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For this purpose, we only require the admission of the following proposition. The circle may be considered as a regular polygon of an infinite number of sides. This may be made intelligible to the mind of any youth, in the following manner. Take a circle and inscribe in it any one of the regular polygons before mentioned, (art. 269, 271, 273). Then proceed to bisect the arcs subtended by the sides, and draw chords to these half arcs. This operation may obviously be continued until the arcs become confounded with their chords; a limit, which on account of the imperfection of our vision and of instruments, will soon be attained. But it is evident that the operation may be extended in imagination beyond this limit, to an indefinite degree. Indeed problems 293, 294, furnish us with a method of approximating the circle to a regular polygon, until they shall not differ by any assignable quantity. It is not therefore absolutely necessary to introduce the idea of infinity; for if we suppose the operation to terminate at some distant finite limit, still the error will be so small that it may be safely neglected, as it is in the quadrature of the circle. We have, however, preferred the use of the term infinite, both because it is more convenient than any other, and because it is the only one which renders the proposition strictly true. We are aware that the above considerations do not amount to a positive demonstration, for this is not practicable. Mathematical infinity, as here presented, is a negative idea, and the only proofs, which propositions involving it admit of, are of a negative or indirect kind. We shall give an indirect demonstration of the rigorous truth of the one in question, by showing in the sequel, that all the conclusions which result from reasoning upon it as a hypothesis, are precisely the same as those which Legendre has obtained with vastly more labor, by a course of strictly geometrical demonstration. Meantime what we have said above, will put the student's mind into a train of reasoning, such as cannot fail to remove whatever doubt the first statement of the proposition might have occasioned. Perhaps, too, it would not be amiss to place before him that negative sort of evidence, which results from the fact, that, whether true or not, its falsity can never be demonstrated; since every attempt to prove a want of coïncidence between the circle and polygon, must suppose the number of sides finite, which would be contrary to the hypothesis.

After all, we expect that the lovers of strict geometry will

object to the introduction here, of a principle which belongs more properly to fluxions. We have already stated our object to be the abridgment of labor; and when we shall have shown that the recognition of this single principle will do away the necessity of many of the longest demonstrations in the work before us, we trust that no one who is not a bigot to particular forms and systems, will find fault with the step we have taken. Let it be remembered, moreover, that the first suggestion of such a principle was made by a man no less profound and sagacious than Kepler, who employed it in a manner very closely resembling the one we propose; that it has already been partially recognised by Legendre himself, in his approximation to the quadrature of the circle; that it forms the entire basis of Trigonometry, as presented in the volume which follows Geometry in the Cambridge course; and finally, that the application of the Differential and Integral Calculus, to geometric magnitudes, through its whole extent, is a recognition at once of its truth and importance. Who then will say that we can avail ourselves too early of that instrument, by which Kepler, Newton, Leibnitz, and Laplace, have so wonderfully facilitated their vast calculations?

We accordingly now proceed, without further apology, to point out the extent to which the limits of the work before us may be abridged, by admitting the principle, that the circle is a regular polygon of an infinite number of sides.

Theorem 287. The circumferences of circles are as their radii, and their surfaces are as the squares of their radii. Admitting our principle, this proposition is included in that of art. 282. It is there demonstrated, that the perimeters of regular polygons of the same number of sides, are as the radii of their inscribed circles; and their surfaces are as the squares of these same radii.' Now the circles in question being regular polygons of an infinite, and therefore the same number of sides, and the radii of the inscribed circles being the radii of the circles in question, as both become coïncident, we have only to substitute for perimeters, the word circumferences, and for the radii of their inscribed circles, the words their radii, and the same theorem is sufficient for both cases. Q. E. D.

Theorem 289. The area of a circle is equal to the product of its circumference by half the radius. It was demonstrated, 280, that the area of a regular polygon is equal to the duct of its perimeter by half the radius of the inscribed circle.'

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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.

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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

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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.

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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.

It is now time to draw our remarks to a close. The alterations 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 re

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