_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.
[Illustration: 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 [Delta]. This will not differ appreciably
from the mean density of the whole. Let the density of the strata of
depth h be [delta]. Denoting the values of gravity above and below by
g_a and g_b we have
[pi]R^3[Delta]
g_b = G (4/3) -------------- = G . (4/3) [pi]R[Delta],
R^2
and
[pi]R^3[Delta]
g_a = G (4/3) -------------- + G . 4[pi]h[delta]
(R + h)^2
(since the attraction of a shell h thick on a point just outside it is
G . 4[pi](R + h)^2h[delta]/(R + h)^2 = G . 4[pi]h[delta]). Therefore
/ 2h 3h [delta] \
g_a = G . (4/3) [pi]R[Delta] ( 1 - -- + -- ------- ) nearly,
\ R R [Delta] /
whence
g_a 2h 3h [delta]
--- = 1 - -- + -- -------,
g_b R R [Delta]
and
[Delta] 3h / / 2h g_a \
------- = -- / ( -1 + -- + --- ).
[delta] R / \ R g_b /
Stations were chosen in the same vertical, one near the pit bank,
another 1250 ft. below in a disused working, and a "comparison" clock
was fixed at each station. A third clock was placed at the upper station
connected by an electric circuit to the lower station. It gave an
electric signal every 15 seconds by which the rates of the two
comparison clocks could be accurately compared. Two "invariable" seconds
pendulums were swung, one in front of the upper and the other in front
of the lower comparison clock after the manner of Kater, and these
invariables were interchanged at intervals. From continuous observations
extending over three weeks and after applying various corrections Airy
obtained g_b/g_a = 1.00005185. Making corrections for the irregularity
of the neighbouring strata he found [Delta]/[delta] = 2.6266. W. H.
Miller made a careful determination of [delta] from specimens of the
strata, finding it 2.5. The final result taking into account the
ellipticity and rotation of the earth is [Delta] = 6.565.
_Von Sterneck's Experiments._--(_Mitth. des K.U.K. Mil. Geog. Inst. zu
Wien_, ii, 1882, p. 77; 1883, p. 59; vi., 1886, p. 97). R. von Sterneck
repeated the mine experiment in 1882-1883 at the Adalbert shaft at
Pribram in Bohemia and in 1885 at the Abraham shaft near Freiberg. He
used two invariable half-seconds pendulums, one swung at the surface,
the other below at the same time. The two were at intervals
interchanged. Von Sterneck introduced a most important improvement by
comparing the swings of the two invariables with the same clock which by
an electric circuit gave a signal at each station each second. This
eliminated clock rates. His method, of which it is not necessary to give
the details here, began a new era in the determinations of local
variations of gravity. The values which von Sterneck obtained for
[Delta] were not consistent, but increased with the depth of the second
station. This was probably due to local irregularities in the strata
which could not be directly detected.
All the experiments to determine [Delta] by the attraction of natural
masses are open to the serious objection that we cannot determine the
distribution of density in the neighbourhood with any approach to
accuracy. The experiments with artificial masses next to be described
give much more consistent results, and the experiments with natural
masses are now only of use in showing the existence of irregularities
in the earth's superficial strata when they give results deviating
largely from the accepted value.