In February 1897 the Indiana House of Representatives passed House Bill 246 by a vote of 67 to nil. The bill offered the state, free of royalties, a "new mathematical truth" devised by a country physician named Edwin J. Goodwin, who believed he had solved the ancient problem of squaring the circle. Buried in its confident prose were several mutually contradictory values of pi, one of which works out to 3.2. The bill sailed on to the Senate, where catastrophe was averted by luck: Clarence A. Waldo, a Purdue mathematics professor, happened to be at the statehouse on budget business, read the text in astonishment, and quietly coached the senators. The bill was postponed indefinitely, and pi escaped the only serious attempt in history to legislate its value.
The episode was doubly absurd because the question had already been closed for good — but to see why, you have to run the whole biography of the most famous number in mathematics: the ratio of a circle's circumference to its diameter, indifferent to the size of the circle and to the opinions of parliaments.
Estimates are as old as measurement itself. A Babylonian clay tablet implies a value of 25/8, which is 3.125. The Rhind Papyrus, an Egyptian mathematical text copied by the scribe Ahmes around 1650 BC, effectively uses the square of 16/9 — about 3.1605. The Hebrew Bible, describing a great bronze basin for Solomon's Temple as ten cubits across and thirty around, implies a workmanlike 3.
Archimedes draws polygons
The first person to trap the number rather than guess it was Archimedes of Syracuse, around 250 BC. He sandwiched a circle between two 96-sided polygons — one drawn just inside it, one just outside — and computed both perimeters, proving that pi lies between 3 10/71 and 3 1/7. That upper bound, 22/7, has been misremembered as "the value of pi" by schoolchildren ever since; it is merely a good ceiling. Archimedes' squeeze remained the world's best method for nearly two millennia, and its greatest practitioner worked far from Greece: the Chinese astronomer Zu Chongzhi, in the fifth century AD, reached the wonderfully accurate fraction 355/113, correct to six decimal places — a record that stood for roughly 900 years. Around 1600 the Dutch-based German fencing master and mathematician Ludolph van Ceulen spent decades grinding polygons to 35 decimal places, a feat commemorated on his tombstone in Leiden; Germans long called pi the "Ludolphine number" in his honour.
The modern era arrived in 1706, twice over. The Welsh schoolmaster William Jones became the first to use the Greek letter π for the ratio in print, and the astronomer John Machin used a new arctangent formula to blow past 100 digits. It was Leonhard Euler's adoption of Jones's symbol a generation later that made π universal. Then came the deeper discoveries about the number's character: Johann Lambert proved in 1768 that pi is irrational — no fraction will ever capture it exactly — and in 1882 Ferdinand von Lindemann proved it transcendental, satisfying no polynomial equation with rational coefficients. Lindemann's theorem killed circle-squaring forever, a full fifteen years before Dr Goodwin persuaded Indiana's lower house otherwise.
The digit hunters
Irrationality made the digits infinite; human nature made hunting them a sport. The Victorian calculator William Shanks devoted years to computing 707 digits by hand, publishing in 1873 — and in the 1940s D. F. Ferguson discovered that Shanks had blundered at the 528th place, invalidating everything after it. Then the machines took over: in 1949 the ENIAC computer produced 2,037 digits in about 70 hours, and the record has fallen almost continuously since. In 2022 a Google Cloud team led by developer advocate Emma Haruka Iwao computed 100 trillion digits, in a calculation that ran for more than 150 days. Such feats are stress-tests for hardware and algorithms rather than mathematics; the digits themselves are, so far as anyone has detected, statistically featureless.
The absurdity of the chase is part of its charm, because almost nobody needs more than a handful of digits. NASA's interplanetary navigators use about fifteen decimal places, and roughly 39 digits would suffice to compute the circumference of the observable universe to within the width of a hydrogen atom. Everything beyond that is pure sport — celebrated every 14 March, when the date writes itself 3.14 and the world eats pie in honour of a ratio.
The empires that first measured pi are dust, and the bill that tried to round it lies dead in a Senate archive. The number itself is exactly what it was before Babylon: unfinished, unrepeating, utterly indifferent — the circle keeps its secret to infinitely many places, and we keep counting.
Quiz nuggets
- Indiana's House Bill 246 of 1897, implying values of pi including 3.2, passed the House 67–0 before Purdue professor C. A. Waldo helped stop it in the Senate.
- The Rhind Papyrus, copied by the scribe Ahmes around 1650 BC, used (16/9)² — about 3.1605 — for pi.
- Zu Chongzhi's fifth-century fraction 355/113 remained the best approximation of pi for roughly 900 years.
- William Jones first used the symbol π in print in 1706; Euler's adoption made it standard.
- Emma Haruka Iwao's Google Cloud team computed 100 trillion digits of pi in 2022.