article Carpentry is the most extended, and we believe we may say the best executed of any in the volume: that part of it, in particular, which relates to the strength and stress of materials, is given at considerable length, and includes an account of most of the experiments that have been instituted in order to reduce the subject to mathematical and uniform principles. These experiments have been made principally by Muschenbroek, Emerson, Parent, Gauthey, Buffon, Du Hamel, &c., many of them with different views, and therefore not comparable with each other; some, however, are of the same kind: but the results of them unfortunately have not that coincidence which is necessary for establishing the theory on any permanent mathematical basis. The experiments of Du Hamel are very curious, and in direct contradiction to the hypothesis on which mathematicians have commonly conducted their investigations; though it does not appear that any very marked difference prevails in the results.

'Du Hamel took 16 bars of willow, 2 feet long, and an inch square, and after supporting them by props under the ends, he sub. jected them to the operation of weights suspended at the middle. Four of them were broken by weights of 40, 41, 47 and 52 pounds; the mean of which is 45 lbs. (464.) He then cut through one-third of four of them, on the upper side, and filled up each cut, with a thin piece of harder wood stuck in tolerably tight. These several pieces were then broken by weights of 48, 54, 50 and 52 pounds; the mean of which is 51 lbs. Four others were then cut through one half, and broken by 47, 49, 50 and 56 lbs; the mean of which is 48 lbs. (52.) The other four were cut through two-thirds, and their mean strength was 42 lbs.

At another time Du Hamel took six battens of willow 36 inches long, and 1 square; after suitable experiments, he found that they were broken by 525 pounds at a medium.

Six bars were next cut through one-third, and each cut was filled with a wedge of hard wood stuck in with a little force, these were broken by 551 pounds on the average.

Six other bars were broken by 542 lbs on the medium, when cut half through, and the cuts were filled up in a similar manner. 'Six other bars were cut three-fourths through, and broken by the pressure of 530 pounds on a medium.

A batten was cut three-fourths through, and loaded until nearly broken, it was then unloaded, and a thicker wedge was introduced tightly into the cut, so as to straighten the batten, by filling up the space left by the compression of the wood, when the batten was broken by 577 pounds.

From these experiments we may perceive that more than twothirds of the thickness, we may, perhaps, with safety say nearly three-fourths contributed nothing to the strength.'

These experiments are very remarkable, and if pursued they might probably lead to some decisive conclusions respecting the effect of the cohesion of fibres differently situated with regard to the fulcrum about which the fracture is formed. At all events, they prove very satisfactorily that the hypothesis, which makes each fibre to act with a force proportional to its distance from that fulcrum, is erroneous; although it appears that the conclusions deduced from it, viz. that the strength is as the area of the section into the depth of the centre of gravity, is nearly correct: but, if so, the author is wrong in what he says relative to a triangular prism having one end fixed in a wall, which he states to be three times stronger with its base downwards than when its vertex is so placed. We suspect, however, that this is merely a slip of the pen, and that it should have been only twice; and we suspect also a similar error at page 176., in the author's illustration of the diminution of strength by boring a solid cylinder in the direction of its axis. Altogether, this article, notwithstanding it displays little taste in the arrangement, and exibits some errors both of the pen and the press, may be advantageously consulted for many practical operations and results.

The next article to which we shall refer is Engineering, which the compiler observes has been almost entirely overlooked in our Encyclopedias, and which he has therefore endeavoured to supply:-but, if we mistake not, much of what is here given may be traced to articles bearing other names in those works to which he alludes, such as Canal, Lock, Dock, Roads, &c.; and therefore the merit of collecting these under one general head, Engineering, is not so great as Mr. Martin seems to imagine. Still, if this article has little to claim with regard to originality, it certainly gives a pleasant history of the progress of those great works which are and will continue to be the glory of the present age. In speaking of the construction and formation of Docks, the writer says:

• Docks, from at first being only a simple contrivance at arsenals for the purpose of building or repairing a single ship, have extended themselves to a magnitude in capacity competent to contain whole fleets. The splendour of the doeks created in London, and at many of the outports, are a monument which excel the famous port of Pireus in Greece, or Alexandria, in Egypt, as much or more than we have excelled the Greeks and Romans in all the facilities to navigation, and the grandeur of our naval architecture. The Greeks and Romans no doubt have far surpassed us in all the elegancies of taste and invention in the fine arts: in these arts

they

they have combined and given form to matter, which could have resulted only from a higher degree of feeling, united to juster notions of nature, than the coldness of our climate and habits can perceive, or hardly give power to copy. But if we are behind in the fine arts, which, as mere copyists, we must be contented to be: in supplying all manner of facilities to commerce, (in which we excel all nations, ancient and modern,) in erecting the immense docks and warehouses inland, which we have done to receive and house safely the produce of the world, and to an extent adequate for that purpose: we have formed a monument at once of our genius, wealth, and skill, which will be as famous in the page of science as the monuments of Athens and Rome are now in the volume of the arts.'

This is one of the best written passages in the article, the author being by no means happy in the combination of words. Thus, in the beginning of the paper, he says, Engineer, civil, in contradistinction to the same profession attendant on military works, is a person of considerable importance in society; his employ embraces pre-eminently canals, and their attendants, reservoirs, locks,' &c. In speaking of the Eddystone lighthouse, he observes: The form of such buildings has involved considerable intricacy of mathematical investigation; Lagrange has calculated that a cylinder is the strongest form in resisting flexure, which is contrary to the known fact, and could only be deduced from the intricacy of the investigation.' In another place, he remarks: Hence cast-iron rail-roads became a second desideratum to canals, excepting only that the invention is due to Englishmen.' In the two former cases, we can guess at the author's meaning, but the latter sentence is to us totally inexplicable.

6

Masonry, Mining, and Watch and Clock-making, are rather long articles, and tolerably executed; at least, as far as practical information is concerned.

We shall now bestow a few observations on the author's Appendix on Practical Geometry, with which he has closed his volume, and we must conclude this article. Here, again, we meet with many of those unfortunate explanations from which it is impossible that a student or a novice can gain any correct information; and, which is still worse, they are sometimes calculated to make an erroneous impression. Thus: "A rhombus has all its sides equal:' ergo, a square is a rhombus. A rhomboid has its opposite sides equal:'-ergo, a rectangle is a rhomboid.- Again; a rhomboid is an oblique prism, whose bases are parallelograms :'-therefore, a rhomboid is both a surface and a solid. A tangent is a straight line drawn so as just to touch against a circle.'

6

In defining the conic sections, an hyperbola is said to be formed by a section parallel to the axis; from which an uninstructed reader would infer that this figure could be made by no other position of the cutting plane. The definitions of the expressions rectangle under two lines, to inscribe, to circumscribe, &c., are of the same kind: but we have quoted enough to shew that science is not the author's forte; nor does he, indeed, assume any scientific pretensions.

The title-page of this volume calls it a second edition, and is dated in 1815; whereas the date of the preface is 1813; while, from references made in the course of the work to publications going forwards at the time of printing the several articles, as also from other circumstances, it is obvious that the volume has not been a second time to the press since 1813; and we do not know that it had previously appeared at a more distant period. We have heard of works that have passed immediately from the first or second edition into the fifth; and of others that have been republished under a new title without even entering the second: but these are paltry deceptions on inexperienced readers, disreputable both to authors and publishers.

ART. XII. An Easy Introduction to the Mathematics; in which the Theory and Practice are laid down and familiarly explained. To each Subject are prefixed, a brief popular History of its Rise and Progress, concise Memoirs of noted Mathematical Authors ancient and modern, and some Account of their Works. The whole forming a complete and easy System of Elementary Instruction in the leading Branches of the Mathematics; designed to furnish Students with the Means of acquiring considerable Proficiency, without the Necessity of verbal Assistance. Adapted to the Use of Schools, junior Students at the Universities, and private Learners, especially those who study without a Tutor. By Charles Butler. 2 Vols. 8vo. il. 118. 6d. Boards. Longman and Co.

OUR

UR introductory books on arithmetic, and all the ele mentary branches of mathematics, are perpetually increasing in number, and already we have many more than are really useful. An author, therefore, ought to have some substantial reason to offer in justification of himself when he adds a farther increase to the already over-stocked market of elementary mathematical treatises. Accordingly, this part of the subject is the first which the present author undertakes to discuss in his preface; and, if he has not proved that his work was absolutely necessary, he has at least shewn that it is

not

not devoid of utility, while we must admit that its mode of arrangement possesses a certain degree of novelty, if not of originality. It appears to have been compiled with a considerable share of care and labour, and the first volume with judgment. It is also printed in a form and with a type that comprise a large portion of matter in a small compass; and it is certainly not ill adapted to the principal purpose which the author intended it to answer, viz. to assist the pupil who is studying without the aid of a master. Its contents are

various, and its notes and illustrations very numerous. Scarcely any name of eminence occurs in the text unattended by a short biographical note at the foot of the page, pointing out to the reader the most important particulars relative to the author in question; such as his inventions, discoveries, improvements, writings, &c.; and most of the principal rules, and methods of solution, as they arise, are referred to their original authors with more minuteness than could with propriety be introduced into the historical sketches which precede the several subjects of arithmetic, algebra, logarithms, geometry, trigonometry, and the conic sections.

The contents of the several parts of Vol. i. are thus enumerated by the author:

Part I. begins with an Historical Account of Arithmetic, explaining, to a considerable extent, the nature and construction of numbers, and proceeds by laying down in a plain and simple manner, what are usually called the four fundamental rules: next follow in order, Reduction, the Compound Rules, Proportion Direct, Inverse, and Compound; the Rules of Practice, the theory and practice of Fractional Arithmetic, Vulgar, Decimal, and Duodecimal; Involution, Evolution, and Progression, both Arithmetical and Geometrical; the whole demonstrated, exemplified, and explained; and as simplicity and clearness were always the objects aimed at, it is hoped no obstacle will be found in the learner's way which may not easily be surmounted. Under these heads, which comprise the whole of Elementary Arithmetic, is given a great number of particular rules and observations, not to be found in any other work, but which are necessary, in order fully to explain the theory, and facilitate the practice of numbers. Besides the examples fully wrought out and explained, several others are introduced under each rule, with their answers only, and a few are given without answers. Part II. contains an Historical Account of Logarithms, the theory and practice of Logarith mical Arithmetic, with numerous examples, problems, and explanations. Part III. contains the History of Algebra, and its fundamental rules; Rules for solving Simple and Quadratic Equations, in which one, two, three, or more unknown quantities are included; and, lastly, a collection of Problems, teaching the application of Simple and Quadratic Equations, in a great variety of

ways;