Construction for an Approximate Quadrature of the Circle

Abstract

DR. ROUSE BALL'S interesting account, in NATURE of May 23, of M. de Pulligny's constructions reminds me of another simple one which I do not think known. If OA, OB are perpendicular radii of a circle of radius 1, and if BCD is a line cutting OA in C and the circle in D and representing the side of the square in question, then OC = √(4/π−1) = 0.52272321, which, put into the form of a continued fraction, has for convergents, etc. The convergent, or 0.52272727, differs (in excess) from, the real magnitude only by 1 in 128750; hence if we take C such that OC = OA, which can be done easily and with great accuracy, the line BCD represents the required side with all the accuracy which any graphic construction can be expected to give. Theoretically, this method is 121 times more accurate than M. de Pulligny's construction with the Archimedes ratio, but thirty-seven times less accurate than that with the Metius ratio. In practice, however, this relative inaccuracy is absolutely unnoticeable, and the method here described is the easier to carry out.