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meridians would be ellipses whose lesser axes would coincide with the axis of rotation, and all the degrees measured between the pole and the equator would give the same compression when combined two and two. That, however, is far from being the case. Scarcely any of the measurements give exactly the same results, chiefly on account of local attractions, which cause the plumbline to deviate from the vertical. The vicinity of mountains has that effect; but one of the most remarkable, though not unprecedented, anomalies takes place in the plains in the north of Italy, where the action of some dense subterraneous matter causes the plumb-line to deviate seven or eight times more than it did from the attraction of Chimborazo during the experiments of Bouguer, while measuring a degree of the meridian at the equator. In consequence of this local attraction, the degrees of the meridian in that part of Italy seem to increase towards the equator through a small space, instead of decreasing, as if the earth was drawn out at the poles, instead of being flattened.
Many other discrepancies occur, but from the mean of the five principal measurements of arcs in Peru, India, France, England, and Lapland, Mr. Ivory has deduced that the figure which most nearly follows this law is an ellipsoid of revolution 'whose equatorial radius is 3962 824 miles, and
the polar radius 3949 585 miles; the difference, or 13 239 miles, divided by the equatorial radius, is from arcs of the meridian, the pendulum, and the true compression is
this fraction is called the compression of the earth, because, according as it is greater or less, the terrestrial ellipsoid is more or less flattened at the poles; it does not differ much from that given by the lunar inequalities. If we assume the earth to be a sphere, the length of a degree of the meridian is 69 British miles; therefore 360 degrees, or the whole circumference of the globe, is 24856 miles, and the diameter, which is something less than a third of the circumference, is about 7912 or 8000 miles nearly. Eratosthenes, who died 194 years before the Christian era, was the first to give an approximate value of the earth's circumference, by the measurement of an arc between Alexandria and Syene.
The other method of finding the figure of the earth is totally independent of either of the preceding. If the earth were a homogeneous sphere without rotation, its attraction on bodies at its surface would be everywhere the same; if it be elliptical and of variable density, the force of gravity, theoretically, ought to increase from the equator to the pole, as unity plus a constant: quantity multiplied into the square of the sine
of the latitude; but for a spheroid in rotation, the centrifugal force varies, by the laws of mechanics, as the square of the sine of the latitude, from the equator, where it is greatest, to the pole, where it vanishes; and as it tends to make bodies fly off the surface, it diminishes the force of gravity by a small quantity. Hence, by gravitation, which is the difference of these two forces, the fall of bodies ought to be accelerated from the equator to the poles, proportionably to the square of the sine of the latitude; and the weight of the same body ought to increase in that ratio. This is directly proved by the oscillations of the pendulum; for if the fall of bodies be accelerated, the oscillations will be more rapid; and in order that they may always be performed in the same time, the length of the pendulum must be altered. By numerous and careful experiments, it is proved that a pendulum which oscillates 86400 times in a mean day at the equator will do the same at every point of the earth's surface, if its length be increased progressively to the pole, as the square of the sine of the latitude.
From the mean of these it appears that the whole decrease of gravitation from the poles to the equator is 0.001457, which, subtracted from T shows that the compression of the terrestrial spheroid is about, which does not differ much
from that given by the lunar inequalities, and from the arcs in the direction of the meridian, as well as those perpendicular to it. The near coincidence of these three values, deduced by methods so entirely independent of each other, shows that the mutual tendencies of the centres of the celestial bodies to one another, and the attraction of the earth for bodies at its surface, result from the reciprocal attraction of all their particles. Another proof may be added: the nutation of the earth's axis, and the precession of the equinoxes, are occasioned by the action of the sun and moon on the protuberant matter at the earth's equator; and although these inequalities do not give the absolute value of the terrestrial compression, they show that the fraction expressing it is comprised between the limits 24 and 15.
It might be expected that the same compression should result from each, if the different methods of observation could be made without error. This, however, is not the case; for, after allowance has been made for every cause of error, such discrepances are found, both in the degrees of the meridian and in the length of the pendulum, as show that the figure of the earth is very complicated; but they are so small, when compared with the general results, that they may be disregarded. The compression deduced from the mean
of the whole appears to be about 2015; that given by the lunar theory has the advantage of being independent of the irregularities of the earth's surface and of local attractions. The regularity with which the observed variation in the length of the pendulum follows the law of the square of the sine of the latitude proves the strata to be elliptieal and symmetrically disposed round the centre of gravity of the earth, which affords a strong presumption in favour of its original fluidity. It is remarkable how little influence the sea has on the variation of the lengths of the arcs of the meridian or on gravitation, neither does it much affect the lunar inequalities, from its density being only about a fifth of the mean density of the earth. For, if the earth were to become fluid after being stripped of the ocean, it would assume the form of an ellipsoid of revolution whose compression is 30, which differs very little from that deter-> mined by observation, and proves, not only that the density of the ocean is inconsiderable, but that its mean depth is very small. There may be profound cavities in the bottom of the sea, but its mean depth probably does not much exceed the mean height of the continents and islands above its level. On this account, immense tracts of land may be deserted or overwhelmed by the ocean, as appears really to have been the case, without any great