For this last method, which in practice is by far the best, we are indebted to Binns’ ‘Orthographic Projection.’
_To construct a Semi-Elliptical Arch._--The span A B (Fig. 26) and rise C D being given, divide C A and C B into any number of equal parts. Through the point D, draw E F parallel to A B, and from the points A and B erect the perpendiculars A E and B F. Divide A E and B F similarly to C A and C B. Produce C D and make C G equal C D. From D draw lines to the points 1, 2, 3, &c., in the lines A E and B F; also from G draw lines through the points 1, 2, 3, &c., in the line A B, and produce these lines until they cut those drawn from D to the corresponding numbers in A E and B F. Through the points thus obtained draw the curve of the ellipse.
_To draw the Gothic Equilateral Arch._--From the points A and B (Fig. 27), with radius A B equal to the span, describe the arcs B C and A C. By joining C to A and B we obtain an equilateral triangle from which this arch derives its name.
_To draw the Gothic Lancet Arch._--In this arch, the centres E and D (Fig. 28) from which the arcs are struck, are situate outside of and in a line with the points of springing A and B; thus it is constructed on an acute-angled triangle, as will be seen by joining C to A and B.
_To draw the Gothic Obtuse Arch._--This arch, called sometimes the Drop-Arch, is constructed on an obtuse-angled triangle; the centres E and D (Fig. 29) being situate below and within the points of springing A and B.
_To draw the Gothic Tudor Arch._--On the line of springing A B (Fig. 30), take any two points as F and G, so that A F is equal to G B. Draw F E and G D cutting each other on the bisecting line through C; from F and G, with radius F A or G B, describe the short arcs, and from E and D, with radius E C or D C, describe the arcs meeting in C.
_To draw the Moorish Horse-Shoe Arch._--The centres E and D (Fig. 31) from which the arcs forming this arch are struck, are situate above and within the points of springing A and B. One of the most graceful forms of this arch is obtained when the height of the points E and D above the line of springing and their distance from the bisecting line through C are equal to one-third of the span A B.