Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/34

is shewn that ${\frac {OF}{FA}}>{\frac {2339{\frac {1}{4}}}{153}}$. Lastly if $OG$ (not shewn in the figure) be the bisector of $FOA$, $AG$ is the half side of a regular 96agon circumscribing the circle, and it is shewn that ${\frac {OA}{AG}}>{\frac {4673{\frac {1}{2}}}{153}}$, and thence that the ratio of the diameter to the perimeter of the 96agon

is $>{\frac {4673{\frac {1}{2}}}{14688}}$, and it is deduced that the circumference of the circle, which is less than the perimeter of the polygon, is < 3 1/7 of the diameter. The second part of the theorem is obtained in a similar manner by determination of the side of a regular 96agon inscribed in the circle.

In the course of his work, Archimedes assumes and employs, without explanation as to how the approximations were obtained, the following estimates of the values of square roots of numbers:

$\textstyle {\frac {1351}{780}}>{\sqrt {3}}>{\frac {265}{153}}$, $\textstyle 3013{\frac {3}{4}}>{\sqrt {9082321}}$, $\textstyle 1838{\frac {9}{11}}>{\sqrt {3380929}}$, $\textstyle 1009{\frac {1}{6}}>{\sqrt {1018405}}$, $\textstyle 2017{\frac {1}{4}}>{\sqrt {4069284{\frac {1}{36}}}}$, $\textstyle 591{\frac {1}{8}}<{\sqrt {349450}}$, $\textstyle 1172{\frac {1}{8}}<{\sqrt {1373943{\frac {33}{64}}}}$, $\textstyle 2339{\frac {1}{4}}<{\sqrt {5472132{\frac {1}{16}}}}$.

In order to appreciate the nature of the difficulties in the way of obtaining these approximations we must remember the backward condition of Arithmetic with the Greeks, owing to the fact that they possessed a system of notation which was exceedingly inconvenient for the purpose of performing arithmetical calculations.

The letters of the alphabet together with three additional signs were employed, each letter being provided with an accent or with a short horizontal stroke; thus the nine integers

1, 2, 3, 4, 5, 6, 7, 8, 9 were denoted by αʹ, βʹ, γʹ, δʹ, εʹ, Ϛʹ, ζʹ, ηʹ, θʹ,

the multiples of 10,

10, 20, 30, ... 90 were denoted by ιʹ, κʹ, λʹ, μʹ, νʹ, ξʹ, οʹ, πʹ, ϟʹ,