the two inclinations of the equator and of the true ecliptic towards the fixed ecliptic. It is, in fact, the excess of the first over the second: it is therefore, the difference of the two preceding results; and it is thence obvious why its expression, which we have developed, should contain the two kinds of variations which characterise them.'

Our author gives an interesting account of the subjects of precession and nutation. But, on comparing his language in the first and second editions of his work, we cannot but notice the singular evidence which they furnish of his progress in national partiality. In his first edition, (speaking of the inferred existence of these phanomena previously to their discovery by observation,) he says,

'L'existence de ces phénomènes est une suite de la théorie de l'attraction; ils ont été découverts et calculés par Newton, avant d'être vus. C'est l'excellent astronome Bradley qui les a le premier reconnus et determinés par l'observation.'

Since that edition was published, however, he seems to have obtained some new light as to these particulars, for his language now is,

La théorie de l'attraction universelle a fait connaître pourquoi les variations périodiques observées par Bradley dans l'obliquité de l'ecliptique et dans la position des equinoxes, &c. sont en rapport avec la position des noeuds de la lune. C'est à d'Alembert que l'on doit cette importante confirmation de la théorie de l'attraction universelle.'

In treating the subject of the motion of the apsides of the sun's apparent orbit, our author presents some particulars worth recording.

'According to the observations of Lacaille, the longitude of the perigee, in 1750, was 309.°5827 (centes.).

When the major axis was perpendicular to the line of the equinoxes this longitude would be 300°.

'The difference is 9°.5827, which at the rate of 191."0668 per year, requires a number of years equal to 958270000÷1910668, or about 500 years.

This phænomenon would therefore take place in the year 1250; when the sun's perigee would coincide with the winter solstice, and the apogee with the summer solstice.

In like manner when the major axis coincided with the line of the equinoxes, the longitude of the perigee was 200°. From that epoch to 1750, it would have advanced 109°.5827. The number of years necessary for this displacement is 10958270000-1910668, or about 5735, which refers this phænomenon to about 4000 years previous to the Christran æra. By a coincidence sufficiently singular it happens that most chronologers refer nearly to this time the first traces of the residence of man upon earth; though it appears by a great number of physical proofs, that the earth itself is much more ancient.

We

We shall not stop to expose the folly of this observation, but leave M. Biot to settle the point with his 'cher et illustre confrère,' Laplace, who, in his 'Exposition,' liv. iv. ch. 4, throws a doubt of a contrary kind upon the Mosiac accounts, and eagerly endeavours to adduce proofs of 'la nouveauté du monde moral, dont les monumens ne remontent guère, au-delà de trois milles ans. Our author, however, goes on:

The same phænomenon will occur again when the solar perigee becomes 400°, that is to say, when it has described 100°-9.°5827, after the year 1750; and, estimating from the preceding results, we shall see that in order to that there will be required a number of years expressed by 5735-1000=4735, which refers this phænomenon to the year 6485. The solar perigee will then coincide with the vernal equinox, while in the opposite position it coincided with the autumnal equinox. In these two cases the line of the solstices, which is always perpendicular to that of the equinoxes, coincides with the minor axis of the solar ellipse.'

M. Biot next proceeds to shew how the position of the apsides affects the relative length of the seasons. Thus, it has been computed that in the year 1800:

'From the vernal equinox to the summer solstice was 92.90588. 'From the summer solstice to the autumnal equinox 'From the autumnal equinox to the winter solstice 'From the winter solstice to the vernal equinox

93.56584.

89d.69954.

89d.07110.

The spring is, therefore, now shorter than the summer, and the autumn longer than the winter.

So long as the solar perigee remains on the side of the equator, on which it is now, the spring and summer taken together, will be longer than the autumn and winter together. In the present age the difference is about 7 days, as appears from the preceding values. These intervals will become equal about the year 6485, when the perigee will reach the vernal equinox; afterwards it will pass beyond it, and the spring and summer taken together, will become shorter than the autumn and the winter.

These phænomena could not obtain if the motion of the sun were circular and uniform; but all the seasons would be equal. The eccentricity of the orbit, therefore, though very small, has a sensible influence on the duration of the seasons; and the displacement of the major axis, though very slow, produces varieties that become perceptible in different ages.'

Book III. on the theory of the moon, contains 21 chapters, and occupies the rest of the second volume. Its subjects are: General phænomena of the lunar motions; theory of the moon's circular motion, (or the first approximation to the true motions); moon's phases; apparent diameter and parallax; theory of the moon's elliptical

motion,

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. valuable part of this book is that which relates to the computations The most 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 a 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 phænomenon 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, disc. If the moon had no real rotatory motion, that is to say, if each apparent

VOL. VII, NO. XIII.

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the

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 its 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 not the same with respect to the centre of the earth, and to the observer placed at its surface. This,

when

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, aeroliths, 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 Mechanics" has shown that we may

*Mec. Céleste, tom. iii. pa. 1.--Rev.
K 2

also