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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 (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 ., 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 pfs and s
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 1.8, 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 2TM 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.
Parallax Lunar Parallax found from Direct Observation
lax deduced from the Transit of Venus Distance of the Sun from the Earth - Annual Parallax Distance of the Fixed Stars.
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.
sun like a black spot. If we could imagine that the sun and Venus had no parallax, the line described by the planet on his disc, and the duration of the transit, would be the same to all the inhabitants of the earth. But, as the semi-diameter of the earth has a sensible magnitude when viewed from the centre of the sun, the line described by the planet in its passage over his disc appears to be nearer to his centre, or farther from it, according to the position of the observer; so that the duration of the transit varies with the different points of the earth's surface at which it is observed (N. 133). This difference of time, being entirely the effect of parallax, furnishes the means of computing it from the known motions of the earth and Venus, by the same method as for the eclipses of the sun. In fact, the ratio of the distances of Venus and the sun from the earth at the time of the transit is known from the theory of their elliptical motion. Consequently the ratio of the parallaxes of these two bodies, being inversely as their distances, is given; and as the transit gives the difference of the parallaxes, that of the sun is obtained. In 1769 the parallax of the sun was determined by observations of a transit of Venus made at Wardhus in Lapland, and at Tahiti in the South Sea. The latter observation was the object of Cook's first voyage. The transit lasted about six hours at Tahiti, and the difference in duration at these two stations was eight minutes; whence the sun's horizontal parallax was found to be 8"-72. But by other considerations it has been reduced by Professor Encke to 8"-5776; from which the mean distance of the sun appears to be about ninety-five millions of miles. This is confirmed by an inequality in the motion of the moon, which depends upon the parallax of the sun, and which, when compared with observation, gives 8′′-6 for the sun's parallax. The transits of Venus in 1874 and 1882 will be unfavourable for ascertaining the accuracy of the solar parallax, and no other transit of that planet will take place till the twenty-first century; but in the mean time recourse may be had to the oppositions of Mars.
The parallax of Venus is determined by her transits; that of Mars by direct observation, and it is found to be nearly double that of the sun, when the planet is in opposition. The distance of these two planets from the earth is therefore known in terrestrial radii, consequently their mean distances from the sun may be computed; and as the ratios of the distances of the planets from the sun are known by Kepler's law, of the squares
of the periodic times of any two planets being as the cubes of their mean distances from the sun, their absolute distances in miles are easily found (N. 134). This law is very remarkable, in thus uniting all the bodies of the system, and extending to the satellites as well as the planets.
Far as the earth seems to be from the sun, Uranus is no less than nineteen, and Neptune thirty times farther. Situate on the verge of the system, the sun must appear from Uranus not much larger than Venus does to us, and from Neptune as a star of the fifth magnitude. The earth cannot even be visible as a telescopic object to a body so remote as either Uranus or Neptune. Yet man, the inhabitant of the earth, soars beyond the vast dimensions of the system to which his planet belongs, and assumes the diameter of its orbit as the base of a triangle whose apex extends to the stars. Sublime as the idea is, this assumption proves ineffectual, except in a very few cases; for the apparent places of the fixed stars are not sensibly changed by the earth's annual revolution. With the aid derived from the refinements of modern astronomy, and of the most perfect instruments, a sensible parallax has been detected only in a very few of these remote suns. a Centauri has a parallax of one second of space, therefore it is the nearest known star, and yet it is more than two hundred thousand times farther from us than the sun is. At such a distance not only the terrestrial orbit shrinks to a point, but the whole solar system, seen in the focus of the most powerful telescope, might be eclipsed by the thickness of a spider's thread. Light, flying at the rate of 190,000 miles in a second, would take more than three years to travel over that space. One of the nearest stars may therefore have been kindled or extinguished more than three years before we could have been aware of so mighty an event. But this distance must be small when compared with that of the most remote of the bodies which are visible in the heavens. The fixed stars are undoubtedly luminous like the sun: it is therefore probable that they are not nearer to one another than the sun is to the nearest of them. In the milky way and the other starry nebulæ, some of the stars that seem to us to be close to others may be far behind them in the boundless depth of space; nay, may be rationally supposed to be situate many thousand times farther off. Light would therefore require thousands of years to come to the earth from those myriads of suns of which our own is but "the remote companion."