Squaring the circle the circular arc b around O with radius , it results S'. Bisect the line segment in D and draw the semicircle b over D. Draw a straight line from O through C up to the semicircle b, it cuts b in E. The line segment is the mean proportional between and , also called geometric mean. Extend the line segment beyond O and transfer twice more, it results F and A, and thus the length of the line segment with the above described approximation value of formula_3, the half circumference of the circle. Bisect the line segment in G
Area of a circle "area" and "circumference". The analytical definitions are seen to be equivalent, if it is agreed that the circumference of the circle is measured as a rectifiable curve by means of the integral The integral appearing on the right is an abelian integral whose value is a half-period of the sine function, equal to . Thus formula_6 is seen to be true as a theorem. Several of the arguments that follow use only concepts from elementary calculus to reproduce the formula formula_1, but in many cases to regard these as actual proofs, they rely implicitly on the fact that one can