Penrose, Francis: An investigation of the principles of Athenian architecture: or the results of a recent survey conducted chiefly with reference to the optical refinements exhibited in the construction of the ancient buildings at Athens (London [u.a.], 1888)

APPENDIX

Many schemes have been proposed for drawing' the Ionic volute
by means of centres of circles ;' the best of these, as it appears
to mc, is that given by Mr Pennethorne in his work on the
Geometry and Optics of the Ancients.2 None of these methods,
however, are quite satisfactory, the multitude of centres are
troublesome, and the curve produced by them is developed in
an intermittent manner, which is not found to be the case in the
best Greek examples.

There is a well-known and beautiful curve, the equiangular
spiral, which admits of being drawn by continued motion ; but it
does not fit the Ionic volute. Nevertheless I believe that the

same end ; but for purposes of calculation the Archimedean Spiral
is the simpler.) The helix should have a groove formed on its
edge, on which a string is wrapped, fastened at the point most
distant from the centre, but having a loose end at the other
extremity with a loop for carrying a pencil or marker. To draw
a capital similar to the Erechtheum or Propylasa, three and three-
quarter complete convolutions of the helix will be required, and
afterwards an addition which will be described. The first two
convolutions will not be wanted except to supply the length of
string necessary to commence the operation.

Let the instrument be now fixed near the point intended for
the centre of E—the eye of the volute—and let two convolutions
of the string be unwound, which will bring the marker to A,

latter was so drawn by the simple expedient of unwrapping a
string from a spiral helix, carved in wood or some other suitable
material, fixed temporarily within the eye of the volute. The
diagram above will explain the process. The small central figure
represents a curve called the Spiral of Archimedes. It was so
named because some of its properties were investigated by that
geometer. It is extremely easy to draw by points measured from
a centre ; for the radii in successive quadrants are to each other
as o, I, 2, 3, 4, 5, etc.; and any intermediate points may be
obtained by subdividing mutually the angles and the increments
of the radii. This spiral ib not in itself fitted for the volute,
because, excepting in the first convolution, it has scarcely any
variety, whereas the volute expands rapidly. The small snail-like
spiral in the middle of the diagram is supposed to be a helix cut
in the form of the Archimedean Spiral. (There is indeed another
spiral, called the involute of the circle, which would answer the

1 Vitr. III. 6, 2 page I33.

continue unwinding until the string arrives at B, about one and
three-quarter convolutions. Thus far, both in the case of the
Erechtheum and Propylasa, the coincidence is such as to leave
nothing to be desired ; but for radii shorter than A and longer than
B, there is a decided discrepance unless the helix itself be varied
as is suggested by the dotted line drawn from H to K. As
respects the curve between B and D, the unadjusted spiral would
have carried the line in the direction BC, and the designer of the
capital no doubt wished for a more expanding figure in that part,
and some modification was also wanted to harmonise the inner
portion of the spiral with the eye of the volute.

To form any of the intermediate fillets or channellings of the
volute, as for instance that shown by the dotted line FG, it is only
necessary to turn the instrument partly round upon its centre as a
pivot, and refix it.

One volute of the capital having been thus drawn, to obtain
the other nothing more was necessary than to reverse the spiral on
its pivot placed symmetrically on the other side of the cap.