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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 plumb-line to deviate from the vertical. The vicinity of mountains produces that effect. One of the most remarkable anomalies of this kind has been observed in certain localities of northern Italy, where the action of some dense subterraneous matter causes the plumbline to deviate seven or eight times more than it did from the attraction of Chimborazo, in the observations 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 nearly* (N. 128). This fraction is called the compression of the earth, and does not differ much from that given by the lunar inequalities. Since the preceding quantities were determined, arcs of the meridian have been measured in various parts of the globe, of which the most extensive are the Russian arc of 250 20 between the Glacial Sea and the Danube, conducted under the superintendence of M. Struve, and the Indian arc extended to 21° 21', by Colonel Everest. The compression deduced by Bessel from the sum of ten arcs is 298, the equatorial radius 3962-802, and the polar 3949-554 miles, whilst Mr. Airy arrives at an almost identical result (3962-824, 3949-585, and 298) from a consideration of all the arcs, measured up to 1831, including the great Indian and Russian ones. If we assume the earth to be a sphere, the length of a degree of the meridian is 6914 English miles. Therefore 360 degrees, or the whole equa
* Sir John Herschel remarks that there are just as many thousands of feet in a degree of the meridian in our latitude as there are days in the year, viz. 365,000.
The Greenwich Observatory is in N. lat. 51° 28′ 40′′.
torial circumference of the globe, is 24,899 English miles. 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.
There is another method of finding the figure of the earth, totally different from the preceding, solely depending upon the increase of gravitation from the equator to the poles. The force of gravitation at any place is measured by the descent of a heavy body during the first second of its fall. And the intensity of the centrifugal force is measured by the deflection of any point from the tangent in a second. For, since the centrifugal force balances the attraction of the earth, it is an exact measure of the gravitating force. Were the attraction to cease, a body on the surface of the earth would fly off in the tangent by the centrifugal force, instead of bending round in the circle of rotation. Therefore, the deflection of the circle from the tangent in a second measures the intensity of the earth's attraction, and is equal to the versed sine of the arc described during that time, a quantity easily determined from the known velocity of the earth's rotation. Whence it has been found that at the equator the centrifugal force is equal to the 289th part of gravity. Now, it is proved by analysis that, whatever the constitution of the earth and planets may be, if the intensity of gravitation at the equator be taken equal to unity, the sum of the compression of the ellipsoid, and the whole increase of gravitation from the equator to the pole, is equal to five halves of the ratio of the centrifugal force to gravitation at the equator. This quantity with regard to the earth is of or T. Consequently, the compression of the earth is equal to diminished by the whole increase of gravitation. So that its form will be known, if the whole increase of gravitation from the equator to the pole can be determined by experiment. This has been accomplished by a method founded upon the following considerations :—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 (N. 127). 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 This is directly the same body ought to increase in that ratio. proved by the oscillations of the pendulum (N. 129), which, in fact, is a falling body; for, if the fall of bodies be accelerated, the oscillations will be more rapid: in order, therefore, 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 86,400 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 0.005.1449, which, subtracted from ris, shows that the compression of the terrestrial spheroid is about 28.2. This value has been deduced by the late Mr. Baily, president of the Astronomical Society, who devoted much attention to this subject; at the same time, it may be observed that no two sets of pendulum experiments give the same result, probably from local attractions. The compression obtained by this method does not differ much from that given by the lunar inequalities, nor from the arcs in the direction of the meridian, and 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 (N. 146) 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 and 33.
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 allow
ance has been made for every cause of error, such discrepancies 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 not to differ much from; 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 elliptical, 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, which differs very little from that determined by observation, and proves, not only that the density of the ocean is inconsiderable, but that its mean depth is very small. There are 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 change in the form of the terrestrial spheroid. The variation in the length of the pendulum was first remarked by Richter in 1672, while observing transits of the fixed stars across the meridian at Cayenne, about five degrees north of the equator. He found that his clock lost at the rate of 2 28 daily, which induced him to determine the length of a pendulum beating seconds in that latitude; and, repeating the experiments on his return to Europe, he found the seconds' pendulum at Paris to be more than the twelfth of an inch longer than that at Cayenne. The form and size of the earth being determined, a standard of measure is furnished with which the dimensions of the solar system may be compared.
THE parallax of a celestial body is the angle under which the radius of the earth would be seen if viewed from the centre of that body; it affords the means of ascertaining the distances of the sun, moon, and planets (N. 130). When the moon is in the horizon at the instant of rising or setting, suppose lines to be drawn from her centre to the spectator and to the centre of the earth these would form a right-angled triangle with the terrestrial radius, which is of a known length; and, as the parallax or angle at the moon can be measured, all the angles and one side are given; whence the distance of the moon from the centre of the earth may be computed. The parallax of an object may be found, if two observers under the same meridian, but at a very great distance from one another, observe its zenith distances on the same day at the time of its passage over the meridian. By such contemporaneous observations at the Cape of Good Hope and at Berlin, the mean horizontal parallax of the moon was found to be 3459", whence the mean distance of the moon is about sixty times the greatest terrestrial radius, or 237,608 miles nearly.* Since the parallax is equal to the radius of the earth divided by the distance of the moon, it varies with the distance of the moon from the earth under the same parallel of latitude, and proves the ellipticity of the lunar orbit. When the moon is at her mean distance, it varies with the terrestrial radii, thus showing that the earth is not a sphere (N. 131).
Although the method described is sufficiently accurate for finding the parallax of an object as near as the moon, it will not answer for the sun, which is so remote that the smallest error in observation would lead to a false result. But that difficulty is obviated by the transits of Venus. When that planet is in her nodes (N. 132), or within 140 of them, that is, in, or nearly in, the plane of the ecliptic, she is occasionally seen to pass over the
*Or more correctly 3422"-325 and 238,793 miles, as deduced from Mr. Adams' more accurate calculations.