SQUARING THE CIRCLE.
Of Mathematical Problems, the most perplexing to ancient and modern mathematicians, although of late years said to be satisfactorily demonstrated, and no longer desiderata of Geometry, are—
1. The Quadrature or Squaring of the Circle;—2. The Duplication, or doubling of the Cube;—and 3. The Trisection of the Angle.
In his "Popular Astronomy,"[11] Professor Arago, treating on the surface of a circle, observes that,—
It is mathematically equal to the product of the length of the circumference, multiplied by half the radius. To square a circle of a given diameter in mètres, is the same as giving the number of squares, of a mètre in each side, of which the surface is the equivalent. If, the diameter being given, the exact circumference were known by a sort of inspiration, the superficial extent of the circular space would be deducible from the two numbers, by the mere multiplication of the numerical length of the circumference by the fourth of the diameter, or half the radius.
But, the circumference being deducible from the diameter only by approximation, the surface alluded to cannot be computed with mathematical rigour; yet the result can be obtained with all desirable precision by the aid of the ratios usually given for such purpose; for instance, the area of the space included within a circle of thirty-eight millions of leagues radius, may be determined within such a degree of precision that the probable error shall not exceed the space of a mite.
"The sect of squarers then," Arago adds,—"are searching after a solution which is proved to be impossible, and which, moreover, would be of no practical use, even if their foolish hopes were crowned with success."
In the "Birds" of Aristophanes, the character is introduced of a geometer, who is going to make a square circle, showing how early this chimerical performance became an object of ridicule.
Thales, Anaxagoras, Pythagoras, Hippocrates, Plato, Apollonius, Ptolemy, with other ancient mathematicians, have given methods for approximating to the area of the circle; and many also among the moderns. In 1775, the Paris Academy of Science determined to discourage papers devoted to this subject, and their course in this respect was soon after adopted also by The Royal Society, it being found that there was among certain geometers a complete mania for settling this and similar problems, the solution of which was either unattainable, or if attained of very questionable value.