Alex's Adventures in Numberland - Alex Bellos [62]
Leibniz had devised the formula using ‘the calculus’, a powerful type of mathematics he had discovered, in which a new understanding of infinitesimal amounts was used to calculate areas, curves and gradients. Isaac Newton had also come up with calculus, independently, and the men spent a good deal of time bickering about who had got there first. (For years, Newton was considered to have won the argument, based on the dates of his unpublished manuscripts, but it now appears that a version of calculus was actually first invented in the fourteenth century by the Indian mathematician Madhava.)
The formula that Leibniz found for pi is what is known as an infinite series, a sum that goes on and on for ever, and it provides a way to calculate pi. First, we need to multiply both sides of the formula by 4 to get:
Starting with the first term and adding successive terms produces the following progression (converted into decimals):
The total gets closer and closer to pi in smaller and smaller leaps. Yet this method requires more than 300 terms to get an answer for pi accurate to two decimal places, so it was impracticable for those wanting to use it to find more digits in the decimal expansion.
Eventually, calculus provided other infinite series for pi that were less pretty but more effective for number-crunching. In 1705 the astronomer Abraham Sharp used one to calculate pi to 72 decimal places, smashing van Ceulen’s century-old record of 35. While this was quite an achievement, it was also a useless one. There is no practical reason to know pi to 72 digits, or to 35 digits for that matter. Four decimal places is enough for the engineers of precision instruments. Ten decimal places is sufficient to calculate the circumference of the Earth to within a fraction of a centimetre. With 39 decimal places, it is possible to compute the circumference of a circle surrounding the known universe to within an accuracy of a radius of a hydrogen atom. But practicality wasn’t the point. Application was not a concern to the pi men of the Enlightenment; digit-hunting was an end in itself, a romantic challenge. A year after Sharp’s effort, John Machin reached 100 digits and in 1717 the Frenchman Thomas de Lagny added a further 27. By the turn of the century, the Slovenian Jurij Vega was in the lead with 140.
Zacharias Dase, the German lightning calculator, pushed the record for pi to 200 decimal places in 1844 in an intense two-month burst. Dase used the following series, which looks more convoluted than the pi formula above but is in fact far more user-friendly. This is because, first, it bears down on pi at a respectable rate. An accuracy of two decimal places is reached after the first nine terms. Second, the , and that reappear every third term are very convenient to manipulate. If is rewritten as and as ××, all the multiplications involving those terms can be reduced to combinations of doubling and halving. Dase would have written out a reference table of doubles to help him with his calculations, starting 2, 4, 8, 16, 32, and carried on for as far as he needed – which, since he was calculating pi to 200 places, would be when the final double is 200 digits long. This happens after 667 successive doublings.
Dase used this series:
After one term this is 3.3
After two terms this is 3.1200
After three terms this is 3.1452
Dase barely had time to rest on his laurels before the Brits decided to covet his achievement, and within a decade William Rutherford had calculated pi to 440 places. He encouraged his protégé William Shanks, an amateur mathematician who ran a boarding school in County Durham, to go even further. In 1853 he reached 607 digits, and by