Incidents in the early life of pi

I have been wondering when and why the Greek letter pi (π) was chosen to represent the ratio of the circumference of a circle to its diameter.

I have learnt that, although the mathematical concept of pi has been known for at least 4,000 years, the use of p as its symbol is only 300 years old. It was introduced in 1706 by a Welsh mathematician, William Jones, who used pi because it was the initial letter of the Greek word for perimeter, perimetros.

His choice of symbol was taken up in 1737 by Leonhard Euler, the pre-eminent mathematician of the 18th century, and with Euler’s endorsement it went on to become the standard symbol for the concept.

I have also looked into the history of estimates of the value of pi. The earliest known approximations date from around 1900BC, when a Babylonian text used 25/8 (3.1250) and an Egyptian text used 256/81 (3.1605). Both are within 1 per cent of the true value.

The first rigorous estimation of the value was made by Archimedes of Syracuse (287–212BC), one of the most brilliant mathematicians of the ancient world. He worked out that pi was between 223/71 (3.1408) and 22/7 (3.1429). His results rested on estimating the area of a circle from the areas of two regular 96-sided polygons, one inscribed within the circle and one circumscribing it.

Later mathematicians refined his method and in about AD480 the Chinese mathematician Zu Chongzhi (AD429–500) calculated the circumference of 12,288-sided polygons to show (correctly) that the value of pi is between 3.1415926 and 3.1415927. He also worked out that 355/113 (3.1415929) was an excellent approximation.

Zu Chonzhi’s values remained the most accurate for centuries. It was not until 1400 that pi was calculated to more than 10 decimal places. The first calculation to 100 decimal places was made by English astronomer, John Machin, in 1706 — coincidentally, the year in which William Jones first used the symbol pi.

Nowadays, of course, computers can calculate pi’s value to schmillions of decimal places.