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motion, (or the second approximation to the true motions,) secular equation of the moon's mean motion; secular equations affecting the elements of the lunar orbit; periodical inequalities in the lunar motions; those which affect the longitude, latitude, and radius vector; libration of the moon, and position of its equator; form and physical constitution of the lunar spheroid ; nature, cause, and computation of solar, and lunar eclipses, transits and occultations; determination of terrestrial longitudes by lunar eclipses, occultations, &c.; relations observed between the age and course of the moon and the tides; explication of some useful periods connected with chronology. The book concludes with two useful notes, one respecting the influence of refraction on the inclined diameters of the moon's disc; and the other exhibiting some ingenious formulæ of M. Olbers for obtaining the elements of the apparent places of the stars in functions of the elements of the true places. The most valuable part of this book is that which relates to the computations of eclipses; but it is not susceptible of abridgment. We have only room for one quotation, which contains the most simple and satisfactory elucidation of the moon's libration, that we remember to have seen.

• The desire to determine the axis of rotation and the plane of the lunar equator, has led to a very careful observation of the lunar spots. Two circumstances facilitate this research : these spots are permanent, and we may in general observe them during the whole course of the same revolution.

*These spots present some varieties in their apparent positions on the lunar disc: they are seen alternately to approach toward and recede from its borders. Those which are near to these edges disappear and re-appear in succession, thus making periodical oscillations. Yet, as the spots themselves do not seem to experience any sensible changes in their respective positions, and as they are always seen again of the same magnitude and under the same form, when they have returned to the same position, it is hence concluded that they are permanently fixed upon the moon's surface. Their oscillations seem, therefore, to indicate à sort of balancing in the lunar globe, to which the name of libration has been given, from a Latin word which signifies to balance.

* But, in adopting this expression, however well it depicts the appearances observed, we must not attach a positive sense to it, for the phanomenon itself has nothing of reality; it is only a complex result of several optical illusions.

"To conceive and separate these illusions, let us recur to some fixed terms. Imagine that a visual ray is drawn from the centre of the earth to the centre of the moon. The plane drawn through the latter centre perpendicular to this ray will cut the lunar globe according to the circumference of a circle, which is, with respect to us, the apparent disc. If the moon had no real rotatory motion, that is to say, if each VOL. VII. NO. XIII.

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point point of its surface remained invariably directed towards the same point. of the heavens, its motion of revolution about the earth alone would discover to us all the points of its surface in succession: the visual ray would therefore meet its surface successively in different points, which would appear to us to pass one after another, to the apparent centre of the lunar disc. The real rotatory motion counteracts the effects of this apparent rotation, and constantly brings back towards us the same face of the lunar globe: whence it is obvious why the opposite face is never revealed to us.

Suppose now, that the rotation of the moon is uniform, as to sense, that is to say, that it does not partake of any periodical inequalities, (this supposition is at least the most natural which can be made, and theory proves that it is correct): then, one of the causes which produce the libration will become evident; for the motion of revolution partaking of the periodical inequalities, is sometimes slower, sometimes more rapid : the apparent rotation which it occasions, cannot therefore, always exactly counterbalance the real rotation, which remains constantly the same, and hence the two effects alternately surpass each other. The points of the lunar globe ought, therefore, to appear turning sometimes in one direction, sometimes in another, about its centre, and the resulting appearance is the same as if the moon had a small balancing from one side to the other of the radius vector drawn from its centre to that of the earth. It is this which is named the libration in longitude.

Several accessary, but sensible causes modify this first result. The spots of the moon do not always retain the same elevation above the plane of its orbit: some of them, indeed, by the effect of its rotation, pass from one side of this plane to the opposite side. These circumstances indicate an axis of rotation, which is not exactly perpendicular to the plane of the lunar orbit; but according as that axis presents to us its greater or its smaller obliquity, it must discover to us successively the two poles of rotation of the lunar spheroid; in like manner as the axis of the earth presents successively its two poles to the sun in the two solstices. Hence we come to perceive, at certain times, some of the points situated towards these poles and lose sight of them afterwards, when they arrive nearer the apparent edge; and it is this which is denominated the libration in latitude. It is but inconsiderable, and therefore indicates that the equator of the moon differs very little from the plane of its orbit.

• Finally, a third illusion arises from the observer's being placed at the surface of the earth, and not at its centre. It is towards this centre that the moon always turns the same face, and the visual ray, drawn from thence to the centre of the moon, would always meet ils surface at the same point, abstracting the preceding inequalities. It is not the same with regard to the visual ray drawn from the surface of the earth; for that ray makes a sensible angle with the preceding one, by reason of the proximity of the moon; an angle which, at the horizon, is equal to the horizontal parallax : in consequence of this difference, the apparent contour of the lunar spheroid is pot the same with respect to the centre of the earth, and to the observer placed at its surface. This,

whea when the moon rises, causes some points to be discovered towards its upper edge, which could not have been seen from the centre of the earth. As the moon rises above the horizon, these points continue to approach the upper edge of the disc, and at length disappear, while others towards its lower edge become visible; the same effect is continued during the whole time that the moon is visible, and, as the part of its disc which appears highest at its rising, is found lowest at its setting, these are the two instants when the difference is most perceptible. Thus, the lunar globe, in its diurnal motion, appears to oscillate about the radius vector drawn from its centre to the centre of the earth. This phænomenon is distinguished by the name of diurnal libration.'

In this book the chapter on the tides is very meagre and defective; but as this is a subject on which we recently had occasion to speak at large, it need not here be resumed.

The fourth book is devoted to the astronomy of planets, comets, and fixed stars; and is divided into fifteen chapters, occupying 243. pages. The following is the distribution of subjects. General phænomena of the planetary motions, mode of determining the positions of the planets' orbits from observation, exact determination of their elements, laws of Kepler, manner of predicting the return of the planets to the same situation with respect to the sun and earth, particularities relative to the physical constitution of the planets, observed rotations, compressions of their axes, &c. satellites of the planets, transmission of light rendered measurable by the retardation of their eclipses, Saturn's ring, comets, determination of their orbits, formulæ for parabolic trajectories, aëroliths, recapitulation of the phænomena which indicate the reality of the earth's motion, aberration of light, stations and retrogradations of the planets, true dimensions of the planetary orbits as deduced from the sun's parallax and other considerations, distances, motions, and annual parallax of the fixed stars, universal gravitation considered as a general fact resulting from the laws of Kepler, masses of the planets, satellites, &c. concluding with a long note on the method of computing the transits of Venus, and making the necessary deductions as to parallax, and the real magnitudes of the planets and their orbits.

This is, on the whole, a valuable book, though the arrangement of its constituent chapters might have been greatly amended. Considering the length to which our article is running, we can only venture upon one quotation from it. After tracing the method of determining the parallax of the sun, from a transit of Venus over the disc of that luminary, M. Biot says,

• The author of the "Celestial Mechanicshas shown* that we may

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also obtain the parallax of the sun after another manner, without observing it immediately, and from the knowledge of an inequality of the lunar motions which is connected with that parallax. To conceive such connection it must be recollected that the inequalities of the lunar motions have determinate relations with the positions of the earth and sun. The calculus makes these relations known; the observations determine the extent of the inequalities; and combining those data, we may deduce the value of the elements on which the inequalities depend, for we have the expression of their dependence and the measure of their action. The whole is reduced to finding inequalities in which that action is, in some sort, comprehended, or in which it is incessantly reproduced, in such manner that it may be inferred exactly by a great number of observations. There exists in the motion of the moon an inequality of this kind, which depends upon the sun's parallax, or upon its distance from the earth ; and on determining that by observation, M. Laplace has thence deduced the value of the parallax equal to 26". 4205 (8."560243 sexiges.) which is nearly the same as the result deduced from the transits of Venus. It is probable that this result of the theory is even more exact than that which has been derived from the observations upon those transits.'

Such coincidences of results, deduced from totally independent methods, are extremely interesting; and every fresh instance has the effect of banishing to a greater distance than ever, all possible doubt of the sufficiency and correctness of the great principle of universal attraction, according to the inverse ratio of the square of the distances. We have long been in possession of a simple and satisfactory method of determining the moon's parallax from the usual theory of gravity, which is brought to our recollection by the preceding quotation; and which, though we know not how to ascribe it to its proper author, we cannot refrain from transcribing from our port-folio, as we think it far too ingenious to remain unknown.

Let S be the space in feet fallen in 1 second, by a heavy body in vacuo at the equator; V the versed-sine of the arc described by the moon, in the same time, to radius 1; R the radius of the equator in feet, ratio of the distance of the moon's and earth's centre, to the semidiameter of the latter that of X to 1: then, by the general law of gravitation, the space descended by the moon in 1", is s = . But the same space is evidently s = VRX. Therefore

S VRX = and X =

Now at the equator,

VR
S = 16.10185, its logarithm 1.2088645
Log. R=-

7.3211900
Log. V =

7.5492882

The

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The sum of the two latter taken from log. S, and the remainder divided by 3, gives 1.7787954 = log. of 60.08906 ; its arithmetical complement is = log. tan, of 57'12."34 the approximate horizontal parallax.

Now, let x + 1 be the distance of the centres of the moon and earth, divided by their centres of gravity in the ratio of x to 1. Imagine a sphere of the saine dimensions as the earth placed at that centre, and to exert the same attractive force on the moon as our earth actually does, the periodic time remaining unaltered : then must the density of this sphere be diminished in the ratio of x? to (x + 1)2 that its nearer distance from the moon may be compensated by the defect of density and attraction. Now, if an inhabitant of this fictitious earth were supposed to compute its distance from the moon in the manner above explained, the quantities V and R would be the same as in the former computation ; but his S would be to our S, as xa to (x + 1)? ; and thence his X' would

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f
x X =

1+1 is the distance from the fictitious earth, or from the common centre

x+1 D

* + X, greater, as was supposed, in the ratio of x+1 to r; that is, It

, estimated the ratio of x + 1 to x, at 40.788 to 39•788 (Princip. lib. iii. prop. 37. cor. 6.) So that the log, of **=0.0035934;

V which added to 1.7787954, the log, of X for an immovable earth gives 1•7823888 = log. of 60-5883 radii of the equator, whence the horizontal parallax there = 56'44"07.

M. Biot having unnecessarily swelled his book by the introduction of extraneous discussions, fiuds, unfortunately, that he has too much matter for two volumes, but not enough for three; he therefore has recourse to his earlier publications, and the communications of his friends, to eke out his last volume. Thus, we are favoured with 216 pages of • Additions, such as, first, a tedious disquisition on the measure of altitudes by the barometer and thermometer, taken from his former work on that subject; then a treatise on dialling, by M. Berroyer, professor of mathematics at the college of Sens; then an essay “Sur le mouvement de translation du systéme planétaire,' by M. Biot himself, who concludes that we have no evidence whatever of any such motion; then, a tract on the rectification of a transit instrument, of course closely connected with physical astronomy; then, an essay on the length of the se

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