er
advances he will realize the importance of this question of exactitude.

[Illustration: FIG. 21.]

[Illustration: FIG. 22.]

In these cutting-out puzzles it is necessary not only to get the
directions of the cutting lines as correct as possible, but to remember
that these lines have no width. If after cutting up one of the crosses
in a manner indicated in these articles you find that the pieces do not
exactly fit to form a square, you may be certain that the fault is
entirely your own. Either your cross was not exactly drawn, or your cuts
were not made quite in the right directions, or (if you used wood and a
fret-saw) your saw was not sufficiently fine. If you cut out the puzzles
in paper with scissors, or in cardboard with a penknife, no material is
lost; but with a saw, however fine, there is a certain loss. In the case
of most puzzles this slight loss is not sufficient to be appreciable,
if the puzzle is cut out on a large scale, but there have been
instances where I have found it desirable to draw and cut out each part
separately--not from one diagram--in order to produce a perfect result.

[Illustration: FIG. 23.]

[Illustration: FIG. 24.]

Now for another puzzle. If you have cut out the five pieces indicated in
Fig. 14, you will find that these can be put together so as to form the
curious cross shown in Fig. 23. So if I asked you to cut Fig. 24 into
five pieces to form either a square or two equal Greek crosses you would
know how to do it. You would make the cuts as in Fig. 23, and place them
together as in Figs. 14 and 15. But I want something better than that,
and it is this. Cut Fig. 24 into only four pieces that will fit together
and form a square.

[Illustration: FIG. 25.]

[Illustration: FIG. 26.]

The solution to the puzzle is shown in Figs. 25 and 26. The direction of
the cut dividing A and C in the first diagram is very obvious, and the
second cut is made at right angles to it. That the four pieces should
fit together and form a square will surprise the novice, who will do
well to study the puzzle with some care, as it is most instructive.

I will now explain the beautiful rule by which we determine the size of
a square that shall have the same area as a Greek cross, for it is
applicable, and necessary, to the solution of almost every dissection
puzzle that we meet with. It was first discovered by the philosopher
Pythagoras, who died 500 B.C., and is the 47th proposition of Euclid.
The y