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

THE APPARENT PROPORTIONS. 47

it becomes impossible. This bas given rise to tbe series of geometrical considerations, which,
in tbe modern System of tbe Matbematics are embodied in tbe general forms of tbe Analytical
Geometry. Now, although tbe Greeks did not possess any analytical forms, we must not conceive
tbat they were Hmited to the mere measuring by scale of their lines and angles. The evidence
is in favour of their having been able to calculate with great precision (and we find in the
ancient Astronomy many elaborate calculations are preserved), and their works of Architecture,
when examined with the proper degree of knowledge, also prove beyond a doubt that it was
requisite for them in their designs to fix points, lines, and surfaces in space by the projection of
visual rays (imaginary fines) often three and four hundred feet in length; yet with all our modern
improvements in the Trigonometry and in the Analytical Geometry, we can discover no error in
their calculations. They arrived, by their ancient methods of calculating, at exactly the same
results as we now do with the aid of our modern Logarithmic Tables, improved Formulae, and
simple Arithmetic. The calculations to them were more laborious, but the results obtained,
either by the ancient or by the modern methods, are exactly the same.

If any evidence were required, beyond the existing works in the Acropolis, to prove the
close connection that subsisted between the ancient Architecture and the ancient Geometry,
we have, mentioned by Vitruvius, and copied most likely from some ancient Greek writer, an
account of those subjects in which an Architect was to be instructed; he says, " an Architect
' should be versed in Geometry, Optics, Arithmetic, Astronomy, Music, a skilful Draughtsman,
' &c." That tbe several branches of the Geometry were early cultivated by the Greeks we have
convincing evidence, both in their Astronomy and in their Architecture; but, although we find
their calculations made with remarkable accuracy, and with reference only to their Architecture
with all the precision that a modern geometrician would require in making similar calculations,
yet we shall rest satisfied by arriving at the same results without following the intricate processes
to which the Greeks were forced to have recourse. I shall, therefore, in all calculations use
the modern Arithmetic in preference to the ancient, tbe modern trigonometrical forms and
tables of sines and tangents instead of the ancient Trigonometry with its tables of chords
and arcs, and similarly with the Geometry. A previous acquaintance with the first elements
of the plane Trigonometry and of the descriptive Geometry will be supposed; but for the
sake of reference there will be tabulated, at tbe commencement of each subject, the few simple
equations that it may be found requisite to employ.

THE GEOMETRY OF THREE DIMENSIONS.

Vitruvius, referring to those things on which Architecture depends, in Book I.,
Chapter II., says, Arrangement is the " disposition in their just and proper places of all the

Note.—All that relates to the Mathematical and Anth- sufficient to refer to their works for a detailed account of

metical forms that were employed hy the Greeks in making the ancient methods of making their Trigonometrical and

their Astronomical calculations (and consequently in their Arithmetical calculations, and also for the ancient theory

Architecture), has heen so ahly condensed hy Mons. of the Conic Sections and of Optics, which will be noticed

Delambre "Histoire de lAstronomie Ancienne, and also later in this work,
by Montucla "Histoire des Mathematiques," that it is