in the time of Cadmus, the Hetrurians, the ancient Grecks and Latins, the Irish, the Teutons, and the Swedes with their Runic: these have one common origin. To the fecond belong the people that spoke the Hanferit, or facred language, now almoft forgotten by the Brahmins, and the Zend and the Pelhvi, which are the ancient Perfian. These two families came at different periods from the fame original flock, as appears from the comparison of their languages, and they brought their languages as well as the worship of fire from the North. Our Author goes on to prove by endless genealogies of words, &c. that all the labours of Hercules were performed in the North, and that the garden of the Hefperides was near the Pole; and that the Reader may not afk impertinently, how the golden apples of that celebrated orchard could grow, bloffom and ripen in the icy nations of the North, M. BAILLY ftops his mouth with the new hypothefis of M. de Buffon, which comes in the luckieft poffible moment to remove the difficulty, by letting us know, that the globe was, at firft, fluid fire, and afterwards all genial warmth even in its polar extremities, and that therefore ruddy apples might have grown where now nothing is discoverable but rocks of ice. It is very unfortunate that the whole discovery of M. BAILLY depends upon the truth of this whimsical and impertinent hypothefis, according to his own confeffion. The twenty-third letter contains our Author's voyage toHell: Its title is Voyage aux enfers: it would not be civil to leave him there, more especially fince he tells us that the fables relative to this region, are of all others the moft curious and interefting, and the most adapted to decide the present question. In imitation therefore of Orpheus, Thefeus, Hercules, Ulyffes, Æneas, and others, down he goes to the fhades. But how come at these infernal regions? For though all nations were agreed that they lay in the bofom of the earth, yet each nation pretended that they were within its domain. The Latins placed them at Baia, near the lake Avernus in Italy-the Greeks in Epirus and Arcadia-and Diodorus Siculus expofed the fraud or folly of thefe pretenfions, and made the accounts of the infernal regions originate in the Egyptian mythology. But Homer knew better, and places them in the country of the Cimmerians, where clouds and darkness, and an eternal night reign. This muft, fays our Author, have been far North of Greece, though the famous bard does not precifely fix the place, and his account was derived from ancient traditions. Numberless etymologies are employed by M. BAILLY to fhove Tartarus and Elyfium towards the Pole, and this letter is fingularly rich in mythologi cal erudition. But now we come to the grand point, the discovery of the Atlantides and the Atlantis of Plato; this is the fubject of in 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 say that Saturn is kept prifoner by Jupiter. These four iflands may, as M. B. conjectures, be Iceland, Greenland, Spitzberg and Nova Zembla, or Rudbeck, a learned Swede, com fome others nearer the Pole. posed a work about a century ago, in which he maintained that Sweden was the Atlantis of Plato; our Author, though he has made good ufe 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 strongly 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 difciple 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 inftead of answering these objections M. BAILLY tells his objectors a fairy tale. ART. II. 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 TH mathematical science propofed by the ingenious Author, is new, eafy, interefting, and remarkable for its order and ac curacy. 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 suppose 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 advises fuch to defer the perusal 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 exercising 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 resolves several problems, relative to chances, which render the application of thefe principles familiar to the ftudent, and alfo fhew him how interefting the queftions 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 tables of Meffrs. Sharp, Briggs, Flack, and Sherwin, are defcribed, illuftrated, and appreciated with refpect 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 firft, 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 first of thefe 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, describes 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 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 seven sections. Thefe contain the most important definitions, difcuffions, problems, folutions of problems, and demonftrations, that regard this interefting 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 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 us :-a 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. ART |