Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/44

XIII. If $AC$ = radius of the circle, then $BL<{\text{arc }}BE$.

XIV. If $G$ is the centroid of the segment, then $BG>GD$ and $\textstyle <{\frac {3}{2}}GD$.

XV. ${\frac {{\text{segment }}ABC}{\triangle ABC}}>{\frac {4}{3}}$ and $<3{\tfrac {1}{3}}{.}{\frac {B'D}{BB'+3OD}}$.

XVI. If $a$ denote the arc (< semi-circle), and $s$, $s'$ its sine and its chord respectively, then $s'+{\frac {s'-s}{3}}<a$$<s'+\left({\frac {s'-s}{3}}\right).\left({\frac {4s'+s}{2s'+3s}}\right),$.

This is equivalent, as Huyghens points out, to $p_{2n}+{\frac {p_{2n}-p_{n}}{3}}<C$$<p_{2n}+\left({\frac {p_{2n}-p_{n}}{3}}\right).\left({\frac {4p_{2n}+p_{n}}{2p_{2n}+3p_{n}}}\right),$