of revolutions. Then
the cutting will go on for ever, and the result will perpetually approach a
circle. It is easily shown that no figure whatsoever, except a circle, has
two axes of symmetry which make an angle incommensurable with the whole
revolution.

Secondly, suppose the angle of the creases commensurable with the
revolution. Find out the smallest number of times which the angle must be
repeated to give an exact number of revolutions. If that number be even, it
is the number of sides of the ultimate polygon: if that number be odd, it
is the half of the number of sides of the ultimate polygon.

Thus, the paper on which I write, the whole sheet being taken, and the
creases made by joining opposite corners, happens to give the angle of the
creases very close to three-fourteenths of a revolution; so that fourteen
repetitions of the angle is the lowest number which give an exact number of
revolutions; and a very few cuttings lead to a regular polygon of fourteen
sides. But if four-seventeenths of a revolution had been taken for the
angle of the creases, the ultimate polygon would have had thirty-four
sides. In an angle taken at hazard the chances are that the number of
ultimate sides will be large enough to present a circular appearance.

Any reader who chooses may amuse himself by trying results from three or
more axes, whether all passing through one point or not.

A. DE MORGAN.

* * * * *

THE BLACK-GUARD.

(Vol. viii., p. 414.)

Some of your correspondents, SIR JAMES E. TENNENT especially, have been
very learned on this subject, and all have thrown new light on what I
consider a very curious inquiry. The following document I discovered some
years ago in the Lord Steward's Offices. Your readers will see its value at
once; but it may not be amiss to observe, that the name in its present
application had its origin in the number of masterless boys hanging about
the verge of the Court and other public places, palaces, coal-cellars, and
palace stables; ready with links to light coaches and chairs, and conduct,
and rob people on foot, through the dark streets of London; nay, to follow
the Court in its progresses to Windsor and Newmarket. Pope's "link-boys
vile" are the black-guard boys of the following Proclamation.

PETER CUNNINGHAM.

At the Board of Green Cloth,
in Windsor Castle,
this 7th day of May, 1683.

Whereas of late a sort of vicious, idle, and masterless boyes and