11. The Priest and the Lady. 12. The Squirrel. 13. In which the Author calls a Spade, a Spade. 14. The Girl Watering a Foal., 15. The Lover to let. 16. Audigier. 17. The Three Thieves. 18. The Crane. 19. The Silly Knight. 20. The Blacksmith of Creil. 21. The Flighty Damsel. 22. The Peck of Oats. 23. Berengier, 24. Chastelaine de Vergy. This interesting story has served for the ground-work of a popular romance, but is perhaps there overloaded with incidents little in the character of the times: here it may again prompt some happier effort at imitation. 25. Pyramus and Thisbe. 26. Florance and Blanche. 27. The Good Woman. 28. The Four Wishes. 29. The Tresses. 30. The Falcon. 31. The Priest and Alice. 32. The Paternoster of Love. 33. The Creed of the Rake. 34. Destourmi. 35. The Saddle-cloth parted. 36. Women, Dice, and Dinners. To this volume also is added an appropriate but incomplete glossary. The entire work merits a place in the library of an English antiquary; since many passages, which Chaucer and others of our early poets knew, recollected, and imitated, will here be detected in their original form. Those stories which found favour in the twelfth century, and which circulated in the European literature of that age among all nations, are here brought together in the best form which they acquired among our forefathers; and those which left most impression, and are repeatedly introduced in their compositions, may deserve to employ the pens of modern poets, and to be consecrated in the regenerating waters of Helicon to a purer immortality. Obscene passages and allusions occur in many of these stories, especially those in the third volume: but allowance is to be made for the simplicity of antient manners, which called many things by plain names without meaning any harm. An arrangement of materials more severely chronological, and additional notices concerning the tributary poets and poems, would have added value to this publication: but we are glad to see so good a selection of the old metrical tales of the French, executed with so faithful an adherence to the original text. ART. VII. Mémoires de la Classe des Sciences, &c.; i. e. Memoirs of the National Institute of France, Vols. VII.-X. [Article continued from the Appendix to Vol. lxx. p. 516-532.] MATHEMATICS, ASTRONOMY, &c. Vol. VII. MEMOIR on the Orbit of the Comet of 1770. By M. BURCKHARDT. Perhaps, scarcely any object of astronomical inquiry has engaged the attention of mathematicians more than Kk3 the the determination of the orbit of this comet. It was first discovered by M. Messier, in the summer of 1770, and was ob served by him with the utmost exactitude and perseverance, during the whole period of its being visible, as well as by many other able astronomers. Its orbit was afterward computed with equal scrupulousness by Pingrè, Laxell, and others, most of whom agreed very nearly in giving to it the same elements; finding it to be an ellipse of comparatively small excentricity, very little inclined to the plane of the ecliptic, and the comet itself performing its revolution in about five years and a half. Astronomers therefore anxiously waited the period at which its re-appearance was to be expected, and particularly in 1781 : but, to their great regret and disappointment, it has never since fallen under their observation. Neither could it be ascertained that any of the comets of which we have the elements, to the amount of sixty-two, previously to 1770, were the same as that which then appeared. Consequently, though it seemed perfectly reasonable to repose an entire confidence in the result of so many independent computations, all agreeing very nearly with each other, yet the circumstance of the comet never having fallen under the eye of any astronomer either before or since 1770 could not but throw some doubt on the accuracy of those determinations. These circumstances induced the National Institute to re-propose the problem as a prize-essay; and M. BURCKHARDT, whose indefatigable exertions in the cause of true astronomical science, are so well known and appreciated, undertook the whole calculation de novo, without availing himself of any former approximation, and on the most general principles: having first procured a copy of the original observations of M. Messier, in order to avoid any errors that might have found their way into the printed copies, from which other astronomers had made their computations:-in short, every possible precaution was taken to secure the due degree of accuracy in every respect. After a long and laborious calculation, the results of it were found to coincide very nearly with those above mentioned; so that no possible doubt can now be entertained on this subject. However difficult, therefore, it may be to account for this comet not having been observed before or after 1770, although its period of revolution is only about five years and a half, yet the fact itself seems to be incontrovertible, and ought to be traced to some competent physical cause. M. BURCKHARDT enters on this question in the sixth section of his memoir; which, as it contains some novel ideas on this interesting subject, we will transcribe. I flatter myself with having proved, on the most incontestible principles, in the preceding sections, that no parabola, nor hyperbola, nor nor any very excentric ellipse, will correspond with the observations made on the comet of 1770; and that this comet actually describes an orbit, agreeing with a period of revolution of five years and a half. The difficult question also, "Why had we not before or have we not since seen a comet, whose period of revolution is of such short duration?" is advanced at least one step; because we are now assured that it is actually to the attractions of Jupiter that we must have recourse for the explanation of this interesting phænomenon in the system of the world. Laxell thought that the attractions of Jupiter in 1767 had very considerably diminished the period of revolution of this comet; and that the same attractions had in 1779 rendered its orbit of greater excentricity: an hypothesis which was afterward adopted by Boscovich. It seems difficult, however, to admit that two such opposite effects were produced by the same cause, acting under nearly the same circumstances. If we allow a total change of orbit, of which astronom furnishes no example, it would be much more probable to suppose that the same effects were produced in both cases by the attractive force of Jupiter; and that this force, which is supposed to have shortened the period of Revolution in 1767, had again diminished it in 1779, and in a manner much more considerable, because the comet approached much nearer to Jupiter in the latter year than in the former. Even this distance might be still farther dimini ed by the attraction itself; and the comet might perhaps have augmented the number of Jupiter's satellites without this being perceived by astronomers. In fact, the distance of this comet from the sun would be, in that case, five times greater than it was at the instant when it ceased to be visible in 1770; and its distance from the earth would be at least four times greater than it was at the same period: so that the comet would have four times less light than in 1770, which would necessarily be still farther diminished by the vicinity of Jupiter. Supposing this change, therefore, to have really taken place, we have no reason to be astonished at its not having been observed even with our best telescopes.' Although, however, M. BURCH RDT thinks that such a change is not impossible, and that it is even more probable than that which was indicated by Laxell, yet he is anxious to account for the comet's non-appearance on principles less repugnant to, the analogy observed in the various bodies of our system; and he has shewn that, with an orbit nearly permanent, it may have repeatedly passed its perihelion without having been perceived. He wishes much to encourage astronomers, and mathematicians, to persevere in completing this part of our astronomical knowlege. If we could once obtain another sight of the comet, we, should never again experience the same difficulty: mathematicians would doubtless unite with astronomers in calculating the instant and place at which it ought to appear. The interval of thirty years,' continues the author, during which period it would be necessary to calculate the attractions, bespeaks a stupendous undertaking. Yet, if the great geometer (La Kk 4 Piace), Place), who has enriched physical astronomy with so many important discoveries, wishes to avail himself of the necessary formula for the accomplishment of this object, and if he thinks that any confidence can be placed in the ultimate result, the author of this memoir will with pleasure attempt the laborious task.' Analysis of Triangles traced on the Surface of a Spheroid. By A. M. LEGENDRE. In all geodetic operations, the calcu lations of the sides and angles of the triangles traced on the terrestrial surface have been made on the same principles as if the earth had been a perfect sphere; and some doubt had therefore been entertained, whether this erroneous hypothesis might not in some measure have affected the truth of the results thence derived. M. LEGENDRE himself felt some hesitation on this subject; which, he observes, seemed to be the better founded, because, we cannot assimilate a spheroidal and a spherical triangle. Even a spheroidal triangle cannot be supposed to be turned about one of its summits, without ceasing to coincide with the spheroidal surface; and much less will it coincide with it if transferred to a different part of the surface which has not the same latitude. In order, therefore, to resolve this difficulty, and to ascertain the amount of the errors which might thus have been introduced into the several geodetical operations, this mathematician undertook a complete investigation of spheroidal triangles; in which, besides the parts requisite for the solution of spherical triangles, it was necessary to introduce the latitude of the summit of the triangle, the azimuth of one of its sides, and the ratio of the two axes of the spheroid. He institutes his investigation with this complete generality, and thence proceeds to deduce an expression for the area of such a triangle the result of which is that no appreciable error will arise, in the triangles employed in geodetical operations, by considering them as spherical; nor in any in which the sides are small in comparison with a great circle of the sphere. It follows,' says the author, from the preceding theorem, that the triangles traced on the surface of a spheroid (and we have principally in view the triangles formed in geodetical operations, of which the sides may extend to a degree or even more,) may be calculated on the same principles as small triangles traced on the surface of a sphere; and both of them may be reduced to rectilinear triangles, by diminishing their angles, each by a quantity equal to one-third of the area of a similar triangle, described on a sphere whose semi axis is 1. All spheroidal trigonometry is comprized in this one principle; and it is also obvious that it extends still more generally to every triangle triangle formed on any surface differing but little from a sphere; as we may suppose that such a surface will nearly coincide with an elliptic spheroid, disposed in such a manner that the vertical sections of the greatest and the least curvature, which always cut each other at right angles in one solid, may coincide with the similar sections and equal arcs in the other solid: wherefore the triangle common to both surfaces will possess the same properties with spherical triangles. The solution of spheroidal triangles, of which the sides are small with regard to the dimensions of the spheroid, depends therefore immediately on that of rectilinear triangles; not only when the spheroid is elliptical and of revolution, but when it is in any manner irregular, provided only that it differs but little from a sphere.' Notes on the Planet discovered by M. Harding. By J. C. BURCKHARDT. Second Correction of the Elements of the new Planet. Bythe Same. -M. BURCKHARDT, who suffers no opportunity of evincing his ardour in the cause of astronomy to escape, availed himself of the first series of observations made on this planet, which has since been named Juno, to compute the elements of its orbit. In the first of the above memoirs, from the want of a greater number of observations, he failed to obtain the most accurate results, which are therefore corrected in his second paper, whence we abstract the elements: 52° 49′ 33′′ On the Comets of 1784 and 1762. By the Same.-The first of these comets was observed by M. Dangos (who discovered it) on the 10th and 14th of April only, bad weather having prevented his farther observations; and it was not seen by any other person: consequently, little confidence can be given to the results obtained from these data, and we think that it is useless to detail them. The comet of 1762 was observed under much more favour able circumstances, by M. Messier, and the determination of its orbit was attempted by five different astronomers, no one of which could avoid errors of four or five minutes. All these orbits differed considerably the one from the other; and great uncertainty prevailed in all respects. Similar difficulties had been experienced with regard to the comets of 1763, 1771, and 1773; and it began to be suspected, that little or no dependance could be placed on our present knowlege of the astronomy of comets. These circumstances induced M. BURCK |