Bouguer’s Experiments.—The earliest experiments were made by Pierre Bouguer about 1740, and they are recorded in his Figure de la terre (1749). They were of two kinds. In the first he determined the length of the seconds pendulum, and thence g at different levels. Thus at Quito, which may be regarded as on a table-land 1466 toises (a toise is about 6.4 ft.) above sea-level, the seconds pendulum was less by 1/1331 than on the Isle of Inca at sea-level. But if there were no matter above the sea-level, the inverse square law would make the pendulum less by 1/1118 at the higher level. The value of g then at the higher level was greater than could be accounted for by the attraction of an earth ending at sea-level by the difference 1/1118 − 1/1331 = 1/6983, and this was put down to the attraction of the plateau 1466 toises high; or the attraction of the whole earth was 6983 times the attraction of the plateau. Using the rule, now known as “Young’s rule,” for the attraction of the plateau, Bouguer found that the density of the earth was 4.7 times that of the plateau, a result certainly much too large.

In the second kind of experiment he attempted to measure the horizontal pull of Chimborazo, a mountain about 20,000 ft. high, by the deflection of a plumb-line at a station on its south side. Fig. 1 shows the principle of the method. Suppose that two stations are fixed, one on the side of the mountain due south of the summit, and the other on the same latitude but some distance westward, away from the influence of the mountain. Suppose that at the second station a star is observed to pass the meridian, for simplicity we will say directly overhead, then a plumb-line will hang down exactly parallel to the observing telescope. If the mountain were away it would also hang parallel to the telescope at the first station when directed to the same star. But the mountain pulls the plumb-line towards it and the star appears to the north of the zenith and evidently mountain pull/earth pull = tangent of angle of displacement of zenith.

Fig. 1.—Bouguer’s Plumb-line Experiment on the attraction of Chimborazo.

Bouguer observed the meridian altitude of several stars at the two stations. There was still some deflection at the second station, a deflection which he estimated as 1/14 that at the first station, and he found on allowing for this that his observations gave a deflection of 8 seconds at the first station. From the form and size of the mountain he found that if its density were that of the earth the deflection should be 103 seconds, or the earth was nearly 13 times as dense as the mountain, a result several times too large. But the work was carried on under enormous difficulties owing to the severity of the weather, and no exactness could be expected. The importance of the experiment lay in its proof that the method was possible.

Maskelyne’s Experiment.—In 1774 Nevil Maskelyne (Phil. Trans., 1775, p. 495) made an experiment on the deflection of the plumb-line by Schiehallion, a mountain in Perthshire, which has a short ridge nearly east and west, and sides sloping steeply on the north and south. He selected two stations on the same meridian, one on the north, the other on the south slope, and by means of a zenith sector, a telescope provided with a plumb-bob, he determined at each station the meridian zenith distances of a number of stars. From a survey of the district made in the years 1774-1776 the geographical difference of latitude between the two stations was found to be 42.94 seconds, and this would have been the difference in the meridian zenith difference of the same star at the two stations had the mountain been away. But at the north station the plumb-bob was pulled south and the zenith was deflected northwards, while at the south station the effect was reversed. Hence the angle between the zeniths, or the angle between the zenith distances of the same star at the two stations was greater than the geographical 42.94 seconds. The mean of the observations gave a difference of 54.2 seconds, or the double deflection of the plumb-line was 54.2 − 42.94, say 11.26 seconds.

The computation of the attraction of the mountain on the supposition that its density was that of the earth was made by Charles Hutton from the results of the survey (Phil. Trans., 1778, p. 689), a computation carried out by ingenious and important methods. He found that the deflection should have been greater in the ratio 17804 : 9933 say 9 : 5, whence the density of the earth comes out at 9/5 that of the mountain. Hutton took the density of the mountain at 2.5, giving the mean density of the earth 4.5. A revision of the density of the mountain from a careful survey of the rocks composing it was made by John Playfair many years later (Phil. Trans., 1811, p. 347), and the density of the earth was given as lying between 4.5588 and 4.867.

Other experiments have been made on the attraction of mountains by Francesco Carlini (Milano Effem. Ast., 1824, p. 28) on Mt. Blanc in 1821, using the pendulum method after the manner of Bouguer, by Colonel Sir Henry James and Captain A. R. Clarke (Phil. Trans., 1856, p. 591), using the plumb-line deflection at Arthur’s Seat, by T. C. Mendenhall (Amer. Jour. of Sci. xxi. p. 99), using the pendulum method on Fujiyama in Japan, and by E. D. Preston (U.S. Coast and Geod. Survey Rep., 1893, p. 513) in Hawaii, using both methods.

Airy’s Experiment.—In 1854 Sir G. B. Airy (Phil. Trans. 1856, p. 297) carried out at Harton pit near South Shields an experiment which he had attempted many years before in conjunction with W. Whewell and R. Sheepshanks at Dolcoath. This consisted in comparing gravity at the top and at the bottom of a mine by the swings of the same pendulum, and thence finding the ratio of the pull of the intervening strata to the pull of the whole earth. The principle of the method may be understood by assuming that the earth consists of concentric spherical shells each homogeneous, the last of thickness h equal to the depth of the mine. Let the radius of the earth to the bottom of the mine be R, and the mean density up to that point be Δ. This will not differ appreciably from the mean density of the whole. Let the density of the strata of depth h be δ. Denoting the values of gravity above and below by ga and gb we have

gb = G · 4⁄3 πR³Δ= G · 4⁄3πRΔ,
R²

and