eveloped according to this method, although of
course it was not so in fact. Some of the arrangements shown in Figure
6 are closely paralleled in the acoustic figures made by means of
musical tones with sand, on a sheet of metal or glass.

[Illustration: Figure 6.]

[Illustration: Figure 7.]

The celebrated Franklin square of 16 cells can be made to yield a
beautiful pattern by designating some of the lines which give the
summation of 2056 by different symbols, as shown in Figure 10. A free
translation of this design into pattern brickwork is indicated in
Figure 11.

If these processes seem unduly involved and elaborate for the
achievement of a simple result--like burning the house down in
order to get roast pig--there are other more simple ways of deriving
ornament from mathematics, for the truths of number find direct and
perfect expression in the figures of geometry. The squaring of
a number--the raising of it to its second power--finds graphic
expression in the plane figure of the square; and the cubing of a
number--the raising of it to its third power--in the solid figure
of the cube. Now squares and cubes have been recognized from time
immemorial as useful ornamental motifs. Other elementary geometrical
figures, making concrete to the eye the truths of abstract number, may
be dealt with by the designer in such a manner as to produce ornament
the most varied and profuse. Moorish ceilings, Gothic window tracery,
Grolier bindings, all indicate the richness of the field.

[Illustration: Figure 8.]

[Illustration: PLATE XII. IMAGINARY COMPOSITION. THE BALCONY]

[Illustration: Figure 9.]

Suppose, for example, that we attempt to deal decoratively which such
simple figures as the three lowest Platonic solids--the tetrahedron,
the hexahedron, and the octahedron. [Figure 12.] Their projection on a
plane yields a rhythmical division of space, because of their inherent
symmetry. These projections would correspond to the network of lines
seen in looking through a glass paperweight of the given shape, the
lines being formed by the joining of the several faces. Figure 13
represents ornamental bands developed in this manner. The dodecahedron
and icosahedron, having more faces, yield more intricate patterns, and
there is no limit to the variety of interesting designs obtainable by
these direct and simple means.

[Illustration: Figure 10.]

If the author has been successful thus far in his exposition, it
should be su