'submultiple of such an angle can be taken; and we have no way of expounding, even in the simplest cases, the ratio 'which one of them bears to another.'

The invention of a theory of solid angles, then, which shall bring them fairly within the limits of geometrical magni. tudes, by shewing that, in all cases, they admit of accurate comparison, is a discovery of no small moment. Had it been effected by a native either of France or Scotland, we doubt not that his merits as an inventor would have been extolled through Europe:-for the philosophers of those two nations, whatever may be their faults in other respects, have certainly the praise (and no small one we esteem it) of permitting the love of country so far to overcome individual feeling, and private rivalship, as generally to stimulate them to a warm and active commendation of the ingenuity and talents of men of their own nation. Dr. Gregory's theory of solid angles, which appears to us at once simple, universal, and unobjectionable, may be best stated in his own words.

A solid angle is defined by Euclid, that which is made by the meeting of more than two plane angles, which are not in the same plane, in one point.

Others define it the angular space comprized between several planes meeting in one point,

It may be defined still more generally, the angular space included between several plane surfaces or one or more curved surfaces, meeting in the point which forms the summit of the angle.

According to this definition, solid angles bear just the same relation to the surfaces which comprize them, as plane angles, do to the lines by which they are included: so that, as in the latter, it is not the magnitude of the lines, but their mutual inclination, which determines the angle ; just so, in the former it is not the magnitude of the planes, but their mu tual inclinations which determine the angles. And hence all those geometers, from the time of Euclid down to the present period, who have confined their attention principally to the magnitude of the plane angles, instead of their relative positions, have never been able to develope the properties of this class of geometrical quantities; but have affirmed that no solid angle can be said to be the half or the double of another, and have spoken of the bisection and trisection of solid angles, even in the simplest cases, as impossible problems.

But all this supposed difficulty vanishes, and the doctrine of solid angles becomes simple, satisfactory, and universal in its application, by as suming spherical surfaces for their measure; just as circular arcs are assumed for the measures of plane angles.* Imagine, that from the sum

* It may be proper to anticipate here the only objection which can be made to this assumption; which is founded on the principle, that quan tities should always be measured by quantities of the same kind. But this, often and positively as it is affirmed, is by no means necessary; nor it many cases is it possible. To measure is to compare mathematically: and if by comparing two quantities, whose ratio we know or can ascertain, with two other quantities whose ratio we wish to know, the point

mit of a solid angle (formed by the meeting of three planes) as a centre, any sphere be described, and that those planes are produced till they cut the surface of the sphere; then will the surface of the spherical triangle, included between those planes, be a proper measure of the solid angle made by the planes at their common point of meeting: for no change can be conceived in the relative position of those planes, that is, in the magnitude of the solid angle, without a corresponding and proportional mutation in the surface of the spherical triangle. If, in like manner, the three or more surfaces, which by their meeting constitute another solid angle, be produced till they cut the surface of the same or an equal sphere, whose centre coincides with the summit of the angle; the surface of the spheric triangle or polygon, included between the planes which determine the angle, will be a correct measure of that angle. And the ratio which subsists between the areas of the spheric triangles, polygons, or other surfaces thus formed, will be accurately the ratio which subsists between the solid angles, constituted by the meeting of the several planes or surfaces, at the centre of the sphere.

Hence, the comparison of solid angles becomes a matter of great ease and simplicity; for, since the areas of spherical triangles are measured by the excess of the sums of their angles each above two right angles (th. 5); and the areas of spherical polygons of n sides, by the excess of the sum of their angles above 2n-4 right angles (th. 6.); it follows, that the magnitude of a trilateral solid angle, will be measured by the excess of the sum of the three angles, made respectively by its bounding planes, above 2 right angles; and the magnitudes of solid angles formed by n bounding planes, by the excess of the sum of the angles of inclination of the several planes above 2n-4 right angles.

As to solid angles limited by curve surfaces, such as the angles at the vertices of cones; they will manifestly be measured by the spheric surfaces cut off by the prolongation of their bounding surfaces, in the same manner as angles determined by planes are measured by the triangles or polygons, they mark out upon the same, or an equal sphere. In all cases, the maximum limit of solid angles, will be the plane towards which the various planes determining such angles approach, as they diverge further from each other about the same summit: just as a right line is the maximum limit of plane angles, being formed by the two bounding lines when they make an angle of 180°. The maximum limit of solid angles is measured by the surface of a hemisphere, in like manner as the maximum limit of plane angles is measured by the arc of a semicircle. The solid right angle (either angle, for example, of a cube) is (2) of the maximum solid angle: while the plane right angle is half the maximum plane angle. pp. 86-88.

in question becomes determined: it signifies not at all whether the magnitudes which constitute one ratio are like or unlike the magnitudes which constitute the other ratio. It is thus that mathematicians, with perfect safety and correctness, make use of space as a measure of velocity, mass as a measure of inertia, mass and velocity conjointly as a measure of force, space as a measure of time, weight as a measure of density, expansion as a measure of heat, a certain function of planetary velocity as a measure of distance from the central body, arcs of the same circle as measures of plane angles; and it is in conformity with this general procedure that we adopt surfaces, of the same sphere, as measures of solid angles.'

Dr. Gregory, having thus traced the analogy between plane and solid angles, proceeds to exemplify his theory by a few examples relative to the solid angles of prisms of different bases, of pyramids, cones, and polyedræ; and deduces some very curious results, for which, however, we must refer to the book itself. Future geometers, we doubt not, will pursue the rich vein of inquiry which is opened in the pages before us, with delight; and while they cannot fail to admire the elegance of the theory proposed by Dr. Gregory, will at the same time be surprised, that a notion so simple and obvious, should never have been educed before.

The chapter on Geodesic Operations, and the Figure of the Earth is, in our estimation, a very valuable one, and contains the only concise and connected view of the nature of trigonometrical surveying in the English language. But we were surprised, that neither the author of it nor his senior colleague, should know who was the real inventor of the rules, in this department of science, erroneously ascribed in this, as in most other works, to General Roy ;-though he was no more the author of those rules than General Mack. General Roy, while possessing a respectable share of scientific information, and a truly philosophical turn of mind, was, nevertheless, but a moderate mathematician. But he had the good fortune to find united to him, as coadjutor in the Trigonometrical Survey, a very excellent mathematicianMr. Isaac Dalby, now Senior Professor of Mathematics in the Royal Military College at High Wycombe. Mr. Dalby was the investigator, or inventor, of most of the theorems and rules given in General Roy's papers in the Philosophical Transactions, and, among them, of the ingenious rule relative to the spherical excess, inserted at p. 135 of the volume before us. This restoration of honour to the person to whom it is really due, we ought to have made in our notice of Mr. Woodhouse's Trigonometry: but we were not then aware that the error was so prevalent as we have since found it to be.

The chapter on Polygonometry exhibits, in small compass, the developement of the principles of that science, and some of the most useful theorems formerly discovered by Lexel and Lhuillier. We could with pleasure quote from it the passage in which the analogy between the three cases of plane trigonometry, and those of polygonometry, is traced: but our account is already swelling under our hands, while the most valuable parts of the volume remain unnoticed.

The problems respecting the Division of Surfaces are all (except the 4th case of Prob. I., which is susceptible of amendment) very neatly solved: and they present a useful

variety to the notice of the student. The chapter on the Nature and Solution of Equations, in general is very perspicuous, but rather too concise. It is, however, far inferior in value to the chapter on Curves, and the Construction of Equations to which it is introductory. The enumeration of Newton's lines of the third order, will be read with great interest and advantage by young mathematicians; and the rule for finding the roots of quadratics is very elegant: we are not prepared to say whether it be new. The chapter on Fluents is principally, and avowedly, borrowed from Mr. Dealtry's very useful chapter on the same subject: and that on the motion of machines and their maximum effects, is taken, with a few alterations and additions, from Gregory's Mechanics.

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Chapter 12, on the Pressure of earth and fluids against walls and fortifications, the theory of Magazines, &c. is neat, but too concise. It is especially defective, in that it offers no solution to the problem respecting the force of running water against banks, &c. But it is followed by a chapter of great originality and importance, on the theory and practice of Gunnery. This is a subject to which Dr. Hutton bas, for many years, paid more attention than any other philosopher in Europe. His public situation gave him admirable opportunities of making experiments; and his own industry and genius qualified him, in a remarkable manner, to conduct such experiments with skill and success. In the present chapter, the Doctor gives, first, a summary of results deduced from his numerous experiments,-as from the series made in 1775, described in a paper honoured with the Royal Society's gold medal in 1778,-from the experiments of 1783, 4, 5, and 6, detailed in the first volume of Dr. Hutton's Tracts,-and from a subsequent course of experiments, the full account of which will, it seems, appear in a collection of Tracts, now printing, by this indefatigable author; from the latter of these sources, he has drawn three tables of resistances for the different sizes of balls and the velocities between 100 feet and 2000 feet per second He then proceeds to the solution of the following problems. 1. To determine the resistance of the medium against a ball of any other size, moving with any of the velocities specified in the three tables. 2. To assign a rule for determining the resistance due to any indeterminate ble velocity of a given ball. 3. To determine the height to which a ball will rise, when fired from a cannon perpendicularly upwards with a given velocity, in a non resisting medium, or supposing no resistance in the air. 4. To determine the height to which a ball projected upwards, as in the last pro

blem, will ascend, being resisted by the atmosphere. 5. To determine the height ascended by a ball projected as in the two foregoing problems; supposing the resistance of the air to be as the square of the velocity. 6. To determine the time of the ball's ascending to the height determined in the last problem, by the same projectile velocity as there given. 7. To determine the same as in prob. 5, taking into the account the decrease of density in the air as the ball ascends in the atmosphere, 8. To determine the time of a ball's ascending, considering the decreasing density of the air as in the last problem. 9. To determine the circumstances of space, time, and velocity of a ball descending through the atmosphere by its own weight. 10. To determine the circumstances of the motion of a ball projected horizontally in the air, abstracted from its vertical descent by gravitation. 11. To determine the ranges of projectiles in the air.

These problems are solved with great ingenuity and comparative simplicity, by blending the results of experiments with physical considerations, and the fluxional analysis, independent of all regard to the real form of the trajectory described by the ball, when the primitive direction is oblique. The solutions furnish some curious specimens in the finding of fluents; and give Dr. Hutton opportunity of shewing his skill in unravelling some intricacies in that branch of mathematics. Towards the conclusion of this chapter, he presents the reader with some very plausible conjectures, relative to the mode by which the French were enabled to project shells so far as three miles, at the late siege of Cadiz.

The concluding chapter of Promiscuous Exercises, contains a most rich and useful variety. They relate to the motion of projected balls with small velocities; the exhaustion of fluids from vessels; the effects of pile engines; the strength and pressure of beams, of different shapes and in various positions; the motion and force of water wheels; the nature and theory of equilibrated arches; the relative strengths of different natural substances; the penetration of balls into blocks of timber, or banks of earth; the vibration of chords; the emptying or filling of fortification-ditches; the fall of water through the arches of bridges; and other subjects connected with the profession of the engineer. Four or five of them are taken from Dr. Hutton's Select Exercises, now out of print; but the remnderare collected from all quarters, and generally appear with much improved solutions. Many of those in which the centre of gravity is introduced, are remarkably simple and elegant. We feel strongly tempted to quote largely from this chapter: but as we are persuaded the volume