In 1686, the ingenious and active philosopher Dr Halley resumed the subject, and discovered the law that connects the elevation of the atmosphere with its density ; of which he gave a clear demonstration, derived from the well known properties of the perbola referred to its asymptotes. Since the height of the mercury indicates the pressure, and come quoit* the elasticity of the external air, it must be proportioned likewise to the density. Where fore the breadth of a given mew of air, or the thickness of a stratum which corresponds to a cer tain prtkn of the mercurial column, will be in vent* as this akitude. Let 0 be the centre of a rectangular hyperbola, of -which OA and 08 are the asymptotes; and conceive the distances OA and OS to represent the heights of the mercury at two stations. The perpendiculars AC and BD, which are reciprocally Oa OA and OB, must hence express the relative thickness of strata corresponding to equal portions or the barometric scale. Divide AB into a multitude of equal segments, and erect the perpendiculars EM, FL, GK, and HI. The inter. eluded spaces, from AC to BD, will denote the sua eessive thickness of the series of strata into which the whole mass of air between the two stations is subdivided. • Consequently the aggregate or ailed. lineal space DBAC, which it proportional 10 the lo garithm of the ratio of 011 to OA, will express the of atinespherin elevation when the mer curial column domes Dem B ae A. Taping equal ascents, therefore, in the atmosphere, the corre sponding densities must form a decreasing geometri cal series.
To apply this elegant theorem, Dr Halley avail ed himself of the best experiments which had been performed to determine the relative densities of air, water, and mercury. In different trials made near the earth's surface, it was found, when the barome ter stood at 29.1 inches, that the air is 840 852, or oven 860 times lighter than water. Taking round numbers, therefore, and assuming the specific gra vity of mercury to be Is*, he reckoned 800 x 13* X 80 = 10,800 inches, or 90 feet, as the altitude of an atmospheric column which, near the surface, would exert a pressure equivalent to that of an inch of mercury. For the coefficient, which answers to the actual constitution of the atmosphere, Halley should have taken the thirtieth part of .4342945, the modulus of the common system of logarithms, or .0144748. But he proceeded less directly, having satisfied himself with taking the arithmetical between the differences of the logarithms of 29 and so, and of those of 80 and 81; a compensation of er rors, which gives .0144768, hardly deviating from the former. Hence he gave this sit* analogy for computing the heights of mountains by the barome ter ; as the constant number .0144765 u to the direr ewe between the logarithms of the barometric co lumns at the tam stetsons, so is 900 feet to the &es ti** required. The result of this operation is evi dently the same as if the logarithmic difference had been multiplied by the number 61t170; a very to lerable approximation at all seasons for a northern climate, and quite accurate, indeed, if the mass of in tervening air had a medium temperature of 46° by Fehrenheit's scale. Dr Halley supposed that the ob.
smations themselves might, from the influence of heat, differ about the fifteenth part between sum mer and winter. But the thermometer was still so i ct an instrument, that it could not be confidence in correcting such variations.
The principle which Halley thus investigated might be derived from a simpler Conceive the atmosphere to be divided into a multitude of equally thin horizontal strata, it is obvious that each successive stratum would, to the pressure of the su perincumbent stratum, add its own weight, which be ing as its density or elasticity, is therefore propor tioned to the collective pressure ; and, consequently, those densities will continually increase in going downwards, exactly in the same way, and after a like progression, as money accumulates at compound in terest, where a constant portion of the aggregate fluid is regularly joined to the capital. Such, in fact, is the distinguishing character of a geometrical progression, that the increase or decrease of each succeeding term is mays proportioned to the term itself. The logarithmic curve is hence the best adapted for exhibiting the relations which connect the densities with the elevations in the annesphere ; the axis of the curve expressing the -elevation, while each ordinate represents the corresponding density of the strata* of aft. It being a fandasaentat pro perty of the logarithmic curve, that every subsea.' gent applied to it has the same length, the exact de. teretbatien of this in the owe of our atmosphere, is the only thing wanted for the final solution of the general problem.
Eleven years after Dr Halley had given his rule for barometrical measurements, this philosopher had an opportunity of applying it to discover the height of Snowdon in North Wales. He found that the barometer which stood at 29.9 inches on the sea-shore near Caernarvon, fell a few hours after, when planted on the summit of the mountain, to 26.1 inches, the altitude having been ascertained previously by a tri gonometrical observation to be 1240 yards.
The year 1687 is memorable as the date of the first publication of the Principle, which was drawn up chiefly at the urgent request of Halley, from disjoint. ed materials that had lain a considerable time in the author's hands. In that immortal work, Newton re sumed the problem of the gradation of atmospheric density, and solved it in that general way which suited his penetrating genius. He demonstrated that, supposing the particles of air, like other bodies, to have their weight or gravitating tendency dimi nished as the squares of their distances from the centre of the earth, if those distances be taken in harmonic progression, the corresponding densities of the atmosphere will form a geometrical one. But since the diminution of attraction at the greatest height we are able to reach, amounts only to the two thousandth part of the whole; this difference is too minute to be admitted into practice ; and the simpler law first established by the sagacity of Halley may be deemed sufficiently accurate for every real purpose.