and in this his anticipations and hopes have been realized. That his naine might be perpetually connected with the memory of his discoveries, he directed that a sphere, inscribed in a cylinder, should be engraved on his tomb, thus making his most brilliant intellectual exploit his only, and, we may add, his most glorious epitaph. It is impossible to contemplate the character of this great man, without feeling that he experienced the same poetical and lofty aspirations after fame, which have always been, we believe, the accompaniments of greatness. The prophetic foresight of Horace, the passionate visions of Cicero, the glowing but solemn confidence of Bacon, justify the spirit of a French philosopher's remark, that, in loftiness of intellectual character, Homer and Archimedes stand upon the same level. But we must break away from these reflections, and enter upon the examination of the work, whose title we have placed at the head of the present article.

Mr Walker is well known as a successful teacher of mathematics, in the celebrated school of Messrs Cogswell and Bancroft, in Northampton. Experience is the only safe test of the merit of an elementary work, in any department of knowledge, designed for the instruction of beginners; and the book before us contains the elements of geometry, moulded to that form, which some years of practical acquaintance with the art of teaching, suggested as the best. The best modern treatise on geometry, compiled from ancient and modern authors, and uniting the excellences of all to an extraordinary degree, is undoubtedly that of Legendre. But on many accounts this is unfit for the use of schools. As an analysis of the science we hold it above all praise. But Legendre, though he departed, in some respects, from the rigid methods of the ancients, did not depart enough from them to avoid a degree of prolixity, which renders his treatise too cumbrous and expensive for a manual in common schools. Without in the least disparaging the merit of that eminent and judicious mathematician, we may assert, and we believe our assertion will be borne out by universal experience, that his work has not supplied the want of an elementary treatise of geometry, for the ordinary and general purposes of instruction; which want, Mr Walker has attempted to supply.

Legendre's work is divided into eight sections, four of which treat of plane geometry, and four of solid geometry. This

division is well enough, but appears a little arbitrary. A somewhat different, and, as we think, a more perspicuous arrangement for beginners, has been adopted by Mr Walker. On this point, let the author speak for himself.

The division of the work into three sections is founded in the nature of the subject. Extension, or the space which matter occupies, has three dimensions, length, breadth, and thickness. These may be considered separately or in connexion. When we consider length alone, its representative is a line. Hence the first section treats of lines and their relations. When we consider length and breadth together, or length in two ways, their representative is a surface. Hence the second section treats of surfaces. Lastly, when we consider length, breadth, and thickness together, or length in three ways, their representative is a solid. Hence the third section treats of solids.'

This arrangement is clear, and the reasons for it are strong. In the first edition, a desire to render every thing perfectly intelligible led the author to omit the use of technical terms as far as possible. We have no partiality ourselves for scientific treatises, overloaded with these ornamental appendages; yet it is very obvious, that as long as facts exist, those facts must have a name; as long as propositions of different forms are to be treated of, it will be very convenient, at least, to have distinguishing terms, which, when their meaning is once settled accurately by definitions, may ever after be employed, in a manner analogous to algebraic signs, instead of the definitions; and if these terms are etymologically significant of their scientific import, so much the better. This defect of the first edition has been corrected in the second, which is, in several other points, a decided improvement upon its predecessor.

It is of great importance, in an elementary work, to bring the subject treated of within as narrow limits as accuracy and perspicuity will admit. This condition has been fulfilled by Mr Walker. As to the scientific strictness of some of his means, we will not now decide, but reserve our remarks for particular instances. We have said that his arrangement differs from M. Legendre's. It differs in several particulars besides what we have already mentioned. For instance, the properties of the circle, of the triangle, of the polygon, &c. are treated of in connexion. The definitions, instead of being given in a body, occur as the nature of the subject demands them. The definitions themselves are given, in some in

6

stances, in a different form, from the introduction of another element, motion, which we have never before seen thus applied. Legendre defines a line thus. A line is length without breadth.' Introducing motion, the definition becomes, A line is the path described by the motion of a point'; and the definition of a straight line, the shortest way from one point to another,' becomes an axiom to the definition, A straight line is the path described by a point moving only in one direction.' This is a simple example, but illustrates the thought. An important use is made of motion, in explaining the meaning of the term angle. After defining it, and illustrating the definition on the plate, we have the following clear summary; the angle may be considered as denoting the quantity, by which a straight line, turning about one of its points, has departed from coincidence with another straight line,'-a perfectly intelligible account of a matter, which, as it is ordinarily explained, is a puzzling mystery to school-boys. Among the original and ingenious demonstrations which we have noticed in this volume, we would instance, particularly, those of Theorems 31 and 32, on perpendicular and oblique lines; and 34, that when two parallels are crossed by a straight line, the alternate internal angles are equal to each other, and the internal-external angles are equal to each other.' The principal demonstration in article 70, in regard to the proportion of lines, is partly original, and partly from Bézout. The approximation to the quadrature of the circle, in article 113, is simple, elegant, and entirely original.

A curved line is defined as 'the path described by a point which changes its direction at intervals so small that they cannot be perceived;' and by corollary, a curved line may be considered as made up of infinitely small straight lines.' This, taken in connexion with Theorem 94, The circle is a regular polygon of an infinite number of sides,' leads to important and curious results. The cylinder becomes a prism of an infinite number of faces, the cone becomes a pyramid of an infinite number of faces, and the sphere becomes a polyedron of an infinite number of faces. By admitting thus much, we have the Fourth Section of Legendre's Second Part reduced something more than one half, and the whole treatise of the 'Elements,' nearly one fourth. We are aware, that the strictness of ancient geometry would reject an aid like this; but for the purposes of practical instruction, we see not the slightest reason for a pertinacious adherence to the rigor of Euclid. Modern

geometers have universally found themselves compelled to depart, more or less, from this ideal severity of demonstration; and we see no objection against wider departures still, if the science may be explained by this means more briefly, and with equal or greater clearness. In such circumstances, nothing short of a blind and bigoted adherence to ancient methods, utterly at war with the spirit of improvement, can persist in following the beaten track.

We have thus cursorily examined Mr Walker's book. His plan is simple and natural; his explanations are clear; his original demonstrations are ingenious; and his illustrations easy and familiar. He has condensed into 102 duodecimo pages more geometrical truth than we had supposed it possible to bring within so narrow limits, and all that is essential to be taught in ordinary mathematical instruction. We recommend this treatise as well adapted to the purpose for which it was designed, and calculated to supply a desideratum in our schools. In parting, we have only one word more to say, which is, that the study of geometry, in our opinion, should precede that of algebra. This latter science is more abstract in its symbols, and requires a greater effort of purely intellectual labor to comprehend it. But geometry starts from notions as simple as the first ideas of arithmetic, and proceeds, step by step, clearly, irresistibly, by a process that cannot, with an ordinary effort of attention, be mistaken, to the most important and striking truths. The imagination is aided by the use of diagrams, and thus a remarkable and happy union of abstract reasoning and sensible perception renders this science an admirable exercise for the yet unfolding intellect. Take that mystery in arithmetic, the doctrine of the square and square root; trace it to algebra, and a faint glimmering of light dawns upon the hitherto impenetrable darkness that enveloped it; but when the pupil advances to geometry, all difficulty vanishes, and the mystery is made as clear as day. And so of others. In geometry there is no such darkness. Let its principles and practice be first understood, therefore, and they will serve as a light to guide the inquirer in the symbolical regions of numbers.

A. H. Everett,

ART. VII.-1. Du Système Permanent de l'Europe à l'égard de la Russie, et des Affaires de l'Orient, par M. DE PRADT, ancien Archevêque de Malines. Paris. 1828.

2. Statistique des Libertés de l'Europe en 1829, par le

Même. Paris. 1829.

In a former article, which appeared in our number for July, 1828, we ventured to offer a few hasty and imperfect suggestions on the political situation of Europe, at the commencement of the late war between Russia and Turkey. We then intimated, that, although the result of the struggle was in a great measure uncertain, the not unfounded jealousy, entertained by Great Britain and the other western powers, of the constantly progressive influence of Russia, would combine with the moderation, for which we were disposed to give credit to the latter government, to limit as much as possible the duration and geographical theatre of the war, and might be expected to bring it pretty early to a close, which would be conformable, in its results, to the policy of Russia, and the wishes of the friends of civilization and humanity throughout the world. These anticipations have been, in the main, confirmed by the progress of events. Although the first campaign in Europe was hardly distinguished by so brilliant a course of triumphs on the part of Russia, as the rivals and the well-wishers of that power had alike foretold; yet, taking the two campaigns in Europe together and including the two in Asia, the exhibition of military power has, upon the whole, quite equalled the most exalted expectations, that either fear or hope could have suggested beforehand. On the other side, the influence of the policy of the western nations, especially Great Britain, in restraining the advances of Russia, and limiting the duration and theatre of the contest, has been distinctly visible at every step; while the facility and good grace with which the Emperor accommodated his proceedings to the successive and not always perfectly reasonable or consistent demands of his anxious allies, and the moderate conditions on which he has granted another term of national existence to an enemy completely at his mercy, evince a spirit of generosity, good faith, and, we may add, good policy, as commendable as it is uncommon in the councils of governments, especially of the form and character of that of Russia. The resistance of the Turks, although at the