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quiry in the twenty-fourth letter. After many other ancient testimonies, which concur in placing this famous isle in the North, our Author quotes that of Plutarch, who confirms these testimonies by a circumstantial description of the isle of Ogygia, or the Atlantis, which he represents as fituated in the North of Europe, and as having near it three islands more, in one of which the inhabitants of the country say that Saturn is kept prisoner by Jupiter. These four islands may, as M. B. conjectures, be Iceland, Greenland, Spitzberg and Nova Zembla, or some others nearer the Pole. Rudbeck, a learned Swede, composed a work about a century ago, in which he maintained that Sweden was the Atlantis of Plato; our Author, though he has made good use both of the hypothesis and of the erudition of Rudbeck, does not, however, adopt his opinion : because it is not conformable with the account of Plato, who represents the Atlantis as an island, which Sweden is not. Adhering still to his system, M. Bailly, persuaded by a variety of plausible cir. cumstances, which he has ingeniously combined, places that famous island among those of the Frozen Ocean. He is strongly seconded by Plutarch, who tells us, that the Atlantis is in a region, where the sun during a whole summer month is scarcely an hour below the horizon, and where that short night has its darkness diminished by a twilight.' This, indeed, is a palpable indication of a Northern climate; but how is this situation reconcileable with the fertility of soil, the mildness of the air, which both Plutarch and Plato mention among the other advantages enjoyed by the Atlantes? And how is it possible to conceive Altronomy cultivated in a frozen and cloudy region, where the observation of the heavenly bodies must have been painful and impracticable? Our Author answers these questions with levity enough: he observes that Plutarch was not the disciple of M. de Buffon, and that these difficulties cannot be removed, but by supposing a change of air and climate in those regions, through the gradual cooling of the earth, and its progressive motion towards

universal congelation.-- This is a bold way of removing difficulties, and it appears to us, that instead of answering these objections M. BAILLY tells his objectors a fairy tale.

A RT. II. Develloppement Nouveau de la Partie Elementaire des Mathematiques,

prise dans tout son etendue, &c.-A New Explanation of the Elementary Part of Mathematics : By Lewis BERTRAND, Professor of Mathematics at Geneva, and Member of the Academy of Sciences, &c. at Berlin. 2 Vols. 4to. Geneva. 1778. Price 36 livres.

T mathematical science proposed by the ingenious Author

,

is new, easy, interesting, and remarkable for its order and accuracy. All the problems, which may be resolved by the circle and the right line, come under the class of elements. But as the properties of the circle and the right line suppose a considerable knowledge of the relations subfisting between quantities, considered in a general view, elements may be divided into two parts ;--the first, arithmetical and algebraical, which furnishes the means of unfolding the properties of the circle and the right line; the second geometrical, containing the explanation of these properties, and their application to the solution of the questions that relate to them, or depend upon them.

curacy. every

The First of these parts is treated by M. BERTRAND in twelve chapters.

In the first he introduces a peasant, who is ignorant of arithmetic, and leads him by a natural and obvious procedure to invent the numbers, and characters, which we have borrowed from the Arabians. In the second, he makes him discover the known methods of addition and subtraction : in the third, the Author puts himself in the place of his disciple, and proposes to himself particular questions of multiplication and division, which lead him to the general rules of these operations, whether they be applied "to whole numbers or to fuch as contain decimal fractions. He always forms, as he proceeds, the theoretical conclusions resulting from his researches, defines the objects presented by the developement of his ideas, and points out the proper ligns for the reprefentation of those ideas.

M. Bertrand begins his fourth chapter by the following propofition, that the product of several factors does not de pend on the order in which they are multiplied :' he fhews the powers and roots of numbers, completes what he had observed with relpect to signs in the preceding chapters, and thus lays down the principles of algebraic notation.

In the fifth chapter our Author treats of broken numbers, and shews how they are to be added, subtracted, multiplied and divided by each other. In the fixth he undertakes the solution of a difficult question in fractions, by a method very different from those which have been employed for that purpose by other analytical writers. But as this chapter, may appear difficult to fome beginners, M. BERTRAND advises such to defer the perufal of it until they have studied the three following chapters, as the truths demonstrated in them do not depend on the propofitions contained in the sixth, and by exercising the fagacity and attention of the young reader, may prepare bim for understanding them with more facility. In chapter the seventh M. BERTRAND points out the methods of extracting the roots of whole and broken numbers of

kind-the eighth contains a complete treatise on arithmetical and geometrical relations and proportions; and the ninth a folution of

determinate

determinate and indeterminate problems of the first degree. The author, in this chapter, explains the four first operations of algebra, and points out the manner of proceeding in order to find out the most complex common divisor of two algebraic quantities. The variety and choice of the problems refolved , in this chapter, as also the reflexions which accompany their solution, are every way proper to excite in the youthtul mind a taste for the science under consideration, and to facilitate remarkably their progress and improvement in mathematical knowledge.

The tenth chapter is employed in the folution of determi. nate problems of the second degree, and the eleventh in displaying the powers of a binomial, whose indices are either broken or negative numbers. In this chapter, among other things, our Author lays down the principles of the science of probabilities, and resolves several problems, relative to chances, which render the application of these principles familiar to the student, and also shew him how interesting the questions are, which depend upon them.—The science of logarithms is amply treated in the twelfth chapter, in which the labours of Lord Naper, the ingenious methods and tabies of Meffis. Sharp, Briggs, Flack, and Sherwin, are described, illustrated, and appreciated with respect to their accuracy, and usefulness in this important branch.

The SECOND PART of this work is fubdivided into two, viz. Elementary Geometry and Trigonometry. The first, which is again subdivisible into three branches, comprehends the properties of the circle and right line,-the application of these properties to the mensuration of plane, rectilinear, and circular surfaces,--and to that of curve surfaces and folids. The first of these branches is largely treated in seven chapters. Here the Author, beginning with the common notion of space, deduces from it the ideas of planes, right lines, angles, triangles, and curves, describes their nature, properties, determinations, cir, cumstances, relations, proportions, &c. folves leveral problems relative to them, and points out the confequences deducible from them. The second branch of elementary geometry occupies two chapters, in one of which the Author compares plane, rectilinear surfaces, one with another; and in the other, gives, nearly, the measure of the area of a circle, and derives from thence, by way of conclusion, the areas of sectors, segments, and, in general, of all figures that are terminated by right lines and the arches of a circle. The third branch is comprehended in fix chapters, in which the Author treats of simple folid angles (for such he calls the angles that are formed by two planes, which meet each other)- of regular folid angles, and 6

their

their principal properties, of regular bodies, their number, con ftruction, &c. -of the definition and construction of prisms, pyramids, concs, and cylinders, of the menfuration of their surfaces, and of their solidity, and of the characters or marks of similarity in folids of every kind.-There is a rich variety of mathematical instruction communicated with great perspicuity and facility in the detail into which M. BERTRAND enters in the illustration of all these subjects.

Trigonometry forms the second branch of geometry, confidered in its general sense. Under this denomination our Author comprehends both Plain and Spherical Trigonometry, as they are branches that spring from the same root, and they are treated in one chapter, which is divided into seven sections. These contain the most important definitions, discussions, problems, folutions of problems, and demonstrations, that regard this interesting branch of mathematical science.

It is proper to observe here, that in treating the great variety of fubjects that naturally require a place in a work of this kind, M. B. has neither employed the differential nor the integral calculus; he has not even made use of the algebraic analysis in all its extent;-he has not gone further than the folution of equations of the second degree. As to his method, it is strictly geometrical, and hence arise the order and precifion that give such relief and encouragement to the student by spreading an air of eafe and facility over laborious discussions, and thus rendering them perspicuous and interesting. For the most part, M. BERTRAND has employed both the analytic and synthetic method, of which he knows perfectly the respective nature, advantages, and resources; the sure progress in knowledge arising from the one, and the expeditious manner of communicating that knowledge, which is the peculiar advantage of the other, are circumstances of which he has happily availed himself in the excellent work now before us :-a work which we think must be of great use, not only in directing the speculations of the student, but in guiding the merchant, the politician, the topographer and geographer, the navigator and astronomer, in the practical duties and occupations of their refpective profeffions.

ART

RIAC

I

ART. III. Histoire Generale de la Chine, ou Annales de cet Empire, &c.-A Gene. sal Hiltory of China, or the Annals of that Empire, translated from Ton; kien, kang-mou, by the late Father J. A. M. de More

DE MAILLA, &c. Vols. V. VI. VII. and VIII. 4to. Paris. 1778. T is time to resume * our accounts of this great work, in

the publication of which the learned Editors + display the moft active diligence, industry, and perseverance. These four volumes contain the history of China from the year 420 of the Christian ära to 1200: the quantity of matter, good, bad, and indifferent, which they contain, will not permit us to give any thing more than a general account of the contents of each,

The fifth volume exhibits the history of the five dynasties Song, Tsi, Leang, Tchin and Soui, in which we find few, if any great princes, and still fewer good ones, though they contain a space of a hundred and nineteen years, and the reigns of twentyseven emperors. After the extinction of the dynasty of Toin, in the year. 420, China was divided into several small fovereignties; besides which, we perceive here a more important division into two great empires, the one northerrt, formed by the entrance of the Tartars into the northern provinces, and the other southern, of which the emperors were Chinese. By the historical series which F. DE MAILLA has followed (confining the attention to the southern empire, and mentioning in the margin only the princes of the dynasty of Song, who reigned in the south), the reader is led to think, that there is only one emperor, and that the northern chief is only a little rebel fovereign: but this is a mistake, the grand annals mention boch the northern and southern emperors (as we learn from the respectable authority of M. de Guignes), and there is no doubt but that their translator ought to have followed the same method. Both this grand division and the smaller ones of the northern districts, pofleffed by Tartar chiefs, introduce confufion into the thread of this history, especially to an European, who is not familiar with these various events and revolutions.

If the dynasties already mentioned exhibit no emperors of great note either for genius or virtue, we are compensated by several displays both of public and private virtue, in inferior ftations. We meet with a Yen-Yen-Tchi, friend and minister to the emperor Ou-ti of the dynasty of Song, who, from a state of extreme poverty and obscurity, rose, by merit alone, to the first posts in the empire, and never forgot himself in any of the

• See our last extract in the Review for December 1777, in the Foreign Correspondence, p. 477.

+ The Abbé GROSSIER and M. le Roux DESHAUTESR AYES, Arabic Profeffor in the Royal College of France, &c. &c.

prosperous

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