郭守敬Guo Shoujing

郭守敬也被称为郭守敬。我们不知道他父母的名字，但他的祖父显然比他的父母更有名，是郭守敬荣，以从古典研究到数学和水力学等广泛领域的专家而闻名。人们只能猜测，郭守敬小时候可能受到祖父的影响，对建造水钟产生了兴趣。我们知道，郭守敬在14岁时确实建造了这样一座钟。他设计了一种莲花漏，即一种顶部有一个莲花形碗、水从中滴入的水钟。到十六岁时，郭守敬正在学习数学。

在我们继续描述郭守敬一生中的事件之前，我们应该简要看看当时的政治局势，因为那是一个动荡不安、战争频仍的时代。成吉思汗死后，他的一个儿子窝阔台于1229年成为大汗。他扩张了蒙古帝国，派遣军队完成了对女真人的征服。到1234年，蒙古人已经完成了对女真帝国的摧毁，并将注意力转向南方。这就是郭守敬成长的中国北方当时的局势。到二十岁时，郭守敬已作为一名水利工程师工作。1251年，作为一名政府官员，他参与了一个修复大活泉河上一座旧桥的项目。这条河位于郭守敬的家乡省份河北，他翻修的这座桥在忻州镇以北不远处。

忽必烈，蒙古领袖成吉思汗的孙子，在1250年代后期开始领导蒙古人进一步推进。1260年5月5日，忽必烈在上都的住所被推举为大汗，并开始组织国家。张文谦是郭守敬的朋友，和他一样也是中央政府官员，1260年被忽必烈汗派往大名，因为据报当地民众发生了动乱。郭守敬陪同张文谦执行这一使命。郭守敬不仅对工程感兴趣，而且还是一位专业天文学家。特别是，他是一位技艺娴熟的仪器制造者，并明白良好的天文观测依赖于制作精良的仪器。他现在开始建造天文仪器，包括用于精确计时的水钟和代表天球的浑仪。

张文谦向忽必烈汗建议，他的朋友郭守敬是水利工程方面的顶尖专家。忽必烈知道水管理对于灌溉、粮食运输和防洪的重要性，于是请郭守敬考察大都（今北京）与黄河之间地区的这些方面。为了给大都提供新的水源，郭守敬在神山找到了白浮泉，并修建了一条30公里长的渠道，将水引到大都。他提议跨不同流域连接供水，修建带有许多水闸以控制水位的新运河，并在他能够进行的改进方面取得了巨大成功。这使忽必烈汗很高兴，并导致郭守敬被要求在全国其他地区开展类似项目。1264年，他被派往甘肃省，修复蒙古人推进该地区期间多年战争对灌溉系统造成的破坏。郭守敬与他的朋友张文谦一起广泛旅行，记录需要完成的工作，以疏通系统受损部分并提高其效率。他直接将报告呈送给忽必烈汗。

蒙古人的推进在忽必烈汗的领导下继续进行，1276年他攻占了今天称为杭州的城市，位于上海以南。多年来，历法改革的需要已被认识到，但忽必烈汗看到了此时引入新历法的政治好处，以强调新政权如何取代旧政权。当被征求建议时，郭守敬解释说：-

... 基本方法是在天文变化的领域进行观测和测试。

郭守敬与王宪钟询一起，被要求设立一个特别局，进行必要的研究并为新历法提出建议。准确的历法取决于用来支持它的天文数据的质量，而郭守敬的第一步是建造十七件新的天文仪器，以便收集准确的数据。在这十七件仪器中，有十三件将安装在忽必烈首都大都（今称北京）的一座天文台中，而另外四件是可携带的仪器，可以从不同地点进行观测。

让我们简要描述郭守敬在建造这样一件仪器时所做的改进。最简单的天文仪器是圭表，不过是一根竖立的杆子，测量其影子的长度。一天中影子的最小长度在夏季比冬季短，在至点时它从变长变为变短，或反之。为了更容易确定这些点，郭守敬在圭表顶部固定了一根横杆，并利用针孔相机的原理将其影子投射到测量刻度上。

忽必烈汗于1279年在北京建立了一座天文台。该年三月开始建造，按照郭守敬提出的设计，工程在两个月内完成。郭守敬的朋友张文谦被任命为台长，郭守敬与他的同事王宪钟询是两位共同台长。理解从仪器收集的数据需要球面三角学的知识，郭守敬设计了一些非凡的公式。我们下面看看他在进行新历法项目时引入的巧妙数学。

这项工作在1280年完成，郭守敬计算出的年长准确到26秒以内，次年忽必烈汗引入了这部极其准确的历法。它使用了364年。张文谦于1283年去世，郭守敬被提升为北京天文台的台长。1292年，除了天文台台长的职务外，他还被任命为都水监的负责人。他现在承担了设计运河系统以连接首都与其他主要城镇的重大项目。他一如既往地取得了成功，甚至在忽必烈汗去世后，尽管郭守敬此时已是一位老人，忽必烈的继任者仍继续征求他的建议。

我们现在应当看看郭守敬在球面三角学和求解方程方面所做的相当卓越的工作。他为三角形给出了若干公式，其中两边是直线，第三边是圆弧。这些公式是近似公式，但郭守敬对此十分清楚。在某种意义上，中国人并不认为近似很重要，他们从未像古希腊人那样痴迷于“化圆为方”这类问题，因为中国人的方法更实用，而且从来不是公理化的。
插图：Guo_Shoujing.gif ↗
图中$d$是圆的直径，$a$是弧$AB$的长度，$x$是$NB$的长度，郭守敬想要计算的就是它。他给出了近似公式

$x^{4} + (d^{2} - 2ad)x^{2} - d^{3}x + a^{2}d^{2} = 0$。

为了求解这个方程，郭守敬使用了一种类似于威廉·乔治·霍纳方法的数值方法。该方程有两个实根，较小的那个是问题的解，而另一个在数值上大于弧长，被郭守敬正确地舍弃了。

与这项工作相关有一个引人入胜的事实，值得探究。方程的两个系数，即常数项和$x^{2}$的系数，都涉及弧长$a$，因此需要为π选取一个值。郭守敬取π = 3，乍看之下这似乎很奇怪，因为在郭守敬进行计算的年代，中国已经知道精确得多的值。例如，已知$\large\frac{22}{7}\normalsize$更精确，而由祖冲之给出的$\large\frac{355}{113}\normalsize$则已知更加精确。郭守敬为什么选择π = 3？令人惊讶的是，这给出的答案比更精确的π值还要好，因为要记住这个公式本身也是基于近似的。人们不得不相信，郭守敬选择π = 3，是因为他知道，他由此针对$O$处不同大小的角所求得的答案，更接近他通过直接测量得到的值。

下面是一个来自计算机代数系统的计算，用以说明这一点。这些值是对半径为100的圆、按递增到90°的角计算出来的。第一列是使用郭守敬的公式并取π的精确现代近似值所得的$x$的值，第二列是取π = 3时该公式给出的结果，而第三列是用三角学（实际上是余弦）计算出的正确答案。

Guo's formula    Guo's formula    Correct 
formula with     formula with     answer  
correct π        π = 3 

  1.2401831       1.1304311       1.2311659
  5.0213452       4.5732560       4.8943483
 11.4521976      10.4253695      10.8993475
 20.5459874      18.7159078      19.0983005
 32.0790854      29.2893218      29.2893218
 45.5479024      41.7514260      41.2214747
 60.2936574      55.5434416      54.6009500
 75.7085631      70.1045733      69.0983006
 91.3723645      85.0063893      84.3565534
107.0799476     100.0000000     100.0000000

郭守敬在历法计算中还使用了其他巧妙的数学技巧。他会在数据点之间插值，并且使用三次插值公式来做这件事。需要插值的原因是，太阳在一年中相对于恒星的运动是不规则的。这是中国天文学家在六世纪发现的。郭守敬考察了累积差，即太阳一天内移动的度数与假定运动恒定时预期移动的度数之差。然后他像艾萨克·牛顿的前向差分插值法那样，列出了累积差的一阶、二阶和三阶差分。

Guo Shoujing is also known as Kuo Shou-ching. We do not know the names of his parents, but his paternal grandfather, who was clearly more famous than his parents, was Guo Yong who was famed as an expert in a wide range of topics from classical studies to mathematics and hydraulics. One can only guess that as a young boy Guo might have been influenced by his grandfather to become interested in constructing water clocks. We know that at age 14 Guo did construct such a clock. He designed a lotus clepsydra, that is a water clock which had a bowl shaped like a lotus flower on the top into which water dripped. By the age of sixteen Guo was studying mathematics.

Before we continue to describe events of Guo's life we should look briefly at the political situation, for it was a troubled time with many wars. After the death of Genghis Khan, one of his sons Ogodei had become the Great Khan in 1229. He had expanded the Mongol empire sending armies to complete the defeat of the Jurchens. By 1234 the Mongols had completed the destruction of the Jurchen empire and turned their attention to the south. This then was the situation in the northern part of China where Guo was growing up. By the age of twenty Guo was working as an hydraulic engineer. In 1251, as a government official, he worked on a project to repair an old bridge over the river Dahuoquan. This was a river in Guo's home province of Hebei and the bridge he renovated was a little way north of the town of Xinzhou.

Kublai, a grandson of the Mongol leader Genghis Khan began leading further Mongol advances in the latter years of the 1250s. On 5 May 1260 Kublai was elected Khan at his residence in Shang-tu and he began to organise the country. Zhang Wenqian, who was a friend of Guo and like him was a central government official, was sent by Kublai Khan in 1260 to Daming where unrest had been reported in the local population. Guo accompanied Zhang on his mission. Guo was not only interested in engineering, but he was also an expert astronomer. In particular he was a skilled instrument maker and understood that good astronomical observations depended on expertly made instruments. He now began to construct astronomical instruments, including water clocks for accurate timing and armillary spheres which represent the celestial globe.

Zhang advised Kublai Khan that his friend Guo was a leading expert in hydraulic engineering. Kublai knew the importance of water management, for irrigation, transport of grain, and flood control, and he asked Guo to look at these aspects in the area between Dadu (now Beijing or Peking) and the Yellow River. To provide Dadu with a new supply of water, Guo found the Baifu spring in the Shenshan Mountain and had a 30 km channel built to bring the water to Dadu. He proposed connecting the water supply across different river basins, built new canals with many sluices to control the water level, and achieved great success with the improvements which he was able to make. This pleased Kublai Khan and led to Guo being asked to undertake similar projects in other parts of the country. In 1264 he was asked to go to Gansu province to repair the damage that had been caused to the irrigation systems by the years of war during the Mongul advance through the region. Guo travelled extensively along with his friend Zhang taking notes of the work which needed to be done to unblock damaged parts of the system and to make improvements to its efficiency. He sent his report directly to Kublai Khan.

The advance of the Monguls was continuing under Kublai Khan and in 1276 he captured the city today named Hangzhou south of Shanghai. For many years the need for calendar reform had been understood but Kublai Khan saw political benefits in bringing in a new calendar at this time to emphasise how the new regime was replacing the old. Asked for his advice, Guo explained that:-

... the fundamental method is to carry out observation and tests in the area of astronomical changes.

Guo, together was Wang Xun, were asked to set up a special bureau to undertake the necessary research and to make proposals for the new calendar. An accurate calendar depends on the quality of the astronomical data used to support it and Guo's first move was to built seventeen new astronomical instruments so that accurate data could be collected. Of these seventeen instruments, thirteen were to be set up in an observatory in Kublai's capital Dadu (today called Beijing or Peking) while the other four were portable instruments which could make observations from different locations.

Let us briefly describe an improvement Guo made in constructing one such instrument. The simplest astronomical instruments was the gnomon, nothing other than a stick which was erected and the length of its shadow measured. The minimum length of shadow during a day is less in summer than in winter and at the solstices it changes from lengthening to shortening or visa versa. To make it easier to determine these points Guo fixed a crossbar to the top of a gnomon and used the principle of the pinhole camera to cast its shadow onto a measuring scale.

Kublai Khan established an Astronomical Observatory in Beijing in 1279. Building began in March of that year and, following a design proposed by Guo, the work was completed in two months. Guo's friend Zhang Wenqian was appointed as director and Guo together with his colleague Wang Xun were the two co-directors. Making sense of the data gathered from the instruments required a knowledge of spherical trigonometry and Guo devised some remarkable formulae. We look below at the clever mathematics which he introduced in undertaking his project on the new calendar.

The work was completed by 1280, Guo having calculated the length of the year correct to within 26 seconds, and in the following year Kublai Khan introduced the use of this extremely accurate calendar. It remained in use for 364 years. Zhang Wenqian died in 1283 and Guo was promoted to be director of the Observatory in Beijing. In 1292, in addition to his role of director of the Observatory, he was made head of the Water Works Bureau. He now undertook major projects designing a canal system to link the capital with other major towns. As always he met with success and even after the death of Kublai Khan, although Guo was by this time an old man, his advice continued to be sought by Kublai's successor.

We should now look at the rather remarkable work which Guo did on spherical trigonometry and solving equations. He produced a number of formulae for triangles, two sides of which were straight lines and the third was the arc of a circle. These formulae are approximate ones, but Guo was well aware of this. In a sense approximation was not regarded as important by the Chinese and they never became obsessed by the "squaring the circle" type of question like the ancient Greeks, since the Chinese approach was more practical and never axiomatic.
插图：Guo_Shoujing.gif ↗
In the diagram $d$ is the diameter of the circle, $a$ is the length of the arc $AB$ and $x$ is the length of $NB$ which Guo wanted to calculate. He gave the approximate formula

$x^{4} + (d^{2} - 2ad)x^{2} - d^{3}x + a^{2}d^{2} = 0$.

To solve this equation Guo used a numerical method similar to Horner's method. The equation has two real roots, the smaller being the solution to the problem while the other, being numerically larger than the length of the arc, was rightly discarded by Guo.

There is a fascinating fact relating to this work which it is interesting to investigate. Two of the coefficients of the equation, namely the constant term and the coefficient of $x^{2}$, involve the length $a$ of the arc, so require a value to be chosen for π. Guo takes π = 3 which at first sight seems strange since much more accurate values were known in China at the time that Guo was doing his calculations. For example $\large\frac{22}{7}\normalsize$ was known to be more accurate and $\large\frac{355}{113}\normalsize$ , given by Zu Chongzhi, was known to be more accurate still. Why did Guo choose π = 3? Surprisingly this gives a better answer than the more accurate values of π, for remember the formula is itself based on approximations. One has to believe that Guo chose π = 3 because he knew that the answers that he then found for different sizes of the angle at $O$ more closely approximated values he found by direct measurement.

Here is a computation from a computer algebra system which illustrates the point. The values are computed for increasing angles up to 90° for a circle of radius 100. The first column is the value of $x$ using Guo's formula taking an accurate modern approximation to π, the second column is the result given by the formula with π = 3, while the third column is the correct answer calculated using trigonometry (in fact the cosine).

Guo's formula    Guo's formula    Correct 
formula with     formula with     answer  
correct π        π = 3 

  1.2401831       1.1304311       1.2311659
  5.0213452       4.5732560       4.8943483
 11.4521976      10.4253695      10.8993475
 20.5459874      18.7159078      19.0983005
 32.0790854      29.2893218      29.2893218
 45.5479024      41.7514260      41.2214747
 60.2936574      55.5434416      54.6009500
 75.7085631      70.1045733      69.0983006
 91.3723645      85.0063893      84.3565534
107.0799476     100.0000000     100.0000000

Guo used other clever mathematical techniques in his calculations for the calendar. He would interpolate values between his data points and he did this using a cubic interpolation formula. The reason that interpolation was required was that the motion of the sun through the stars throughout the year is irregular. This was discovered by Chinese astronomers in the sixth century. Guo looked at the accumulated difference, namely the difference in degrees moved by the sun in a day compared with the expected degrees moved if the motion was constant. He then tabulated first, second, and third differences of the accumulated difference as in Newton's forward difference interpolation method.