equal to unity on the same scale; then AK the normal at A is parallel to Sr. Produce Sz to meet the curve at P, and the tangent at P is parallel to DC. Also produce SD to u, and let Su be 5:67 on any scale, draw uv perpendicular thereto and equal to unity on the same scale; then

the tangent at A is parallel to Sv; it is also at right-angles to the normal A K. Svu represents the angle which the tangent and radius-vector make with each other at every point on this spiral.

As another example, let A B (fig. 158) be given as 30

on any scale, AS equal to

20, BS to 10, the ratio of AS

to BS being that of 2 to 1; then (190) we have

which is found by the tables to be the tangent of 77.55°, and this is the angle between the tangent and radius-vector at every point on the spiral. Take Sy equal to 4:53 on any scale, draw x y z perpendicular thereto, and let x y or yz be unity on the same scale; then the normal A K at A is parallel to Sa; and if the line Sz is produced to meet the curve at P, that will be the point where the tangent is parallel to the axis DC. If we take Su equal to Sy, or to 4.53 on any scale, and draw u v perpendicular and equal to y z or to unity on that scale; then the line Sv is parallel to the tangent at A. The angle Svu or Sxy represents the angle (77.55°) which the tangent and radius-vector make with each other at all points of the curve. The length of SC is a mean-proportional between 20 and 10, or is the square-root of their product, 200; this gives SC= 14.14; Sb is the square-root of 20 × 14·14, or 16-8; Sa is the square-root of 20 × 16.8, or 18.34. The value of Sa can also be found directly from the formula previously given (189);

which is found in the tables to be the logarithm of 18.34.

When SA and Sa (fig. 158) are determined, the other points b, c, &c., can be found by drawing OA and OA' (fig. 159) making an acute-angle, OA being taken equal

M

to SA (fig. 158), and O A' equal to S A' and less than SA.

Take Oa equal to Sa, draw A A', A'a; then proceed as before to draw successive parallels to those two lines, and Ob, O c, &c., will be equal to Sb, Sc, &c. (fig. 158). Also O a', Ob', &c., will be equal to Sa', Sb', &c., on the dotted spiral.

If it is desired to draw the arcs Aa, ab, &c., as arcs of circles, the normals must be drawn at those points, making an angle of 77.55° with the radii SA, S a, Sb, &c., and then the point where the normals at A and a intersect will be the centre for the arc A a; the point where the normals at a and b meet will be the

centre for the arc ab; and so on.

192. To apply the equiangular-spiral to the formation of volutes and other architectural ornaments.-In drawing volutes the depth A B (fig. 154, 156, 158) and the ratio of AS to BS are usually given. The volute can therefore be drawn in the manner above described (189).

The point A will not, however, be the highest point of the spiral which rises until it reaches P found in the manner before described (190), and at this point the tangent is parallel to the axis DC. Also if PS is produced to meet the curve again at p. the tangent at p will be also parallel to DC, and the point p will be the lowest on the spiral.

The spiral given in figure 154 is a very near approximation to the usual form of Ionic volute, having SA to SB as 10 to 7. If another spiral A' a' b', as shown by the dotted

line, is required, as is the case when there is a fillet or moulding on the face, we must proceed as in figure 155, taking OA equal to S A, O A' to SA'; Oa, Ob, &c., equal to Sa, Sb, &c., and drawing the parallels we obtain Oa', O b', &c., equal to the required lengths of S a', Sb', &c., and the spiral can be drawn in as before. In the same way any number of lines consisting of similar spirals can be obtained.

In volutes it is usual to let the innermost convolution of the spiral die away into a circular eye (fig. 160). To find

the centre of this circle draw EP, LP, the normals at two consecutive points, meeting in P; then P is the centre, and PE the radius of the required circle which has a common tangent with the spiral at E, and PE is very nearly equal to the radius of curvature at E.

An application of this spiral to form the contour of a classical ornament is shown in figure 161, in which two equal spirals are turned reversed ways against each other, S being the focus, CSX the horizontal axis, A S B perpendicu

lar thereto. In this case the ratio of SA to SB is that of 4 to 1, and the value of a, or the tangent of the angle between the radius-vector and tangent, is 2.2. Taking Sy as unity on any scale, yz perpendicular thereto, and equal to 2.2 on the same scale, Sz will be parallel to AT the tangent at A, and also to the tangent at B; z Sy will represent the angle Fig. 161.

between the tangent and radius-vector drawn from S at every point on the curve.

Make the angle BSP equal to the angle yz S, and p will be the lowest point on the curve, the tangent at p being parallel to CSX.

The equiangular-spiral being capable of any amount of variation, is the most valuable of all the known spirals for decorative purposes.