New Elementary Geometry; Ith Practical Applications; A Shorter Course Upon the Basis of the Larger Work (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. 1873 Excerpt: ...arc D E XiR. 244. Cor. 2. Let the circumference of the circle whose diameter is unity be denoted by-(which is called pi), the radius by R, and the diameter by D; and the circumference of any other circle by C, and its area by A. Then, since circumferences are to each other as their diameters (Theo. XXX. Cor. 1), we shall have, C: D:: n: l; therefore C=D Xn = 2i?Xi. Multiplying both numbers of this equation by R, we have Cxr = R2xt, or A = R X; that is, the area of a circle is equal to the product of the square of its radius by the constant number n. 245. Cor. 3. The circumference of every circle is equal to the product of its diameter, or twice its radius, by the constant number w. 246. Cor. 4. The constant number n denotes the ratio of C the circumference of any circle to its diameter; for--= jr. 247. Scholium. The exact numerical value of the ratio denoted by Jt can be only approximately expressed. The approximate value found by Proposition XXIX. is 3.1415926; but, for most practical purposes, it is sufficiently accurate to take rr = 3.1416. The symbol n is the first letter of the Greek.word ept'nerpov, perimetron, which signifies circumference. PROBLEMS IN CONSTRUCTION. 248. The following Problems depend, upon the application of principles established in what has preceded, for solution: --Problem I. To construct a rectangle that shall be equivalent to a given triangle. Let A B C be the given triangle. Draw the indefinite straight line C D parallel to the base A B; bisect A B by the perpendicular E F, and make E G equal to F B. Then, by drawing G B, the rectangle E F B G is equivalent to the triangle ABC. For the rectangle E F B G has the same altitude, E F, as the triangle ABC, and half its base (Theo. II. Cor. 1, Bk. IV.). Problem II. To divide a given st