In the last number of the Athenæum for 1855 a correspondent says "the thing is no longer a problem but an axiom." He makes the square equal to a circle by making each side equal to a quarter of the circumference. As De Morgan says, he does not know that the area of the circle is greater than that of any other figure of the same circuit.
Such ideas are evidently akin to the poetic notion of the quadrature. Aristophanes, in the "Birds," introduces a geometer, who announces his intention to make a square circle. And Pope in the "Dunciad" delivers himself as follows:
Mad Mathesis alone was unconfined,
Too mad for mere material chains to bind,—
Now to pure space lifts her ecstatic stare,
Now, running round the circle, finds it square.
The author's note explains that this "regards the wild and fruitless attempts of squaring the circle." The poetic idea seems to be that the geometers try to make a square circle.
As stated by all recognized authorities, the problem is this: To describe a square which shall be exactly equal in area to a given circle.
The solution of this problem may be given in two ways: (1) the arithmetical method, by which the area of a circle is found and expressed numerically in square measure, and (2) the geometrical quadrature, by which a square, equal in area to a given circle, is described by means of rule and compasses alone.
Of course, if we know the area of the circle, it is easy to find the side of a square of equal area; this can be done by simply extracting the square root of the area, provided the number is one of which it is possible to extract the square root. Thus, if we have a circle which contains 100 square feet, a square with sides of 10 feet would be exactly equal to it. But the ascertaining of the area of the circle is the very point where the difficulty comes in; the dimensions of circles are usually stated in the lengths of the diameters, and when this is the case, the problem resolves itself into another, which is: To find the area of a circle when the diameter is given.
Now Archimedes proved that the area of any circle is equal to that of a triangle whose base has the same length as the circumference and whose altitude or height is equal to the radius. Therefore if we can find the length of the circumference when the diameter is given, we are in possession of all the points needed to enable us to "square the circle."
In this form the problem is known to mathematicians as that of the rectification of the curve.
In a practical form this problem must have presented itself to intelligent workmen at a very early stage in the progress of operative mechanics. Architects, builders, blacksmiths, and the makers of chariot wheels and vessels of various kinds must have had occasion to compare the diameters and circumferences of round articles. Thus in I Kings, vii, 23, it is said of Hiram of Tyre that "he made a molten sea, ten cubits from the one brim to the other; it was round all about * * * and a line of thirty cubits did compass it round about," from which it has been inferred that among the Jews, at that time, the accepted ratio was 3 to 1, and perhaps, with the crude measuring instruments of that age, this was as near as could be expected. And this ratio seems to have been accepted by the Babylonians, the Chinese, and probably also by the Greeks, in the earliest times. At the same time we must not forget that these statements in regard to the ratio come to us through historians and prophets, and may not have been the figures used by trained mechanics. An error of one foot in a hoop made to go round a tub or cistern of seven feet in diameter, would hardly be tolerated even in an apprentice.