from the theory of the moon; and not one of between is 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 Céleste, 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.' Astronomy 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 litle 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. LAGRANGE. 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, sequently, the final expressions of these variables ought to include six constant arbitrary quantities, which we may regard as the elements of rotation, of which three relate to the rotation itself, and the three others to the plane to which we refer the rotation, as in the case of the movement of translation. These elements become variable by the perturbating forces of the other planets; and the determination of their variations, a problem which had never been attempted, is that which the learned author has proposed to investigate in some future memoir. Supplement to the preceding Memoir. - We have stated that the author, in his last paper, had very considerably simplified as well as generalized his formula: still, however, it occupied forty pages, and required an uncommon degree of attention in the reader to follow the author through all the steps of the investigation: but the memoir was scarcely printed before he discovered that his general result, at which he had arrived only by means of a long and complicated analysis, might be deduced immediately from the primitive equations, in such a manner as not to occupy more than two or three pages. This deduction is made the subject of the present supplement. General Formula for the higher Order of the Planetary Perturbations. By J. C. BURCKHARDT.-The purport of this memoir is sufficiently indicated in the title; and it is impossible to enter into any analysis of it within the limits of our article. Memoir on various Means proper for perfecting Lunar Tables. By the Same.-We know that theory alone is not sufficient for determining, with the required degree of accuracy, the co-efficients of the lunar inequalities; and astronomers have therefore preferred to arrive at them by comparing a number of observations in which the inequality is the greatest; or, rather, the sum of the errors of the tables furnished by these observations, with a similar sum, where their greatest value is negative. The difference of the two sums, divided by the number of observations, will give the co-efficient sought: but, in order to use a greater number, we employ even those which are a little distant from the maximum, and search at the same time for each observation the sines of the argument; dividing the sum of the errors by the sum of the sines, instead of dividing by the number of observations. This is necessarily a very laborious computation; and M. BURCKHARDT's theorem for abridging it is all that we can give of the contents of this memoir; wiz. The mean sum of the sines of an infinite series of arcs in arithmetical progression, from 90° to 90°-y, is equal to the sine of the arc y divided by the arc itself. Vol. X. will be reported in our next Appendix. To the REMARKABLE PASSAGES in this Volume. N. B. To find any particular Book, or Pamphlet, see the A ACHILLES TATIUS, obs. Acne, description of that com- Addison, obs. on the style of, 30S. Aino, the native of Jesso, account Alcohol, in fermented liquore, re- Ali Pacha, visit to, 340. velling in, 490. American Presidents, 253-5. Attention, its effect on intellectual power, 403.405. Austin, Mr., on a new condenser B Bacon, Sir Francis, life of, by Bank of England, its discounts not of Ireland, amount of its Barlow, Joel, account of, 254. 221. Beattie, Dr., obs. on his style, 308. the Ureters, 301. account of, 528. His ode to |