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quiry in the twenty-fourth letter. After many other ancient teftimonies, which concur in placing this famous ifle in the North, our Author quotes that of Plutarch, who confirms thefe teftimonies by a circumftantial description of the ifle of Ogygia, or the Atlantis, which he reprefents as fituated in the North of Europe, and as having near it three islands more, in one of which the inhabitants of the country fay that Saturn is kept prifoner by Jupiter. These four islands may, as M. B. conjectures, be Iceland, Greenland, Spitzberg and Nova Zembla, or fome others nearer the Pole. Rudbeck, a learned Swede, compofed 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 hypothefis 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 ftill to his fyftem, M. BAILLY, perfuaded by a variety of plaufible circumftances, which he has ingeniously combined, places that famous ifland among thofe of the Frozen Ocean. He is ftrongly feconded by Plutarch, who tells us, that the Atlantis is in a region, where the fun during a whole fummer month is scarcely an hour below the horizon, and where that fhort night has its darkness diminished by a twilight.' This, indeed, is a palpable indication of a Northern climate; but how is this fituation reconcileable with the fertility of foil, the mildness of the air, which both Plutarch and Plato mention among the other advantages enjoyed by the Atlantes? And how is it poffible to conceive Aftronomy cultivated in a frozen and cloudy region, where the obfervation of the heavenly bodies must have been painful and impracticable? Our Author answers these questions with levity enough he obferves that Plutarch was not the disciple of M. de Buffon, and that these difficulties cannot be removed, but by fuppofing a change of air and climate in thofe regions, through the gradual cooling of the earth, and its progreffive motion towards univerfal 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. ART. II.

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Develloppement Nouveau de la Partie Elementaire des Mathematiques, prife dans tout fon etendue, &c.-A New Explanation of the Elementary Part of Mathematics: By LEWIS BERTRAND, Profeffor of Mathematics at Geneva, and Member of the Academy of Sciences, &c. at Berlin. 2 Vols. 4to. Geneva. 1778. Price 36

livres.

HIS is a work of great merit, as the method of treating mathematical science proposed by the ingenious Author, is new, eafy, interesting, and remarkable for its order and ac

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curacy. All the problems, which may be refolved by the circle, and the right line, come under the clafs of elements. But as the properties of the circle and the right line fuppofe a confiderable knowledge of the relations fubfifting between quantities, confidered in a general view, elements may be divided into two parts;-the firft, arithmetical and algebraical, which furnishes the means of unfolding the properties of the circle and the right line; the fecond geometrical, containing the explanation of these properties, and their application to the folution of the queftions that relate to them, or depend upon them.

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

In the first he introduces a peafant, 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 fecond, he makes him discover the known methods of addition and fubtraction: in the third, the Author puts himself in the place of his difciple, and proposes to himself particular queftions of multiplication and divifion, which lead him to the general rules of thefe operations, whether they be applied to whole numbers or to fuch as contain decimal fractions. He always forms, as he proceeds, the theoretical conclufions refulting from his refearches, defines the objects prefented by the developement of his ideas, and points out the proper figns for the reprefentation of thofe ideas.

M. BERTRAND begins his fourth chapter by the following propofition, that the product of feveral factors does not depend on the order in which they are multiplied: he fhews the powers and roots of numbers, completes what he had obferved with refpect to figns in the preceding chapters, and thus lays down the principles of algebraic notation.

In the fifth chapter our Author treats of broken numbers, and fhews how they are to be added, fubtracted, multiplied and divided by each other. In the fixth he undertakes the folution of a difficult queftion in fractions, by a method very different from thofe which have been employed for that purpose by other analytical writers. But as this chapter, may appear difficult to fome beginners, M. BERTRAND advifes fuch to defer the perufal of it until they have ftudied the three following chapters, as the truths demonftrated in them do not depend on the propofitions contained in the fixth, and by exercifing the fagacity and attention of the young reader, may prepare him for understanding them with more facility. In chapter the feventh M. BERTRAND points out the methods of extracting the roots of whole and broken numbers of every kind-the eighth contains a complete treatife on arithmetical and geometrical relations and proportions; and the ninth a folution of determinate

determinate and indeterminate problems of the fift 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 divifor of two algebraic quantities. The variety and choice of the problems refolved, in this chapter, as alfo the reflexions which accompany their folution, are every way proper to excite in the youthful mind a tafte for the fcience under confideration, and to facilitate remarkably their progrefs and improvement in mathematical knowledge.

The tenth chapter is employed in the folution of determinate problems of the fecond degree, and the eleventh in displaying the powers of a binomial, whofe indices are either broken or negative numbers. In this chapter, among other things, our Author lays down the principles of the fcience of probabilities, and refolves feveral problems, relative to chances, which render the application of these principles familiar to the ftudent, and alfo fhew him how interefting the queftions are, which depend upon them.-The fcience of logarithms is amply treated in the twelfth chapter, in which the labours of Lord Naper, the ingenious methods and tables of Meffis. Sharp, Briggs, Flack, and Sherwin, are defcribed, illuftrated, and appreciated with refpect to their accuracy, and ufefulness in this important branch.

The SECOND PART of this work is fubdivided into two, viz. Elementary Geometry and Trigonometry. The first, which is again fubdivifible into three branches, comprehends the properties of the circle and right line, the application of these properties to the menfuration of plane, rectilinear, and circular furfaces, and to that of curve furfaces and folids. The firft of these branches is largely treated in feven chapters. Here the Author, beginning with the common notion of space, deduces from it the ideas of planes, right lines, angles, triangles, and curves, defcribes their nature, properties, determinations, circumftances, relations, proportions, &c. folves feveral problems relative to them, and points out the confequences deducible from them. The fecond 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 conclufion, the areas of fectors, fegments, 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 fimple folid angles (for fuch he calls the angles that are formed by two planes, which meet each other)—of regular folid angles, and

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their principal properties, of regular bodies, their number, conftruction, &c.—of the definition and conftruction of prisms, pyramids, concs, and cylinders, of the menfuration of their furfaces, and of their folidity, and of the characters or marks of fimilarity in folids of every kind.-There is a rich variety of mathematical inftruction communicated with great perfpicuity and facility in the detail into which M. BERTRAND enters in the illuftration of all thefe fubjects.

Trigonometry forms the fecond branch of geometry, confidered in its general fenfe. Under this denomination our Author comprehends both Plain and Spherical Trigonometry, as they are branches that fpring from the fame root, and they are treated in one chapter, which is divided into feven sections. Thefe contain the most important definitions, difcuffions, problems, folutions of problems, and demonftrations, that regard this interefting branch of mathematical fcience.

It is proper to obferve 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 ufe of the algebraic analyfis in all its extent; -he has not gone further than the folution of equations of the fecond degree. As to his method, it is ftrictly geometrical, and hence arife the order and precifion that give fuch relief and encouragement to the ftudent by fpreading an air of eafe and facility over laborious difcuffions, and thus rendering them perfpicuous and interefting. For the moft part, M. BERTRAND has employed both the analytic and fynthetic method, of which he knows perfectly the refpective nature, advantages, and refources; the fure progrefs in knowledge arifing from the one, and the expeditious manner of communicating that knowledge, which is the peculiar advantage of the other, are circumftances of which he has happily availed himself in the excellent work now before usa work which we think must be of great ufe, not only in directing the fpeculations of the ftudent, but in guiding the merchant, the politician, the topographer and geographer, the navigator and aftronomer, in the practical duties and occupations of their refpective profeffions.

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À R T. III.

Hiftoire Generale de la Chine, ou Annales de cet Empire, &c.-A Gene ral History of China, or the Annals of that Empire, tranflated from Ton, kien, kang-mou, by the late Father J. A. M. DE MOYRIAC DE MAILLA, &c. Vols. V. VI. VII. and VIII. Paris. 1778.

4to.

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T is time to refume* our accounts of this great work, in the publication of which the learned Editors + display the moft active diligence, industry, and perfeverance. Thefe four volumes contain the hiftory of China from the year 420 of the Christian æra 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 hiftory of the five dynasties Song, Tfi, Leang, Tchin and Soui, in which we find few, if any great princes, and ftill fewer good ones, though they contain a fpace of a hundred and nineteen years, and the reigns of twentyfeven emperors. After the extinction of the dynafty of Toin, in the year 420, China was divided into feveral small fovereignties; befides which, we perceive here a more important divifion into two great empires, the one northern, formed by the entrance of the Tartars into the northern provinces, and the other fouthern, of which the emperors were Chinese. By the hiftorical feries which F. DE MAILLA has followed (confining the attention to the fouthern empire, and mentioning in the margin only the princes of the dynafty of Song, who reigned in the fouth), 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 both the northern and fouthern emperors (as we learn from the refpectable authority of M. de Guignes), and there is no doubt but that their tranflator ought to have followed the fame method. Both this grand divifion and the fmaller ones of the northern diftricts, poffeffed by Tartar chiefs, introduce confufion into the thread of this hiftory, efpecially 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 compenfated by feveral difplays both of public and private virtue, in inferior ftations. We meet with a Yen-Yen-Tchi, friend and minifter to the emperor Ou-ti of the dynafty of Song, who, from a state of extreme poverty and obfcurity, rofe, by merit alone, to the firft pofts in the empire, and never forgot himfelf 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 DESHAUTESRAYES, Arabic Profeffor in the Royal College of France, &c. &c.

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