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 $\textstyle {a_{2n}}^{2}=a_{2}n'{\,.\,}{\frac {1}{2}}a_{n}$

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.