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knowledge, he may find mathematically the smalt corrections which reduce the science to its present state.
. To the observations made, more than half a century ago, by Lacaille and Bradley, we shall join those which Dr. Maskelyne published regularly for more than forty years, and the work in which all the recent observations of M. Piazzi are registered; and, finally, those of the Board of Longitude, published annually in the Connaissance des Tems.'
• According to this plan, we shall admit nothing which is not decisively proved; we shall even vary the proofs as often as we shall judge necessary. Thus we shall cause to pass in review all the parts of astronomy; we shall present them in a different order from the authors who have preceded us; but the form alone will be changed.
Some authors justly celebrated, have pursued a method nearly similar to those in treatises of geometry or algebra, and have attempted to invent the science for their readers. Thus they became exposed to the reproach of giving long treatises but little complete. The reason probably is, that in geometry and analysis, if all the theorems are essentially connected with some preceding theorem, we do not always see the necessity of passing from the first to those which are corollaries; since the same theorem may have a great number of consequences, which have little analogy to one another, and of which we do not see the utility: while in astronomy the phenomena to be explained occur continually as we proceed. Our treatise, therefore, will be complete when the whole is explained, and when we possess rules of computation for every particular. Thus we shall treat of nothing useless; we shall omit nothing essential; and we shall not be detained longer upon the subject, than if, after the example of Lacaille, we had at once supposed the observer at the centre of the sun.
• Our demonstrations generally commence by the manner of synthesis; the purely analytical method not being always either the easiest or the shortest.' When the problems appear susceptible of an easy construction, which will speak to the eyes, we shall employ it in preference; such construction may furnish us with the fundamental equations: but if analysis can afterwards simplify that formula, and present it in a shape better fitted for computation, or should facilitate the combinations and lead to more general alid fertile results, we shall not permit those advantages to escape.
· That this word' analysis, however, may not alarm any of my readers; let it be remarked, that astronomy, if we omit the planetary perturbations, requires only the knowledge of the most elementary theorems of geometry, the simplest rules of algebra, á few of the chief properties of the conic sections, the two fundamental theorems of the differential and integral calculus, and above all; spherical trigonometry, which astronomy itself has called into existence, and which we shall deduce even from our obserdations with the aid of rectilinear trigonometry.'
We have made this long extract unhesitatingly, because it will be interesting, not only as it serves to develop the plan of Delambre's work, but as it explains the means which, in the estimation of this experienced astronomer, may best be pursued to attain a knowledge of his favourite science. We shall now proceed to examine, with as much minuteness as our limits will allow, the several parts of the treatise ; first presenting an outline of the contents of each volume, and then pointing to the more ingenious and valuable portions of it.
The first volume is divided into nineteen chapters, from the first of which, containing an introductory sketch of the plan, the preceding quotation has been translated. In the following chapters the Author treats, in succession, of the observations which first appear requisite, the pendulum and astronomical telescope, observation of the sun, gnomonics, ancient and modern instruments, plumb-line and level, vernier, micrometer and reticle, circles, quadrants, and transit instruments : to these suoceed a sketch of spherical trigonometry, with its application to gnomonics, and an explication of the trigonometry of the Greeks : and these again are einployed in the investigation of refraction, twilight, and parallax, in the formation of a catalogue of stars, in tracing the annual course of the sun, the diurnal motion, and the method of corresponding altitudes.'
In this volume we find many particulars worthy of notice, but can specify only a few. Thus, on the subject of trigonometry, , the Author exhibits a very perspicuous view of that of the Greeks, and demonstrates the celebrated formulæ of Napier with great simplicity and elegance. He also deduces a variety of formulæ presenting the relations between four, five, and six parts of spherical triangles, and tending to simplify the differential expressions of these triangles. Of those differentials he exhibits a more complete and methodical collection than we have hitherto seen ; and he adds a very curious table for the verification of trigonometrical formulæ. He also lays before the reader some ingenious rules to facilitate trigonometrical mnemonics.
From the application of trigonometrical theorems to the observations of the stars, the general uniformity of their motion is inferred, at the same time that some minor irregularities lead to the detection and determination of what is denominated refraction. This subject our Author treats copiously and elegantly. The construction given originally by Cassini, leads immediately to the formula of Bradley, namely, r =p tan (2-qr), r being the refraction that corresponds to the zenith distance 2, p. and q co-efficients to be determined by observation. He examines the different formulæ of Simpson, Boscovich, Laplace, &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 well 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 indeel of drawing a table of refractions from observations alone, without recurring to any abstract theory.
Thus far the Author has proceeded as though the astronomia cal 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 investi-gates 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 indicate 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 rigbi 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 precession : 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 star. These formulæ, deduced solely from observation, are explicable by à 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 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 corre sponding altitudes.
In the course of the preceding induction, he introduces a simple but elegant synthetical solution of the problem of the shortest troilight. 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
les; 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 severalmethods 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, all 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 cælestium 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,