Chapters on the modern geometry of the point, line, and circle Volume 1; being the substance of lectures delivered in the University of Dublin to the candidates for honors of the first year in arts - 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. 1863 Excerpt: ...the same four lines (108), one is always convex, one always reentrant, and one always intersecting. 110. The sum of the external angles of a polygon of any order, convex, reentrant, or intersecting, regard being had to their signs as well as their magnitudes,. = + 4m right angles, m being some integer of the natural series 0, 1, 2, 3, &c. less than half-the order of the polygon. For, conceiving the polygon described by the motion of a point setting out from any position on one of its sides, and traversing its entire perimeter, returning again to the point of starting; the several external angles of the polygon are then evidently the several deviations to the right vr left, in the direction of its motion, made by the describing point in passing during the circuit from the several sides to their successors, which for convex polygons universally (fig. a), and for others too occasionally (figs. /8' and 7'), take place all in the same direction, and have therefore all the same sign, but which for reentrant and intersecting polygons generally (figs. #, 7, and a') take place some in one and others in the opposite direction, and have therefore some one and others the opposite sign; but since, on the completion of the entire circuit, the original direction of the motion is always finally regained, therefore the total amount of deviation however made up, that is the sum with their proper signs of the external angles of the polygon, = 0, or = 4 right angles, or = im right angles, m however being always less than half the order of the polygon, the deviation at each angle being necessarily limited to two right angles. In the three polygons represented in figs, a, 8, and 7, and in all convex polygons universally, m = 1; in the three represented in figs, a.', 8', and ...
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