This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1901 edition. Excerpt: ... no danger of crushing from the horizontal thrust around "the second crown from the top, as stone will bear on a square unit a pillar of a square unit section and several thousand units high. Similarly for resultants on all the joints. We conclude that with the backing used and by increasing the depth of arch ring at the springing about onethird, the arch will be stable. FORMULAS FOB VOLUME. 75. When the soffit and exterior surface of the dome are both surfaces of spheres having the same centre, the volumes of the voussoirs are easily obtained. Thus, by geometry, the area of the zone formed by revolving an arc as rs (Fig. 22) about the axis ac is equal to the altitude h of rs multiplied by 2w, r being the radius as of the sphere. Pass now two meridian planes through ac, whose included angle is--of a circumference. The part of the zone included between them has an area so that the pyra n mid formed on this base with a vertex at a, 27tr'h na$ a volume----. 3n Similarly the pyramid having the part and h' = altitude of arc tv. Therefore the volume of the voussoir rstv included between the meridian planes and the conical joints rv and st is, where r and h are the radius and altitude of the exterior arc, r' and h' of the interior. If the altitudes of the type h are made equal in successive arcs, the values h' will all be equal. This can be shown as follows: Draw horizontal lines through r and v, Fig. 22, and drop verticals h and h' from s and t to them, and call the chords rs and vt, c and c', . from similar triangles, .-. When h is constant for successive arcs of the extrados, h' is similarly constant for the corresponding arcs of the intrados. Similarly the centre line of the arch ring will be divided into arcs of the same altitude when...