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ART.
XII. Asiatic Researches, or Transactions of the Society in-

stituted in Bengal for inquiring into the History,
&c. &c. of Asia. Vol. IX.

175 XIII. 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 XIV. Publications respecting the Conduct of the War 197 Quarterly List of New Publications

237 Appendix

245

ERRATA.

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R 93. 1. 28. for KINGS, read THINGS.

126. 1. 29. for Geralde, read Gerarde.
- 131. 1. 25. for comes, read come.
. 135. note. for

freigos,
read

peegos.
- 138. l. 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 Senabrir. ---- 236. 1. 6. for indisputable, read indispensable.

THE

EDINBURGH REVIEW,

OCTOBER 1809.

No. XXIX.

ART. I. Discours sur les Progre's 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.

AFTER 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, coulsisting 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. Boygainville, 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 VOL. XV. NO. 29.

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aujourd'hui 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 qae 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 peu. ples, 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 assuining 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 sime 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 cqual 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 equai. 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 different kinds, and

n have

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 me, moir 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 llot 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

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