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

By some such means as these Archimedes found that if the diameter of a circle be called 7, then the circumference will be nearly 22; and that if the square of the diameter be 14, then the area of the circle will be equal to about 11; but this computation was slightly in error, and gave to the area of the circle too great a measure by about one three-thousandth part of the whole. At a later period, however, a European mathematician, named Metius, discovered a method which makes an extraordinary approach to accuracy, and is at the same time easily remembered. He found that if the diameter be considered equal to 113, then the circumference would equal 355; or if we multiply the square of the radius by 355, and divide it by 113, the area will be given. Now this method is so very nearly correct, that the area of a circle one foot in diameter is given within the fifty-thousandth part of a square inch.

Other mathematicians have carried the approximation still further. Ludolph Van Ceulen worked it out to 36 places of figures, showing that if the diameter be 1, the circumference will be

3.14,159,265,358,979,323,846,264,338,327,950,288.

or that if the last figure be 8, the result will be a little below the truth, and if 9, a little above it.

Since this, Mr. Sharp, an English mathematician, carried the approximation to 72 places of figures; Mr. John Machin to 100 figures, and eclipsed all others. M. de Lagny worked it out to 128 places of figures, and of the degree of nearness to which this computation brings the proportion, Montucla says, "If we suppose a circle, the diameter of which is a thousand million times greater than the distance between the sun and the earth, the error in the proportion of the circumference would be a thousand million times less than the thickness of a hair."

But after all, none of these computations are quite correct; they all deviate from the truth, and bring us to the conclusion that there are no numbers or collection of numbers which will give the exact ratio of the circumference, or of the area of a circle to its diameter. We offer this explanation on the subject to our young friends that they may not be puzzled by the question; and that should they be asked to square the circle, or hear any one assert that he can do so, they may be able to show that they are "awake" to the question, and know how to explain it.

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