Let a d b be the opening, and c D its spring or rise. In the middle of a b, at D, erect a perpendicular D e, equal to twice C D, its rise; and from E draw E A and E b, and divide a e and b e into any number or equal parts, as a, b, c, and 1, 2, 3. Join b a, 3 c, 2 b, and 1 a, and it will form the arch required.

Fig. 39.

The more parts a e and b e are divided into, the greater will be the accuracy of the curve.

Many curves may be made in the same manner, according to the position of the lines a e and e b; and if instead of two lines drawn from a and b, meeting in e, a perpendicular be erected at the same points, and two lines be then drawn from the ends of these perpendiculars meeting in an angle, and these lines be divided into any number of equal parts, the points of the adjacent lines may be joined, and a curve will be formed resembling a Gothic arch. The demonstration already given is therefore very useful to the workman, as he may vary the form of the curve by altering the position of the lines, either with respect to the angles which they make with each other, or their proportional lengths.