The Geometrical Companion; In Which the Elements of Abstract Geometry Are Familiarised, Illustrated, and Rendered Practically Useful (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. 1828 Excerpt: ... circle, cutting Dce, as at F, and join Cf; (or, which is the same thing, take the chord Cf equal to that diameter). Finally, join the given point with the point G, where Cf is intersected by the given circle. Then, Ag will be the tangent required. For: as Cg is half of Cf, the radius being half the diameter, therefore the chord Cf is divided equally at G by a right line from the centre, A. Hence, by Art. 52, Ag is perpendicular to Cg, and by Df.t. XX. is a tangent to the given circle at a. In this method the practical student will observe that the problem is solved without the help of any intermediate problem, There are other oval arches, more properly called elliptical; but a description of these does not fall within the limits of the present work. In Architecture, the doctrine of Tangents is of almost incessant use and application. The Greeks and Romans scarcely used any other form of arch, or any other curve, than the circular, in their buildings; and however beautifully varied their architectural ornaments may appear, the outlines of these are nothing more than simple com-' binations of circular arches and their tangents, or of right lines merely. which is not the case with either Euclid's method or the preceding one; Prod. VII. being employed in the former, and Prob. V. in the latter. The construction of a basket-handle arch mentioned in the above Article will serve to evince the practical utility of this problem. An Architect proposes to himself the following object: On a given base to construct a basket-handle arch of a given height. How is he to accomplish this undertaking? Let Ab represent the given base, and St the given height. Divide Ab equally at c, by Prob. V., and draw Cl perpendicular to Ab, by Prob. VII., and take Cl equal to St. Through L...