AV • 11" : \1" : NV" • \V" : \V"', &c.
Now, W W' : .:3`, and W' : W" : : P : .:/`" Hence a`: P P' : P", kc. Q. E. D.
Now it has been ascertained by the experiments of Pe rier upon the Puy. de Dome, that by ascending about 3000 feet the barometer fell from 28 to 24.7 inches. The den sity, consequently, at the earth's surface is to the density at this height in the ration of 28 to 24J. By taking 12 terms, in a decreasing geometrical progression with 28 and 24 7, the least will be the height at which the barome ter should stand at an elevation of 36,000, or nearly seven miles. To find this, we know, from the principles of geo / metrical progression, that the least term, or a = r (n— 1) 28 28\ 11= 3.972, or four inches. At this height; 124.7/ therefore, the density is only 4- of what it is at the earth. From this fact the following table of the atmospheric den sity at different elevations is easily found.
Altitude in miles. Corresponding density.
0 1 7 14 21 28 T.13" 35 42_„ 49 17 ITT 56 3T3€ From this it would appear that, even an at infinite height, the density of the atmosphere, though small, does not be come nothing. This curious circumstance was first re marked by Dr. Taylor, who thought it so improba ble, that he was ir• lined to believe, that when the air be came highly rarefied, its density decreased in a much high.
er proportion than the compressive force; which would have the effect of circumscribing the whole within narrow er limits. Were the density of the air to be concluded from the principles laid down by Cotes so great, as sensi bly to disturb the motion of the planets, it would be neces sary to have recourse to such an hypothesis.
This, however, is by no means the case. The following remarks have been made on this subject by an ingeni ous philosopher. know not for what reason mathe maticians have been afraid to admit the infinitude of the atmosphere of the earth, whether they thought it would bear hard upon the Newtonian doctrine of a void, or im pede the planetary motions. But neither the one nor the other of these consequences is to be apprehended; for neither the phenomena of nature nor the principles of the Newtonian philosophy, require that there should be any where a chasm in the universe, or that the whole material world should be actually circumscribed within any finite space. A large proportion of intersper
sed vacuity is sufficient for all purposes." The air, even at a comparati% el) small elevation, becomes exceedingly rare, and could not produce a sensible resistance upon the motions of a planet after very many ages. The table an nexed, and which was calculated by Bishop Horsley, who made the observations quoted above, will show this dis tinctly.
In order to illustrate the manner in which the density of the atmosphere diminishes, as we recede from the earth's surface, let CD, Fig. 2, Plate CCCCLXV, be a curve, so connected with the abscissa AB, that the ordinates CA, FE, which are removed from each other by any common distance AE, are in geometrical progression.
Then if CA be taken to represent the density of the surface of the earth, and one-fourth CA the densitty at L, distant seven miles from A, the ordinate at any other point whatever will represent the density at that point. This curve has received from mathematicians the name of the Logarithmic Curve, because the abscissa are the loga rithms of the ordinates; and it will benecessary to investi gate a few of its properties, in order to understand the ap plication which has been made of Cotes' discovery to the measurement of heights by the barometer.
Proposition I. If any three ordinates be drawn to the curve which are equidistant, the intercepted areas are to each other in the same ratio as the middle ordinate is to either of the extreme ordinates. Let DF, FE, Plate CCCCLXV. Fig. 3. be each divided into any number of small portions, each equal to D m or F n. Then if A, B, D, be the ordinates, drawn from those points which are situated between D, F, and A', B', C', be the ordi nates drawn from the corresponding points between FE: it follows that the space HDKF = (A+B+C+D+ &c.) D And the space KFEL=(A'A-B'-l-C-FD'+, &c.) FN. But from the nature of the curve