The Magician's Own Book, or, the Whole Art of Conjuring / Being a complete hand-book of parlor magic, and containing over one thousand optical, chemical, mechanical, magnetical, and magical experiments, amusing transmutations, astonishing sleights and subtleties, celebrated card deceptions, ingenious tricks with numbers, curious and entertaining puzzles, together with all the most noted tricks of modern performers.

by George Arnold; Frank Cahill

Take a piece of card of the shape and size or proportions of the subjoined, and cut it into three parts, and with these three form a perfect square.

To do this, cut it in the direction of the dotted lines, and it will then be easy to lay down the pieces to form a perfect square.

SQUARING THE CIRCLE.

"Squaring the circle," as it is called, is the puzzle of puzzles, and there are many persons who fancy this can be accomplished, as there are also many who believe that they can discover "perpetual motion."

The meaning of this phrase squaring is scientifically expressed by the term finding the quadrature of the circle; that is, the act of producing a square equal to a given circle; and many persons but slightly acquainted with mathematics have puzzled their brains to effect this object. The Cardinal de Cusa rolled a cylinder over a plane, till the point which was first in contact with the plane touched it again; and then, by a train of reasoning very unmathematical, he endeavored to determine the length of the line thus described. Oliver de Serras worked a circle, and also a triangle equal to an equilateral triangle, inscribed within the circle, and imagined that the former was exactly equal to two of the latter, forgetting that the double of this triangle is equal to the hexagon inscribed within the circle, and therefore smaller than the circle itself. A Frenchman challenged the world, and deposited 10,000 livres as a stake, that he could accomplish the feat. He reduced the problem to the mechanical process of dividing a circle into four quarters, and then turning these with their angles outwards, so as to form a square, which he asserted to be equal to the circle; this however was soon proved to be ridiculous.

Some persons have taken a piece of pasteboard, and cutting it out into a circular form, and by cutting that circular disc into pieces of a square form and definite dimensions, and fitting the same turned pieces one into the other, have come near to a notion of the superficial area of a circle. But this kind of demonstration is purely mechanical, and is neither geometrical nor scientific, and, is in fact, no demonstration according to mathematics. For if we take the pieces of card, however exactly they may appear to be formed, and examine them with a microscope, we shall soon find that none of them are geometrically true, nor of the same length or breadth, and therefore the conclusion arrived at is a false one.

The early mathematicians, in their attempts to solve this problem, generally proceeded on the following plan. If we draw a square exterior to a circle, that is, touching the square in four points, each side of the square being equal to the diameter of the circle, we can soon convince ourselves that the boundary of the square will be greater than the circumference of the circle, and the area of the former greater than that of the latter. But if the square be drawn within the circle, so that only the four corners touch it, then it is equally evident that the circle is larger, both in boundary and area, than the square. By this proceeding, we arrive at the conclusion that a circle is smaller than a square external to it, and larger than one internal to it. Let us next suppose that we draw a regular pentagon, that is, a figure of five equal sides, exterior to the circle, and touching it on five points; then it is evident that as the circle is wholly contained within the pentagon, it must be smaller than that which contains it. But if the pentagon be described within the circle, touching it at the five angular points, then of course the circle is larger than the pentagon which it contains.

Now, in geometry, my young readers must bear in mind, the exact periphery or circumference, and the exact area of any figure bounded by straight lines, may be determined with rigorous accuracy; and if we draw two polygons—say of one hundred sides, one within and one without the circle—we can ascertain the exact area of those polygons, and affirm that the area of a circle is greater than a certain amount, and less than another certain amount. These two amounts, if the number of the sides of the polygon be so large as we here suppose, may be so very nearly alike, that either one will give the area of the circle with great closeness.