example:-Humboldt found that at the level of the sea, near the foot of Chimborazo, the barometer stood at exactly 30 inches, while at the summit of the mountain it was only 14.85. The logarithm of 30 is 1.4771213, and the logarithm of 14.85 is 1.1717237; then subtracting 1.4771213 1.1717237 0.3053976 Multiply this by 63,946, which produces 19,539 for the elevation in feet. If the mean temperature of the two stations were 69.68° no correction is necessary for temperature. This is a tolerably close approximation: the most careful calculation has given 19,332 for the real height, and this was probably estimated for a lower temperature. A method has been given by Leslie for measuring heights without the use of logarithms. This rule is as follows:-Note the exact barometric pressure at the base and at the summit of the elevation, and then make the following proportion:-As the sum of the two pressures is to their difference, so is the constant number 52,000 feet to the answer required in feet. Suppose for example the two pressures were 29.48 and 26.36; then As 29.48+26.36: 29.48-26.36 :: 52000 ft.: 2905.4 ft., the answer required. This rule has been found applicable to the mean temperature of our climate for all heights under 5000 feet, and is, therefore, available for all the elevations in Britain. The barometer should be furnished with a vernier, for reading off hundredths of an inch, as a difference of th of an inch will indicate from 88 to 100 or 110 feet, according to the density or pressure. SCALE OF HEIGHTS AND DEPTHS. 87 Of course the results are not perfectly but only approximatively true. 57. The frontispiece to this treatise contains a scale of the most remarkable heights and depths, compared with the probable height of the atmosphere. The horizontal lines show the heights to which a barometer must be carried to make the mercury sink to the number of inches indicated in the right-hand column. Thus, supposing the pressure at the sea-level to be equivalent to 30 inches of mercury, at the height of two miles (as shown on the left-hand scale) the pressure will be reduced to about 20 inches (as seen by the right-hand scale), showing that at this height the barometer is relieved of rd of the ordinary pressure, or, in other words, that 3rd of the whole mass of air is situated below this level. It will be seen also that at five miles (which is about the height of the highest peaks of the Andes and Himalaya) the pressure is only equivalent to 11 inches; and at 5 miles to about 10 inches; so that only 3rd of the atmosphere (as regards quantity) is situated above that level. The lines marked 10 and 20 inches, therefore, divide the whole atmosphere into three layers, containing equal quantities of air; and, by the same reasoning, it will appear that all the horizontal lines in the figure divide it into 30 layers of equal mass, so that their extreme inequality of space will give a correct idea of the enormous compression of the lower strata by the weight of the upper; the upper 30th part, for instance, occupying more space than all the remaining 8ths. The plane which divides the atmosphere into two equal halves (or which is marked 15 inches in our figure) will be observed to be at the height of about 3 miles, or 18,000 feet. Humboldt and Bonpland ascended on the side of Chimborazo a little above this level; and in balloons the barometer has sunk sometimes to 12 inches, showing that the aeronauts had risen above 3ths of the atmosphere. By applying a pair of compasses to the right-hand scale, it will be found that the distance from the 30 to the 15 inch level is the same as from the 20 to the 10 inch, from the 10 to the 5, from the 2 to the 1, from the 4 to the 2, or from any other number to its half or double. So also the distance between any number and its triple is the same as between any other number and its triple; and the same is true of any other multiple: the distance between 30 and 6, for instance, is equal to that between 20 and 4, between 15 and 3, or 5 and 1. Now, this is the property of a logarithmic scale (such as that engraved on the carpenter's sliding rule), viz., that numbers having equal ratios are found at equal distances apart; just as in a table of logarithms, any pairs of natural numbers that have equal ratios are found opposite to pairs of logarithms that have equal differences, so that numbers in geometrical progression have their logarithms in arithmetical progression. This law, then, that the elasticity of the air diminishes upwards in a geometrical progression, for heights that increase in arithmetical progression; or that the pressures at different heights vary as the numbers of which those heights are the logarithms, is the foundation of the method of measuring heights by the barometer. This law, although preserved at the greatest accessible heights, cannot remain true throughout the atmosphere, because in such case it could have no limit, for there would be no height at which the pressure could be reduced absolutely to 0, so that air in an inconceivably rarefied state would extend even to the moon and the planets, which is certainly PRODUCTION OF SOUND. 89 not the case; and indeed various reasons lead to the conclusion that it ceases altogether under the height of 100, and, in all probability, under 50 miles *. Even at 20 miles, it must be rarer than the vacuum produced by the best air-pump, and at 5 miles probably too thin to support animal life. 58. In addition to the aerial currents of varying degrees of intensity, which constitute all the varieties of wind, there are other movements carried on conjointly with the former, not less important to us, and even more wonderful. These are the vibratory motions by which sound is propagated; and there is something very astonishing in the precision and distinctive character of these aerial pulses. Amidst the multiplicity of sounds which fill the air there is no difficulty in naming the source of each. The ringing of bells, the hum of insects, the song of birds, the lowing of cattle, the rattle of cart-wheels, the roar of the cataract, and the rolling of thunder; all these and a thousand other sounds are, as it were, daguerreotyped in the air, and represent to us their source with characteristic distinctness. They do not confuse or bewilder us, for although apparently mingled together, we can separate them and attend to any one; we can lay that down and attend to another; and what is perhaps of far greater consequence, we can recognise our friends and acquaintances by the sounds * Calculations founded on the duration and appearances of twilight (a phenomenon due entirely to the atmosphere) give for its height values varying from 45 to 90 miles. No certain conclusion can, therefore, be drawn from these; but by a different calculation, depending on a more careful collation of the observations made on the Andes and in balloons respecting the upward decrease of temperature and pressure, M. Biot has been led to infer that the elasticity and density become O at a height not exceeding 30 miles. Other philosophers calculate the height at from 40 to 50 miles. The phenomena of twilight may also be accounted for without supposing so great a height as 45 miles, which is therefore more probably above than below the truth. of their voices alone, for no two friends have the same voice, any more than the same countenance. And then how wonderful is the power we possess of shaping air into words, by which we express our thoughts, our wants, our instructions, our promises, our affections to others, by which we regulate the actions and influence the judgment of others. All this is very wonderful, and science can take us a very little way indeed in explaining the timbre or characteristic qualities of different sounds. We have also no means of measuring the different intensities of sounds. We have numerous instruments for measuring the effects of heat, electricity, atmospheric pressure and vapour; but we have no Sonometer for measuring the intensity of sound, because we do not know what effect of sound can be taken as the true measure of its intensity. A similar objection applies to Photometers, or light-measurers. The conditions necessary to the production and propagation of sound were noticed in a former treatise *. A body in a state of stable equilibrium is disturbed therefrom by a certain impulse. It should here be remarked, that mechanicians distinguish two kinds of equilibrium, stable and unstable. Both alike are produced by the marked neutralization of different forces, but with this difference-that if a body in unstable equilibrium be moved ever so little, so as to leave a portion of force unbalanced, this force will urge the body further and further from its original position, and it will not rest until it has found a new position of equilibrium. But a body in stable equilibrium not only tends to preserve its position unaltered, but to return thereto when disturbed within certain limits, because the force thus brought into activity does not tend to drive it farther, but to bring it back to its * Introduction to Natural Philosophy. Sec. 15, 43, 44. |