(2) The relations for the successive calculation of the perimeters and areas of polygons inscribed and circumscribed to a circle.
Denoting by , , the perimeter and area of an inscribed regular polygon of sides, and by , the perimeter and area of a circumscribed regular polygon of sides, these relations are
, ,
, .
Thus the two series of magnitudes
,
,
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 145 and 160 for sin 1°.
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 12° 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 more exact than that of Archimedes; this was expressed in sexagesimal measure by 3° 8′ 30″ which is equivalent to
or .
We have now to pass over to the Indian Mathematicians. Áryabhatta (about 500 A.D.) knew the value
for .
The same value in the form 39271250 was given by Bhâskara (born 1114 A.D.)