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

46 THE APPABENT PBOPOBTIONS,

the arcs of great circles, of which the eye is the centre, are incommensurable one with another,
the whole design would appear inharmonious, and the ratios between the several apparent
magnitudes would all be irrational quantities; consequently, the perfect idea of proportion
that had existed in the mind of the Architect would be apparently lost in the executed work,
and as a work of Art it would entirely fail.

This primary idea, that the executed design should reflect the original idea of the
Architect, with all the perfection of form and proportion with which it was first conceived in
his mind, gave rise to the several corrections we meet with in ancient Architecture, and led
to the study and cultivation of the several branches of the ancient mathematics.

To produce this effect of an apparent harmony between the several portions of the work
upon the minds of those who viewed the executed design, it was requisite that certain
corrections should be made upon the several visual angles, so that all the apparent magnitudes
should be commensurable one with another; this change in the visual angles produced a
corresponding change in the dimensions of the primary figure, and it is the altering of the
positions of the points and lines of the design as originally conceived, and the determining of
the true dimensions at the angle of the design nearest to the point of sight, which will constitute
the second part of the Theory of Proportion. The subject may be considered under four
distinct heads—

First. A few observations on the ancient method of laying down the design upon three given
horizontal and vertical planes XY, XZ, YZ, passing through the point of sight 0, so
as to fix each point and line of the design correctly in space, and the few simple
trigonometrical forms of calculation that are required for the determination of the
sides and angles of the visual angles.

Second. To determine by calculation the point of sight 0, and the position of the design upon
the horizontal plane XY and vertical planes XZ, YZ.

Third. The law for determining upon the given plane HOZ the true apparent perpendicular
magnitudes measured by the visual angles, and the true heights of the executed
design at the angle nearest to the given point of sight 0.

Fourth. The application of the theory explained in Part II. to the designing of the Parthenon,
the Erechtheium, the Propylaea, and other Athenian Works, showing the perfect
mathematical agreement between the dimensions obtained from observation, and
those arrived at by the trigonometrical calculation.

FIRST.

OBSERVATIONS ON THE GEOMETRY OF THREE DIMENSIONS, AND ON
THE TRIGONOMETRICAL CALCULATIONS APPLIED TO ARCHITECTURE.

In the most simple questions that occur in the Geometry, it is seldom that we can by any
direct means measure any line or surface; if we imagine ourselves at a distance from the body