The Arithmetic of Beauty 
ot materials—brick, 
stone, iron and 
wood. All of the 
arts, and particularly 
music and architec¬ 
ture, portray in 
different manners 
and degrees the 
truths ot number. 
Architecture does 
this in two ways, 
esoterically, as it 
were, in the form of 
harmonic proportions, and exoterically in the 
form of symbols which represent numbers 
and groups of numbers. The fact that a 
series of threes and a series of fours mutually 
conjoin in i 2, finds an architectural expres¬ 
sion in the Tuscan, the Doric, and the Ionic 
orders according to Vignole, for in them all 
the stylobate is four parts, the entablature 
3, and the intermediate column 12. The 
affinity between 4 and 7 revealed in the fact 
that they express the ratio between the base 
and the altitude of a right-angled triangle of 
60 degrees, and 
the musical inter¬ 
val of the dimin¬ 
ished seventh, is 
architecturally 
suggested in the 
Gothic chapels of 
Windsor and Ox¬ 
ford whose widths 
and lengths are in 
the proportion of 
4 to 7, and it finds 
complete objec¬ 
tive expression 
in the Palazzo 
Giraud which is 
four stories in 
height with seven openings in each story. 
Every building is a symbol of some number 
or group of numbers ; and other things being 
equal, the more perfect the numbers involved, 
the more beautiful will be the building. 3, 
5, and 7 are the numbers of most frequent 
occurrence, and they are the most satisfactory, 
because being of small quantity they are easily 
grasped by the eye, and being odd they have 
a center or axis, so necessary in every archi¬ 
tectural composition. Next in value are 
lowest multiples of 
these numbers and 
the least common 
multiples of any two 
of them, namely : 6, 
9, 12; 10, 15, 20; 
14, 21, 28 ; etc., be¬ 
cause the eye, with 
a little assistance, 
is able to resolve 
them into their con¬ 
stituent factors. It 
is part of the art of 
architecture to render such assistance, for the 
eye counts always, consciously or uncon¬ 
sciously, and when it is confronted with a 
number of units greater than it can readily 
resolve, it is refreshed and rested if these 
are so grouped and arranged that these units 
reveal themselves as factors of the higher 
quantity. 
I here is a raison d'etre for string courses 
other than to mark the position of a floor on 
the interior of a building, and for quoins and 
pilasters other than to betray the presence of 
a transverse wall. These sometimes serve 
the useful purpose of so subdividing a faqade 
that the eye estimates the number of its open¬ 
ings without conscious effort and consequent 
fatigue. "The tracery of Gothic rose-windows 
forms perhaps the highest and finest archi¬ 
tectural expression of number. Just as thirst 
makes water more sweet they confuse the eye 
with their complexity only to more greatly 
gratify it by revealing the inherent simplicity 
in which this complexity had its origin. 
A NUMERICAL ANALYSIS OF GOTHIC TRAGLILY 
ARCHITECTURAL ORNAMENT 
CONSIDERED AS THL 
052ECTIF1CATI0N 
OF NUME>LR___, 
MULTIPLICATION 
IN GROUPS' OF FIVE 
ALTERNATION OF 
THREE AND SEVEN 
TWO 
THREE 
NUMERATION IN GROUPS EXPRESSED ARCHITECTURALLY 
A 5UND ARCADE IN THE- SOUTH 
TRANSEPT OPTHE CHAPEL OP 
LINCOLN CATHEDRAE 
PALAZZO PRAXEEES1' 
GIMIGNANO 
■SAN 
RENAISSANCE 
ORNAMENT 
264 
