A Brief History of π

The history of π is one involving over 3,500 years and countless hours of menial (and often unnecessary) labor.

Way back over 3,000 years ago before the before the advent of modern miracles like sliced bread and aerosol cans are Ï€’s roots available for finding. The Egyptians after countless years of losing in battle due to having square wheels on their chariots decided to solve this wheel thingy once and for all so after many committee meeting and much arguing Ï€ was decided to be 3 1/7 and a certain bored scribe known as Ahmed recorded it in 1650 B.C.. The Egyptians remained the most powerful country in the area for many years mainly due to the fact that their neighbors, when they did decide to have round wheels, insisted that the ratio of circumference of chariot wheel to the length of spokes was exactly 3.

However it was the Greeks who really had too much time on their hands and devoted what time they were not watching soap operas to various arcane mathematical amusements. It was thus that around 500 B.C. a certain unknown would be burger flipper came up with the idea that by putting a polygon inside a circle and calculating the polygon’s area and dividing by its circumference they could get a rough estimate as to the ratio of circumference to diameter of a circle (aka. that irrational, Ï€) and eventually if the polygon had enough sides it would be a circle. In the 4th century B.C. a certain oddball who like to chant ‘Eureka!’ in the streets of Syracuse by the name of Archimedes estimated Ï€ to be between 3 10/71 and 3 1/7 using a 96 sided polygon. Nearly 500 years elapsed before a ‘civilization’ had enough of a breather to devote human resources to the Ï€ problem. So it was Claudius ‘the sun revolves around me’ Ptolemy that discovered Ï€ was darned close to 377/120. Two century later the Indian, Arbhyata estimated Ï€ to be 3.1416.

It took 1100 years for the value ascribed to π to make progress to the fact that people kept unaccountably dying from plagues and the people who survived were far too busy farming sugar beets to pay attention to mathematical matters and it took a brilliant high school dropout named Adriaen Romanus to calculate 15 correct digits of π. In 1610 Ludolph Van Ceulen, perhaps the most mind numbingly unnoccupied person of all time, published 35 digits of π. He had spent most of his life on the calculations and used a polygon with over 4 billion sides to make them. However there were simply not enough people who had the time to calculate π so fortunately for the history of π a new method of π calculation was found.

First John Wallis discovered an infinite rational (though not entirely sane) product method for calculating Ï€. Then James Gregory discovered the arctangent series. Wilhelm Leibniz put these two together to come up with an irrational infinite product of arctangent formula for Ï€ involving arctangent calculations and lots of good hot coffee. However this method rapidly gained popularity due in part to the fact that the coffee and arctangent formula was a lot quicker than the inscribed polygon and morphine approach. In 1699 Abraham Sharp calculated 72 places of Ï€ using this newfangled method; fortunately he didn’t burn himself with the coffee.

Soon the simplicity of arctangent formulas attracted other impoverished lawyers to the race. John Machin calculated 100 digits of Ï€ in 1706 whil Thomas Fantet de Lagny followed suit with 127 digits in 1719. After a brief respite in calculations Georg Vega calculated 140 digits of fruitcake… er Ï€ in 1794. In 1844 200 digits of Ï€ were calculated by Johann Dase and L.K. Stassnitzky and barely 11 years later 500 digits were calculated by Richter. However when in 1874 William Shanks published his calculation of 707 digits of Ï€ in England he was too zonked out to notice a small error at the 527th place making the last 25% of his digits as very close to random as was possible without a computer.

It took until 1945 before anyone had the stamina or technology to check Shanks calculation so with the aid of a desk calculator he came up with 808 correct digits of Ï€ in 1947. This draft dodger was D.F. Ferguson. However the developement of the ENIAC soon rendered this work a waste of time. It wasn’t much of a computer but it did manage to calculate 2,037 digits of Ï€ in 70 hours. The amazing thing was not the fact that the ENIAC was so slow, or the fact that they could fit it inside retail office space, but that it did the calculation all by itself.

With the arrival of computers pen and paper calculations were rendered obsolete and Ï€ calculators became programmers rather than mathematicians. In 1955 3,000+ digits were calculated at the Naval Ordnance Research Center in only 13 minutes, that’s almost 500 times faster than the ENIAC only 6 years later. In 1959 an IBM 704 calculated 16,000+ digits of Ï€. In 1961 an IBM 7090 calculated over 100,000 digits of Ï€ in around 9 hours, in 1966 an IBM 7030 calculated 250,000 digits of Ï€, a year later a CDC 6600 calculated 500,000 digits and in 1973 a CDC 7600 calculated 1,000,000 digits of Ï€ in 23 hours.

However the techniques for calculating Ï€ still used arctangents which have an n^2 growth rate. In 1976 Eugene Salamin rediscovered a formula by Gauss that while too calculation intensive in Gauss’ time was well suited for modern super computers the size of the Whitehouse and had a much lower growth rate than arctangents formulas did. The results was in 1982 a HITAC M-280H calculated 16 million digits of Ï€ in 30 hours, in 1988 a Hitachi S-820 calculated 201 million digits of Ï€ in 6 hours, in 1989 both the 500 million and 1 billion Ï€ calculation marks were broken, in 1995, 6 billion digits were calculated, in 1996 8 billion were and finally in 1997 a Hitachi SR2201 calculated 51 billion digits of Ï€ in 29 hours. This Hitachi machine had 1024 processors and 212 Gigabytes of RAM. However in September of 1999 a new record of 206,158,430,000 was announced. The calculation was again by Yasumasa Kanada and the University of Tokyo. The calculations took over 37hours and 43 more to verify. The machine used contained 817GB of main memory and consisted of 128 Hitachi SR8000 processors. In 2002, taking some 400 hours on a newer Hitachi machine, Kanada again broke the record, this time with 1.24 trillion digits. This is the current standing record.

(for further details on Ï€, see “The Joy of Pi“, by David Blatner, a truly useless… err remarkable book).