Elements of geometry and the first principles of modern geometry

by William H.H. Phillips

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. 1878 ...to all the vertices, the polygon will be divided into n equal triangles, and the area of each will be (III., 7, Cor.). Cor. 1. Applying this formula to the inscribed square, and taking the perimeter instead of the side, we have, W2--S V4r'--S2 4 Let n = 4096, and 2r = a" = 1; then I. and II. give P = 3.141592, and Psm, = 8.141593; but the circumference, C, is greater than the perimeter of the inscribed polygon, and less than that of the circumscribed hence C = 3.141592 is correct to within one-millionth of the diameter d, which was taken as the unit. Cob. 2. From I. it is evident that if the radius, or diameter, were increased, the value of P would be increased in the same ratio, and consequently the circumference hence, for any circle whose diameter is d, we have, C = 3.14159 Xd = 7td, where n denotes not the approximate value 3.14159, but the exact ratio of the diameter to the circumference. Cor. 3. It was shown (11) that the area of a regular Pa polygon =---; but the circle is a regular polygon whose Scholium. It may be shown by higher analysis that n is irrational; and hence the circumference and area of a circle cannot be expressed hx numbers in terms of the radius There is no known method of constructing a square exactly equal to a circle, or a line equal to a circumference; nor has the impossibility of this problem been demonstrated. See (14.) xnr. Problem. The area of a regular inscribed polygon being given, and that of a similar circumscribed polygon, to find the areas of the regular inscribed and circumscribed polygons of doicble the number of sides. Let p denote the area of the regular inscribed polygon; P, that of the similar circumscribed polygon; j, and P15 the areas of the regular inscribed and circumscribed polygons of doubl...

50 pages, Paperback

Published January 1, 2012