First Case. - In Fig. 722, let A B C represent the outer curve of an arch in a circular wall corresponding to A' H C of plan, and let F B D represent the inner opening in the wall, as shown by E' F' D' in plan. Then AEBCD will represent the soffit of the arch in elevation and A' H C D' F' E' the same in plan. In the engraving the outer curve of the arch is a perfect semicircle, and the inner curve is stilted, as shown, so as to make the soffit level at B. Instead of the stilted arch, the inner curve may, if desired, be drawn as a semi ellipse of which E D is the minor axis and F B one-half of the major axis.

Divide A B of elevation into any convenient number of equal parts, shown by the small figures. With the T-square parallel with the center line B B', drop lines from the points in A B. cutting A' H of plan, as shown. Since that portion of the inner arch from E to 12 is drawn vertical, as above explained, divide 12 B into the same number of parts as was A B, and, with the T-square parallel with the center line B B'. drop lines to E' F', as shown. Connect opposite points in A' H with those in E' F'. as shown by the solid lines in plan. Also divide the four-sided figures thus produced by means of the diagonal dotted lines 6 8, 5 9, etc., as shown. The several triangles thus produced will represent in plan the triangles into which the soffit, or under side, of the arch is divided for the purpose of obtaining its pattern. In order to ascertain the real distances across the surface of the arch which the solid and dotted lines represent, it will be necessary to construct a series of sections of which these lines are the bases, as shown in Figs. 723 and 724.

In constructing the diagram shown Fig. 723, the several solid lines of the plan, though not exactly equal in length (because they are not drawn radially from the center of the curve A' H C'), may be considered as of the same length. Draw the right angle P Q R as in Fig. 723, and from Q set off horizontally the distance H F' of plan, as shown by Q E. Draw R S parallel with Q P, and, measuring from Q, set off on Q P the length of lines dropped from points in A

Fig. 722. - Plan and Elevation of Arch in a Circular Wall. - First Case.

B to A F, as shown by corresponding figures 2 to 6. Likewise set off from R on R S the length of lines dropped from points in E B to E F, as shown by the figures 12 to 7, and connect the points in P Q with those in S R, as indicated by the solid lines in plan. Thus connect 1 with 12, 2 with 11, 3 with 10, etc. To construct the diagram based upon the dotted lines of the plan, draw the right angle MNO in Fig. 724, and, measuring in each instance from N, set off on N M the same distances as in Q P of Fig. 723. Starting from N, set off on N O the lengths of dotted lines in plan, as shown by the small figures in N O. With the T-square parallel with M N, draw lines from the points in N O, and, in each instance measuring from N O, make these lines of the same length as lines of similar number dropped from points in E B of elevation to E F. Connect the points in these lines with points in M N, as indicated in plan by the dotted lines. Thus connect 6 with 8, 5 with 9, 4 with 10, etc.

Fig. 723. - Diagram of Sections on Solid Lines of Plan, Fig. 722.

Fig. 724. - Diagram of Sections on Dotted Lines of Plan, Fig. 722.

The next step is to obtain the distances between points in A B of elevation as if measured on the outer opening in the curved wall. To do this, on F A extended set off a stretchout of A' H of plan, as shown by the small figures 5', 4', etc., and with the T-square at right angles to the stretchout line J F, draw the usual measuring lines. With the T-square parallel with J F, carry lines from the points in A B to lines of similar number drawn from the stretchout line. A line traced through these points, as shown by J B, will give the true distances desired between the points in the outer curve of the arch.

The distances between points in E B, the inner curve, are obtained in a similar manner. To avoid a confusion of lines, the stretchout of F' E' of plan is set off on F C of elevation, as shown by the small figures, 7', 8', 9', etc. B D is also divided into the same parts as was E B, and from the points thus obtained lines are drawn to the right parallel with F C. With the T-square parallel with B F, carry lines from the stretchout points in F K, cutting lines of similar number drawn from the points in B D. A line traced through the points thus obtained, as shown by B K, will give the distance between points as if measured on the inner curved line of the wall.

Fig. 725. - One-Half Pattern of Soffit of Arch Shown in Fig. 722.

From the several sections now obtained the pattern may be developed in the following manner: At any convenient place draw the line a e in Fig. 725, making it in length equal to Q R of Fig. 723, or A' E' of plan. At right angles to a e draw e 12, in length equal to R 12 of Fig. 723, and connect a with 12 if it is desired to show the triangle. From a as center, and J 2 of elevation as radius, describe a small arc, 2, which intersect with one struck from point 12 of pattern as center, and 12 2 of Fig. 724 as radius, thus establishing the point 2 of pattern. With 2 of pattern as center, and 2 11 of Fig. 723 as radius, describe another arc, 11, which intersect with one struck from 12 of pattern as center, and 12 11 of B K as radius, thus establishing the point 11 of pattern. Continue in this way, using the tops of the sections in Figs. 728 and 724 for measurements across the pattern, the spaces in J B for the distances along the edge a h of pattern, and the spaces in B K for the distances along the inner edge e f, establishing the several points, as shown. Through the points in a h and e f lines are to be traced, while f h is to be connected by a straight line, thus completing one-half the pattern. The other half of pattern can be obtained by the same method or by any convenient means of duplication.