pring from a necessity which is numerical. It is clear that the
arrangement of numbers in a magic square is necessitous--they must be
placed in a certain way in order that the summation of every column
shall be the same. The problem then becomes to make that necessity
reveal itself to the eye. Now most magic squares contain a _magic
path_, discovered by following the numbers from cell to cell in
their natural order. Because this is a necessitous line it should not
surprise us that it is frequently beautiful as well.

[Illustration: Figure 3.]

The left hand drawing in Figure 4 represents the smallest aggregation
of numbers that is capable of magic square arrangement. Each vertical,
horizontal, and corner diagonal column adds up to 15, and the sum of
any two opposite numbers is 10, which is twice the center number. The
magic path is the endless line developed by following, free hand, the
numbers in their natural order, from 1 to 9 and back to 1 again. The
drawing at the right of Figure 4 is this same line translated into
ornament by making an interlace of it, and filling in the larger
interstices with simple floral forms. This has been executed in white
plaster and made to perform the function of a ventilating grille.

Now the number of magic squares is practically limitless, and while
all of them do not yield magic lines of the beauty of this one, some
contain even richer decorative possibilities. But there are also other
ways of deriving ornament from magic squares, already hinted at in the
discussion of the Colonial quilt.

[Illustration: Figure 4.]

[Illustration: Figure 5.]

Magic squares of an even number of cells are found sometimes to
consist of numbers arranged not only in combinations of the ordinary
and the reverse ordinary orders of counting, but involving two others
as well: the reverse of the ordinary (beginning at the upper right
hand, across, and down) and the reversed inverse, (beginning at the
lower left hand, across, and up). If, in such a magic square, a simple
graphic symbol be substituted for the numbers belonging to each order,
pattern spontaneously springs to life. Figures 5 and 6 exemplify the
method, and Figures 7 and 8 the translation of some of these squares
into richer patterns by elaborating the symbols while respecting their
arrangement. By only a slight stretch of the imagination the beautiful
pierced stone screen from Ravenna shown in Figure 9 might be conceived
of as having been d