[Fig. 36] To draw Goldman’s Volute, the cathetus C F being given.
Divide C F into 15 equal parts. With C as centre describe a circle A E B to form the eye of the volute, making the diameter 3⅓ of these parts. Bisect A C and C B in 1 and 4. On 1 4 draw a square, 1, 2, 3, 4. Produce the sides 1 2, 2 3, and 3 4 to G, H, and I respectively.
Divide 1 C into three equal parts. Draw lines parallel to 1 G through the points of division to P and L, which cut the line C 2 in the points 6 and 10. Through these points (6 and 10) draw lines to M and Q parallel to E H, cutting C 3 in 7 and 11. In the same way draw lines parallel to 3 I from 7 and 11 to N and R. The points 1, 2, 3, 4, 5, &c., will then form the centres of the series of quadrants which are to form the outer spiral that begins with the radius 1 F. To describe the inner spiral. A´ F´ in [Fig. 36] (a) is equal to A F ([Fig. 36]). F´ S´ is made equal to the breadth of the fillet at the top F S. V´ F´ is drawn at right angles to F´ A´ and equal to C 1. By joining V´ A´ and drawing T´ S´ parallel to V´ F´, then T´ S´ is obtained which will be the length of half the side of the square for drawing the inner spiral. The method for obtaining the inner spiral is the same as for the outer.
[Fig. 37] There is no geometric means of drawing a perfect catenary curve; at best we can only obtain it by an approximation in geometry. The curve is formed by suspending a chain from two points and pricking points along the curve of the chain. These
Fig. 36.—Goldman’s volute.
points will mark the path of the catenary. In the accompanying figure three catenary curves are drawn from a chain suspended from points A and B.
[Fig. 38]—To draw a cycloid curve when the generating circle is given. In order to find the length of the line A B on which the circle rolls, and which must be the length of the circumference of the given circle, we must first find approximately that length by