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gravitation, which is again diminished by this tendency. Thus the waters immediately under the moon are drawn from the earth at the same time that the earth is drawn from those which are diametrically opposite to her; in both instances producing an elevation of the ocean of nearly the same height above the surface of equilibrium; for the diminution of the gravitation of the particles in each position is almost the same, on account of the distance of the moon being great in comparison of the radius of the earth. Were the earth entirely. covered by the sea, the water thus attracted by the moon would assume the form of an oblong spheroid, whose greater axis would point towards the moon, since the columns of water under the moon and in the direction diametrically opposite to her are rendered lighter in consequence of the diminution of their gravitation; and in order to preserve the equilibrium, the axes 90° distant would be shortened. The elevation, on account of the smaller space to which it is confined, is twice as great as the depression, because the contents of the spheroid always remain the same. The effects of the sun's attraction are in all respects similar to those of the moon's, though greatly less in degree, on account of his distance; he therefore only modifies the form of this spheroid a little. If the waters were capable of assuming the form of equi
librium instantaneously, that is, the form of the spheroid, its summit would always point to the moon, notwithstanding the earth's rotation; but on account of their resistance the rapid motion produced in them by rotation, prevents them from assuming at every instant the form which the equilibrium of the forces acting upon them requires. Hence, on account of the inertia of the waters, if the tides be considered relatively to the whole earth, and open sea, there is a meridian about 30° eastward of the moon, where it is always high water both in the hemisphere where the moon is and in that which is opposite. On the west side of this circle the tide is flowing, on the east it is ebbing, and on every part of the meridian at 90° distant, it is low water. These tides must necessarily happen twice in a day, since the rotation of the earth brings the same point twice under the meridian of the moon in that time, once under the superior, and once under the inferior, meridian.
In the semidiurnal tides there are two phenomena particularly to be distinguished, one occurring twice in a month, and the other twice in a year.
The first phenomenon is, that the tides are much increased in the syzigies, or at the time of new and full moon. In both cases the sun and moon
are in the same meridian, for when the moon is new they are in conjunction, and when she is full, they are in opposition. In each of these positions their action is combined to produce the highest or spring tides under that meridian, and the lowest in those points that are 90° distant. It is observed that the higher the sea rises in full tide, the lower it is in the ebb. The neap tides take place when the moon is in quadrature; they neither rise so high nor sink so low as the spring tides. The spring tides are much increased when the moon is in perigee, because she is then nearest to the earth. It is evident that the spring tides must happen twice in a month, since in that time the moon is once new and once full.
The second phenomenon in the tides is the augmentation, which occurs at the time of the equinoxes, when the sun's declination is zero, which happens twice every year. The greatest tides take place when a new or full moon happens near the equinoxes while the moon is in perigee. The inclination of the moon's orbit on the ecliptic is 5° 8' 47"-9; hence, in the equinoxes, the action of the moon would be increased if her node were to coincide with her perigee. The equinoctial gales often raise these tides to a great height. Besides these remarkable variations, there are others arising from the declination of the sun and moon,
which have a great influence on the ebb and flow of the waters. The moon takes about twenty-nine days and a half to vary through all her declinations, which sometimes extend about 28 degrees on each side of the equator, while the sun requires about 365 days to accomplish his motion from tropic to tropic through about 23 degrees, so that their combined motion causes great irregularities, and, at times, their attractive forces counteract each other's effects to a certain extent; but, on an average, the mean monthly range of the moon's declination is nearly the same as the annual range of the declination of the sun; consequently the highest tides take place within the tropics, and the lowest towards the poles.
Both the height and time of high water are thus perpetually changing; therefore, in solving the problem, it is required to determine the heights to which the tides rise, the times at which they happen, and the daily variations. Theory and observation show, that each partial tide ncreases as the cube of the apparent diameter or of the parallax of the body which produces it, and that it diminishes as the square of the cosine of the declination of that body.
The periodic motions of the waters of the ocean, on the hypothesis of an ellipsoid of revolution entirely covered by the sea, are very far from
according with observation; this arises from the very great irregularities in the surface of the earth, which is but partially covered by the sea, from the variety in the depths of the ocean, the manner in which it is spread out on the earth, the position and inclination of the shores, the currents, and the resistance the waters meet with, causes it is impossible to estimate, but which modify the oscillations of the great mass of the ocean. However, amidst all these irregularities, the ebb and flow of the sea maintain a ratio to the forces producing them sufficient to indicate their nature, and to verify the law of the attraction of the sun and moon on the sea. La Place observes, that the investigation of such relations between cause and effect is no less useful in natural philosophy than the direct solution of problems, either to prove the existence of the causes or to trace the laws of their effects. Like the theory of probabilities, it is a happy supplement to the ignorance and weakness of the human mind. Thus the problem of the tides does not admit of a general solution; it is certainly necessary to analyse the general phenomena which ought to result from the attraction of the sun and moon, but these must be corrected in each particular case by local observations modified by the extent and depth of the sea, and the peculiar circumstances of the place.