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

56

THE APPARENT PROPORTIONS.

whole apparent height, = H, will always be found to have been made commensurable with the
apparent length and breadth of the solid exactly containing within it the whole design.

Fig.l. This solid may be con-
ceived as bounded by planes
containing the Middle Step and
the fascia above the Corona, then
the whole will be exactly em-
braced within these given planes.

Let 8 = the height of the
Middle Step above the horizon-
tal plane XY,then by calculation
we must determine the apparent
length and breadth of the
Middle Step at the given height

z above the horizontal plane XY, measured by the visual angles MOM' + MOM" = apparent

length and breadth of the given solid.

To determine the angle MOM',
Fig. 2—

On the plane XZ let mm1 = z,

pmni of/right

Om = 'r Om'2 + mm'2, let Om"-
Om, then the angles M'm"0 and
Mm"0 are both right angles, and
the angle MOM' - m"OM' —
m"OM. By Form 1, Log. tan.
A = log. r + log. a. — log. b, the
angles m"OM' and m;/OM are

determined, and consequently also the angle MOM'.

The angle on the return side, MOM" is similarly determined.

MOM' + MOM" = M, a constant quantity.

The apparent length and breadth of the
solid containing within it the whole design,
measured by the visual angles at the height z
above the horizontal plane XY.

Let the apparent given height hOk = H.

As H : M :: 1 : x, this ratio is generally incommensurable ] A reference to the calculated

let A : B, be the nearest commensurable ratio . ,designs of the Parthenon will r en-

As B : A :: M : H = true apparent height......J der clear all that is here stated.

These simple corrections, 1st, in the whole height (H), and 2nd, in the height of the