another axis, which soon afterwards is discovered to be that of the ecliptic. But the knowledge of that is not here necessary: for, though the student is not yet in a state to apply the complete formula, he sees that the known part suffices for the relative positions, which may be determined at all times from observasions made in a space of six months The positions of the fixed stars thus determined for the day of each observation, serve to ascertain those of the sun for every day in a year. From this determination it is shown, that the apparent annual course of that luminary is a great circle inclined to the equator: the inclination of this circle to the equator, and the stars near which the common intersection falls, are ascertained for the year 1800: the same particulars are determined, from Lacaille's tables, for 1750 and the comparison of the two sets of results shows the retrogradation of the equinoctial points; proves, also, that the axis of the equator turns about the pole of the ecliptic; and furnishes a complete knowledge of the precession, and of the formulæ by which it may be computed. Here the Chevalier completes the explication of spherical astronomy, and of the diurnal motion both of the sun and of the stars. He then computes their risings and settings, the seasons and climates; and terminates both the first volume and this branch of his admirable induction, by an ingenious theorem for the correction of corresponding altitudes.

In the course of the preceding induction, he introduces a simple but elegant synthetical solution of the problem of the shortest twilight. But upon this, being a matter of pure speculation, we cannot dwell: it is time we should turn to the second volume. The order observed in this volume will be evident from the contents of its several subdivisions. The subjects here treated in succession, are, the sun and its principal inequality; elliptical motion; the hypotheses of the sun's motion, and of the earth's motion about the sun, with reasons for preferring the latter; different species of time; risings and settings of the planets; equation of time; the construction of solar ables; the moon; eclipses; the planets in their order, with a general table of the planetary system.

When tracing the inequalities of the sun's annual motion, M. Delambre first explains them after the manner of the ancients by an eccentric or an epicycle, and then deduces from those theories expressions which are found of the same form as those of the elliptical motion, and which both enable the student to estimate the errors of the ancient hypotheses, and lead him to the true elliptic theory and the Keplerean laws. He exhibits several methods of computing tables of the equation of the centre, the

radius vector and its logarithm, true and mean anomalies, &c. one of which is new, simple, and proceeds directly to its object with all requisite precision. Here, also, he presents some valuable formulæ by Gauss, Oriani, Lagrange, &c. which, we believe, are as yet but little known in England; and he exhibits several comprehensive and useful tables. Other valuable tables are given in the disquisitions on the equation of time, and on the solar reductions to the meridian and the solstice.

The three last chapters in this volume abound with elaborate and excellent investigation. The theory of the moon is presented with great perspicuity and elegance; and a very ingenious method is given for finding, by observation and classifying, all the perceptible inequalities in the motion of that luminary. The determination of the lunar revolutions, or months, lead naturally to the theory of eclipses. The Author exhibits a very simple graphical construction, by which the principal circumstances of eclipses may be determined with sufficient accuracy for most practical purposes; furnishing, indeed, as we have ascertained by trial, the times of the beginning, middle, and end of an eclipse, each within a minute. Here it is that the great utility of the theorems concerning parallaxes is evinced. But the Author, at the same time that he shows how advantageously they may be employed, shows also how the student may attain his object without having recourse to them. He proposes a new and ingenious trigonometrical method of computing, more simply and more exactly than by any other process we have hitherto seen, ali the circumstances of an eclipse of the sun, moon, star, or planet, the lines of commencement and termination, the phases, &c. for all parts of the earth. The whole is reduced to the computation of two triangles, the one spherical, the other rectilinear; the same formulæ serving for all the phenomena, which is a peculiar advantage of this method. Our Author elucidates the method by a detailed example.

Among the interesting matter relating to the planets, in the copious chapter of 176 pages which terminates the second volume, we find some curious formulæ for the computation of rare and important phenomena, by Delambre himself; and farther theorems applicable to the motion of newly discovered planets and comets, extracted from a work by M. Gauss, entitled, "Theoria Motús Corporum cœlestium in Sectionibus conicis solem ambientium."

The subject of transits of inferior planets over the sun's disk, is treated with considerable perspicuity, and the use of the transits of Venus especially, in determining the parallax of the sun, is shown by a very full account of the observations, processes, and deductions, in the case of the celebrated transit of 1769,

The Author gives us the medium result of fourteen separate determinations of the sun's parallax 8" 57, the extremes being 8" 41 and 8/ 75. He also presents the reader with two tables, in one of which he exhibits the principal circumstances of all the transits of Venus, from the year 902 to the year 2984, and all the transits of Mercury from 1605 to 1894. From these tables we shall extract all which relates to future transits, beginning with that which is to occur in the present year, .but which, from some singular omission, is neither mentioned in the Nautical Almanac, nor the Connaissance des Tems. These results cannot but be interesting to men of science; and possess this peculiar advantage, that being computed from modern tables of the sun and planets, they are much more correct than the results of Dr. Halley, which have usually been presented in our Encyclopædias and other general repositories of scientific information.

The reader will observe that the times of conjunction, and of the middle of the transits, are given in the following tables for Paris. They will be reduced to the corresponding times for the meridian of London, by deducting 9 minutes, and 43 seconds, from each.

The third volume, to which we must now proceed, comprehends eleven chapters, and treats of the following subjects: viz. stations and retrogradations of the planets; rotations of the planets; aberration and annual parallax of the stars; nutation; displacing of the ecliptic, and different motions of the stars; comets; satellites; magnitude and figure of the earth; nautical astronomy; projections of the sphere; the calendar.

This volume, like the preceding two, abounds with elegant investigation, comprehensive deductions, and useful tables. We can, however, select only a few particulars. The subject of aberration is important, by reason of the striking confirmation of the Copernican hypothesis which it furnishes, and of the way in which correct formulæ for this species of reduction tend to give accuracy to astronomical observations. M. Delambre ex

hibits many theorems for aberration which are both simple and new; at least new to us, and to astronomers generally, although he assures us he has employed them for thirty years. We regret much that they are not of such a kind as can easily be presented in this analysis.

To the subject of comets the Chevalier devotes 275 pages. Besides the methods of Lambert, Olbers, Lagrange, Laplace, and Legendre, which he exhibits with considerable perspicuity, he gives an entirely new method of his own. He gives the ex

pression for the anomaly and the radius vector, on the elliptic hypothesis, and all the theorems for cometary orbits, under a form of which the first term is the only one to be retained when the orbit is regarded as parabolic. Thus the student may always see what may be safely neglected, and if the parabola is insufficient, he may attempt several ellipses.

• Cette méthode,' he remarks, n'emploie que des opérations les plus usuelles de l'astronomie; elle n'offre aucun calcul difficile ni long, les erreurs y sont presque impossibles, et quand on a trouvé une parabole approximative, on en peut corriger à la fois tous les élémens sur la totalité des observations, par le moyen des équations de condition, comme on fait pour les planètes. Ce moyen de rectification me paraît plus simple, plus direct, et plus satisfaisant qu'aucun de ceux qu'on a proposés jusqu'ici, et qui sont tous fondés sur les méthodes de fausse position.'

The Author next presents a few speculations upon the nature of comets, and their tails; upon which, however, as if conscious he could throw no new light on that obscure subject, he does not dwell. He gives, what is much more valuable, some excellent tables for the orbits of comets, occupying 40 pages, and serving greatly to simplify both the direct and inverse problem concerning these bodies, which has so long perplexed astronomers. Here he acknowledges his obligations to the preceding labours of Barker and Zach, and seems by a comparison of their tables to have detected some errors in those of the latter astronomer.

The thirty-fifth chapter, on the figure and magnitude of the earth, may be regarded as a very comprehensive and valuable abridgement of the principal theorems and deductions in the celebrated Base du Systeme metrique.' M. Delambre gives first a succinct history of attempts at measuring the earth; then traces the plan of operation, and the best methods of computation, in reference to the triangles, azimuths, latitudes, compression of the terrestrial spheroid, terrestrial refraction, reduction to the level of the sea, &c. He also points out the means of confirming or correcting the measurements of meridians by experiments on the lengths of pendulums, in different latitudes. We regard this as, altogether, one of the most interesting portions of Delambre's work.

The two last chapters contain an elegant treatise on projections of the sphere, and a dissertation on the calendar, in which some curious theorems are investigated by means of the indeterminate analysis. Among other ingenious rules and formulæ, we noticed those which have been proposed by M. Gauss, for the determination of Easter. They differ from all other rules we have seen, in this respect, that they are independent. We shall