Quadrature. — Chambers's Encyclopaedia

Quadrature. The 'quadrature' of a plane curve is effected when a square is found which has the same area as the given curve. Practically it is effected when any rectilinear figure of equal area has been found, for it is easy then to obtain the equivalent square. The quadrature, regarded as an arithmetical process, consists in finding the area of the curve in terms of any square unit.

The great problem in quadrature has been the Quadrature of the Circle. The workers in this subject may be divided broadly into two classes: (1) trained mathematicians, who clearly understand the nature of the problem and the difficulties which surround it; (2) those who do not understand the nature of the problem or its difficulties, and who think that they may, by good fortune, succeed where others have failed. The number of the workers of the second class became greatly diminished when the search for 'perpetual motion' became general. And, at the present day, the ranks (now fortunately small) of the perpetual-motionists and the circle-squarers are almost entirely composed of unfortunate individuals whose mental capacities are small, in too many cases the impairment of their faculties having been brought about by a development of their fruitless idea into monomania. Apart from its great historical interest to the mathematician, the subject scarcely merits detailed notice, except in so far as such notice may be useful in preventing further waste of mental energy by some who, were their energies properly directed, might succeed in increasing the sum of useful knowledge.

The nature of the problem may be understood from the following brief account. Let an equiangular $n$ -gon be inscribed in a circle, and let its corners be joined to each other and to the centre. The area of each triangle so formed is $\frac{1}{2}ar \cos \theta$ , where $a$ is the base of the triangle, $r$ is the radius of the circle, and $\theta$ is one-half of the vertical angle. Hence the area of the polygon is $\frac{1}{2}nar \cos \theta$ ; and this can be made as nearly equal to the area of the circle as we please by making $n$ sufficiently large. In the limit, when $n$ is infinite, the two areas are equal. But, when $n$ is infinite, $\theta$ vanishes and $na$ becomes the circumference, $c$ , of the circle. Hence the area of the circle is $\frac{1}{2}cr$ —that is to say, it is equal to the area of a triangle erected on the radius of the circle as base and of height equal to the circumference of the circle.

The arithmetical quadrature of the circle would therefore be effected if we could find the value of the ratio of the circumference to the diameter—that is, the value of $\pi$ in the equation $c = 2\pi r$ . The geometrical quadrature would be effected by finding a geometrical method of drawing a straight line equal in length to the circumference.

It has long been known that the arithmetical solution of the problem is impossible, for it has been proved that the quantity $\pi$ is incommensurable. And proofs have been advanced that the geometrical quadrature is also impossible; but these proofs are by no means simple, and do not always convince those who are able to judge of their accuracy. Still, apart from such proofs, the mere consideration of the fact that (discounting incapable workers) the question has been fruitlessly attacked by the ablest mathematicians of past centuries should be sufficient to deter any reasonable person from engaging in the quest: for it follows that the probability of a solution being possible is excessively small—too small to justify the staking of a man's sanity, or at least the usefulness of his life, upon the result. Any mathematician who now considers the question seeks not for a solution, but for a simple and convincing proof that a solution is impossible. (It must be remembered that a 'geometrical' solution means a solution which involves no more postulates than those of Euclid.)

James Gregory, in 1668, gave a proof of the impossibility of the geometrical quadrature which Huygens, although he at first objected to it, finally admitted in so far as it applied to any sector of a circle. Newton also gave a proof of this limited problem, but his proof is not conclusive.

Archimedes was the first to give a practical measurement of the quantity $\pi$ . By a consideration of the inscribed and escribed 96-gons he proved that it lies between $3\frac{1}{6}$ and $3\frac{1}{7}$ . This result is correct only to the second decimal figure. Two Hindu measurements are 3.1416 and 3.1623. Ptolemy gives 3.141552. A great improvement on previous results was made by Peter Metius in the 16th century. His result was correct to the sixth decimal place inclusive; but its correctness was accidental, for he gave two fractions between which the result lay and took the arithmetical means of the numerators and the denominators in order to obtain his final numerator and denominator—a totally unwarranted method. Vieta gave the result correct to the ninth decimal place inclusive; Adrianus Romanus gave it correct to the fifteenth; and Van Ceulen gave it to the thirty-sixth. Snell introduced considerable improvements in the method, and gave 55 decimal figures. Abraham Sharp gave 75, Machin 100, De Lagny 128, Vega 140. The latter result is only correct to 136 places. Montucla cites an Oxford manuscript in which the result (given to 154 places) is correct to 152 places. In 1846 Dase gave a result with 200 decimals, and, in the following year, Clausen gave 250. In 1851 Shanks gave 315, which were extended by Rutherford to 350; and, shortly afterwards, Shanks gave 527, which he extended to 607. An interesting experimental method was adopted by R. A. Smith. He tossed a thin rod upon a uniformly planked floor, the length of the rod being three-fifths of the breadth of a plank. If $l$ be the length of the rod, while $b$ is the breadth of a plank, the probability of the rod intersecting a seam is $2l/\pi b$ . From the result of 3204 tosses, he found $\pi = 3.1412$ . The true value to 20 places is 3.14159265358979323846.

Any one who is desirous of a more detailed historical account may consult De Morgan's article on the subject in his Budget of Paradoxes (1872).