Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/42

which is a model of geometrical reasoning, he undertakes by improved methods to make a careful determination of the area of a circle. He establishes sixteen theorems by geometrical processes, and shews that by means of his theorems three times as many places of decimals can be obtained as by the older method. The determination made by Archimedes he can get from the triangle alone. The hexagon gives him the limits 3.1415926533 and 3.1415926538.

The following are the theorems proved by Huyghens:

I. If $ABC$ is the greatest triangle in a segment less than a

semi-circle, then

$\triangle ABC<4(\triangle AEB+\triangle BFC)$,

where $AEB$, $BFC$ are the greatest triangles in the segments $AB$, $BC$.

II. $\textstyle \triangle FEG<{\frac {1}{2}}\triangle ABC$,

where $ABC$ is the greatest tri- angle in the segment.

III. ${\frac {{\text{segment }}ACB}{\triangle ACB}}>{\frac {4}{3}}$,

provided the segment is less than

the semi-circle.

This theorem had already been given by Hero.

IV. ${\frac {{\text{segment }}ACB}{\triangle ATC}}<{\frac {2}{3}}$.