ISBN: 1146543131 / ISBN-13: 9781146543132
by John Perry
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from ... Show 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. 1893 Excerpt: ... least two points. H a line passes through three points or more we must consider its segments as distinct sides. Thus, Oabciii (Fig. 88) is a figure which satisfies the conditions; it consists of seven points, joined by twelve lines, and it contains seven closed polygons or triangles. Also A B c D o o' (Fig. 89) is a figure consisting of six points joined by twelve lines; to satisfy condition (2) given above, Chap. XIV. FIGURES FORMED OF STRAIGHT LINES. 147 we must not consider A B C D as one of the polygons, we only take into account the eight triangles. It will be found that in all figures satisfying the above conditions the number of closed polygons plus the number of points equals the number of sides plus two. This is proved by taking any suitable figure and adding a new side; it is found that the sum of the number of closed polygons and the number of points is also increased by unity. This is, indeed, the relation between the number of faces, summits, and edges of a polyhedron, and all the figures of which we speak may be regarded as projections of polyhedra. 145a. Straight-line figures generally may be divided into: --1. Deformable figures, or those which may alter in shape, the lines retaining their original lengths. 2. Figures perfectly stiff. 3. Figures which would be perfectly stiff, even if we removed one or more lines. It may be shown that a figure belongs to class 1, 2, or 3, according as the number of its sides is less than, equal to, or greater than, double the number of points minus three. It is evident that in a figure of the third class the lengths of the extra lines may be expressed mathematically in terms of the other sides. Thus, if there are a points, b sides, and e polygons in any figure, we generally find that there exist b-2 ...