the greater part of the people; and the Major justifies and ap plauds them for being as merry as they can the rest of the day, and for keeping clear of what he calls a sullen countenance, gloomy subjects, a set form of speech, and a stiff behaviour.' He insists they shall by all means have a religion, a pure and rational religion,' he says, such as is contained in the sublime pages of revelation;' for, it is of infinite use to mankind in a temporal sense.' But not even for the sake of this, the most important of all the benefits of religion, will he consent to have the Indians disturbed, in their devout and laudable adherence to the creed of their forefathers. The book contains a variety of passages in which the writer appears to take considerable credit to himself, as a philosopher, for placing religion in the light in which it is regarded by politicians of the very inferior rank. There is a desultory entertaining description of the Character,' taken in a general and comprehensive sense,' of the Louisianians.' The representation of the Aborigines' too much resembles that in Guthrie's Grammar, and in Robertson. To be sure, it forms a striking picture, ready for the use of every successive exhibition. But if a man pretends to paint in the sobriety of truth, in the very scene where the reality is displayed, and absolutely from the life, it is unpardonable to play off again on our imaginations the horrible visions of the long courses of torture and cannibalism. Why cannot we obtain, at last, the mere plain truth as to the degrees and modes of cruelty which captive enemies are condemned to suffer? There is an ineffectual attempt to revive, under some appearance of probability, the notion of there being a Welch tribe of Indians, somewhere in North America. The Major compensates to himself the extreme penury of his religious credence, by believing such a proposition as that it would be easy enough for Prince Madoc to make three successful voyages to America before the invention of the compass, and two straight back to Wales. The most curious and interesting chapter of all, (but it admits not of abridgement) is that on the rivers of North America. We will transcribe the description of the confluence of the two noblest of them, the Missouri and the Mississippi, the former of which, he says, is decidedly the greater river. The junction of the two great rivers is in north latitude thirtyeight degrees, and forms an interesting spectacle. The two islands in the mouth of the Missouri oblige him to pay his tribute to what is denominated the father of rivers, through one large, and two small channels. As if he disdained to unite himself with any other river, however respectable and dignified, he precipitates his waters nearly at right angles across the Mississippi, a distance of more than twentyfive hundred yards. The line of separation between them, owing to VOL. X. L the difference of their rapidity and colours, is visible from each shore, and still more so from the adjacent hills. The Mississippi, as if astonished at the boldness of an intruder, for a moment recoils and suspends his current, and views in silent majesty the progress of the stranger. They flow nearly twenty miles before their waters mingle with each other.' For an American the composition is tolerable; but the Major has a good share of those words and phrases, which his literary countrymen must, however reluctantly, relinquish before they will rank with good writers. The standard is fixed; unless it were possible to consign to oblivion the assemblage of those great authors on whose account the Americans themselves are to feel complacency in their language, to the latest ages. Art. II. Tracts on Mathematical and Philosophical Subjects; comprising, among numerous important articles, the Theory of Bridges, with several Plans of recent Improvement. Also, the Results of numerous Experiments on the Force of Gunpowder, with Applications to the Modern Practice of Artillery. By Charles Hutton, LL.D. F.R.S. &c. Late Professor of Mathematics in the Royal Military Academy, Woolwich. 3 vols. 8vo. pp. xii. 1252, with plates. Wilkie and Robinson. Price 2l. 8s. boards. 1812. WE E are persuaded there is no need to introduce this work to the notice of our readers by any prefatory observations. Dr. Hutton has now occupied a most eminent station among British mathematicians, for full half a century: and we rejoice that, at an advanced period of life, he has found himself sufficiently in circumstances of ease and leisure, to be able to prepare for the press a connected and uniform edition of such of his works, whether previously published or not, as possess most originality. These he now gives to the world under the modest designation of "Tracts." The number included in this collection is thirty-eight. A few of them have been already printed in the Philosophical Transactions, and in other works; but most of them are quite new; and such as are not so, having been recast and greatly improved, may be also considered in some measure as original compositions.' Being of a miscellaneous nature, they are here arranged nearly according to the order of time in which they were composed.' The subjects to which they relate are very various. We shall describe them briefly in their order. The first tract is a republication, with very considerable improvements, of Dr. Hutton's little treatise on the theory of arches and piers, first published at Newcastle, in the year 1772. The five succeeding tracts relate also to the subject of bridges, and comprehend: 1. Queries concerning London bridge, proposed by the city magistrates in 1746, and answered by Mr. George Dance, then surveyor-general of the city works. 2. Report of a committee of members of the Royal Society, respecting London bridge, addressed to the common-council of the city. 3. Mr. John Robertson's opinion in 1754, of the consequences to the tides in the river Thames, by erecting a new bridge at London. 4. Dr. Hutton's answers to questions proposed, in 1801, by the Select Committee of Parliament, relative to a project for erecting a new iron bridge, of a single arch, over the river Thames at London, instead of the old London bridge. 5. A very interesting history of iron bridges, illustrated by neat elevations of those at Colebrook Dale, Buildwas, Bristol Harbour, Telford and Douglass's proposed iron bridge at London, and the elegant aqueduct at Pontcysylte, in Wales. These six tracts doubtless contain, together, the most valuable body of information, both theoretical and practical, on the subject of bridge building which has yet been published. The seventh, eighth, and ninth tracts relate to infinite series. Of these the first clearly marks their nature, and defines their equivalent or radix, so as to be free from metaphysical objections: the second exhibits a simple and ingenious method of valuing numerical infinite series, which have their terms alternately plus and minus: and the second explains a method of summing the series a + bx + c x2 + dx3 + ex*, &c. in the case when it converges very slowly. 3 In the tenth tract we have the investigation of Dr. Hutton's well-known approximating rules for the extraction of roots of numbers; and in the eleventh, a new method for finding the roots of such equations as have their terms alternately plus and minus. The contents of the remainder of the first volume, may be detailed in the Doctor's own words: • Tract xii treats of the binomial theorem; exhibiting a demonstration of the truth of it in the general case of fractional exponents. The demonstration is of this nature, that it proves the law of the whole series in a formula of one single term only: thus P, Q, R, denoting any three successive terms of the series, expanded from the gived binomial (1 + x), and if I g PQ, then is-n Q=R, which h + n denotes the general law of the series, being a new mode of proving the law of the coefficients of this celebrated theorem. But, besides this law of the coefficients, the very form of the series is, for the first time, here demonstrated, viz. that the form of the series for the developement of the binomial (1 + x), with respect to the exponents, will be 1 + ax + bx2 + cx3 + dx1 + &c. a form which has heretofore beer assumed without proof. 6 • Tract xiii treats on the common sections of the sphere and cone: with the demonstration of some other new properties of the sphere, which are similar to certain known properties of the circle. The few propositions which form part of this tract, is a small specimen of the analogy, and even identity, of some of the more remarkable properties of the circle, with those of the sphere. To which are added some properties of the lines of section, and of contact, between the sphere and cone : both of which can be further extended as occasions may offer. Tract xiv, on the geometrical division of circles and elipses into any number of parts having equal perimeters, and areas either all equal or in any proposed ratios to each other: constructions which were never before given by any author, but which, on the contrary, had been accounted impossible to be effected. • Tract xv contains an approximate geometrical division of the circumference of the circle. ⚫ Tract xvi treats on plane trigonometry, without the use of the common tables of sines, tangents, and secants: resolving all the cases in numbers, by means of certain algebraical formulæ only. Tract xvii is on Machin's quadrature of the circle; being an investigation of that learned gentleman's very simple and easy series for that purpose, by help of the tangent of the arc of 45 degrees; which series the author had given without any proof or investigation. Tract xviii, a new and general method of finding simple and quickly converging series; by which the proportion of the diameter of a circle to its circumference may easily be computed to a great many places of figures. By this method are found, not only Machin's series, noticed in the last tract, but also several others that are much more simple and easy than his. ‹ Tract xix, the history of trigonometrical tables, &c.: being a critical description of all the writings on trigonometry made before the invention of logarithms. • Tract xx, the history of logarithms; giving an account of the inventions and descriptions by several authors on the different kinds of logarithms. Tract xxi, on the construction of logarithms; exhibiting the various and peculiar methods employed by all the different authors, in their several computations of these very useful numbers. ‹ Tract xxii, treats on the powers of numbers; chiefly relating to curious properties of the squares, and the cubes, and other powers of numbers. ‹ Tract xxiii, is a new and easy method of extracting the square roots of numbers; very useful in practice. • Tract xxiv, shows how to construct tables of the square-roots, and cube-roots, and the reciprocals of the series of the natural numbers; being a general method, by means of the law of differences of such roots and reciprocals of numbers. • Tract xxv, is an extensive table of roots and reciprocals, con structed in the above manner, accompanied also with the series of the squares and cubes of the same numbers.' Vol. I. pp. v―vii. All these tracts may be perused with great advantage by young mathematicians: but the contents of tracts 8, 12, 13, 14, 18, 19, 20, 21, will be found the most curious and instructive. Those on the history and construction of logarithmic and trigonometric tables are elaborate and extremely valuable: they were originally given in Dr. Hutton's separate work on those subjects, in 1785. The second volume commences with tract the twenty-sixth, which is on the mean density of the earth, being an account of the calculations made from the survey and measures taken at Mount Shichallin, in order to aseertain the mean density of the earth; improved from the Philosophical Transactions, vol. 68, for the year 1778.' This paper describes, in 68 pages, a series of most intricate and laborious investigations and computations, such as we believe no man of genius living would have undertaken besides Dr. Hutton, and such as we apprehend hardly any other man could have carried through, had he made the attempt. The observations and measurements relative to Mount Shichallin, conducted under the direction of the late excellent Astronomer Royal, were undertaken, as all our philosophical' readers will recollect; for the purpose of determining, experimentally, whether the earth actually exerted, as the Newtonian theory of universal gravitation supposed it to exert, an attraction upon other bodies, whether at or remote from its surface. If the earth exert an aggregate attraction, it will be constituted of the attraction of its several parts, and will be manifested more or less by such of those parts as are so circumstanced that the effects of the attraction become appreciable: a large mountain, for example, may be so situated that its attraction shall in some measure oppose the aggregate attraction of the earth, and its effect may become appreciable by its drawing a plumb-line from the vertical position in which it would hang if operated upon solely by the attraction of the earth towards its centre. The first experiment made for the determination of this point, was by the French academicians, who, in 1735, were sent to measure a degree near the equator: their observations were made upon Chimboraco, which they found to occasion a deviation of 7 in the plumb-line. The effect fell greatly short of what had been expected; but the observers afterwards learnt that this mountain had formerly been a volcano, and they actually found some calcined stones upon it. This experiment then, though it proved that mountains do act at a distance, did not determine the point with all the satisfaction that was looked for. Nor was any thing farther attempted in the long interval from |