ART. XIII. XIV. XII. Asiatic Researches, or Transactions of the Society instituted in Bengal for inquiring into the History, &c. &c. of Asia. Vol. IX. Histoire des deux derniers Rois de la Maison de Stuart. Par Ch. J. Fox. Ouvrage traduit de l'Anglais; auquel on a joint une Notice sur la Vie de l'Auteur 190 Publications respecting the Conduct of the War Quarterly List of New Publications 175 197 237 -138. 1. 33. for Angiscarpi, read Angiocarpi. - 198. 1. 22. from bottom, for battles, read battle. - 218. 1. 6. from bottom, for measures, read means. - 228. note, 1. 10. from bottom, for Senalvia read Senabria. 236. 1. 6. for indisputable, read indispensable. THE EDINBURGH REVIEW, OCTOBER 1809. No. XXIX. ART. I. Discours sur les Progrès des Sciences, Lettres et Arts, depuis 1789 jusqu'à ce jour (1808); ou, Compte rendu par l'Institut de France à S. M. l'Empereur et Roi. En Hollande. 1809. AF FTER the intercourse of England with the nations of the Continent has been so long and so unhappily interrupted, it cannot but be acceptable to our readers, to receive, from the most enlightened of those nations, an account of the scientific and literary improvements that have taken place in Europe during the last nineteen years. This account is of high authority, consisting of reports made to the Emperor of the French by Committees of the National Institute, about the beginning of the year 1808. These reports, made by command of the Emperor, are mere abstracts or skeletons of more extensive memoirs, which we may expect hereafter to be published. Even the abstracts, however, are interesting; not only on account of the information they contain, but as belonging to a ceremonial, which, if not quite singular, is certainly very uncommon in the courts of princes. They are accompanied with very useful notes by the editor J. L. Kesteloot, a Dutch physician of the University of Leyden. We are told, that on the 6th of February, his Majesty being in his Council, a deputation from the mathematical and physical classes of the National Institute was introduced by the Minister of the Interior, and admitted to the bar of the Council. M. Bougainville, the oldest member, and therefore the president of the Class, then addressed the Emperor in a short speech; which we shall give in his own words. SIRE, Votre Majesté Imperiale et Royale a ordonné que les classes de l'Institut viendraient dans son conseil lui rendre Compte de l'Etat des Sciences, des Lettres et des Arts, et de leur progres depuis 1789. La classe des Sciences Physiques et Mathématiques s'acquitte aujourd'hui aujourd'hui de ce devoir; et si je me présente à la tête des savan la composent, c'est à mon âge que je dois cet honneur. Mais, SIRE, telle est la diversité des objets dont cette c s'occupe, que même avec la precision dont un savoir profor l'esprit d'analyse donnent la faculté, le rapport qui en contient posé exige une grande etendue. Ce n'est donc que de l'esquisse, et pour ainsi dire, de la pr de leur ouvrage, que MM. Delambre et Cuvier vont faire la lec Je ne me permets qu'une seule observation; c'est que l'ép de 1789 à 1808, en même temps qu'elle sera pour les événemens tiques et militaires une des plus mémorables dans les fastes des ples, sera aussi une des plus brillantes dans les annales du m savant. La part qui est due aux Français pour le perfectionnemen methodes analytiques qui conduisent aux grandes découverte systême du monde, et pour les découvertes même dans les régnes de la nature, prouvera que si l'influence d'un seul hom fait des héros de tous nos guerriers, nos savans, honorés par la p tion de votre Majesté qu'ils ont vue dans leurs rangs, sont en d'ajouter des rayons à la gloire nationale. ' After this address from M. Bougainville, which is cert commendable for its simplicity, though the compliment in last paragraph might have been better turned, Delambre, s tary of the class of Mathematics, proceeded to read his re from which we shall select such passages as appear to us the interesting. The report begins with the elementary branches of the m matics, and takes notice of two treatises which have appeared in department within the limits of the period above mentioned,of Legendre and Lacroix. That of Legendre, it is said, is des to recal geometry to its antient severity, at the same time that it gests some new ideas concerning an analytical mode of tre several of the elementary parts of that science. To under these two remarks, it must be observed, that the French m maticians, having long since abandoned Euclid, had departed in many things, from the rigour of strict demonstration; a tice which, in the Elements, where the foundation of the sc is to be laid, was surely much to be condemned. Bossut's ments of Geometry, which appeared about the year 1775, most the only one in the French language, except the two mentioned, where geometrical accuracy is aimed at throug The work of Legendre, however, has accomplished its o more completely, we think, than that just mentioned, or, in than any other modern treatise of elementary geometry. It ry extensive, including the properties of the sphere, together the cubature and complanation of the solids bounded by pl and also of the sphere, cylinder and cone. At the same time, the propositions contained in it are purely elementary, that is, such as, by their simplicity and generality, deserve to be considered as the fundamental truths of the science of geometry. Among those analytical methods of demonstration, to which an allusion is made above, we were long since particularly struck with one, of which, as it happens, we can convey some idea without the assistance of a diagram. It is well known to those who have compared different treatises of elementary geometry, that one of the greatest difficulties which they present, is the doctrine of parallel lines. Euclid was not able to extricate himself from this difficulty, otherwise than by the introduction of a proposition as an axiom, which certainly is by no means self-evident. Later writers have uniformly experienced the same difficulty; and some of them, trying to avoid the introduction of a new axiom, have fallen into downright paralogisms. Legendre, in his Elements, has given two demonstrations of the properties of parallel lines, without assuming any new axiom. One of these, which is contained in the text, is prolix and less simple than the nature of the theorem to be proved entitles us to expect. The other demonstration, however, which is in the notes, possesses the most perfect simplicity, at the same time that it is new; proceeding on a principle that has been long recognized, but from which no consequence, till now, has ever been deduced. If we could demonstrate, independently of all consideration of parallel lines, that the three angles of a triangle are together equal to two right angles, the object in view would be accomplished, and the difficulty, in this part of the elements, would be entirely overcome. Now, the theorem just mentioned would be easily demonstrated, if we had proved, when two angles of one triangle are equal to two angles of another, that the third angles are also equal, whatever may be the inequality of the bases, or of the triangles themselves. Of this proposition, Legendre gives the following demonstration. If the third angle of a triangle depend not on the other two angles alone, but on these angles and also on the base, then is there some function of these angles, and of the base, to which the third angle is equal. But, if this is true, an equation exists between the angles of a triangle and one of its sides; and, if so, a value of that side may be found in terms of the three angles; that is, the side has a given ratio to the angles; which is impossible; for they are quantities of dif ferent kinds, and can have no ratio to one another. Whenever, therefore, two angles of one triangle are equal to two of another, A 2 their their third angles are also equal, whatever their bases may be. This reasoning appears to us extremely ingenious and satisfactory. It takes for granted nothing but that an angle and a line are magnitudes which admit of no comparison; a proposition, of which no one who understands the terms can entertain the smallest doubt. The reasoning, however, will not be readily followed by those who are unacquainted with algebra, or to whom the nature of functions and equations is not tolerably familiar. It is curious, that a principle which all the world knew, and which was received into geometry so long ago as the days of Plato, was never made subservient to the purposes of reasoning, till in the instance just mentioned, where it is found actually to involve in it the solution of a great difficulty. We must, however, take leave of Legendre's treatise, which we cannot sufficiently recommend. The Elements of Lacroix are also extremely valuable, though not marked, so strongly as the preceding, with the characters of originality and invention. Delambre goes on to remark, that the fine collection of the Greek mathematicians was completed in 1791, by the Archimedes of Torelli. We suppose that he has here in view the splendid edition of Torelli's Archimedes printed at Oxford, not indeed in 1791, but in the year following. He makes further mention of a translation of the same into French by M. Peyrard, with a memoir by Delambre himself on the Arithmetic of the Greeks. Before this memoir,' he adds, of which your Majesty yourself condescended to furnish the subject, it was difficult to conceive how the Greeks, with a notation so imperfect in comparison of ours, could possibly execute the arithmetical operations indicated by Archimedes and Ptolemy.' This translation of Archimedes, so far as we know, has not yet reached England. The memoir of Delambre must be peculiarly interesting to mathematicians. On the subject of the antient geometers and their writings, we must be indulged in a few more remarks. What the collection of the Greek geometers is to which Delambre refers, we do not perfectly understand; but of one thing we are certain, that that collection can never be considered as complete, while the collections of Pappus, one of the most valuable remains of antient science, are known only by a very imperfect translation, and while the original continues shut up in great libraries with other unpublished manuscripts. The most perfect MS. of Pappus, we believe, is at Oxford, and is particularly described by Dr Horseley, in his restoration of the Inclinations of Apollonius. The late Professor Simson of Glasgow was the man of all others who had studied Pappus with the greatest care, as well as the greatest intelligence; and all the commentaries on that author |