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the longitude is determined by the occultations of stars and lunar distances. The occultation of a star by the moon is a phenomenon of frequent occurrence: the moon seems to pass over the star, which almost instantaneously vanishes at one side of her disc, and after a short time as suddenly reappears on the other; and a lunar distance is the observed distance of the moon from the sun, or from a particular star or planet, at any instant. The lunar theory is brought to such perfection, that the times of these phenomena, observed under any meridian, when compared with those computed for Greenwich in the Nautical Almanac, give the longitude of the observer within a few iniles. The accuracy of that work is obviously of extreme importance to a maritime nation: we have reason to hope that the new Ephemeris, now in preparation, will be by far the most perfect work of the kind that ever has been published.
From the lunar theory, the mean distance of the sun from the earth, and thence the whole dimensions of the solar system, are known; for the forces which retain the earth and moon in their orbits are respectively proportional to the radii vectores of the earth and moon, each being divided by the square of its periodic time; and as the lunar theory gives the ratio of the forces, the ratio of the distances of the sun and moon from the earth is obtained; whence it appears that the sun's mean distance from the earth is nearly 396 times greater than that of the moon. The method, however, of finding the absolute distances of the celestial bodies in miles, is in fact the same with that employed in measuring the distances of terrestrial objects. From the extremities of a known base, the angles which the visual rays from the object form with it are measured; their sum subtracted from two
right angles gives the angle opposite the base; therefore, by trigonometry, all the angles and sides of the triangle may be computed-consequently the distance of the object is found. The angle under which the base of the triangle is seen from the object is the parallax of that object; it, evidently increases and decreases with the distance; therefore the base must be very great indeed to be visible at all from the celestial bodies. The globe itself, whose dimensions are obtained by actual admeasurement, furnishes a standard of measures, with which we compare the distances, masses, densities, and volumes of the sun and planets.
The theoretical investigation of the figure of the earth and planets is so complicated, that neither the geometry of Newton nor the refined analysis of La Place have attained more than an approximation: it is only within a few years that a complete and finite solution of that difficult problem has been accomplished by our distinguished countryman Mr. Ivory. The investigation has been conducted by successive steps, beginning with a simple case, and then proceeding to the more difficult; but in all, the forces which occasion the revolutions of the earth and planets are omitted, because, by acting equally upon all the particles, they do not disturb their mutual relations. fluid mass of uniform density, whose particles mutually gravitate to each other, will assume the form of a sphere when at rest; but if the sphere begins to revolve, every particle will describe a circle, having its centre in the axis
of revolution; the planes of all these circles will be parallel to one another, and perpendicular to the axis, and the particles will have a tendency to fly from that axis in consequence of the centrifugal force arising from the velocity of rotation. The force of gravity is everywhere perpendicular to the surface, and tends to the interior of the fluid mass, whereas the centrifugal force acts perpendicularly to the axis of rotation, and is directed to the exterior; and as its intensity diminishes with the distance from the axis of rotation, it decreases from the equator to the poles, where it ceases. Now it is clear that these two forces are in direct opposition to each other in the equator alone, and that gravity is there diminished by the whole effect of the centrifugal force, whereas, in every other part of the fluid, the centrifugal force is resolved into two parts, one of which, being perpendicular to the surface, diminishes 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 resolved 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 centrífugal 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 mass in rotation assumes the form of an ellipsoid of revolution, whose compression is. Such, however, cannot be the form of the earth, because the strata increases 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; that the particles of equal density would arrange themselves in concentric elliptical strata, the most dense being in the centre; but that the compression 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 coinciding with the axis of the earth's rotation. The intensity of the centrifugal force is measured by the deflection of any point from the tangent in a second, and is determined froin the known velocity of the earth's rotation: the force of gravitation at any place is measured by the descent of a heavy body in the first second of its fall. At the equator the centrifugal force is equal to the 289th part of gravity, and diminishes towards the poles as the cosine of the latitude, for the angle between the directions of these two forces, at any point of the surface, is equal to its latitude. But whatever the constitution of the earth and planets may be, analysis proves that, 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 1 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. But there is another method of ascertaining the figure of our planet. It is easy to show, in a spheroid whose strata are elliptical, that the increase in the length of the radii, the decrease of gravitation, and the increase in the lengths of the arcs of the meridian, corresponding to angles of one degree, from the pole to the equator, are proportional to the square of the cosine of the latitude. These quantities are so connected with the ellipticity of the spheroid, that the total