Modern geometry; a new elementary course of plane geometry. With an appendix by W.Brydone Jack - Softcover
Synopsis
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. 1882 Excerpt: ...are called regular polygons. The sides of a figure may be equal and the angles unequal, but such a figure will not be regular. Take a folding pocket-measure, formed of a number of equal rules jointed together, and join the two ends; a polygon will be formed, the sides of which are equal (Fig. n8), but it will not be regular unless it be also equi-angular. If the two conditions be fulfilled, the figure will be a regular polygon (Fig. 119). Relation to the Circle.--The two most prominent features of a regular polygon are, first, its roundness, and, second, its symmetry about certain lines through a point within it which may be regarded as the centre. A regular polygon and a circle are thus intimately related. If a circle be made to pass through three angular points of a regular polygon, it will pass through all; if a circle be made to touch three of the sides, it will touch all. If a circumference be divided into equal parts, and the points of division joined, or if tangents be drawn through them, a regular polygon will be formed. Symmetrical Properties of Polygons.--The following are the principal symmetrical properties of regular polygons:--1. The diagonals which join the opposite vertices in a regular polygon of an even number of sides are diameters of the circumscribing circle. These vertices (Fig. 120) contain between them the same number of sides; moreover the sides, and consequently the subtended arcs, are all equal; hence the sum of these arcs in one direction is equal to the sum of the arcs in the other direction: and therefore the vertices bisect the circumference, and are at the opposite extremities of a diameter. The point of intersection of two diagonal diameters is the centre of a regular polygon of an even number of sides. 2. The straight line-...
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