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the force of gravity; but the other, being at a tangent to the surface, urges the particles towards the equator, where they accumulate till their numbers compensate the diminution of gravity, which makes the mass bulge at the equator, and become flattened at the poles. It appears, then, that the influence of the centrifugal force is most powerful at the equator, not only because it is actually greater there than elsewhere, but because its whole effect is employed in diminishing gravity, whereas, in every other point of the fluid mass, it is only a part that is so employed. For both these reasons, it gradually decreases towards the poles, where it ceases. On the contrary, gravity is least at the equator, because the particles are farther from the centre of the mass, and increases towards the poles, where it is greatest. It is evident, therefore, that, as the centrifugal force is much less than the force of gravity-gravitation, which is the difference between the two, is least at the equator, and continually increases towards the poles, where it is a maximum. On these principles Sir Isaac Newton proved that a homogeneous fluid (N. 119) mass in rotation assumes the form of an ellipsoid of revolution (N. 120), whose compression is. Such, however, cannot be the form of the earth, because the strata increase in density towards the centre. The lunar inequalities also prove the earth to be so constructed; it was requisite, therefore, to consider the fluid mass to be of variable density. Including this condition, it has been found that the mass, when in rotation, would still assume the form of an ellipsoid of revolution (N. 120); that the particles of equal density would arrange themselves in concentric elliptical strata (N. 121), the most dense being in the centre; but that the compression or flattening would be less than in the case of the homogeneous fluid. The compression is still less when the mass is considered to be, as it actually is, a solid nucleus, decreasing regularly in density from the centre to the surface, and partially covered by the ocean, because the solid parts, by their cohesion, nearly destroy that part of the centrifugal force which gives the particles a tendency to accumulate at the equator, though not altogether; otherwise the sea, by the superior mobility of its particles, would flow towards the equator and leave the poles dry. Besides, it is well known that the continents at the equator are more elevated than they are in higher latitudes. It is also necessary for the equilibrium of the ocean that its density
should be less than the mean density of the earth, otherwise the continents would be perpetually liable to inundations from storms and other causes. On the whole, it appears from theory, that a horizontal line passing round the earth through both poles must be nearly an ellipse, having its major axis in the plane of the equator, and its minor axis coincident with the axis of the earth's rotation (N. 122). It is easy to show, in a spheroid whose strata are elliptical, that the increase in the length of the radii (N. 123), the decrease of gravitation, and the increase in the length of the arcs of the meridian, corresponding to angles of one degree, from the poles to the equator, are all proportional to the square of the cosine of the latitude (N. 124). These quantities are so connected with the ellipticity of the spheroid, that the total increase in the length of the radii is equal to the compression or flattening, 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 courses 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 would be greater,
exactly in proportion as the curvature is less. A comparison of the length of a 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 determining the latitude of its extreme points by astronomical observations (N. 125), 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 (N. 126), the sides and angles of which are either measured or computed, so that the length of the arc is ascertained with much laborious calculation. 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. The officers who conducted the trigonometrical survey, in measuring 500 feet of a base in Ireland twice over, found that the difference in the two measurements did not amount to the 800th part of an inch; and in the General Survey of Great Britain, five bases were measured from 5 to 7 miles long, and some of them 400 miles apart, yet, when connected by series of triangles, the measured and computed lengths did not differ by more than 3 inches, an unparalleled degree of accuracy; but such is the accuracy with which these operations are conducted.
Arcs of the meridian have been measured in a variety of latitudes in both hemispheres, as well as arcs perpendicular to the meridian. From these measurements it appears that the length of the degrees increases from the equator to the poles, nearly in proportion to the square of the sine of the latitude (N. 127). 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 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 25° 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 . 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