oung reader who knows nothing of the elements of geometry will get
some idea of the fascinating character of that science. The triangle ABC
in Fig. 27 is what we call a right-angled triangle, because the side BC
is at right angles to the side AB. Now if we build up a square on each
side of the triangle, the squares on AB and BC will together be exactly
equal to the square on the long side AC, which we call the hypotenuse.
This is proved in the case I have given by subdividing the three squares
into cells of equal dimensions.

[Illustration: FIG. 27.]

[Illustration: FIG. 28.]

It will be seen that 9 added to 16 equals 25, the number of cells in the
large square. If you make triangles with the sides 5, 12 and 13, or with
8, 15 and 17, you will get similar arithmetical proofs, for these are
all "rational" right-angled triangles, but the law is equally true for
all cases. Supposing we cut off the lower arm of a Greek cross and place
it to the left of the upper arm, as in Fig. 28, then the square on EF
added to the square on DE exactly equals a square on DF. Therefore we
know that the square of DF will contain the same area as the cross. This
fact we have proved practically by the solutions of the earlier puzzles
of this series. But whatever length we give to DE and EF, we can never
give the exact length of DF in numbers, because the triangle is not a
"rational" one. But the law is none the less geometrically true.

[Illustration: FIG. 29.]

[Illustration: FIG. 30.]

Now look at Fig. 29, and you will see an elegant method for cutting a
piece of wood of the shape of two squares (of any relative dimensions)
into three pieces that will fit together and form a single square. If
you mark off the distance _ab_ equal to the side _cd_ the directions of
the cuts are very evident. From what we have just been considering, you
will at once see why _bc_ must be the length of the side of the new
square. Make the experiment as often as you like, taking different
relative proportions for the two squares, and you will find the rule
always come true. If you make the two squares of exactly the same size,
you will see that the diagonal of any square is always the side of a
square that is twice the size. All this, which is so simple that anybody
can understand it, is very essential to the solving of cutting-out
puzzles. It is in fact the key to most of them. And it is all so
beautiful that it seems a pity that it should not be familiar to