Excerpt from Journal of the Franklin Institute of the State of Pennsylvania, Vol. 8: Devoted to the Mechanic Arts, Manufactures, General Science, and the Recording of American and Other Patented Inventions

A cube and a sphere were to be made, and their relative masses ascertained hydrostatically. An approximation as inferior to that which the mathematician has obtained, as matter is inferior to mind. Others have proposed to cut a square and a circle out of a plate of metal, or other substance, both equal in thickness, and then to weigh them, ignorant of the fact that neither the weight or dimensions of such articles is capable of being ascertained with a millionth part of the correctness with which the mathematician has squared the circle.

The subjoined account of what has been proposed, and done, upon this subject, is abstracted entirely from Montucla, as contained in his "Recreations in Mathematics, " translated by Hutton.

Montucla divides those who have pursued this inquiry into two classes, the first, geometricians, who, aware of the difficulty or impossibility of the problem, have not been led away by illusions, but have confined themselves to the finding out the most exact methods of approximation. The other class, those who scarcely acquainted with the elements of geometry, and hardly knowing on what principle the problem depends, have twisted and turned the circle in every direction, and have laboured, like the unfortunate Ixion, eternally rolling the heavy burden, without bringing it any nearer to its place of destination. When one error is pointed out to them, they soon return with their propositions in a new, but equally contemptible form, and unhesitatingly contest the best established truths in the elements of geometry, appearing to believe themselves specially appointed by Heaven to reveal truths to mankind, the discovery of which is withheld from the learned, that it may be bestowed upon idiots.

In the time of Aristophanes, the question of the quadrature had already become celebrated, as in order to ridicule Meto, he introduces him on the stage, promising to square the circle.

We first find in the writings of Archimedes, the announcement of the truth, that the circle is equal to the rectangle of half the circumference by the radius. Still something more was necessary, namely, to determine the proportions between the circumference and the diameter, and although he was unable to accomplish this with mathematical precision, he showed that the diameter being 1, the circumference would be more than 3 10/1, and less than 3 10/70 or 3 1/7.

Since that time, if great exactness be not required, in order to find the ratio of the diameter to the circumference, the proportion of 1 to 3 1/, or of 7 to 22 is employed, that is to say, the diameter is tripled, and 1/7 of it is added: this seventh is never neglected, but by the most ignorant workmen.

Among the modern geometricians, the first who made any addition to our knowledge on this subject was Peter Metius, of the Netherlands, whose name was mentioned in connexion with the discovery of the telescope in our last number.

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