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increase in the length of the radii is equal to the compression, and the total diminution in the length of the arcs is equal to the compression multiplied by three times the length of an arc of one degree at the equator. Hence, by measuring the meridian curvature of the earth, the compression, and consequently its figure, become known. This, indeed, is assuming the earth to be an ellipsoid of revolution, but the actual measurement of the globe will show how far it corresponds with that solid in figure and constitution.
The course of the great rivers, which are in general navigable to a considerable extent, prove that the curvature of the land differs but little from that of the ocean; and as the heights of the mountains and continents are inconsiderable when compared with the magnitude of the earth, its figure is understood to be determined by a surface at every point perpendicular to the direction of gravitation, or of the plumb-line, and is the same which the sea would have if it were continued all round the earth beneath the continents. Such is the figure that has been measured in the following manner :—
A terrestrial meridian is a line passing through both poles, all the points of which have their noon contemporaneously. Were the lengths and curvatures of different meridians known, the figure of the earth might be determined; but the length of one degree is sufficient to give the figure of the earth, if it be measured on different meridians, and in a variety of latitudes; for if the earth were a sphere, all degrees would be of the same length, but if not, the lengths of the degrees will be greatest where the curvature is least, and will be greater exactly in pro
portion as the curvature is less; a comparison of the lengths of the degree in different parts of the earth's surface will therefore determine its size and form.
An arc of the meridian may be measured by observing the latitude of its extreme points, and then measuring the distance between them in feet or fathoms: the distance thus determined on the surface of the earth, divided by the degrees and parts of a degree contained in the difference of the latitudes, will give the exact length of one degree, the difference of the latitudes being the angle contained between the verticals at the extremities of the arc. This would be easily accomplished were the distance unobstructed, and on a level with the sea; but on account of the innumerable obstacles on the surface of the earth, it is necessary to connect the extreme points of the arc by a series of triangles, the sides and angles of which are either measured or computed, so that the length of the arc is ascertained with much laborious computation. In consequence of the irregularities of the surface, each triangle is in a different plane; they must therefore be reduced by computation to what they would have been, had they been measured on the surface of the sea; and as the earth may in this case be esteemed spherical, they require a correction to reduce them to spherical triangles.
Arcs of the meridian have been measured in a variety of latitudes north and south, as well as arcs perpendicular to the meridian. From these measurements it appears that the lengths of the degrees increase from the equator to the poles, nearly in proportion to the square of the sine of the latitude; consequently the convexity of the earth diminishes from the equator to the poles.
Were the earth an ellipsoid of revolution, the 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 plumb-line to deviate from the verticle. 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 discrepances 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, gives for the compression deduced from arcs of the meridian, from the pendulum, and the true compression is 200615 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, shows that the compression of the terrestrial spheroid is about 292-90, 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 and
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,