Historical Figures in Mathematics In reading about these areas of mathematics that we often take as obvious. it often was, once someone had worked it out beforehand. This series is about some of the lesser-known of those people.

Zu Chongzhi: the calculation of PI (π) (conclusion) Last time we met Chinese mathematician and polymath Zu Chonzhi and his much more accurate calendar whose adoption was opposed by an influential minister of the Emperor

Zu replied that his calendar was: 12 1691772624 /4593632611 ... not from spirits or from ghosts, months in a year. But Zu would but from careful observations and know how to reduce fractions to accurate mathematical their lowest terms by dividing top calculations. ... people must be and bottom by the greatest willing to hear and look at proofs in common divisor. Doing this gives order to understand truth and facts. 1691772624 /4593632611 = 144 /391 Despite having such a powerful (the common devisor being opponent as Tai Faxin, Zu won 11748421, of course) approval for his calendar from and hence the extra month in 144 Emperor Xiaowu and the Taming out of 391 years. calendar was due to come into use in 464. However, Xiaowu died in Before we leave our discussion of 464 before the calendar was Zu's astronomical work we give introduced, and his successor was further details of his work in this persuaded by Tai Faxin to cancel area. He was not the first Chinese the introduction of the new astronomer to discover the calendar. Zu left the imperial precession of the equinoxes (Yu Xi service on the death of Emperor did so in the fourth century) but he Xiaowu and devoted himself was the first to take this into entirely to his scientific studies. account in calendar calculations. Because of the precession of the Of course, it is not unreasonable to equinoxes the tropical year is ask where the numbers 144 and shorter by about 21 minutes than 391 came from. Having accurate the sidereal year (the time taken by knowledge of the lengths of the year the Sun to return to the same place and the month were necessary, but against the background stars). Zu's it is still not clear how Zu calculations of the length of the translated this into a cycle of 391 year were well within the range that years. It has been suggested that allowed him to differentiate between Zu found that there were 365 the tropical and sidereal year. 9589 /39491 days in a year and Jupiter takes about 12 years to 116321 /3939 days in a month. This complete its orbit but Zu was able gives to give a much more accurate value than that. He discovered that in 7 If a/b ≤ c/d then a/b ≤ (a+c) /(b+d ) ≤ c/d cycles of 12 years, Jupiter had for any integers a, b, c, d. He then completed seven and one twelfth knew that orbits, giving its sidereal period as 11.859 years (accurate to within 3 ≤ π ≤ 22 /7 one part in 4000). so, approximately, He gave the rational approximation π = 3.1415926 = (3x + 22y) /(x + 7y) 355 /113 to π in his text Zhui shu (Method of Interpolation), which is giving y = 16x approximately, so correct to 6 decimal places. He also proved that π = (3x + 22 × 16x) /(x + 7 × 16x) = 355 /113 . This accuracy in the calculation of 3.1415926 < π < 3.1415927 π was not achieved in the west for a remarkable result about which it over a millennium, being discovered would be nice to have more details. in 1585 by the Dutch Sadly Zu Chongzhi's book is lost. It mathematician Adrian is reported in the History of the Sui Anthoniszoom. dynasty , compiled in the 7th century by Li Chunfeng and others, Martzloff presents another possible that: way that Zu might have found 355 /113 by luck rather than Zu Chongzhi further devised a mathematical skill. However, given precise method [of calculating]. that Zu's work was considered very Taking a circle of diameter difficult and advanced, it is 10,000,000 chang, he found the doubtful that it was found by a circumference of this circle to be less lucky numerical accident. than 31,415,927 chang and greater than 31,415,926 chang. He deduced In 656, after editing by Li from these results that the accurate Chunfeng, the treatise Zhui shu value of the circumference must lie (Method of Interpolation) became a between these two values. Therefore text for the Imperial examinations the precise value of the ratio of the and it became one of The Ten circumference of a circle to its Classics when reprinted in 1084. diameter is as 355 to 113, and the However, the Zhui shu was too approximate value is as 22 to 7. advanced for the students at the Imperial Academy and it was To compute this accuracy for π, Zu dropped from the syllabus for that must have used an inscribed reason. This almost certainly regular polygon of 24,576 sides and explains why the text has not undertaken the extremely lengthy survived, being lost in the early calculations, involving hundreds of twelfth century. square roots, all to 9 decimal place In the latter part of his life Zu accuracy. Since his book is lost we Chongzhi collaborated with his son, will never know exactly how he Zu Geng (or Zu Xuan), who was found the rational approximation also an outstanding 355 / from the decimal 113 mathematician. approximation. Historians believe, however, that he knew that