祖冲之Zu Chongzhi

祖冲之的名字有时写作祖冲之 Ch'ung 祖冲之。他来自一个著名的家族，原籍中国北方的河北省。他的曾祖父是东晋朝廷的官员，东晋朝廷建立于建康（今南京）。由于朝廷阴谋而削弱，东晋在420年的一次叛乱后被刘宋王朝取代。祖冲之的祖父和父亲都曾担任刘宋王朝的官员，刘宋王朝的朝廷也在建康（今南京）。

祖氏家族是一个极具天赋的家族，连续几代人除了担任朝廷官员外，还是对历法有特殊兴趣的天文学家。在古代中国，有一种信仰，认为皇帝从上天获得统治权。为新皇帝专门制定历法，建立了从上天到特定统治者的联系。这意味着天文学家在朝廷中扮演重要角色，因为他们的技能可以导致皇帝的成功统治。祖氏家族将他们的数学和天文技能从父亲传给儿子，事实上，这是这些技能传播的主要方式之一。

祖冲之按照家族传统，在成长过程中学习了各种技能。特别是，他从才华横溢的父亲那里学习了数学、天文学和历法科学。他从多个来源学习数学，但主要来自刘徽对九章算术的注释。祖还学习了其他技能，因为他在工程方面表现出色，并且擅长文学创作，写了十部小说。祖冲之遵循家族传统为皇帝服务。他被孝武帝（在位454年至464年）任命为扬州（江苏省的一个城市）的官员，然后担任建康（今南京）军事参谋部的官员。

在此期间，祖从事数学和天文学工作。特别是，他正在研究一种新的、更准确的历法。一直使用的历法基于19年周期，年份由12个月组成，每月29或30天。在19年中有7年插入一个额外的月份，使其成为基于太阳和月亮的历法，19年中有235个月。这在412年改为基于600年周期的历法，在221年中插入一个额外的月份。这个历法对祖来说不够准确。

公元462年，祖冲之向皇帝提出了一种新历法——《大明历》，它以391年为一个周期。在这391年中有144年插入了一个额外月份，因此391年中共有4836个月。他之所以能制定出如此精确的历法，是因为他计算出回归年（连续两次春分之间的时间）的长度为365.24281481日（与真值365日5小时48分46秒仅差50秒），以及月球的交点月为27.21233日（现代值为27.21222日）。

然而，就他的历法而言，祖在朝廷有一个对手。这是戴法兴，皇帝的一位大臣，他宣称祖：-

……歪曲天理，违反经典教导。

祖回答说，他的历法：-

……不是来自神灵或鬼魂，而是来自仔细的观察和精确的数学计算。……人们必须愿意听取并审视证明，才能理解真理和事实。

尽管有像戴法兴这样强大的对手，祖冲之还是获得了孝武帝对其历法的批准，大明历原定于464年启用。然而，孝武帝在464年历法推行前去世，他的继任者被戴法兴说服，取消了新历法的推行。祖冲之在孝武帝去世后离开朝廷，全身心投入科学研究。

当然，询问数字144和391从何而来并非不合理。准确掌握年和月的长度是必要的，但祖冲之如何将其转化为391年的周期仍不清楚。在5中提出，祖冲之发现一年有$365 \large\frac{9589}{39491}\normalsize$天，一个月有$\large\frac{116321}{3939}\normalsize$天。这给出

$12 \large\frac{1691772624}{4593632611}\normalsize$

一年中的月数。但祖冲之会知道如何通过分子分母同除以最大公约数来将分数化为最简形式。这样做得到

$\large\frac{1691772624}{4593632611}\normalsize = \large\frac{144}{391}\normalsize$

因此391年中有144年多出一个月。

在结束对祖冲之天文工作的讨论之前，我们再给出他在这一领域工作的更多细节。他并不是第一个发现岁差的中国天文学家（虞喜在四世纪就发现了），但他是第一个在历法计算中考虑到这一点的人。由于岁差，回归年比恒星年（太阳返回到相对于背景恒星的同一位置所需的时间）短约21分钟。祖冲之对年长的计算精度足以让他区分回归年和恒星年。木星绕其轨道一周约需12年，但祖冲之能给出比这精确得多的值。他发现，在7个12年的周期中，木星完成了七又十二分之一圈轨道，得出其恒星周期为11.859年（精确到四千分之一以内）。

他在其著作Zhui shu（《缀术》）中给出了有理的近似值$\large\frac{355}{113}\normalsize$，精确到小数点后6位。他还证明了

$3.1415926 < \pi < 3.1415927$

一个引人注目的结果，若能了解更多细节就好了。遗憾的是，祖冲之的著作已失传。据History of the Sui dynasty记载，该书由李淳风等人于7世纪编纂，其中提到（不同译本见[1]或[3]）：-

祖冲之进一步设计了一种精确的方法[计算]。取直径为10,000,000丈的圆，他发现该圆的周长小于31,415,927丈，大于31,415,926丈。他从这些结果推断出圆周长的准确值必定介于这两个值之间。因此，圆周长与其直径之比的精确值为355比113，近似值为22比7。

为了将π计算到这一精度，祖必定使用了内接正24,576边形，并进行了极其冗长的计算，涉及数百次平方根运算，全部达到9位小数精度。由于他的著作已失传，我们永远无法确切知道他如何从小数近似值得到有理的近似值$\large\frac{355}{113}\normalsize$。然而，历史学家认为他知道

如果$\Large\frac a b\normalsize \le \Large\frac c d$则$\Large\frac a b\normalsize \le \Large\frac {a+c}{b+d}$

对任意整数$a, b, c, d$。他进而知道

$3 ≤ \pi ≤ \large\frac{22}{7}\normalsize$

因此，近似地，

$\pi = 3.1415926 = (3x + 22y)/(x + 7y)$

给出$y = 16x$的近似值，所以

$\pi = \Large\frac{3x + 22 \times 16x}{x + 7 \times 16x}\normalsize = \Large\frac {355}{113} \\$

Martzloff在[3]或[4]中提出了另一种可能的方式，即祖可能凭运气而非数学技巧发现了$\large\frac{355}{113}\normalsize$。然而，鉴于祖的工作被认为非常困难且高深，它是由幸运的数值巧合所发现的可能性不大。

公元656年，经李淳风编辑后，Zhui shu（Method of Interpolation）成为科举考试的教材，并在1084年重印时成为数学十大经典之一。然而，Zhui shu对国子监的学生来说过于高深，因此被从教学大纲中删除。这几乎可以肯定地解释了为什么该文本未能流传下来，在十二世纪初就失传了。

在他的晚年，祖冲之与他的儿子祖暅（或祖暅）合作，后者也是一位杰出的数学家。

Zu Chongzhi's name is sometimes written as Tsu Ch'ung Chi. He came from a famous family who were originally from Hopeh province in northern China. His great grandfather was an official at the court of the Eastern Chin dynasty which had been established at Jiankang (now Nanking). Weakened by court intrigues, the Eastern Chin dynasty was replaced after a revolt by the Liu-Sung dynasty in 420. Zu Chongzhi's grandfather and father both served as officials of the Liu-Sung dynasty which also had its court at Jiankang (now Nanking).

The Zu family was an extremely talented one with successive generations being, in addition to court officials, astronomers with special interests in the calendar. In ancient China there was a belief that an emperor received his right to rule from heaven. Producng a calendar specifically for a new emperor established a link from the heavens to the particular ruler. This meant that astronomers had important roles at court for their skills could result in an emperor's successful rule. The Zu family handed their mathematical and astronomical skills down from father to son and, indeed, this was one of the main ways that such skills were transmitted.

Zu Chongzhi, in the family tradition, was taught a variety of skills as he grew up. In particular he was taught mathematics, astronomy and the science of the calendar from his talented father. He learnt mathematics from a number of sources, but mainly from Liu Hui's commentary on the Nine Chapters on the Mathematical Art. Zu learnt other skills too for he excelled in engineering and was skilled in literary composition writing ten novels. Zu Chongzhi followed in the family tradition of serving the emperors. He was appointed by the Emperor Xiao-wu (who ruled from 454 to 464) first as an officer in Yang-chou, a city in Kiangsu, and then as an officer in the military staff in Jiankang (now Nanking).

During this time Zu worked on mathematics and astronomy. In particular he was working on a new, more accurate calendar. The calendar which had been in use was based on a 19 year cycle with years consisting of 12 months of 29 or 30 days. In seven of the 19 years an extra month was inserted making it a calendar based both on the sun and the moon with 235 months in 19 years. This had been changed in 412 to a calendar based on a 600 year cycle with an extra month inserted in 221 of the years. This calendar was not accurate enough for Zu.

In 462 Zu proposed a new calendar, the Tam-ing Calendar (Calendar of Great Brightness), to the Emperor which was based on a cycle of 391 years. In 144 of the 391 years an extra month was inserted, so there were 4836 months in 391 years. He was able to make a calendar with this degree of accuracy since he had calculated the length of the tropical year (time between two successive occurrences of the vernal equinox) as 365.24281481 days (an error of only 50 seconds from its true value of 365 days 5 hours 48 minutes 46 seconds), and a nodal month for the moon of 27.21233 days (compare the modern value of 27.21222 days).

Zu, however, had an opponent at the court as far as his calendar was concerned. This was Tai Faxin, one of the Emperor's ministers, who declared that Zu was:-

... distorting the truth about heaven and violating the teaching of the classics.

Zu replied that his calendar was:-

... not from spirits or from ghosts, but from careful observations and accurate mathematical calculations. ... people must be willing to hear and look at proofs in order to understand truth and facts.

Despite having such a powerful opponent as Tai Faxin, Zu won approval for his calendar from Emperor Xiao-wu and the Tam-ing calendar was due to come into use in 464. However, Xiao-wu died in 464 before the calendar was introduced, and his successor was persuaded by Tai Faxin to cancel the introduction of the new calendar. Zu left the imperial service on the death of Emperor Xiao-wu and devoted himself entirely to his scientific studies.

Of course, it is not unreasonable to ask where the numbers 144 and 391 came from. Having accurate knowledge of the lengths of the year and the month were necessary, but it is still not clear how Zu translated this into a cycle of 391 years. In [5] it is suggested that Zu found that there were $365 \large\frac{9589}{39491}\normalsize$ days in a year and $\large\frac{116321}{3939}\normalsize$ days in a month. This gives

$12 \large\frac{1691772624}{4593632611}\normalsize$

months in a year. But Zu would know how to reduce fractions to their lowest terms by dividing top and bottom by the greatest common divisor. Doing this gives

$\large\frac{1691772624}{4593632611}\normalsize = \large\frac{144}{391}\normalsize$

and hence the extra month in 144 out of 391 years.

Before we leave our discussion of Zu's astronomical work we give further details of his work in this area. He was not the first Chinese astronomer to discover the precession of the equinoxes (Yu Xi did so in the fourth century) but he was the first to take this into account in calendar calculations. Because of the precession of the equinoxes the tropical year is shorter by about 21 minutes than the sidereal year (the time taken by the Sun to return to the same place against the background stars). Zu's calculations of the length of the year were well within the range that allowed him to differentiate between the tropical and sidereal year. Jupiter takes about 12 years to complete its orbit but Zu was able to give a much more accurate value than that. He discovered that in 7 cycles of 12 years, Jupiter had completed seven and one twelfth orbits, giving its sidereal period as 11.859 years (accurate to within one part in 4000).

He gave the rational approximation $\large\frac{355}{113}\normalsize$ to in his text Zhui shu (Method of Interpolation), which is correct to 6 decimal places. He also proved that

$3.1415926 < \pi < 3.1415927$

a remarkable result about which it would be nice to have more details. Sadly Zu Chongzhi's book is lost. It is reported in the History of the Sui dynasty, compiled in the 7th century by Li Chunfeng and others, that (see [1] or [3] for a different translation):-

Zu Chongzhi further devised a precise method [of calculating ]. Taking a circle of diameter 10,000,000 chang, he found the circumference of this circle to be less than 31,415,927 chang and greater than 31,415,926 chang. He deduced from these results that the accurate value of the circumference must lie between these two values. Therefore the precise value of the ratio of the circumference of a circle to its diameter is as 355 to 113, and the approximate value is as 22 to 7.

To compute this accuracy for π, Zu must have used an inscribed regular 24,576-gon and undertaken the extremely lengthy calculations, involving hundereds of square roots, all to 9 decimal place accuracy. Since his book is lost we will never know exactly how he found the rational approximation $\large\frac{355}{113}\normalsize$ from the decimal approximation. Historians believe, however, that he knew that

if $\Large\frac a b\normalsize \le \Large\frac c d$ then $\Large\frac a b\normalsize \le \Large\frac {a+c}{b+d}$

for any integers $a, b, c, d$. He then knew that

$3 ≤ \pi ≤ \large\frac{22}{7}\normalsize$

so, approximately,

$\pi = 3.1415926 = (3x + 22y)/(x + 7y)$

giving $y = 16x$ approximately, so

$\pi = \Large\frac{3x + 22 \times 16x}{x + 7 \times 16x}\normalsize = \Large\frac {355}{113} \\$

Martzloff, in [3] or [4], presents another possible way that Zu might have found $\large\frac{355}{113}\normalsize$ by luck rather than mathematical skill. However, given that Zu's work was considered very difficult and advanced, it is doubtful that it was found by a lucky numerical accident.

In 656, after editing by Li Chunfeng, the treatise Zhui shu (Method of Interpolation) became a text for the Imperial examinations and it became one of The Ten Classics when reprinted in 1084. However, the Zhui shu was too advanced for the students at the Imperial Academy and it was dropped from the syllabus for that reason. This almost certainly explains why the text has not survived, being lost in the early twelfth century.

In the latter part of his life Zu Chongzhi collaborated with his son, Zu Geng (or Zu Xuan), who was also an outstanding mathematician.