If this process can be
indefinitely continued, the cuttings must become smaller and smaller, for
the following reason. Suppose, at the outset, the boundary point nearest to
the intersection of the axes is distant from that intersection by, say four
inches; it is clear that we cannot, after any number of cuttings, have a
part of the boundary at less than four inches from the intersection. For
there never is, after any cutting, any approach to the intersection except
what there already was on the other side of the axis employed, before that
cutting was made. If then the cuttings should go on for ever, or
practically until the pieces to be cut off are too small, and _if this take
place all round_, the figure last obtained will be a good representation of
a circle of four inches radius. On the suppositions, we must be always
cutting down, at all parts of the boundary; but it has been shown that we
can never come nearer than by four inches to the intersection of the axes.

But it does not follow that the process _will_ go on for ever. We may come
at last to a state in which both the creases are axes of symmetry at once;
and then the process stops. If the paper had at first a curvilinear
boundary, properly chosen, and if the axes were placed at the proper angle,
it would happen that we should arrive at a {15} _regular_ curved polygon,
having the two axes for axes of symmetry. The process would then stop.

I will, however, suppose that the original boundary is everywhere
rectilinear. It is clear then that, after every cutting, the boundary is
still rectilinear. If the creases be at right angles to one another, the
ultimate figure may be an irregular polygon, having its four quarters
alike, such as may be inscribed in an oval; or it may have its sides so
many and so small, that the ultimate appearance shall be that of an oval.
But if the creases be not at right angles, the ultimate figure is a
perfectly regular polygon, such as can be inscribed in a circle; or its
sides may be so many and so small that the ultimate appearance shall be
that of a circle.

Suppose, as in MR. INGLEBY'S question, that the creases are not at right
angles to each other; supposing the eye and the scissors _perfect_, the
results will be as follows:

First, suppose the angle made by the creases to be what the mathematicians
call _incommensurable_ with the whole revolution; that is, suppose that no
repetition of the angle will produce an _exact_ number