neither defects nor vices appear in their genuine light; as, in the paintings of great artists, the different tints of light are fo blended as to leave no particular colour in its primitive and natural state.' This is by much the finest morceau in the volumes before us, and discovers an acquaintance with hiftory and politics. America enjoyed a degree of temporary confequence and celebrity from its having been confidered by all Europe as the rival of Britain. Accordingly the American revolution has been the fubject of much philofophical theory and political fpeculation. After the conclufion of a difaftrous war, in which the infant States of America obtained the victory over the most powerful nation of Europe, the projector and the patriot looked to the new hemifphere as an afylum for the oppreffed and perfecuted part of mankind, and as the chofen feat of freedom and tranquillity of the liberal and commercial arts, in the long annals of future ages. The most flattering expectations were formed, and the finest predictions delivered, concerning the profperity of a government planned in an enlightened age, which was to have liberty for its bafis, and general happiness for its object. It. was believed that the golden age would be reftored, or the millennium introduced. Thefe vifions have vanifhed, and a dark cloud, pregnant with thunder and tempeft, hangs over America. The Marquis de Chaftellux is an agreeable writer, but defultory, inaccurate, and fuperficial. He facrifices truth to` the graces. He looks at every object through a prifmatic glafs; and fcatters his gaudy colours with an indifcriminating hand. He has found the moft convenient tranflator that ever an author was bleft with. He is always at hand to confirm, by a note, any fingular or unfupported decifion in the text. He feems to have made the tour of the world in an air-balloon; to have acquired a knowledge of all countries, and cultivated an acquaintance with all nations. If he fometimes feems to have forgot the idiom of the Englifh language, and even to violate the rules of univerfal grammar, fuch deficiencies may be fuppofed to have taken place in the variety of his adventures, and the extent of his excurfions. He indeed appears to be un grand Charlatan; which expreffion we fhall leave him to tranflate. ART. By ART. II. Tracts, Mathematical and Philofophical. Charles Hutton, LL.D. F.R.S. and Profeffor of Mathematics in the Royal Military Academy, Woolwich. 4to. 14s. boards. Robinfons. London, 1786. THE fubjects of these tracts, which are miscellaneous, are chiefly on the abstruser topics of mathematical and philofophical inquiry; yet fuch as, with a very few exceptions, are of very material and extenfive utility in their applica tion. The mathematical tracts, of the greatest importance, relate to the history and theory of feries; the philofophical, to the theory and practice of projectiles: the whole may be included under this divifion, excepting a few excursions into the regions of fpeculative geometry, which the author has intermixed as matters of curiofity rather than use. : As an introduction to the methods of the author's invention for finding the values of infinite feries, he has premised fome definitions relating to the different kinds of series, and their diftinctions into converging, diverging, and neutral (or those which neither converge nor diverge) and having particularly fhewn the mifconceptions which have fometimes arifen from the common acceptation of the term, fum of a feries, he has thought it more proper to fubftitute that of radix; fince every infinite feries may be confidered as arifing from the evolution of fome finite expreffion, which may be fubftituted inftead of the feries itfelf in thefe preliminaries the author has, in fome meafure, followed the ideas of M. Euler (Com. Pet. Vol. V.) in his differtation upon this fubject. The firft of the author's methods is extremely general, and extends to the valuation of all numeral infinite feries, - whofe terms are alternately and ; whether converging, diverging, or neutral. It is derived from an obvious and well-known property of infinite feries with alternate figns, viz. that, by collecting their terms in order, we obtain a feries of fucceffive fums, which, in converging feries, approach nearer to, in diverging, recede farther from, the true value of the feries propofed; and by analogy, in neutral feries, remain always at the fame diftance, By arithmetical means, interpofed between these fums, a new feries is formed, the collected terms of which approach ftill nearer; and, as the feries thus formed has the fame properties and incidents as the original one propofed, new leries of arithmetical means may be perpetually interpofed, till a general formula is at laft found, arifing from the law which the numeral co-efficients in these repeated approximations obferve, which will, in fome cafes, be accurately true, viz. when n, the power of the denominator, is a finite number; and will always be the nearer as n is greater. The author has been copious in exemplifying the use and application of this method in a variety of inftances; and particularly in the fummation of a very difficult feries, on which M. Euler has exprefsly written a differtation. The coincidence of this method with the famous differential theorem fhews both its truth and its derivation from the fame principle, though its demonstration is here independently given: but it appears to have an obvious advantage in the greater facility of its application. To this follows a method for fumming a very flowly converging feries, with its terms all pofitive, viz. afbxt cx2+dx3, &c. when x is nearly equal to 1. The investigation of this method is derived from the series itself, and, what has hitherto been much fought for, the fum is given with a confiderable degree of accuracy, much greater, as it should seem, than by any other methods hitherto generally known or made public. In the fourth tract there is a very general method given. for extracting all roots out of any numbers propofed. This method is chiefly remarkable for its great fimplicity and generality, as it includes all the particular rational formulæ of Halley and De Lagny. The fifth tract is of more importance, being an original method of inveftigating the roots of the higher equations; in which, incidentally, many new lights are thrown on the nature of equations in general, and fome curious properties of numbers remarked, as it fhould feem, for the first time; in particular, that when the three roots of a cubic equation are in arithmetical progreffion, the author's theorem here laid down will affign the middle root exactly, otherwise by approximation only. To this follows a very curious hiftory, as well as a general demonstration, without fluxions, of the famous binomial theorem. The hiftorical part of this tract is new, and tends to difcriminate how much of the invention of this celebrated theorem was antecedent to the time of Newton, The author has fhewn fufficiently, that the law of the co-efficients, even of the terms in the higher powers, was known very early; and that, in fact, though he has no doubt of its being re-invented by Newton, that the fubftance of the binomial theorem, as far as it relates to integral powers, is given at length, without fymbols, by Briggs, in his "Trigonometria Britannica." The The great improvement which was unquestionably made by Newton to this method was the application of fractional exponents, and thereby making it univerfal for the extrac tion of roots, as well as the railing of powers, and with the fame fimplicity. : From this hiftory of the invention, the author proceeds to a history equally curious, of its demonftration; which was never given by Newton, otherwife than indirectly, and by induction, but has fince been given by various authors, both fluxionally and algebraically but in the latter, and only legitimate method, commonly with fome restrictions in point of generality; moft of the algebraical inveftigations of it having either been adapted to integral powers alone, er, if extended to fractional, being inadequate to exprefs the law of the continuation of the terms. The demonftration which the author has fubjoined aníwers very fatisfactorily all these conditions; and, what feems quite new in it, is, moreover, a ftrict demonftration of the form of the feries itfelf, which has hitherto been taken for granted in demonfrations of the truth of the theorem. The feventh and eighth fections are concerning fome new properties of the fphere and cone, inveftigated by the author; and a divifion, hitherto deemed impracticable, of circles and ellipfes. Thefe have doubtlefs confiderable curiofity, as geometrical exercifes; but having no immediate connection with the fubjects of the preceding or following inquiries, we fhall proceed to that which takes up the larger part of this volume, and which the author gives as the most important in this publication, viz. his New Experiments in Artillery in the years 1783, 4, 5. After briefly premifing, that the first attempts made to elucidate the true theory of military projectiles, founded on the air's refiftance, by experiments with cannon balls, were thofe made by the author in 1775, which were honoured by the Royal Society with their gold medal; the author obferves, that, in the courfe of thefe experiments, the effects of different quantities of powder, different weights of fhot, and different degrees of windage, were compared, and found to preferve conftant and regular laws; from which, though the experiments were made with the fame pieces of ordnance, fo many important conclufions were derived, that has fince been his great defire to extend them, with the help of fome additional machinery and accommodations. Of these additions to his machinery, and other accommodations for thefe experiments, which were amply furnished by the direction of the mafter-general of the ordnance, he has has given an accurate defcription, which is farther illuftrated by_plates. Five brafs pieces were caft, and bored with great exactnefs, for this purpose, of one pound fhot, but of different lengths and weights; and one of them was made to be diminished after firing, by fawing off a part of its length; and was also fufpended in a particular manner, by a contrivance in its cafting, as near as poffible to its varying. centres of gravity, after its fucceffive diminutions; this was to fhew the effect of length alone, abstracted from other cir→ cumftances, in a gun. Balls of iron were also caft, and rounded with great exactnefs; and no lefs care was taken of the quality and proportion of the powder. It is not poffible, in this place, to recite the hiftory of a feries of experiments of this kind, and of this length of time in the performance; it must fuffice to state with brevity their general object; to examine how far it has been attained; and the uses to which it may be applied. The object of these experiments, as well as the former ones made by the author in 1775, feems to have been to determine, with as much precifion as poffible, fuch data as may afford eafy rules for the feveral cafes of practical gunnery, upon the true theory of the air's refiftance. Thefe data, the concomitant effect of which on the air's refiftance is to be ascertained, are, the weight of powder, the weight and diameter of the ball, the initial velocity, the elevation and length of the gun, the time of the flight, and horizontal range of the ball. The extreme care and accuracy with which thefe experiments appear to have been made, gives reafon to believe that the relation of them is as faithful as it is circumftantial and minute. A table of them, as concurrent in their operation, is fubjoined as the refult of thofe inquiries. To have afcertained them with reafonable exacnefs, is as much as could be hoped for from these experiments; and the completeness of the apparatus, as well as the attention in the performance, gives reafon to hope that they have been done at leaft more fatisfactorily than in any experiments hitherto made with cannon balls. To erect upon them a fyftem of practical gunnery, founded on the genuine and true theory of projectiles, is proposed by the. author as the fubject of future labours. The vast importance of the air's refiftance is now generally known and acknowledged, as well as the difficulty of reducing a theory, fo complex in its principle, to that facility in practice which may render it useful in the.common management of artillery. Thofe who are the best judges of the |