As already noted, the great difficulty with most circle-squarers is that they are ignorant both of the nature of the problem to be solved and of the results which have been already attained. Sometimes we see it explained as the drawing of a square inside a circle and at other times as the drawing of a square around a circle, but both these problems are amongst the very simplest in practical geometry, the solutions being given in the sixth and seventh propositions of the Fourth Book of Euclid. Other definitions have been given, some of them quite absurd. Thus in France, in 1753, M. de Causans, of the Guards, cut a circular piece of turf, squared it, and from the result deduced original sin and the Trinity. He found out that the circle was equal to the square in which it is inscribed, and he offered a reward for the detection of any error, and actually deposited 10,000 francs as earnest of 300,000. But the courts would not allow any one to recover. In the last number of the Athenaeum 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.