"The Birds", a comedy written by the ancient Greek playwright Aristophanes, records a reference to Anaxagoras, a Greek philosopher noted for his scientific inquiries, and his attempt to find the value of pi, which Anaxagoras referred to as "squaring the circle". Although Anaxagoras apparently failed in his efforts, this is the first record of the Ancient Greek quest to determine the value of pi, a recurring theme in ancient Greek mathematics and philosophy.
Euclid's "Elements" provided a published "text book" of geometry, which provided axioms and methods many of which remain in use today, and which provided tools for those who would ultimately calculate pi using the methods of the geometrer.
In the third century BCE Archimedes of Syracuse, combined geometry with logical thinking that was a precursor of the methods of calculus, to determine a remarkbly accurate value for pi.
Archimedes inscribed and circumscribed regular polygons in and around a circle, using a method of upper and lower "bounds" which is conceptually similar to differential calculus.
Archimedes value for pi, using these "bounds" was that 3 10/71< pi < 3 1/7, or 223/7 < pi < 22/7. In numeric terms, this would equate to 3.14084... < pi < 3.14825... While Archimedes did not claim to have determined a precise value for pi, if we calculate the average of the upper and lower bounds for pi determined by Archimedes, the calculated value is pi - 3.1418. This is within approximately two ten-thousandths of the value of pi determined today.