Elements of Geometry, Plane and Spherical Trigonometry and Conic Sections (Paperback)

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1854 Excerpt: ...a side of the lower base, and s a side of the upper base, and a the slant hight; then the surface of one face is measured by ia(S+s). There are just as many of these surfaces as the frustum has sides. Let m represent the number of sides; then the whole surface must be fyt(mS--ms). But (mS--ms), is the perimeter of the two bases; and-Ja is one-half of the slant hight. Therefore, &c. Q. E. D. Scholium. Let circles be described round the bases of the frustum, as represented in the last figure; and conceive the number of sides to be indefinitely increased; then S and will be indefinitely small, and m indefinitely great; but however small S and s may be (the corresponding number to m being as much increased), the expression (mS--ms) will still represent the perimeters of the two bases. But, when S and s are indefinitely small, while OA, and DH, that is, the distances from the axis of the frustum from its edges being constant, the perimeter, mS, will become the perimeter of the circle of which OA is the radius; and ms will be the perimeter of the circle of which DH is the radius; that is, mS=2rt(A 0), and ms=2x(DH); and by addition, mS+ms=2x(A O+DH) But, in this case, a becomes AD, one-half the edge of the frustum; and the frustum of the pyramid becomes the frustum of a eone, and its surface is measured by iADX2(AO+J)E); hence, The convex surface of a frustum of a cone, is equal to half its sides, multipled by the sum of the circumferences of its two bases. The above expression is the same as JT% /AO+DE ADX2-( 1 J If we take the middle point, P, between O and B, and draw PM parallel to OA and ED, gives.... ADXiPM That is, the convex surface of the frustum of a cone, is equal to its side, multiplied by the circumference of a circle which is exactly midway...