emi-side of a circumscribed regular 12-gon; then as AB:BO:OA::1:
sqrt(3):2 he sought an approximation to sqrt(3) and found that AB:BO >
153:265. Next he applied his theorem[4] BO + OA:AB::OB:BD to calculate
BD; from this in turn he calculated the semi-sides of the circumscribed
regular 24-gon, 48-gon and 96-gon, and so finally established for the
circumscribed regular 96-gon that perimeter:diameter < (3-1/7):1. In a
quite analogous manner he proved for the inscribed regular 96-gon that
perimeter:diameter > 3-(10/71):1. The conclusion from these therefore
was that the ratio of circumference to diameter is < 3-1/7 and >
3-(10/71). This is a most notable piece of work; the immature condition
of arithmetic at the time was the only real obstacle preventing the
evaluation of the ratio to any degree of accuracy whatever.[5]

No advance of any importance was made upon the achievement of Archimedes
until after the revival of learning. His immediate successors may have
used his method to attain a greater degree of accuracy, but there is
very little evidence pointing in this direction. Ptolemy (fl. 127-151),
in the _Great Syntaxis_, gives 3.141552 as the ratio[6]; and the Hindus
(c. A.D. 500), who were very probably indebted to the Greeks, used
62832/20000, that is, the now familiar 3.1416.[7]

It was not until the 15th century that attention in Europe began to be
once more directed to the subject, and after the resuscitation a
considerable length of time elapsed before any progress was made. The
first advance in accuracy was due to a certain Adrian, son of Anthony, a
native of Metz (1527), and father of the better-known Adrian Metius of
Alkmaar. In refutation of Duchesne(Van der Eycke), he showed that the
ratio was < 3-(17/120) and > 3-(15/106), and thence made the exceedingly
lucky step of taking a mean between the two by the quite unjustifiable
process of halving the sum of the two numerators for a new numerator and
halving the sum of the two denominators for a new denominator, thus
arriving at the now well-known approximation 3-(16/113) or 355/113,
which, being equal to 3.1415929..., is correct to the sixth fractional
place.[8]

The next to advance the calculation was Francisco Vieta. By finding the
perimeter of the inscribed and that of the circumscribed regular polygon
of 393216 (i.e. 6 X 2^16) sides, he proved that the ratio was >
3.1415926535 and < 3.1415926537, so that its value became known (in
1579) correctly to 10 fr