the nature of it; although the want of diagrams may prevent us from doing it complete justice.

The principle on which the demonstration rests is simply this; that, if any reasons can be assigned why a body shall move in any direction, and if the same reasons will apply to make it move in some other direction, then it follows that it will take neither of these, but will remain at rest, or move in some direction different to the two former. Hence, if two forces act on a point, the resultant of those forces will be in the plane of the former; since any reasons, which we may assign to prove that it ought to follow a direction above that plane, may be applied also to prove that it would follow a direction symmetrically situated below the same: therefore the resultant will follow neither the one nor the other of these; and the same may be demonstrated of any direction that is not situated in the plane of the two original forces. Again, if two equal forces act at the same point, their resultant will be in the line which bisects the angle formed by the direction of the two equal forces. In the first place, the resultant will be in the plane of these two forces by the preceding proposition; and, whatever reasons may be brought to shew that the resultant is situated on the one side of the line which bisects the angle, the same reasons may be employed to prove that it is symmetrically situated on the opposite side of the bisecting line, and the same may be shewn of every resultant that is not in the line which bisects the original angle.

Having thus determined the resultant of two equal forces, that of two unequal but commensurable forces is readily found; and ultimately that in which the two are incommensurable.

The author next reduces these results into general formulæ; illustrates what is commonly denominated the parallelopipedon of forces; examines the case of parallel forces; of forces situated in a plane, and applied to different points connected together in any invariable manner; and, lastly, of forces acting in any manner whatever in space. - The succeeding chapter treats of the centre of gravity of bodies, and the method of determining it both analytically and by graphical construction, with a variety of problems by way of illustration. Among others, it is required to find the centre of gravity of a curve of double curvature; the centre of gravity of any area comprized between a curve, its abscess, and its ordinate; of the sector of a circle; of an area comprized between two branches of a curve; of any solid of revolution, &c. &c. - In the next chapter, the author treats of Guldin's theorem, or, as it is commonly called, the centro-baryc method; in which, after a

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few

few transformations, he deduces this very curious and remarkable result; viz. "Every figure, whether superficial or solid, generated by the motion of a line or surface, is equal to the product of the generating magnitude into the path of its centre of gravity." - The tenth chapter treats of the seven mechanical powers, the funicular machine being considered as one of them, and its properties are illustrated at considerable length; after which, a chapter on Friction terminates the first part, or the Elements of Statics.

Part II., on Dynamics, commences with an illustration of the laws of inertia; the subjects of uniform and variable motion; the descent of bodies in a vacuum, in resisting mediums, and on inclined planes; of the motion of bodies in curves; the equation of the trajectories which they describe, the determination of their velocities, &c. &c. - The ninth chapter treats of the doctrine of projectiles, in the same general manner as in the other parts to which we have already referred: but we doubt whether the author has succeeded so well in this; his investigations here being somewhat tedious, and several pages employed in deducing formulæ which might certainly have been much more directly obtained. It is remarkable that this theory, which is extremely simple in itself, is generally rendered so abstruse and difficult as we find it in most treatises on mechanics.

Let us, for example, conceive any line, A B, drawn to represent the range of a projectile; A 2 its direction, AH a line parallel to the horizon; and B 2 a perpendicular to the same. Let the angle of elevation be denoted by a, the angle of inclination of the plane by b, the time of flight = t, the velocity = v, and the range = r; also g = the descent of a body in one second. Then A2 = tv, by the laws of uniform motion; 2 B=gt, by the laws of falling bodies; and AB = r, by hypothesis: whence we have immediately the following proportions:

cos. b: sin. (a + b) :: tv :

cos. b: cos. a

tvsin. (a+b)=gt2

cos. b
tv cos. a
:: tv:
cos. b
gt

whence, again, by comparison

..........

=

r

sin. (a + b) cos. a

which three equations are sufficient for every case relating to the time, velocity, and range of a projectile, its angle of elevation, and the angle of inclination of the plane, while that plane passes through the point of projection. It is singular, as we have before observed, that results which are so readily obtained

13

obtained should commonly be made to occupy so many pages as are allotted to them by most authors on this subject.

To proceed in our analysis. Chapter xi. is employed in discussing the different modes of measuring forces, in which the author glances slightly at the dispute, now almost forgotten, relative to forces wives, momentum, &c.: he then treats of centrifugal forces; the centre of oscillation; the simple and compound pendulum; and the mechanical properties of the cycloid. -The xxth chapter is employed in demonstrating and illustrating the principle of D'Alembert; and the remainder of this part in shewing its application to the solution of several general problems relative to the motion of bodies about fixed axes, and of bodies in space.

Part III. treats of the Theory of Fluids; viz. the pressure of fluids; general equations of their equilibrium; the application of the same in the case of incompressible fluids, and of those that are elastic. -The sixth chapter contains a description of several machines: as the hydrostratic balance; the areometer; the syphon; pumps, &c.; and lastly, of the barometer, and the method of measuring heights with that instrument.

Having thus given a concise abstract of the principal contents of this volume, we shall conclude by recommending it to the perusal of such English students as wish to qualify themselves for entering on the more important works of the French mathematicians.

ART. VIII. Mélanges d'Analyse Algébrique, et de Géométrie, &c.; i. e. Miscellanies in Algebraical Analysis and Geometry. By M.J. DE STAINVILLE. 8vo. pp. 680. Paris. 1815. Imported by De Boffe. Price 14s.

have read over this thick volume with some attention, it that we have not before frequently seen in other authors; though, at the same time, it is but justice to M. DE STAINVILLE to add that we have seldom observed the several subjects on which it treats exhibited in a manner so clear and intelligible. The theory of equations, for example, is nearly the same with that which is given by La Croix, but it is much easier to be followed; M. DE S. entering more into illustration, while he gives a greater developement to his principles.

Wand must say that we have found very little in

In his first theorem, the author proves that, if a rational polynomial of any degree can be rendered equal to zero, by the substitution of a instead of the unknown r, this poly nomial

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nomial will be divisible by x-a.' - This is obvious; for, if we have

Ax + B + Ca2+ &c. + Tx+V=0 and A a" + Ba" + Ca2+ &c. +Ta+V=0

we have by subtraction

I

A(x-a") + B (x-1-a-1) + &c. T(x-a)=o

which is divisible by r-a, each of its terms being so: but, as the quantity which we have subtracted from the first is equal to zero, the remainder is in fact equal to the quantity from which the subtraction was made; and this therefore will have the same divisor with the latter, viz. x-a. This division being performed, the polynomial is reduced one degree lower; where, again, the same operation may be repeated, and the quantity again depressed one degree, and so on till it be reduced to a simple factor: whence it follows that every polynomial of this form may be resolved into m simple factors, and no more than m; and whence also every equation of the mth degree has m roots, and can have but m.

We find nothing new, as we have before observed, in this manner of considering the subject of equations: but, on the whole, we think that it is rendered rather more easy to be comprehended by a student than we have commonly seen it; and the same remark will apply to the greater part of the volume. As to the arrangement which the author has adopted, it will perhaps be better understood by transcribing the heads of his several theorems relating to this theory, viz.-1. If a polynomial of any degree become zero by the substitution of a, instead of r, the polynomial will be divisible by x-a. - 2. Every rational polynomial of any integral degree may be considered as the product of as many factors, of the first degree, as equal the number of units in the highest power of the unknown quantity of which it is a function. After these two theorems, of which we have indicated the mode of demonstration, several problems follow; viz. to change the signs of the roots of equations; to multiply them by any quantity k; to increase or diminish them by any proposed quantity; to find the sum of any similar powers of the roots, &c. &c. 3. In the third theorem, it is demonstrated that, if in any polynomial which contains only whole and positive powers of we substitute for that variable two numbers, and the results of the two substitutions have contrary signs, there will be at least one real root comprized between those limits. - 4. When a rational polynomial has only one variation of sign, it will have one real positive root, and it can have but one. - 5. Every poly

nomial of an odd degree has necessarily one real root of a contrary sign to that of its last term; and every polynomial of an even degree, of which the last term is negative, has at least two real roots, the one positive and the other negative. -6. When a polynomial of an even degree admits of no real root, there is still some quantity, with some angle 4, which makes

r (cos. + sin. -1)

a root of this polynomial.

The succeeding seventy or eighty pages are employed in discussions relative to the solution of cubic equations and the irreducible case; and the next forty pages are allotted to the solution of equations of the fourth degree. It is therefore obvious that the author has not aimed much at condensation; and the demonstrations, as we have before said, are rendered very satisfactory: but it requires some patience to wade through the several pages over which they are spread.

M. DE S. allots the latter part of the volume, arranged exactly in the same style with that which we have examined, to the developement of exponential quantities; as logarithms, sines, cosines, arcs, &c.; the decomposition of the difference of two exponentials into factors; of the theorem of Leibnitz; the theorem of Taylor; and finally, an analytical demonstration of the area of a rectangle.

The above must in course be considered as a very imperfect sketch of the contents of M. DE STAINVILLE'S Miscellanies. They doubtless include a great variety of subjects, but nothing that is sufficiently novel to excite any great degree of interest beyond that which belongs to the principles and theorems themselves; though many of these are highly curious, and are certainly placed by the author in their most obvious light.

ART. IX. Tableau de la Grande Bretagne, &c.; i. e. A Picture of Great Britain, or Observations on England, viewed in London and in the Country, by Major-General Pillet, with a Supplement by M. SARRAZIN, Major-General in the Armies of the King, Commandant of the Legion of Honour, &c. 8vo. pp.333. Paris. 1816. Imported by De Boffe. Price gs. sewed.

THE author of this volume is already known by several works which display military criticism, rapidity of glance, and more freedom from national prejudice than is common among Frenchmen. He was the son of an innkeeper, was educated in Gascony, entered young into military service, and rose by his merits to the rank of a general officer in the French army. He was also distinguished by the friendship of the present ruler

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