aujourd'hui de ce devoir; et si je me présente à la tête des savans qui la composent, c'est à mon âge que je dois cet honneur. Mais, SIRE, telle est la diversité des objets dont cette classe s'occupe, que même avec la precision dont un savoir profond et l'esprit d'analyse donnent la faculté, le rapport qui en contient l'exposé exige une grande etendue. • Ce n'est donc que de l'esquisse, et pour ainsi dire, de la preface de leur ouvrage, que MM. Delambre et Cuvier vont faire la lecture. Je ne me permets qu'une seule observation; c'est que l'époque de 1789 à 1808, en même temps qu'elle sera pour les événemens politiques et militaires une des plus mémorables dans les fastes des peuples, sera aussi une des plus brillantes dans les annales du monde savant. La part qui est due aux Français pour le perfectionnement des methodes analytiques qui conduisent aux grandes découvertes du systême du monde, et pour les découvertes même dans les trois régnes de la nature, prouvera que si l'influence d'un seul homme a fait des héros de tous nos guerriers, nos savans, honorés par la protection de votre Majesté qu'ils ont vue dans leurs rangs, sont en droit d'ajouter des rayons à la gloire nationale. ' After this address from M. Bougainville, which is certainly commendable for its simplicity, though the compliment in the last paragraph might have been better turned, Delambre, secretary of the class of Mathematics, proceeded to read his report, from which we shall select such passages as appear to us the most interesting. The report begins with the elementary branches of the mathematics, and takes notice of two treatises which have appeared in that department within the limits of the period above mentioned,—those of Legendre and Lacroix. That of Legendre, it is said, is destined to recal geometry to its antient severity, at the same time that it suggests some new ideas concerning an analytical mode of treating several of the elementary parts of that science. To understand these two remarks, it must be observed, that the French mathematicians, having long since abandoned Euclid, had departed also, in many things, from the rigour of strict demonstration; a practice which, in the Elements, where the foundation of the science is to be laid, was surely much to be condemned. Bossut's Elements of Geometry, which appeared about the year 1775, is almost the only one in the French language, except the two here mentioned, where geometrical accuracy is aimed at throughout. The work of Legendre, however, has accomplished its object more completely, we think, than that just mentioned, or, indeed, than any other modern treatise of elementary geometry. It is very extensive, including the properties of the sphere, together with the cubature and complanation of the solids bounded by planes, and 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. Euciid 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, |