Is from 10592 to 10000, because the differ ence 592 is the 18th part of the whole factor 10592, and because 18 is the 24th part of 435 ; therefore the change of temperature, answering to the change of the factor Is, is which reduces the 55° to 31°. So that, a — 10000 x log. of—, At fathoms, is the easiest expression for the altitude, and answers to the temperature of 31°, or very nearly the freezing point : and for every degree above that, the re sult must be increased by so many times its 435th part, and diminished when be low it.