On the diameter through take , and draw parallel to . Then ; therefore ; thus , so that is less than the circumference of the circle by less than two millionths of the radius.
The rectangle with sides equal to and half the radius has very approximately its area equal to that of the circle. This construction was given by Specht (Crelle's Journal, vol. 3, p. 83).
(4) Let be the diameter of a given circle. Let
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Describe the semi-circles , with and as diameters; and let the perpendicular to through cut them in and respectively. The square of which the side is is approximately of area equal to that of the circle.
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Fig. 19.
We find that , and since we see that is greater than the side of the square whose area is equal to that of the circle by less than two hundred thousandths of the radius.