Pennethorne, John; Robinson, John [Ill.]: The geometry and optics of ancient architecture: illustrated by examples from Thebes, Athens, and Rome (London [u.a.], 1878)

140

THE COLUMNS,

Then construct the square, EFGH, and bisect each side of the square in the points
1, 2, 3, 4, and from each of these points of bisection as centres, and with radii equal to la,
2b, 3c, Ad, describe the four arcs of the circles combining to form one revolution of the spiral
line, abode, within the given parallelogram ABDC.

Repeating a similar operation upon the parallelogram, A'B'D'O, we describe a second
revolution of the No. 1 spiral line, and the third revolution is also contained within the
parallelogram, Al'BVC", and is similarly described.

By the same method, the spiral lines Nos. 2 and 3 are traced, as likewise the remaining
spiral lines which form the finished design of the volute. (See Plates VII. and VIII.)

The elliptical curved lines connecting the volutes together, namely, the lines hbh, etc.,
Fig. 2, might be composed of the arcs of circles, combined with either the arcs of hyperbolas
or of parabolas ; but in the examples of the Athenian Ionic capitals we find from observation
that these curved lines are composed of the arcs of circles, combined with the arcs of ellipses.

Fig. 2.—The Elliptical Curved Lines connecting the Volutes.

In the two examples of the Ionic volutes, in the Vestibule of the Propylsea, Plate VII.,
Fig. 3, and in the North Portico of the Erechtheium, Plate VIII., Fig. 3, these elliptical curved
lines appear to be laid down in the following manner—

From the centre point, No. 1, the arc of the circle, hh1, is described, and from some
point, No. V, in the same vertical line the arc of the circle, hh", is traced; then bisect HC in
P, and draw the vertical lines PW and PV, and let Bb be the minor axis of the ellipse. Bisect