秦九韶Qin Jiushao

秦九韶，又名秦秦九韶，生于南宋时期。他的祖先来自山东省鲁郡，一些传记错误地将此引为他的出生地。他的父亲秦九韶九韶（或秦九韶）是一名毕业生，在地方行政中担任官员。大约在1219年，当秦九韶大约十七岁时，他的父亲正担任巴州知州。此时秦九韶自愿参军，参与平定叛乱，并服役了一段时间。秦九韶的父亲大约在1224年搬到南宋都城杭州，秦九韶随父前往。他在其著名著作Shushu Jiuzhang的序言中写道（例如见[3]）：-

我年轻时住在京城，因此能够在太史局学习；后来，一位隐士学者教我数学。

我们知道秦九韶是一个叛逆的青年，以众多风流韵事闻名，不喜欢权威[1]：-

在他父亲举行的宴会上，一块石头突然落在宾客中间，引起了一阵骚动；调查显示，这枚投掷物来自秦九韶的方向，他正在向一个年轻女孩展示如何用弓作为投石索来投掷弹丸。

事实上，秦九韶并没有在都城杭州住很久，因为1226年他的父亲被派往四川的铜川（今三台），秦九韶也随他去了那里。遗憾的是，我们不知道是哪位隐士学者教了秦九韶数学，但我们确实知道他学习了九章算术。到1233年，秦九韶自己已是四川省一个县的知县，此时他师从成都（四川中部）的一位官员学习写诗。值得一提的是，秦九韶不仅是数学天才和诗歌能手，还精通击剑、射箭、骑马、音乐和建筑。然而，他的性格还有另一面。一位同时代人在给皇帝的信中形容他为[1]：-

……像虎狼一样凶猛，像毒蛇或蝎子一样恶毒。

他好斗的天性无疑适合军旅生活，在四川省服役期间他成为了一名指挥官兼防御者。蒙古首领成吉思汗于1227年去世，但蒙古人于1230年恢复了对南宋的进攻。他们的军队于1234年入侵四川省，秦九韶被迫离开。他在Shushu Jiuzhang的序言中写道（例如见[3]）：-

在与蛮夷的动乱时期，我在遥远的边境度过了数年，在箭矢和石弹中不顾自身安危，忍受了十年的危险和不幸。

然而，我们不必为秦九韶感到太难过，因为他是个不诚实的恶棍，随时准备毒害他不喜欢的人。他在湖北省蕲州（今蕲春）担任行政官员，但他在那里的行为如此恶劣，以至于引发了一场军事叛乱。然后他被任命为安徽省徽州（今歙县）的知州，但在这里他从事非法的盐业交易，从而致富。接着他搬到浙江省吴兴，在那里安顿下来，挥霍他非法获得的财富。1244年中，他被派往南京担任高级行政官员。担任此职三个月后，他的母亲于1244年9月去世，秦九韶离开南京的职位守丧，回到他母亲一直居住的安徽省徽州。在徽州守丧期间，秦九韶写下了他著名的数学论著Shushu Jiuzhang（Mathematical Treatise in Nine Sections），该书于1247年问世。这是一部非凡的著作，促使George Sarton写道，秦九韶是（见[1]或[3]，其中引用了这句话）：-

……他所属民族、他那个时代，乃至所有时代最伟大的数学家之一。

在我们考察Shushu Jiuzhang的内容之前，继续描述秦九韶的生平。1254年，他在京城杭州再次担任文官职务，但几个月后便辞职。1259年，他被任命为海南琼州知州，上任百日后因贪污和剥削被免职，再次非法获取巨额财富后返回家乡。鉴于其犯罪交易的记录，人们或许会认为此时秦九韶已无法被任用，但他随后设法在浙江鄞县（靠近宁波）获得了一个助理职位，他的朋友吴潜已被任命为那里的海军军官。或许吴潜和他的朋友秦九韶一样腐败，因为他被从鄞县免职，而在1260年，秦九韶也被派往广东省梅州（今梅县），并在那里去世。

我们已经看到，秦九韶是一个极不守原则的人，但他也是一位几乎无人能及的数学天才。在考察他的数学成就之前，让我们再讲述一些关于他品行的故事。据记载，秦九韶欺骗了他的朋友吴潜，从而占有了后者的一些土地；还有记载说，秦九韶惩罚了家中的一名女性成员，将她关起来不给食物。

Shushu Jiuzhang (Mathematical Treatise in Nine Sections)在一定程度上是以九章算术为蓝本的，尽管秦九韶的论著要精深得多。第1章讲的是不定分析；其中包含了关于中国剩余定理的卓越工作，这出现在全书的一开始。我们在下面讨论它。第2章称为Heavenly phenomena，处理历法问题以及关于雨或雪的问题。例如，有一个问题问的是：已知雨水在一个上口为直径$a$的圆、底面为直径$b$的圆、且$a > b$的容器中达到某一高度$h$，求雨水在平地上会积聚到的高度。第3章称为Boundaries of fields，考察测量术。这一章给出了一个卓越的公式，它把一个图形的面积表示为一个四次方程的根。这里的新颖之处在于，系数不是数，而是图形中一些未加指定的长度的函数。这一章的另一个新颖之处是给出了用三角形三边表示其面积的公式，本质上就是海伦公式。

第四章名为Telemetry，探讨涉及测量到不可达点距离的问题。再次出现了高次方程，其中一个问题涉及求解十次方程。问题如下：-

给定一个未知直径的圆形城墙城市，有四座城门，分别位于四个基本方位。一棵树位于北门以北三里处。如果从南门出发立即向东走九里，树恰好进入视野。求城墙的周长和直径。

秦九韶得到方程（实际上是关于$x^{2}$的五次方程，其中$x^{2}$是城市的直径）：-

$x^{10} + 15x^{8} + 72x^{6} - 864x^{4} - 11664x^{2} - 34992 = 0$

[解：$x = 3$，所以城市直径为$x^{2}= 9$里]

在第五章至第九章中，分别命名为Taxes, Money and grain, Fortifications and buildings, Military affairs和Commercial affairs，问题需要更标准的数学方法，秦九韶没有引入进一步的创新。在整个文本中，除了上述十次方程外，秦九韶还将某些问题的解归结为三次或四次方程，他用标准中国方法（即今天所谓的保罗·鲁菲尼-威廉·乔治·霍纳方法）求解。例如以下两个方程

$4608x^{3} - 3000000000\times 30\times 800 = 0$

[答案：$x = 2500$]
和

$-x^{4} + 1534464x^{2} - 526727577600 = 0$

[答案：$x = 720$。注意解$x = 1008$未给出]

秦九韶也解线性联立方程组，特别是该方程组

$140x +  88y +  15z = 58800\newline792x + 566y + 815z = 392000\newline 64x +  30y +  75z = 29400$

被解出。
[答案：$x = 300, y = 180, z = 64$]

文本的另一个显著特点是秦九韶用0表示零，因此他不仅使用了一个表示零的符号，而且这个符号是一个小圆圈。他写到以前对零的使用：-

……在所有古书中我们都发现空位。

正如我们已经提到的，文本中最引人注目的方法是求解联立整数同余式的方法，即中国剩余定理。秦九韶考虑了以下类型的问题

$x ≡ r_{k} (\mod m_{k}), k = 1, 2, ..., n$。

今天，我们开始处理这类问题时假设$m_{k}$是整数。然而，秦九韶乐于研究所涉数字为有理的的问题。因此，他的第一步是将问题转化为$m$为整数的情形。

尽管自孙子的工作以来800年间，中国在这类问题上没有进展的证据，但秦九韶现在展示了如何处理$m_{k}$不两两互素的情形。因此，他的下一步是对$m_{k}$进行多轮处理，以某种方式进行替换，使得最终新的模数两两互素，但原问题的解在替换后保持不变。然后他求解每个同余式，最后将答案重新组合，给出联立同余式组的解。沈在[15]中全面讨论了秦九韶的求解方法；另见[3]和[4]。Libbrecht写道[2]：-

我们不应低估[秦九韶]的革命性进步，因为从[孙子]的单个余数问题，我们一下子来到了求解余数问题的一般程序，甚至比卡尔·弗里德里希·高斯的方法更先进，而且没有丝毫逐渐演变的迹象。

这是一项如此杰出的工作，以至于我们不禁要问秦九韶是如何做到的。我们确实知道秦九韶是个乐于偷窃的无赖，那么他可能偷了他的数学吗？当然，如果他偷了，那他是从谁那里偷的呢？这似乎不太可能，因为它引发的问题比解决的还多。一些历史学家怀疑秦九韶是否真的能解决如此深刻的问题，暗示他可能是从答案倒推出来的。这一暗示似乎没有什么说服力，因为这些问题不容易反向推导，而且无论如何，秦九韶确实看起来知道自己在做什么。也许他是通过印度处理这类问题的方法学到了这个方法。尽管秦九韶使用符号0暗示了可能的印度知识，但印度处理这类同余问题的方法足够不同，使得这种可能性极小。人们只能得出结论，接受秦九韶是有史以来最伟大的数学天才之一。

有趣的是，尽管秦九韶性格如此，他并没有将这个杰出的方法据为己有。他说他是在杭州天文局学习时从历法专家那里学到的。然而，他指出这些历法专家使用这个规则却不理解它。这其中必定有道理，因为毫无疑问，计算历法是研究一阶同余理论的重要动机。不过，看来秦九韶必定将这些思想推进得更远，并在他的数学工作中表现出一种谦逊，这在他生活的其他方面肯定是缺乏的。

这项工作有多令人印象深刻？好吧，只需说莱昂哈德·欧拉未能为这些问题提供令人满意的解，而重新发现这种求解同余式组的方法则留给了卡尔·弗里德里希·高斯、昂利·勒贝格和汤姆斯·斯蒂尔吉斯。

Qin Jiushao, also known as Ch'in Chiu-Shao, was born at the time of the Nan (Southern) Sung dynasty. His ancestors came from Lu-chun in Shantung province and some biographies quote this incorrectly as his birthplace. His father, Qin Jiuyu (or Ch'in Chiu-yu), was a graduate who worked as an official in local administration. In around 1219, when Qin was about seventeen years old, his father was working as a prefect of Bazhou. At this time Qin volunteered for the army, which was putting down a rebellion, and served for a while. Qin's father moved to Hang-chou, the capital of the Nan Sung, in around 1224 and Qin went with his father. He wrote in the preface to his famous work Shushu Jiuzhang (see for example [3]):-

In my youth I was living in the capital, so that I was able to study in the Board of Astronomy; subsequently, I was instructed in mathematics by a recluse scholar.

We know that Qin was a rebellious youth, famous for his many love affairs, who disliked authority [1]:-

During a banquet given by his father, a commotion was created when a stone suddenly landed among the guests; investigations disclosed that the missile had come from the direction of Qin, who was showing a young girl how to use a bow as a sling to hurl projectiles.

In fact Qin did not live in the capital Hang-chou for long since his father was posted to Tongchuan (now Santai) in Szechwan province in 1226 and Qin went there with him. Sadly, we do not know which recluse scholar taught Qin mathematics, but we do know that he studied the Nine Chapters on the Mathematical Art. By 1233 Qin was himself the sheriff of a subprefecture in Szechwan province and at this time he was instructed in writing poetry by an official from Chengdu, in central Szechwan province. It is worth noting at this point that as well as being a genius in mathematics and accomplished in poetry, Qin was expert at fencing, archery, riding, music and architecture. However, there was another side to his character. He was described by a contemporary in a letter to the Emperor as [1]:-

... as violent as a tiger, or a wolf, and as poisonous as a viper or a scorpion.

His aggressive nature no doubt suited army life and he became a commander defender while serving in Szechwan province. Genghis Khan, the Mongol leader, died in 1227 but the Mongols resumed their attacks on the Han Sung in 1230. Their armies invaded Szechwan province in 1234 and Qin was forced to leave. He wrote in the preface of Shushu Jiuzhang (see for example [3]):-

At the time of the troubles with the barbarians, I spent several years on the remote frontier, without care for my safety among the arrows and stone missiles, I endured danger and unhappiness for ten years.

We need not feel too sorry for Qin, however, for he was a dishonest rogue who was quite prepared to poison those whom he disliked. He served as an administrator in Qizhou (now Qichun) in Hupeh province, but his behaviour there was so bad that it cause a military revolt. Then he was appointed governor of Hui-chou (now She-hsien) in Anhwei province but here he undertook illegal dealings in salt which made him rich. He then moved to Wu-hsing in Chekiang province where he settled down to spend his illegally acquired riches. In the middle of 1244 he was posted as a senior administrator to Nanking. After holding this post for three months, his mother died in the September 1244 and Qin left his post in Nanking for the mourning period and returned to Hui-chou in Anhwei province where his mother had been living. During his period of mourning in Hui-chou, Qin wrote his famous mathematical treatise Shushu Jiuzhang (Mathematical Treatise in Nine Sections) which appeared in 1247. This is a remarkable work which led to George Sarton writing that Qin was (see [1] or [3] where this is quoted):-

... one of the greatest mathematicians of his race, of his time, and indeed of all times.

Before we look at the contents of the Shushu Jiuzhang we continue our description of Qin's life. He took up his work in the civil service again in 1254 in the capital Hang-chou but resigned after a few months. Appointed governor of Qiongzhou in Hainan in 1259 he was dismissed for corruption and exploitation after a hundred days in office and returned home having again acquired immense wealth illegally. One might expect that by this time Qin would be unemployable, given his record of criminal dealings, but he next managed to gain an appointment as an assistant in the district of Yin (near Ningpo) in Zhekiang where his friend Wu Qian had been appointed as a naval officer. Perhaps Wu Qian was as corrupt as his friend Qin, for he was dismissed from Yin and, in 1260, Qin was also sent away to Meizhou (now Meixian), in Guangtong province where he died.

We have seen that Qin was a highly unprincipled character but he was also a mathematical genius with few equals. Before looking at his mathematical achievements, let us recount further stories of his character. It is recorded that Qin cheated his friend Wu Qian so that he became the owner of some of his land, and also that Qin punished a female member of his household by confining her without food.

The Shushu Jiuzhang (Mathematical Treatise in Nine Sections) is to some extent modelled on the Nine Chapters on the Mathematical Art although Qin's treatise is far more sophisticated. Chapter 1 is on indeterminate analysis; it contains remarkable work on the Chinese remainder theorem which occurs right at the beginning of the text. We discuss it below. Chapter 2 is called Heavenly phenomena and it deals with questions on the calendar and also questions about rain or snow. For example one problem asks for the height that rainwater would collect on level ground given that it reaches a certain height $h$ in a vessel with a circular top of diameter $a$ and circular base of diameter $b$ where $a > b$. Chapter 3 is called Boundaries of fields and looks at surveying. There is a remarkable formula given in this Chapter which expresses the area of a figure as the root of an equation of degree 4. The novelty here is that the coefficients are not numbers but are functions of lengths in the figure which are left as unspecified. Another novelty in this chapter is a formula for the area of a triangle given in terms of its sides, essentially Heron's formula.

Chapter 4, called Telemetry, looks at problems involving measuring the distance to inaccessible points. Again equations of high degree appear, one problem involving the solution of the equation of degree 10. The problem is:-

Given a circular walled city of unknown diameter with four gates, one at each of the four cardinal points. A tree lies three li north of the northern gate. If one turns and walks eastwards for nine li immediately on leaving the southern gate, the tree just comes into view. Find the circumference and the diameter of the city wall.

Qin obtains the equation (really an equation of degree 5 in $x^{2}$, where $x^{2}$ is the diameter of the city):-

$x^{10} + 15x^{8} + 72x^{6} - 864x^{4} - 11664x^{2} - 34992 = 0$

[Solution: $x = 3$, so diameter of city is $x^{2}= 9$ li]

In Chapters 5 to 9 named Taxes, Money and grain, Fortifications and buildings, Military affairs, and Commercial affairs the problems require more standard mathematical methods and Qin does not introduce further innovations. Throughout the text, in addition to the tenth degree equation above, Qin also reduces the solution of certain problems to a cubic or quartic equation which he solves by the standard Chinese method (namely that which today is called the Ruffini-Horner method). For example the following two equations

$4608x^{3} - 3000000000\times 30\times 800 = 0$

[Answer: $x = 2500$]
and

$-x^{4} + 1534464x^{2} - 526727577600 = 0$

[Answer: $x = 720$. Note the solution $x = 1008$ is not given]

Qin also solves linear simultaneous equations, in particular the system

$140x +  88y +  15z = 58800\newline792x + 566y + 815z = 392000\newline 64x +  30y +  75z = 29400$

is solved.
[Answer: $x = 300, y = 180, z = 64$]

One further remarkable feature of the text is that Qin uses 0 for zero, so not only does he use a symbol for zero, but that symbol is a little circle. He writes about previous uses of zero:-

... in all old books we find empty places.

As we have mentioned, the most remarkable method in the text is the method for solving simultaneous integer congruences, the Chinese Remainder Theorem. Qin considers problems of the type

$x ≡ r_{k} (\mod m_{k}), k = 1, 2, ..., n$.

Today we start such problems assuming that the $m_{k}$ are integers. However, Qin is happy to look at problems where the numbers concerned are rational. His first step is therefore to convert to a situation where the $m$ are integers.

Although there is no evidence of progress on such problems in China since the work of Sun Zi which was 800 years earlier, Qin now shows how to handle the case where the $m_{k}$ are not pairwise coprime. His next move, therefore, is to make various passes through the $m_{k}$ making replacements in such a way that eventually the new moduli are pairwise coprime but the solution to the original problem remains unchanged by the replacements. He then solves each congruence and finally reassembles the answers to give the solution to the system of simultaneous congruences. Shen discusses Qin's method of solution fully in [15]; see also [3] and [4]. Libbrecht writes [2]:-

We should not underestimate [Qin's] revolutionary advance, because from [Sun Zi's] single remainder problem, we come at once to the general procedure for solving the remainder problem, even more advanced than Gauss's method, and there is not the slightest indication of gradual evolution.

This is such a brilliant piece of work that we are left with asking how Qin could have achieved it. We certainly know that Qin was a rogue who was happy to steal, so could he have stolen his mathematics? Of course if he stole it then whom did he steal it from? This does not seem likely, for it raises more questions than it solves. Some historians have wondered whether Qin could have really solved such a deep problem, suggesting that perhaps he worked back from the answer. There seems little that is convincing in that suggestion since these problems are not readily worked in reverse and, anyway, Qin really does appear to know what he is doing. Perhaps he learnt of the method through Indian approaches to such problems. Although Qin's use of the symbol 0 suggests possible Indian knowledge, the Indian approach to such congruence problems is sufficiently different to make this highly unlikely. One is left with no conclusion other than accepting that Qin was one of the great mathematical geniuses of all time.

Interestingly, despite Qin's character, he does not claim this brilliant method as his own. He says that he learn it from the calendar experts when he was studying at the Board of Astronomy in Hang-chou. He states, however, that these calendar experts used the rule without understanding it. There must be something in this for, without a doubt, calculating calendars was an important motivation for studying the theory of first-order congruences. It would appear though that Qin must have taken these ideas much further and be showing a modesty in his mathematical work which was certainly lacking in other aspects of his life.

How impressive is this work? Well suffice to say that Euler failed to provide a satisfactory solution to these problems and it was left to Gauss, Lebesgue and Stieltjes to rediscovered this method of solving systems of congruences.