The point was to find a general rule for forming a
perfect square out of another square combined with a "right-angled
The triangle to which geometricians give this
high-sounding name is, of course, nothing more or less than half a
square that has been divided from corner to corner.
The precise relative proportions of the square and
triangle are of no consequence whatever.
is only necessary to cut the wood or material into five pieces.
Suppose our original square to be ACLF in the
above diagram and our triangle to be the shaded portion CED.
first find half the length of the long side of the triangle (CD) and
measure off this length at AB.
Then we place the triangle in its
present position against the square and make two cuts, one
from B to F, and the other from B to E.
Strange as it may seem, that is
all that is necessary!
If we now remove the pieces G, H, and M to their
new places, as shown in the diagram, we get the perfect square BEKF.
Take any two square pieces of paper, of different
sizes but perfect squares, and cut the smaller one in half from corner
to corner. Now proceed in the manner shown, and you will find that the
two pieces may be combined to form a larger square by making these two
simple cuts, and that no piece will be required to be turned over.
The remark that the triangle might be "a little
larger or a good deal smaller in proportion" was intended to bar cases
where area of triangle is greater than area of square. In such cases
six pieces are necessary, and if triangle and square are of equal area
there is an obvious solution in three pieces, by simply cutting the
square in half diagonally.