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 $$p_{2n} + \frac{p_{2n}-p_n}{3} < C$$$$ < p_{2n} + \left(\frac{p_{2n}-p_n}{3}\right).\left(\frac{4p_{2n}+p_n}{2p_{2n}+3p_n}\right),$$