measure, which is the ten-millionth part of that quadrant of the meridian (N. 151) passing through Formentera and Greenwich, the middle of which is nearly in the forty-fifth degree of latitude. Should the national standards of the two countries be lost in the vicissitude of human affairs, both may be recovered, since they are derived from natural standards presumed to be invariable. The length of the pendulum would be found again with more facility than the mètre. But, as no measure is mathematically exact, an error in the original standard may at length become sensible in measuring a great extent, whereas the error that must necessarily arise in measuring the quadrant of the meridian is rendered totally insensible by subdivision in taking its ten-millionth part. The French have adopted the decimal division, not only in time but also in their degrees, weights, and measures, on account of the very great facility it affords in computation. It has not been adopted by any other people, though nothing is more desirable than that all nations should concur in using the same standards, not only on account of convenience, but as affording a more definite idea of quantity. It is singular that the decimal division of the day, of degrees, weights, and measures, was employed in China 4000 years ago; and that at the time Ibn Junis made his observations at Cairo, about the year 1000 of the Christian era, the Arabs were in the habit of employing the vibrations of the pendulum in their astronomical observations as a measure of time. SECTION XIII. Tides-Forces that produce them-Three kinds of Oscillations in the Ocean -The Semidiurnal Tides-Equinoctial Tides-Effects of the Declination of the Sun and Moon-Theory insufficient without Observation-Direction of the Tidal Wave-Height of Tides-Mass of Moon obtained from her Action on the Tides-Interference of Undulations-Impossibility of a Universal Inundation-Currents. ONE of the most immediate and remarkable effects of a gravitating force external to the earth is the alternate rise and fall of the surface of the sea twice in the course of a lunar day, or 24h 50m 28s of mean solar time. As it depends upon the action of the sun and moon, it is classed among astronomical problems, of which it is by far the most difficult and its explanation the least satisfactory. The form of the surface of the ocean in equilibrio, when revolving with the earth round its axis, is an ellipsoid flattened at the poles; but the action of the sun and moon, especially of the moon, disturbs the equilibrium of the ocean. If the moon attracted the centre of gravity of the earth and all its particles with equal and parallel forces, the whole system of the earth and the waters that cover it would yield to these forces with a common motion, and the equilibrium of the seas would remain undisturbed. The difference of the forces and the inequality of their directions alone disturb the equilibrium. It is proved by daily experience, as well as by strict mathematical reasoning, that, if a number of waves or oscillations be excited in a fluid by different forces, each pursues its course and has its effect independently of the rest. Now in the tides there are three kinds of oscillations depending on different causes, and producing their effects independently of each other, which may therefore be estimated separately. The oscillations of the first kind, which are very small, are independent of the rotation of the earth; and, as they depend upon the motion of the disturbing body in its orbit, they are of long periods. The second kind of oscillations depends upon the rotation of the earth, therefore their period is nearly a day. The oscillations of the third kind vary with an angle equal to twice the angular rotation of the earth, and consequently happen twice in twenty-four hours (N. 152). The first afford no particular interest, and are extremely small; but the difference of two consecutive tides depends upon the second. At the time of the solstices this difference, which ought to be very great according to Newton's theory, is hardly sensible on our shores. La Place has shown that the discrepancy arises from the depth of the sea; and that, if the depth were uniform, there would be no difference in the consecutive tides but that which is occasioned by local circumstances. It follows therefore that, as this difference is extremely small, the sea considered in a large extent must be nearly of uniform depth; that is to say, there is a certain mean depth from which the deviation is not great. The mean depth of the Pacific Ocean is supposed to be about four or five miles, that of the Atlantic only three or four, which, however, is mere conjecture. From the formulæ which determine the difference of the consecutive tides, it is proved that the precession of the equinoxes, and the nutation of the earth's axis, are the same as if the sea formed one solid mass with the earth. Oscillations of the third kind are the semdiurnal tides so remarkable on our coasts. They are occasioned by the combined action of the sun and moon; but, as the effect of each is independent of the other, they may be considered separately. The particles of water under the moon are more attracted than the centre of gravity of the earth, in the inverse ratio of the square of the distances. Hence they have a tendency to leave the earth, but are retained by their gravitation, which is diminished by this tendency. On the contrary, the moon attracts the centre of the earth more powerfully than she attracts the particles of water in the hemisphere opposite to her; so that the earth has a tendency to leave the waters, but is retained by 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 waters 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. If the waters were capable of assuming the form of equilibrium 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 seas, 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. This great wave, which follows all the motions of the moon as far as the rotation of the earth will permit, is modified by the action of the sun, the effects of whose attraction are in every respect like those produced by the moon, though greatly less in degree. Consequently a similar wave but much smaller raised by the sun tends to follow his motions, which at times combines with the lunar wave, and at others opposes it, according to the relative positions of the two luminaries; but, as the lunar waves is only modified a little by the solar, the 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 (N. 153). 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 occurring at the time of the equinoxes when the sun's declination (N. 154) is zero, which happens twice every year. The greatest tides take place when a new or full moon |