Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/36

(2) The relations for the successive calculation of the perimeters and areas of polygons inscribed and circumscribed to a circle.

Denoting by $p_{n}$, $a_{n}$, the perimeter and area of an inscribed regular polygon of $n$ sides, and by $P_{n}$, $A_{n}$ the perimeter and area of a circumscribed regular polygon of $n$ sides, these relations are

$p_{2n}={\sqrt {p_{n}P_{2n}}}$, $a_{2n}={\sqrt {a_{n}A_{n}}}$,
$P_{2n}={\frac {2p_{n}P_{n}}{p_{n}+P_{n}}}$, $A_{2n}={\frac {2a_{2n}A_{n}}{a_{2n}+A_{n}}}$.

Thus the two series of magnitudes

$P_{n},p_{n},P_{2n},p_{2n},P_{4n},p_{4n},\ldots$,
$A_{n},a_{n},A_{2n},a_{2n},A_{4n},a_{8n},\ldots$,

are calculated successively in accordance with the same law. In each case any element is calculated from the two preceding ones by taking alternately their harmonic and geometric means. This system of formulae is known as the Archimedean Algorithm; by means of it the chords and tangents of the angles at the centre of such polygons as are constructible can be calculated. By methods essentially equivalent to the use of this algorithm the sines and tangents of small angles were obtained to a tolerably close approximation. For example, Aristarchus (250 B.C.) obtained the limits 1/45 and 1/60 for sin 1°.

The works of the later Greeks

Among the later Greeks, Hipparchus (180—125 B.C.) calculated the first table of chords of a circle and thus founded the science of Trigonometry. But the greatest step in this direction was made by Ptolemy (87—165 A.D.) who calculated a table of chords in which the chords of all angles at interval of 1/2° from 0 to 180° are contained, and thus constructed a trigonometry that was not surpassed for 1000 years. He was the first to obtain an approximation to $\pi$ more exact than that of Archimedes; this was expressed in sexagesimal measure by 3° 8′ 30″ which is equivalent to

$\textstyle 3+{\frac {8}{60}}+{\frac {30}{3600}}$ or $3{\frac {17}{120}}\equiv 3{\cdot }14166\ldots$.

The work of the Indians

We have now to pass over to the Indian Mathematicians. Áryabhatta (about 500 A.D.) knew the value

${\frac {62832}{20000}}\equiv 3{\cdot }1416$ for $\pi$.

The same value in the form 3927/1250 was given by Bhâskara (born 1114 A.D.)