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ed to, the result will come out more favourable to the elliptic the→ ory than he supposes. There are, however, even after that correction is admitted, considerable deviations from the elliptic figure, such as the attraction of mountains is hardly sufficient to explain. The degrees that have been lately measured in France with so much exactness, compared with one another, give an ellipticity of about To, and the same ellipticity corresponds well to the degrees measured in the trigonometrical survey of England, whether of the meridian, or the perpendicular to it. At the same time, the measures in France compared with those in Peru, give for the ellipticity of the meridian, which is less than half the former quantity. The observations of the lengths of the pendulum give the same nearly; so that this may be taken as the mean result.

The Fourth book of the Mechanique Céleste, treats of the tides;a subject on which much new light has been thrown by the investigations of La Place.

The first satisfactory explanation which was given of the flux and reflux of the sea, was that of Newton, founded on the principle of attraction. The force of the moon acting on the terrestrial spheroid, supposing this last to be covered with water, must tend, as Newton demonstrated, to diminish the gravity of the waters toward the earth, both at the point where the moon was vertical, and at the point diametrically opposite; and this in such a ratio, that the waters would assume the figure of an oblong elliptic spheroid, with its greater axis directed to the moon. The sun must affect the great mass of the waters in a similar manner, and produce an aqueous spheroid, that at the time of new and full moon would coincide with the former, and therefore augment its effect; while at the quarters it would be at right angles to it, and in part destroy that effect.

The subject, however, was not so fully handled by Newton, but that great room appeared for improvements; and accordingly, the subject of the tides was proposed as the prize-question by the Academy of Sciences in the year 1740. This produced the three excellent dissertations of Daniel, Bernoulli, Euler, and Maclaurin, which shared the prize; but shared it, we must confess, with another essay, that of Father Cavalleri a Jesuit, who endeavoured to explain the tides by the system of vortices. It is the last time that the vortices entered the lists with the theory of gravita


Many excellent dissertations on the same subject have appeared since; but they are all defective in this, that they suppose the waters of the ocean in a state of equilibrium, or to be brought, by the action of gravitation, toward the earth, and to


ward the two other bodies just mentioned, into the figure of an aqueous spheroid, where the particles of the water, by the ac tion of these different forces, were maintained at rest.

This, however, is by no means the case: the rotation of the earth does not allow time to this spheroid ever to be accurately formed; and, long before the three attractions are able to produce their full effect, they are changed relatively to one another, and disposed to produce a different effect. Instead, therefore, of the actual formation of an aqueous spheroid, the tendency to it produces a continual oscillation in the waters of the ocean, which are thus preserved in perpetual movement, and never can attain a state of equilibrium and of rest. To determine the nature of these oscillations, however, is a matter of extreme diffi culty, and is a problem which neither Newton, nor any of the three geometers who pursued his tract, was able, in the state of mechanical and mathematical science which then existed, to resolve. The best thing which they could do, was that which they actually accomplished, by inquiring into the nature of the spheroid, which, though never actually attained, was an ideal mean to which the real state of the waters made a periodical and imperfect approach. Neither the state of mechanical or mathematical science was such as could yet enable any one to deter mine the motions of a fluid, acted on by the three gravitations above mentioned, and having, besides, a rotatory motion. The nature of fluids was not so well known as to admit of the differe ential equations containing the conditions of such motions to be exhibited; and mathematical science was not so improved as to be able to integrate such equations. The first man who felt himself in possession of all the principles required to this arduous investigation, and who was bold enough to undertake a work, which, with all these resources, could not fail to involve much difficulty, was La Place; who, in the years 1775, 1779 and 1790, communicated to the Academy of Sciences a series of memoirs on this subject, which he has united and extended in the Fourth book of the Méchanique Céleste.

Considering each particle of water as acted on by three forces, its gravitation to the earth, to the sun and to the moon, and also as impelled by the rotation of the earth, he inquires into the nature of the oscillations that will be excited in the fluid. He finds, that the oscillations thus arising may be divided into three classes. The first do not depend on the rotation of the earth, but only on the motion of the sun or moon in their respective orbits, and on the place of the moon's nodes. These oscillations vary periodisally, but slowly; so that they do not return in the same order,

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till after a very long interval of time. The oscillations of the second class, depend principally on the rotation of the earth, and return in the same order, after the interval of a day nearly. The oscillations of the third class, depend on an angle that is double the angular rotation of the earth; so that they return after the interval nearly of half a day. Each of these classes of oscillations, proceeds just as if the other two had no existence; a circumstance that tends very much to simplify the investigation into their combined effect.

The oscillations of the first kind are proved to be almost entirely destroyed by the resistance which any motion of the whole sea must necessarily meet with; and they amount nearly to the same as if the sea were reduced at every instant to an equilibrium under the attracting body.

The oscillations of the second class involve, in the expression of them, the rotation of the earth; and they are also affected by the depth of the sea. The difference of the two tides in the same day, depend chiefly on these oscillations; and it is from thence that La Place determines the mean depth of the sea to be about four leagues.

The oscillations of the third kind, are calculated in the same manner; and from the combination of all these circumstances, the height of the tides in different latitudes, in different situations of the sun and moon,-the difference between the consecutive tides, the difference between the time of high water and the times when the sun and moon comes to the meridian,-all these circumstances, are better explained in this method than they have ever been by any other theory. La Place has instituted a very elaborate comparison between his theory and observations on the tides, made during a succession of years at Brest, a situation remarkably favourable for such observations.

1. Between the laws by which the tides diminish from their maximum at the full and change, to their minimum at the first and third quarters, and by which they increase again from the miniinum to the maximum, as deduced from the observations at Brest, and as determined by the theory of gravitation, there is an exact coincidence.

2. According to theory, the height of the tides, at their maximum, near the equinoxes, is to their height in similar circumstances at the solstices, nearly as the square of the radius to the square of the cosine of the declination of the sun at the solstice; and this is found to agree nearly with observation.

3. The influence of the moon on the tides increases as the cube of her parallax; and this agrees so well with observation, that the law might have been deduced from observation alone.

4. The retardation of the tides from one day to another, is but half as great at the syzigies as at the quadratures. This is the conclusion from theory; and it agrees well with observation, which makes the daily retardation of the tide 27' in the one case, and 55' in the other.

Many more examples of this agreement are mentioned; and it is highly satisfactory to find the genuine results of the theory of gravitation, when deduced with an attention to all the circumstances, and without any hypothetical simplification whatsoever, so fully confirmed in the instance that is nearest to us, and the most obvious to our senses.

La Place has treated a subject connected with the tides, that, so far as we know, has not been touched on by any author before him. This is the stability of the equilibrium of the sea. A fluid surrounding a solid nucleus, may either be so attracted to that nucleus, that, when any motion is communicated to it, it will oscillate backwards and forwards till its motion is destroyed by the resistance it meets with, when it will again settle into rest; or it may be in such a state, that when any notion is communicated to it, its vibrations may increase, and become of enormous magnitude. Whether the sea may not, by such means, have risen above the tops of the highest mountains deserves to be considered; as that hypothesis, were it found to be consistent with the laws of nature, would serve to explain many of the phenomena of natural history. M. La Place, with this view, has inquired into the nature of the equilibrium of the sea, or into the possibility of such vast undulations being propagated through it. The result is, that the equilibrium of the sea must be stable, and its oscillations continually tending to diminish, if the density of its waters be less than the mean density of the earth; and that its equilibrium does not admit of subversion, unless the mean density of the earth was equal to that of water, or less. As we know, from the experiments made on the attraction of mountains, as well as from other facts, that the sea is more than four times less dense than the materials which compose the solid nucleus of the globe are at a medium, the possibility of these great undulations is entirely excluded; and therefore, says La Place, if, as cannot well be questioned, the sea has formerly covered continents that are now much elevated above its level, the cause must be sought for elsewhere. than in the instability of its equilibrium.

With the questions of the figure of the earth, and of the flux and reflux of the sea, that of the precession of the equinoxes is closely connected; and La Place has devoted his Fifth book to the consideration of it. This motion, though slow, being always in the same direction, and therefore continually accumulating, ha



early been remarked, and was the first of the celestial appearances that suggested the idea of an annus magnus, one of those great astronomical periods by which so many days and years are circumscribed. As it affects the whole heavens, and as the changes it produces are spread out over the vast extent of 25,000 years, it has proved a valuable guide amid the darkness of antiquity, and has enabled the astronomer to steer his course with tolerable certainty, and here and there to discover a truth in the midst of the traditions and fables of the heroic ages.

Newton was the first who turned his thoughts to the physical cause of this appearance; and it required all the sagacity and penetration of that great man to discover this cause in the principle of universal gravitation. The effect of the forces of the sun and moon on that excess of matter which surrounds the earth at the equator, must, as he has proved, produce a slow angular motion in the plane of the latter, and in a direction contrary to that of the earth's rotation. The accurate analysis of the complicated effect that was thus produced, was a work that surpassed the power, either of geometry or mechanics, at the time when Newton wrote; and his investigation, accordingly, was founded on assumptions that, though not destitute of probability, could not be shown to be perfectly conformable to truth; and it even involved a mechanical principle, which was taken up without due consideration. Nevertheless, the glory of having been first in the career, is not tarnished by a partial failure, and is a possession which the justice of posterity does not suffer Newton to share even with those who have since been more successful in their researches.

The first of these was D'Alembert. That excellent mathematician gave a solution of this problem that has never been surpassed for accuracy and depth of reasoning, though it may have been, for simplicity and shortness. He employed the principle already ascribed to him of the equilibrium among the forces destroyed when any change of motion is produced; and it was by means of the equations that this proposition furnished, that he was enabled to proceed without the introduction of hypothesis. Solutions of the same problem have since been given by several mathematicians, by Thomas Simpson, Frisi, Walmsley, &c. and many others; not, however, without some difference (such is the difficulty of the investigation) in the results they have obtained. La Place has gone over the same ground, more that he might give unity and completeness to his work, than that he could expect to add much to the solution of D'Alembert. As he has proceeded in a more general manner than the latter, he has obtained some conclusions not included in his solution. He has shown,


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