6 6 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 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 VOL. XXX.-No. 67. 51 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 ef fort 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. 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 first moment somewhat obstinate, turned out, on the whole, to be as feeble and as badly directed as there was every reason to suppose that it would be, from the well-known decrepitude. of that ruined and tottering empire. The terms of the peace, were dictated, as we have just remarked, in a spirit of moderation towards Turkey, and good faith towards the other powers; but are yet decidedly favorable to the future advancement of Russia ;—a great deal more so, probably, than if they had evinced, and for the moment gratified, the most inordinate and grasping ambition. The general result places in strong relief the relative strength of the two belligerents, and completely settles the question, already free from doubt in the minds of most judicious men, of the military preponderance of Russia in the southeast of Europe, and indirectly, by a necessary consequence, over the whole continent. This result, while it involves consequences of the deepest interest to the western nations of the old world, is by no means a matter of indifference to us. The state of the international relations among the great powers of Europe, constitutes regularly one of the principal elements to be taken into view in making up an opinion on our own foreign policy; and it is of high moment that our statesmen, and the public at large, should possess correct information respecting the changes that successively occur in the nature of these relations. We therefore venture to hope, that we shall not be considered as deviating too far from the line of observation, which seems to be naturally marked out for an American Journal, by offering a few hints on this subject. We shall first briefly notice, chiefly under a political point of view, some of the events of the war, and shall then indulge in a few conjectures on its probable effects upon the political situation of Europe and the world. The general object of our remarks will be to develope and substantiate the suggestions contained in the preceding paragraph. We must here premise, that, in our opinion, as we have already in fact intimated, the result of the war, while it has been highly agreeable to the policy of Russia, is also conformable to the wishes of the friends of civilization and humanity throughout the world. We have seen at times, with surprise, the idea thrown out in some of our most respectable journals, that the sympathies of the people of this country were on the side of the Turks in this struggle, because the Russians had carried the war, so wantonly provoked by the former, back into their territory. With all our respect for the writers alluded to, we cannot but think that they have mistaken the feelings of the people on the point in question. If indeed they merely mean by sympathy the sentiment of commiseration which naturally springs up in the mind, at the view of a fellow-being in a state of suffering, however obviously the result of his own fault or crime, there can of course be no objection to the use of the term in this connexion. But if, as seems more probable, it be intended to convey the impression, that there was anything wrong on the part of Russia, in invading the territory of the Ottoman empire for the purpose of obtaining satisfaction for the manifold injuries and insults which she had received from that power during the series of years that has elapsed since the last treaty, or that there has been any exhibition of an overbearing spirit in her deportment during the course of the war, or of the negotiations that preceded and terminated it, we must, for our part at least, dissent entirely from the opinion. So far indeed are we from taking this view of the subject, that we consider the Russian government as having shown a reluctance to enter on the war, and an anxiety to employ every possible method for escaping from it, which, in a weaker power, would have been looked upon as actually dishonorable, which unquestionably contributed to raise the pretensions and increase the insolence of the Turks, and which nothing but the extreme delicacy of the relations between Russia and the other great powers of Europe would have at all justified. There cannot be a doubt, that the reckless levity with which the Turks broke the treaty of Akerman, before the ink with which they had signed it was fairly dry, and the contemptuous frankness with which they avowed, in their official manifesto, that they signed it merely to gain time and overreach the Russians, were the effect of the long delay and unexampled patience exhibited by the latter during the negotiations, and which the Turks, whose vocabulary contains no expression for such ideas as those of good faith and good policy, could only attribute to bodily fear. The people of the United States know how to commiserate the distress, which the comparatively innocent population of Turkey must have suffered in consequence of the invasion of their territory, however rightful in itself, and however moderately and judiciously conducted; as they would commiserate the innocent and suffering |