Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/43

 V. If $$A_n$$ is the area of an inscribed regular polygon of $$n$$ sides, and $$S$$ the area of the circle, then $$\textstyle S > A_{2n} + \frac{1}{3}(A_{2n}-A_n)$$.

VI. If $${A_n}'$$ is the area of the circumscribed regular polygon of $$n$$ sides, then $$\textstyle S < \frac{2}{3}A_n' + \frac{1}{3}A_n$$.

VII. If $$C_n$$ denotes the perimeter of the inscribed polygon, and $$C$$ the circumference of the circle, then $$\textstyle C > C_{2n} + \frac{1}{3}(C_{2n}-C_n)$$.

VIII. $$\textstyle \frac{2}{3}CD + \frac{1}{3} EF > \text{arc }CE$$, where E is any point on the circle.

FIG. 9.

IX. $$\textstyle C < \frac{2}{3}C_n + \frac{1}{3}C_n'$$

where $$C_n'$$ is the perimeter of the circumscribed polygon of $$n$$ sides.

X. If $$a_n$$, $$a_n'$$ denote the sides of the in- and circum-scribed polygons, then $$\textstyle {a_{2n}}^2 = a_2n'{\,.\,}\frac{1}{2}a_n$$

XI. $$C <$$ the smaller of the two mean proportionals between $$C_n$$ and $$C_n'$$.

$$S <$$ the similar polygon whose perimeter is the larger of the two mean proportionals.

XII. If $$ED$$ equals the radius of the circle, then $$BG > \text{arc }BF$$.

FIG. 10.