But the regular polygon of an infinite number of sides becomes confounded with its inscribed circle. Therefore it must have for its area, its perimeter or circumference multiplied by half the radius. Q. E. D.

Theorem 312. The circle is greater than any polygon of the same perimeter. We would vary this phraseology conformably with our principle, by saying, 'the circle is greater than any other polygon of equal perimeter.' Then it becomes a corollary to proposition 311, of two regular isoperimetrical polygons, that is the greater which has the greater number of sides. For the number of sides in the circle being infinite, must be greater than that of any other polygon. Q. E. D.

We now come to Part Second. This is divided into four sections. The first three treat of planes, polyedrones, and figures described on the surface of a sphere. In no part of these, therefore, could our mode of reasoning be substituted. But by applying it to the Fourth Section, which treats of the sphere, the cylinder, and the cone, the limits of that section will be reduced more than one half. We proceed to point out, as briefly as possible, in what manner this application is to be made.

In the first place, Lemmas, 513, 514, 515, may be omitted, being only subsidiary to theorem 516. The solidity of a cylinder is equal to the product of the base by its altitude. This proposition is involved in that of the solidity of the prism, 406. It is there shown that the solidity of a prism is equal to the product of its base by its altitude. But the cylinder is a prism of an infinite number of faces. This may be illustrated by the same kind of reasoning as that employed with regard to the circle. Indeed it is a necessary consequence of admitting that the circle which forms the base of the cylinder, is a polygon. Hence a second demonstration is unnecessary. The solidity of a cylinder must have the same measure as that of a prism. Q. E. D.

Again, lemma 522 may be omitted, being only introduced for the sake of demonstrating theorem 523. The convex surface of a cylinder is equal to the circumference of its base multiplied by its altitude. Now it has already been demonstrated, lemma 520, that the convex surface of a right prism, is equal to the perimeter of its base multiplied by its altitude.' Admitting, then, our principle, the convex surface of a cylinder will consist of an infinite number of rectangles, each having for its alti

tude, the altitude of the cylinder, and the sum of their bases, forming the circumference of the base of the cylinder. In other words, the cylinder is a right prism, of an infinite number of faces. Hence the convex surface of a cylinder must be the same as the convex surface of a prism. Q. E. D.

Theorem 524. The solidity of a cone is equal to the product of its base by a third part of its altitude.' It was before demonstrated, 416, that every pyramid has for its measure, a third part of the product of its base by its altitude." But according to our principle, a cone is a pyramid, having an infinite number of triangular faces. This necessarily follows from its base being a polygon of an infinite number of sides. Hence the measure of a cone must be the same as that of a pyramid. Q. E. D.

Theorem 527, enunciated in general terms, is as follows. 'The frustum of a cone is equal in solidity to the sum of three cones, having for their common altitude, the altitude of the frustum, and for their respective bases, the inferior base of the frustum, its superior base, and a mean proportional between these two.' In art. 422, it was demonstrated that the frustum of a pyramid has the same measure for its solidity. Admitting our principle, a second demonstration becomes entirely unnecessary; for the frustum of a cone becomes the frustum of a pyramid of an infinite number of faces. Therefore it must have the same measure of solidity. Q. E. D.

Theorem 528. The convex surface of a cone is equal to the circumference of its base multiplied by half its side." This results necessarily from the definition of a cone just given. All the triangles forming its surface, have the side of the cone for their common altitude, and the sum of their bases forms the circumference of the base. Hence the sum of all these triangles, or the convex surface of the cone, is equal to the circumference of the base into half the side. Q. E. D.

Theorem 530. The convex surface of the frustum of a cone is equal to its side multiplied by half the sum of the circumferences of the two bases.' The two bases being parallel polygons of an infinite number of sides, it follows that the convex surface is composed of an infinite number of trapezoids, each having the side of the frustum for its altitude. Therefore the sum of all the trapezoids, or the convex surface, must be equal to the side of the frustum, that is, the altitude of the trapezoids, multiplied by half the sum of all the parallel sides, or half the sum of the two circumferences. Q. E. D.

Theorem 535. The surface of a sphere is equal to the product of its diameter by the circumference of a great circle.' This is incuded in the preceding corollary. The entire surface described by the revolution of a semipolygon about its axis, is equal to the product of the axis into the circumference of the inscribed circle.' Here we have only to substitute for semipolygon the word semicircle, and for axis the word diameter, and to remember that the great circle and the inscribed_circle are one and the same, in order to perceive that the surface of the sphere must have the measure enunciated. Q. E. D.

Theorem 538. • The surface of any spherical zone is equal to the altitude of this zone multiplied by the circumference of a great circle.' This is involved in lemma, 533, as may be shown by considerations precisely like the preceding, and therefore unnecessary to be mentioned.

Theorem 546. Every spherical sector has for its measure the zone which serves as a base multiplied by a third of the radius, and the entire sphere has for its measure its surface multiplied by a third of the radius. This long demonstration may be avoided, and the truth inferred directly from theorem 545, by substituting, as our principle justifies, the words spherical sector, in place of the words polygonal sector. The same demonstration will then suffice for both cases.

The altera

It is now time to draw our remarks to a close. tions which we have suggested, will reduce the limits of the work about one fifth. Had we time we might mention several other propositions which might be omitted altogether, on the ground of standing isolated or leading to no practical results. Whether the increased popularity and practical utility, which might thus be insured, be an object worthy of consideration to the publishers, when another edition shall be called for, as we understand will soon be the case, it does not concern us to inquire. Of this, however, we are assured, that the wants of the public do really require a work on geometry less amplified than Legendre, and at the same time rendered more practical; and we know of no treatise which would so well serve for the basis of such a work, as that which we have attempted to review.

XI.-1. De l'Opposition dans le Gouvernement, et de la Liberté de la Presse, par M. LE VICOMTE DE BONALD, Pair de France. Paris, 1827.

2. Debates in the British Parliament on the Change of Ministry, and on the Battle of Navarino. London, 1828. 3. Manifesto of the Sublime Ottoman Porte of December 20, 1827.

ALTHOUGH the period which has elapsed since the close of the last general war, is commonly spoken of as a season of tranquillity, and may be justly viewed as such when compared with the five and twenty years immediately preceding, it has nevertheless been filled up by an almost uninterrupted series of important events. The foundation of a numerous brotherhood of new nations in Spanish and Portuguese America, the establishment of representative governments in various parts of the continent of Europe, the four military revolutions in the Italian and Spanish Peninsulas, and finally the desperate and glorious struggle for national existence in Greece, are occurrences hardly inferior, in permanent interest and actual importance, to those which constituted the successive stages of the French Revolution. Of these great political movements some have already nearly reached, in one way or another, their natural termination, and have ceased in consequence to attract, so strongly as before, the attention of the world. Notwithstanding,the fatal dissensions that have lately distracted the councils of some of our sister Republics of the South, and the clouds that overshadow to a certain extent their immediate future prospects, we cannot permit ourselves to doubt for a moment, that the independence and freedom of the whole of Spanish America are substantially secure; and we are firmly persuaded that this grand consummation will open a new and most auspicious chapter in the history of human affairs. On the other hand, the temporary triumph of liberal principles in several quarters of the south of Europe, which was attended from the beginning with many ominous and unsatisfactory circumstances, was soon succeeded by such decisive reverses, as to leave for the moment, in that quarter, no farther ground of hope. But while the general interest in these two courses of events has been thus diminished by success on the one hand, and by failure on the

other, the establishment and progress of representative government on the Continent, and the war of Independence in Greece, continue to offer from year to year new incidents of constantly augmenting importance. The last few months in particular have been marked in both by occurrences of signal moment. The sea-fight of Navarino, if its influence on the fortunes of the Turkish Empire should at all correspond with present probabilities, can hardly fail to form an epoch in the history of the Christian world in general, as well as in that of the Greek Revolution; while the late dissolution of the French House of Deputies, followed by the unexpected triumph of liberal principles at the elections, and a consequent change of ministry, is perhaps the most curious incident that has yet occurred in the operation of the new representative constitutions of the continent. Either of these events, considered in all its details and consequences, would furnish ample materials for a long article, and might be separately treated without inconvenience. But at this distance from the scene of action, the public curiosity does not require, perhaps, so copious a supply of particulars, or so minute an investigation of their character and results, as might be suitable elsewhere; and when political events of this magnitude, though not in themselves directly connected, proceed from and exercise their influence upon a cluster of neighboring powers, so closely united by relations of every kind, as those which make up the European commonwealth, it is in some respects convenient and advantageous to review them together. We accordingly propose in this article to survey, in the very rapid and imperfect manner, which only our limited resources and space will allow, the present aspect of the general politics of Europe, and shall digest our remarks under the two leading heads which we have already specified. The fluctuations and changes which have successively occurred in the British ministry, since the retirement of Lord Liverpool, although highly important, as well from the immense interests immediately affected by every movement in the main spring of so vast a machine, as from the high celebrity of many of the names that appear in these transactions, are nevertheless not likely, we think, to produce any very material effects, either on the internal condition or foreign relations of the country. It would however be improper to leave them entirely out of view on the present occasion, and before we proceed to examine the other two subjects, alluded to above, we shall briefly con