A Refined Value of Pi
The irrational number pi can be computed to an infinite number of decimal places. It expresses the ratio of the circumference of a circle to its diameter, a relationship which cannot be framed in terms of whole numbers. (Pi is needed to compute the area of a circle or volume of a sphere.) The value of pi was computed by Archimedes to three decimal places, and by Ptolemy to four decimal places. But after that, for 1450 years, no greater accuracy was achieved in the Western world. The Chinese, however, made great strides forward in computing pi.
One way in which the ancient mathematicians tried to approach an accurate value for pi was to inscribe polygons with more and more sides to them inside circles, so that the areas of the polygons(which could be computed) would more and more closely approach the area of the circle. Thus, they could try to find a value for pi, since the circle's area was found by using the formula containing it. (They could measure the diameter, and squeeze a polygon whose area they knew into the circle; the only unknown number would be pi, which could then be calculated.) Archimedes used a 96-sided polygon, and decided that pi had a value between 3.142 and 3.140.
The Chinese tried to sneak up on pi in the same fashion, but they were better at it. Liu Hui in the third century AD started by inscribing a polygon of 192 sides in a circle, and then went on to inscribe one of 3072 sides which 'squeezed' even closer. He was thus able to calculate a value of pi of 3.14159. At this point, the Chinese overtook the Greeks.
But the real leap forward came in the fifth century AD, when truly advanced values for pi appeared in China. The mathematicians Tsu Ch'ung-Chih and Tsu Keng-Chih (father and son), by means of calculations which have been lost, obtained an 'accurate' value of pi to ten decimal places, as 3.1415929203. The circle used for the inscribing of the polygons is known to have been 10 feet across. This value for pi was recorded in historical records of the period, but the actual books of those mathematicians have vanished over the centuries. Nine hundred years later, the mathematician Chao Yu-Ch'in (about 1300 AD) Set himself to verify this value of pi. He inscribed polygons in a circle with the enormous number of 16,384 sides. By this means he confirmed the value given by the Tsu family. The Tsu family had a lead in the computation of pi of about 1200 years. Even by 1600 AD in Europe, the celebrated calculation of the value of pi by Adriaen Anthoniszoon and his son only gave 3.1415929, an approximate value extending to seven places, which still fell three short of the value found by the Tsu family.