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Amusements in Mathematics

"To fret thy soul with crosses."
"But, for my part, it was Greek to me."
Julius Cæsar, i. 2.
Many people are accustomed to consider the cross as a wholly Christian symbol. This is
erroneous: it is of very great antiquity. The ancient Egyptians employed it as a sacred
symbol, and on Greek sculptures we find representations of a cake (the supposed real
origin of our hot cross buns) bearing a cross. Two such cakes were discovered at
Herculaneum. Cecrops offered to Jupiter Olympus a sacred cake or boun of this kind. The
cross and ball, so frequently found on Egyptian figures, is a circle and the tau cross. The
circle signified the eternal preserver of the world, and the T, named from the Greek letter
tau, is the monogram of Thoth, the Egyptian Mercury, meaning wisdom. This tau cross is
also called by Christians the cross of St. Anthony, and is borne on a badge in the bishop's
palace at Exeter. As for the Greek or mundane cross, the cross with four equal arms, we
are told by competent antiquaries that it was regarded by ancient occultists for thousands
of years as a sign of the dual forces of Nature—the male and female spirit of everything
that was everlasting.
The Greek cross, as shown in Fig. 5, is formed by the assembling together of five equal
squares. We will start with what is known as the Hindu problem, supposed to be upwards
of three thousand years old. It appears in the seal of Harvard College, and is often given
in old works as symbolical of mathematical science and exactitude. Cut the cross into
five pieces to form a square. Figs. 6 and 7 show how this is done. It was not until the
middle of the nineteenth century that we found that the cross might be transformed into a
square in only four pieces. Figs. 8 and 9 will show how to do it, if we further require the
four pieces to be all of the same size and shape. This Fig. 9 is remarkable because,
according to Dr. Le Plongeon and others, as expounded in a work by Professor Wilson of
the Smithsonian Institute, here we have the great Swastika, or sign, of "good luck to you
"—the most ancient symbol of the human race of which there is any record. Professor
Wilson's work gives some four hundred illustrations of this curious sign as found in the
Aztec mounds of Mexico, the pyramids of Egypt, the ruins of Troy, and the ancient lore
of India and China. One might almost say there is a curious affinity between the Greek
cross and Swastika! If, however, we require that the four pieces shall be produced by
only two clips of the scissors (assuming the puzzle is in paper form), then we must cut as
in Fig. 10 to form Fig. 11, the first clip of the scissors being from a to b. Of course
folding the paper, or holding the pieces together after the first cut, would not in this case
be allowed. But there is an infinite number of different ways of making the cuts to solve
the puzzle in four pieces. To this point I propose to return.
It will be seen that every one of these puzzles has its reverse puzzle—to cut a square into
pieces to form a Greek cross. But as a square has not so many angles as the cross, it is not
always equally easy to discover the true directions of the cuts. Yet in the case of the
examples given, I will leave the reader to determine their direction for himself, as they
are rather obvious from the diagrams.