MATHEMATICS, and ASTRONOMY.

In the History of the Class, M. DELAMBRE pays some wellmerited compliments to La Place and Lagrange, for their respective solutions of a very interesting problem, relating to the stability of the solar system; or rather of the permanent magnitude of the major axis of the planetary orbits, and consequently of the mean periods of revolution. He observes that it is particularly worthy of attention that both these memoirs have been occasioned by another, not less interesting, lately read to the class by a young geometer, (Poisson,) their worthy pupil; who, in the very first steps of his career, has placed himself in the rank of the most distinguished masters.

The acceleration of the moon, although it had been proved to be restrained within limits, after which it will experience a similar retardation, made it suspected that the same fact might have place with regard to the earth and the other planets: whereas astronomers had always conducted their calculations on the supposition of its uniformity. This problem, therefore, viz. to ascertain whether or not the earth was subject to any such acceleration, became one of great importance in the present advanced state of astronomical science; and La Place had accordingly occupied himself some years ago on this subject, and had indeed demonstrated approximatively the permanent magnitude of the major axis: but, having, for the sake of simplifying the calculation, employed only the first powers of the masses, and the third of the excentricities and inclinations, some doubt still remained as to the rejected quantities. This circumstance gave rise to a memoir by Lagrange,. which was published in the memoirs of the Berlin Academy, in which he proved the same as La Place, retaining all the successive powers of the two latter quantities, but still employing only the first powers of the masses. In this state, the problem remained till Poisson demonstrated the same, includ¬ ing the second power of the masses. Some ideas thrown out by the author of this paper (of which he did not, as it frequently happens, see the full advantage,) again drew the attention both of Lagrange and La Place to the same problem; both of whom succeeded in giving it a full and complete demonstration, but on principles totally different from each other, and by which the stability of the solar system under the present order of things is placed beyond every possible doubt.

The above is the only subject introduced into the History, under the head of mathematical discovery. The writer then passes in review a few new works published in the course of the year; beginning with the new edition of Lagrange's tract on the solutions of numerical equations; and remarking some

of

of its additions, particularly a note in which the author has shewn the application of his method to Gauss's problems relating to the solution of binomial equations having prime indexes of the form 2m+1. We shall avail ourselves of this opportunity to offer a few observations on the nature of this work, which seems to be much misunderstood in England. It is commonly thought to offer an infallible practical method for the solution of numerical equations, which, though admitted to be long and tedious, may (it is supposed) still be employed for that purpose; whereas we much question whether, since its first publication, a single equation higher than a cubic has ever been solved by it. In fact, it can only be viewed as a very ingenious and elegant theory of numerical equations, but which at the same time furnishes no practical solution of those of high dimensions. When we consider that it requires us, first, to find the equation of differences, which rises to the degree _m (m—19), that of the original equation

I. 2

being m, and afterward to obtain the limits of the roots of this new equation, before we begin to enter on the solution of the one proposed ; - that we have then to find the nearest integral root of each new resulting equation, for every new figure obtained in our approximation;-after which, the continued fraction is to be converted into a series of converging fractions, and these again into decimals, and that the same is to be repeated for each of the possible roots of the proposed equation;- when, we say, these processes are properly appreciated, it will not be denied that, though Lagrange's method offers a rule for the general solution of numerical equations, it must be regarded as merely theoretical, and not as supplying any practical solution of the higher equations.

M. DELAMBRE next gives a short notice of the third edition of La Place's Exposition of the System of the World, and the second edition of Legendre's Theory of Numbers; and, lastly, an account at some length of the Exposé des resultats des grandes opérations Geodesiques, faites en France et Espagne, par MM. Biot et Arago. Nothing very particular occurs in this report, except the astonishing coincidence of the measured arc, or that which was obtained from the geodetic operations, with the same as deduced from astronomical observation; the latter giving 1374439.13 and the former 1374438.72 metres, being a difference of only 0.41 metres, or about half a yard in an arc comprehending nearly 14 degrees of latitude. It may not, however, be amiss to observe that this coincidence arises from assuming a compression of as deduced

16

from

from the theory of the moon; and not one of between and as assumed by Don J. Rodriguez in his memoir, which fell under our strictures in the Rev. for April last, p. 385., and which we have since seen noticed by DELAMBRE himself who, like us, and on the same principles, denies the legality of that author's conclusion.

A short notice of a memoir by La Place, on the double refraction of light in diaphanous crystals, concludes this part of the present volume.

:

Historical Account of the Life and Writings of F. Berthoud, by M. DELAMBRE. This paper presents nothing very interesting but the subject of it was highly celebrated for his mechanical talents, particularly in his construction of chronometers, being the first who succeeded in this respect in France, as Harrison was the first in England. He was also the author of a valuable treatise de la Mesure des Temps;" in which he seems to have had the laudable desire of doing strict justice to all those who, by their mechanical talents and ingenuity, have been the means of bringing horology to its present state of perfection. His extreme predilection for his art, however, and his enthusiasm for great artists, led him into some rather illiberal comparisons with regard to the latter and men of science. Berthoud was born in Swisserland in 1727, and died in 1807, at Groslay in the valley of Montmorency.

Report on a reflecting Sextant of M.Lenoir, by M.BURCKHARDT. This report relates to a sextant of rather a new construction, for the convenience of taking terrestrial elevations: but it is not of sufficient importance to require our particular notice.

Presentation of the Report of the Progress of the Mathematical and Philosophical Sciences from 1789 to 1807, to his Imperial and Royal Majesty in his Council of State, Feb. 16. 1808. - We have here a very interesting report, though the altered condition of the individual to whom it was addressed renders some of the compliments bestowed on him rather bizarre.

The deputation of the classes was composed of their late worthy president Bougainville, Tenon vice-president, Delambre, and Cuvier, secretaries, and Lagrange, Monge, Messier, Fleurieu, C. Berthollet, Haüy, Lamarck, Thouin, Lacepède, and Dessessarts, members. M. Delambre read his report relative to the mathematical sciences; mentioning slightly the elementary treatises of geometry by Lacroix and Legendre, the translation of the Greek mathematicians into French by Peyrard, the new species of geometry (geometry of the compasses) by Mascheroni, (which, he observes, was brought into France with the treaty of Campo Formio,) and the descriptive geometry of Monge. He next enters at some length into the extensive geodetic operations

which had been carrying on in France during nearly all the stormy period of the revolution, and which he states have spread a taste for Geodesia in almost every nation; and he concludes by observing that we may now soon hope to see the whole surface of Europe covered with triangles, so that sovereigns will hereafter know the extent of their territories more accurately than individual proprietors know their own estates. He then alludes to the decimal division of the circle; and the immense tables which have been computed for the application of it to trigonometry are mentioned with the eulogium which they so well deserve. Analysis is the next subject brought under review, in which the discoveries of La Place, Lagrange, and Gauss, stand most conspicuous; and thence the speaker passes to the works of Lacroix, Legendre, Poisson, and Carnot, paying merited compliments to the talents of their respective authors. - From analysis, he proceeds to mechanics; where again La Place, Lagrange, Prony, and Poisson, are introduced with great applause. Of the Mécanique Celeste, the reporter observes that, in this great work, in which every page glows with the true genius of analysis and the most important of all its applications, we perceive throughout entirely new theories of the author's own invention, or others which can be scarcely viewed in any secondary light, on account of the new forms which they have received in his hands.'

Ástronomy furnishes a distinct head in this report, of which our limits will admit a very imperfect sketch. The astonishing progress made in that pursuit, during the period included in this paper, is certainly now, and probably ever will be, the most brilliant in the history of the sciences. Dr. Herschel, a short time before 1789, had discovered the Georgium Sidus; and he has since observed its six satellites, and two others belonging to Saturn. Olbers, on his part, has discovered two new planets; Piazzi, one; and lastly a fourth has been found by M. Harding; by which the permanent bodies of our system have been nearly doubled; while a great number of comets have likewise been observed during the same period. These discoveries, however, great as they are, and much as they are calculated to excite our admiration of the persevering industry and indefatigable exertions of their respective authors, are by far the least important of the astronomical improvements of the above period. The singular perfection which the necessary instruments have now attained, the minute accuracy of observation, and the profound investigations of modern astronomers, have brought the science to that state which leaves scarcely any thing to be desired, except simplification.

Having done ample justice to the science and professors of astronomy, the author passes rapidly over the subjects of mathematics, physics, and geography, in which the details are scanty.

M. Cuvier next read his report of the physical sciences, which we have already noticed under that head in the present article. MEMOIRS.

On the Theory of the Variations of the Elements of the Planets, and in particular on the variation of major axes of their orbits. By J. L. LAGRANGE. We have had occasion to notice the purport of this memoir in the preceding part of this article: but it is impossible, within the limits which we are bound to observe, to give any adequate idea of the profound nature of the investigations of the author.

Third Memoir on the Measurement of Heights by the Barometer. By M. RAMOND.-This may be characterized as a great memoir on a little problem, containing about 100 quarto pages on the practical operations of levelling planes by means of the barometer. It gives the detail of several operations connected with this subject, and probably many good practical as well as theoretical maxims.

Memoir on the general Theory of the Variation of the arbitrary constant Quantities in all mechanical Problems. By J. L. LA GRANGE. We have here a generalization of the author's. preceding memoir. In that paper, he had applied his calculus merely to the perturbation of the planets, in which point his solution was complete, but it was confined to that problem only. In the present article, he shews its general application to any system of bodies, acting on each other according to any law; by which his former investigation becomes only a particular case of the general problem. Besides, also, the universality of the present calculus, it is much simplified; so that many intermediate steps being useless are omitted, and the author arrives at his general result by a much more direct and elegant method, In consequence, he has announced his intention of shewing its application, in some future memoir, to a problem equally interesting, but more difficult than the former. The system of the world, besides the perturbations of the planets to which the theory of the variations of the elements naturally apply, presents another highly important problem, susceptible of being submitted to the same theory; viz. the circumstances attending the rotation of the planets about their centres of gravity, considering them as not spherical, and having regard to the attraction of the other planets. This problem, like the former, depends on three differential equations of the second order, between three independent variable quantities; and, consequently,