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&c. ; and with regard to that of Simpson, first published in his “ Dissertations,” in 1743, he remarks, that though it is only an approximation, it is one of the best; that it will serve very weH for observations upon all such heavenly bodies as do not go beyond 78° in zenith distance; and that when it ceases to be exact, all others, even the most refined, become doubtful. He suggests ready means of comparing other formulæ with this of Simpson, furnishes a valuable comparative table of refractions according to a variety of theorems; and, lastly, points out convenient means of deducing from observation the requisite constant quantities, and indeeil of drawing a table of refractions from observations alone, without recurring to any abstract theory,
Thus far the Author has proceeded as though the astronomical observer were posited at the centre of the celestial motions. But may an astronomer assume this as a probable hypothesis ? or must he abandon it ? In order to free the student from the delusions of sense, and lead him to the discovery of the true state of things, Chevalier Delambre pursues, through the latter half of his first volume, a most masterly train of induction, of which we would fain give a perspicuous sketch. He investigates the formulæ which relate to parallax, giving them the requisite developments to ensure exactness and facilitate computations. The theory he here presents is entirely trigonometrical, the parallax depending solely upon the distances either of the observer, or of the heavenly body, from the centre of motion. The formulæ at once indieate the circumstances which best conduce to the discovery of the relation which subsists between those two distances, and this relation is all which their use requires. Hence the student is taught to infer, with certainty, that the fixed stars have not any diurnal parallax; and is prepared to form and arrange a catalogue of them by their right ascensions and declinations.
This catalogue, however, is not to be regarded as possessing all possible precision, since the observer has not yet any idea of aberration, of nutation, or even of the preeession : nevertheless, the precautions suggested ensure the relative positions of the fixed stars from all but almost imperceptible errors, and these may be removed, and the catalogue perfected, by means of the method of reductions. To this, our Author proceeds by comparing two well-known and authentic catalogues, the one prepared by Piazzi, in 1800, the other by Lacaille, in 1750. From this comparison he deduces the precession, and even the general formulæ which may afterwards be applied to each particular śtar. These formulæ, deduced solely from observation, are explicable by a conical motion of the axis of the equator about 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 ascertaived for the year 1800 : the same particulars are determined, froin 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 ab les ; the moon ; eclipses ; the planets in their order, ith 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 low 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 coelestium 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 81 57, the extremes being 811 41 and 8/1 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 Almunac, 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 havc 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.
TRANSITS OF MERCURY.
Conjunc- Mean Geocentric Middle
Semi-du- Shortest ration. distance.
h m 8 14 46 18 2 13 52 9 14 N. 14 39 34 1 21 37 14 OS. 0 27 21 3 28 2 8 16 N. 8 21 42 2 33 53 5 37 S. 7 42 18 3 22 33 8 58 S. 1 59
S 2 41 33 2 36 N. !9 29 34 2 0 23 10.52 N. 19 27 411 45 21 12 20 S. 7 4 34 3 53 31 4 31 N. 13 8 53 2 39 2 3 57 S. 14 23 53 2 34 20 12 21 N. 6 45 49 2 37 36 4 20 N.
h m S 8 Dec. 16 17 44 1882 6 Dec. 4 25 44 2004 7 June 21 044 2012 5 June 13 27 0 2117 10 Dec. 15 6 37 2195 8 Dec. 3 18 40 2247 Jil June 0 30 23 2255 8 Juue 16 53 56 2360 12 Dec. 13 59 9 2368 10 Dec. 2 10 2 2490 12 June 3 58 35 2498 9 June 20 21 2603 15 Dec. 12 54 16 2611 13 Dec. 1 11 12 2733 15 June 7 23 56 2741 12 June 23 43 59 2846 16 Dec. 11 53 15 2864 14 Dec. 0 13 29 2984 14 June 3 2 22
8 30 13 38 52 2 47 26 3 23 19 20 30 19 1235 15 1 49 51
43 13 23 47 59 11 35 55 0 53 41 3 1 13
1 43 10 29 S.
15 14 S.
13 20 S
17 9 N. 3 53 23 2 35 N. 3 7 24 9 56 N. 1 54 10 14
12 S. 3 56 9
0 45 N.
The third volume, to which we must now proceed, comprehends eleven chapters, and treats of the following subjects : viz. stations and retrogradation 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; Dautical 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, hy 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 exhibits 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 pre- : sented 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, be gives an entirely new method of his own. He gives the ex