The Ratio Between Diameter and Circumference in a Circle Demonstrated by Angles, and Euclid's Theorem, Proposition 32, Book 1, Proved to Be Fallacious (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. 1870 Excerpt: ...will be equal to four right angles. In either case the angles at the centre of the circle will be equal to the arcs by which they are subtended. Hence: Euclid's 19th definition is positively fallacious. J. Radford Young, a living "recognised Mathematician" saw this, for, in his "Elements of Euclid" he observes: --" The term polygon, however, is often employed as a general name for rectilineal figures of all kinds, without regard to the number of sides; so that the rectilineal figures defined above (definitions XVII. and XVIII.), may, without impropriety, be called polygons of three and of four sides respectively." This brings to light some remarkable facts, with reference to Mathematics as applied to Geometry. If a regular polygon of any number of sides greater than 4 be inscribed in a circle, the ratio between the sides of the polygon and their subtending arcs is constant, and-expresses this ratio, whatever be the value of It; but this ratio does not hold good for a polygon of 3 or of 4 sides inscribed in a circle. But, if the circumference of a circle be divided into any number of equal arcs, and from one of these arcs T'sth part be deducted, and the remainder be multiplied by the sum of the arcs, the product is constant, and equal to the perimeter of a regular hexagon or six-sided polygon inscribed in the circle: and this is true, whether the inscribed polygon to the circle, be a polygon of 3 sides or of 4 sides, or of any other number of sides. Any one conversant with the simple rules of arithmetic "can follow out and test this." Geometry is an exact science, and Geometers by excluding arithmetical considerations from their study of Geometry, are led into all sorts of blunders; and I am glad to find, that "rec...