沈括Shen Kuo

沈括也被称为沈括或沈括 Gua。他的父亲是沈括 Chou，母亲的娘家姓是Hsu。沈括出生时两人似乎都已相当年迈；他的父亲大约53岁，母亲大约44或45岁。沈括 Chou是一位政府官员，曾在各省之间辗转任职，最后在首都开封从事法律职业结束了他的职业生涯。沈括的母亲来自中国的苏州地区，在儿子的成长中扮演了极其重要的角色，因为他所有的早期教育都来自他的母亲。

当他大约九岁时，沈括开始跟随父亲在全国各地奔波，从一个职位调任到下一个职位。像沈括 Chou这样的官员必须承担的工作种类非常多样，因为他涉及财务事务，涉及技术事务，如照看水道和运河以及监督大型建筑项目，而且他还要对所任职地区的农业负责。年幼的沈括通过观察父亲的工作学到了很多，而且由于头脑极其敏锐，他充分利用了呈现在他面前的机会。

沈括 周认识并钦佩的另一位地方官员是 王宪钟 安石。他比 沈括 周的儿子 沈括 大10岁，很快 沈括 也开始钦佩 王宪钟 安石的行政才能，以及他的理想主义。当 沈括 的父亲于1051年去世时，王宪钟 安石为他撰写了墓志铭。作为官员之子，沈括 本人在父亲去世后无需参加其他人必须参加的正式考试即可成为官员。他于1054年被任命为第一个低级官职，随后又担任了若干其他职务。

沈括出生时的皇帝是仁宗。然而，他的政府内部正在出现问题。有来自北方的行政官员，他们大多非常保守，跟随父亲进入文官系统，并遵循传统方法。也有来自南方的新官僚，在通过科举考试后进入政府。王宪钟安石是这种新式行政官员之一，他通过能力而非庇护晋升，并遵循儒家理想。一系列危机，一些内部叛乱，一些来自越境入侵者的攻击，以及繁荣的下降，都意味着新官僚想要的改革开始被实施。

然而，行政机构中的传统派并不轻易放弃，当改革取得成功后，行政机构又故态复萌。到 沈括 开始在行政机构中逐级晋升时，改革派已经回归，此时他们已学会了政治手腕。改革派取得了一些进展，但根本问题依然存在：官僚机构日益昂贵，而税收却在下降。然而，沈括 正以非凡的才能崭露头角。他参与了多项治水方案，这在当时的中国始终是一个重大问题，而这些方案结果非常成功。通过修建排水渠和堤坝，他开垦了大片土地，使其成为高产农田。

1061年，沈括 在安徽省宁-沈括 任通判时，进行了一次地图测绘，随后又实施了另一项重大的土地开垦计划。这同样非常成功，在 沈括 于1063年通过全国考试并迁往扬州后，知州意识到这里有一位才能出众的人，便推荐他到河南省都城开封的朝廷任职。王宪钟 安石已于1060年进入中央政府，但在仁宗皇帝在位期间他并不特别受宠。然而仁宗于1064年去世，英宗短暂在位至1068年，随后 沈括 宗继位。在 沈括 宗治下，改革步伐加快，1069年 王宪钟 安石被任命为参知政事。他随即推出了一项名为“新法”的重大改革计划，沈括 被认定为 王宪钟 在朝廷内强大的改革集团十八名成员之一。

当 沈括 在地方担任官员时，他承担了大量非常成功的项目，但这并未完全占据他的精力，他利用业余时间研究天文学、历法科学以及这些背后的数学。1072年，沈括 被任命为司天监长官。按照惯例，新皇帝继位时应制定新历法。更换统治者后，新皇帝会寻求新的官方历法，从而以新的天象影响确立新的统治。沈括 宗于1068年成为皇帝时并未这样做，现在 沈括 被赋予了这一任务。他制定了雄心勃勃的计划，要重组司天监并收集必要的天文数据。他设立了一个项目，测量月亮和行星的位置，每晚三次绘制精确坐标，持续五年。然而，司天监的许多工作人员不称职，沈括 不得不解雇六名他发现伪造数据的人。工作人员的低效使历法不如应有的准确，但它于1075年投入使用，持续了约20年后被取代。

沈括 现在承担了若干项目以支持 王宪钟 的改革议程。他监督治水项目、军事组织，并广泛巡行各省，为地方官员提供建议。他用木板制作了他所访问地区的三维地图，木板上使用了混合胶水的锯末和熔化的蜡。王宪钟 在政府中的对手将 沈括 视为周游全国为 王宪钟 的改革计划争取支持的人，他们开始密谋反对他。然而 沈括 在所赋予的任务中如此成功，以至于他继续被要求承担具有重大国家意义的项目。1075年，他承担了修订国家防御战略的任务，并与北方威胁入侵的契丹部落进行了非常成功的谈判。他利用自己的历史知识驳斥契丹部落对中国领土的主张，为该地区带来了持续数年的和平。回到都城开封后，沈括 被任命到国子监，并于1077年被任命为财政专员。他在担任此职期间撰写的关于供求理论的著作非常卓越。他撰写了关于价格预测方法、货币供应、价格控制、市场干预以及其他主题的内容，这些主题直到现代才再次得到如此深入的研究。

他在政府中达到了权力极大的地位，成功推行了有效的改革，却在1077年秋被指控有不诚实行为。这些指控完全虚假，是由 王宪钟 改革计划的政治对手提出的。沈括 是一个才华横溢的人，但他的才能在于完成任务，而非政治谋划。由于政治手腕高超的 王宪钟 已心灰意冷并于1076年秋退休，他突然变得孤立无援。正是 王宪钟 的继任者对 沈括 提出了这些虚假指控。

沈括被派往陕西路担任知州和军事长官。1081年，他在那里组织了对唐古特的进攻，取得了重大胜利，并扩大了宋朝对新地区的控制。皇帝很高兴，沈括向他提出了加强该地区防御的建议。然而，仍有人反对他，他的计划被推翻。相反，他认为无用的防御计划却被实施。唐古特的进攻导致宋军战败并遭受重大损失。尽管他对这次失败毫无责任，但他被王宪钟的继任者指责，并被解除指挥权并流放。

就科学而言，这似乎是幸运的，因为在他被放逐之后，他写下了自己的科学著作。他被禁止参加官方活动，并被迫基本上处于软禁之中。然而，他可以自由从事科学工作，并且他从事了一项他于1076年应皇帝要求开始的项目，即绘制所有中国领土的地图。他制作了23幅地图，全部按1:900,000的比例尺绘制。沈括知道地图往往不能长期保存，而书籍则有更好的长期保存机会。因此，他用方向坐标和距离来编码他的地图，写道[1]：-

因此，尽管后世这些地图可能失传，但有了我的书，地域划分可以按照二十四个方向铺展开来，地图可以迅速重建，而不会有丝毫差异。

地图具有重大的军事意义，也具有科学价值，因此他的项目得到了政府的支持。在经历了接近软禁的六年之后，沈括被允许住在他自己选择的地方。十年前他还是领军人物时就已经购置了一处产业，但他从未见过自己在江苏（镇江）郊外的这处产业。1086年，他获准前往参观，发现了一个极其美丽的地方，这是他多年来梦寐以求的。他将其命名为梦溪，并于1086年获得赦免、国家养老金以及在他选择的地方居住的许可。他在梦溪孤独地度过了生命的最后七年。

他创作了许多著作，但最著名的是Meng ch'i pi t'an (Brush talks from Dream Brook)。他写道[1]：-

因为我只有毛笔和墨砚与我对话，所以我称它为《笔谈》。

这是一份非凡的科学文献，包含他在数学、音乐、天文学、历法、地图学、地质学、光学和医学方面的工作。其中还提到了磁罗盘及其在地图学中的使用。

沈括声称植物化石是气候变化的证据。他在远离海洋的岩石中识别出某些海洋生物的化石，并理解了这意味着什么。在太行山地层中观察到贝壳，他推断这些山脉虽然现在远离海洋，但一定曾经是海岸[1]：-

我看到双壳类贝壳和卵形岩石像带子一样水平穿过悬崖。这曾经是海岸，尽管海洋现在在数百英里以东。我们所称的大陆是淤泥的淹没……

让我们简要看看他的一些成就。首先让我们看看他在数学方面取得了什么成就。首先要注意的是，沈括对数学的看法比中国人通常更为抽象。大多数中国数学是由实际问题驱动的，但沈括乐于为数学问题本身而研究它们。让我们看看他研究的三个具体问题。

他考虑了围棋盘的问题，它有19×19条线，给出361个交叉点。在每个交叉点上，有三种可能性之一：没有棋子、白子或黑子。沈括问有多少种可能性。对我们来说这是一个简单的计算，即$3^{361}$。当然，这是一个非常大的数字，沈括的问题是在中国数字系统中表达这样的数字。答案大约是$1.74 \times 10^{172}$，沈括将其写为wan wan wan ... wan。这里wan = $10^{4}$，他写的是，据推测，43万的乘积。我们说推测，因为抄写员在复制有多少万时犯了错误，可能不理解沈括在做什么的意义。

我们将给出的第二个例子来说明沈括的数学是一个单位大小物品的堆。如果堆的顶部是一个矩形$a \times b$，堆的底部是一个矩形$c \times d$，并且堆中有$h$层物品，那么他表明堆中物品的数量是$((2b + d)a + (2d + b)c + (c - a))h/6$。Needham [5]将此视为迈向体积积分的一步，如果这有点牵强，人们肯定可以说这里有一种抽象。沈括没有指定物品是什么，但保持公式一般，他取它们为单位体积，并且（故意）忽略物品之间留下的体积。

在他的天文工作中，沈括必须从赤道坐标转换为黄道坐标。这至少需要对球面几何和三角学有一定的理解。事实上，沈括展示了观察空间排列的非凡能力，他给出了圆弧长度的近似公式，用弧所对的弦表示。在这项工作中有证据表明他接近三角学，实际上拥有正弦函数供他使用。

沈括在天文学方面的工作在许多方面都很出色。他试图解释为什么行星在天空中运动时会出现逆行运动周期，正如所观察到的那样。其他中国天文学家只关注行星的位置，从未试图解释所观察到的现象。与试图将所有运动解释为圆周运动的希腊天文学家不同，沈括提出行星以柳叶形运动与绕地球的圆周运动复合的方式运动。据说他还建造了浑仪、水钟和青铜圭表，这是一种指针，其影子可给出正午的时间。

除了Brush talks from Dream Brook之外，沈括还写了Supplement to Brush Talks和Sequel to Brush Talks。他还写了一部医学著作Good prescriptions和Record of longings forgotten，后者是关于山中生活的观察，但已失传。他关于历法改革的论著也已失传，他未发表著作的遗稿汇编也失传了大约一半。

Sivin 在 [1] 中写道：-

仅凭兴趣的广博并不能解释 沈括 对中国科学思想研究的重要性。另一个方面是他深刻的技术好奇心。……最重要的是，在 沈括 身上，如同在其他伟大的科学人物身上一样，人们能感受到一种特殊的直接性。作为一个身处过去的重负始终沉重地压于思想工作之上的社会中的成员，他 nevertheless 常常径直越过根深蒂固的结构与假设。

Shen Kua is also known as Shen Kuo or Shen Gua. His father was Shen Chou and his mother's family name was Hsu. Both seem to have been quite old when Shen Kua was born; his father being around 53 years old, and his mother about 44 or 45 years old. Shen Chou was a government official who moved around between various posts in the provinces before ending his career in the legal profession in the capital Kaifeng. Shen Kua's mother came from the Soochow region of China and played an extremely important role in her son's upbringing since all his early education came from his mother.

When he was about nine years old, Shen Kua began to move around the country with his father as he went from one post to the next. The type of work that officials such as Shen Chou had to undertake was very varied, for he was involved in financial matters, in technical matters such as looking after the waterways and canals and overseeing major building projects, and he would have responsibilities for agriculture in the district to which he was posted. The young boy Shen Kua learnt much from observing his father at work, and having an extremely sharp mind, he took full advantage of the opportunities presented to him.

Another local administrator whom Shen Chou came to know and to admire was Wang Anshi. He was 10 years older than Shen Chou's son Shen Kua and soon Shen Kua also came to admire Wang Anshi's administrative skills, and also his idealism. When Shen Kua's father died in 1051, Wang Anshi composed his epitaph. As the son of an official, Shen Kua himself could become an official after the death of his father without taking the formal examinations which others were required to sit. He was appointed to his first minor official post in 1054 and this was followed by a number of other postings.

The emperor at the time of Shen Kua's birth was Jen-tsung. However problems were developing in the ranks of his administration. There were administrators from the north who were mostly highly conservative, following their fathers into the civil service, and followed traditional methods. There were also new bureaucrats from the south entering the administration after passing the civil service examinations. Wang Anshi was one of this new breed of administrator, who progressed through ability rather than through patronage, and followed Confucian ideals. A number of crises, some internal rebellions, some attacks from invaders crossing borders, and falling prosperity, all meant that reforms which the new bureaucrats wanted began to be implemented.

However, the traditionalists in the administration did not give up easily and when success followed reforms, the administration fell back into its old ways. By the time that Shen Kua started to progress through the ranks of the administration, reformers had returned now having learnt political skills. The reformers made some progress but fundamental problems remained with an increasingly costly bureaucracy yet declining tax revenues. Shen, however, was making a name for himself as an extraordinarily able man. He was involved in a number of schemes to control water, always a major problem in China at this time, which turned out to be very successful. Building drains and embankments, he reclaimed large areas of land which then became productive farmland.

While subprefect of Ning-kuo in Anhwei province in 1061, Shen undertook a cartographic survey and then undertook another major land reclamation programme. Again it was very successful and after Shen passed national examinations in 1063 and moved to Yang-chou, the governor realised that here was someone of outstanding ability and recommended him for an appointment at court in the capital Kaifeng in Honan province. Wang Anshi had entered the central government in 1060 but he was not particularly favoured while Jen-tsung was emperor. However Jen-tsung died in 1064 and Ying-tsung reigned briefly until 1068 when Shen-tsung took over. Under Shen-tsung reforms gathered pace and in 1069 Wang Anshi was appointed second privy councillor. He now launched a major reform program known as the "New Policies" and Shen was identified as one of eighteen members of Wang's strong reforming group within the court.

While Shen had worked as an official in the provinces he had undertaken a large number of highly successful projects but this had not fully occupied him and he had spent his spare time studying astronomy, calendar science, and the mathematics behind these. In 1072 Shen was made Director of the Astronomy Bureau. It was customary for a new calendar to be set up on the succession of a new emperor. After a change of ruler the new Chinese emperor would seek a new official calendar thus establishing a new rule with new celestial influences. This had not been done when Shen-tsung became emperor in 1068 and Shen Kua was now given the task. He made ambitious plans to reorganise the Astronomy Bureau and to collect the necessary astronomical data. He set up a programme which would measure the positions of the moon and planets, plotting exact coordinates three times a night for five years. However, many of the staff at the Astronomy Bureau were incompetent and Shen had to dismiss six whom he found were falsifying data. The inefficiency of the staff made the calendar less accurate than it should have been but it came into use in 1075 and lasted about 20 years before being replaced.

Shen now undertook a number of projects to support Wang's reform agenda. He supervised water control projects, military organisation, and travelled widely in the provinces advising local administrators. He created three dimensional maps of areas he visited made from wooden plates on which sawdust mixed with glue and melted wax were used. Wang's opponents in government saw Shen as someone travelling round the country obtaining support for Wang's reform programme and they began to conspire against him. However Shen was so successful in the tasks he was given that he continued to be asked to undertake pojects of major national importance. In 1075 he undertook the task of revising the defence strategy of the country, and he undertook very successful negotiations with the Khitan tribes in the north who were threatening to invade. Using his knowledge of history to refute the claims of the Khitan tribes to Chinese territory, he brought peace to the area which lasted for a number of years. Returning to the capital Kaifeng, Shen was appointed to the Imperial Academy and appointed Finance Commissioner in 1077. His writings on the theory of supply and demand, made while holding this office, are quite remarkable. He wrote on methods of forecasting prices, the currency supply, price controls, market intervention and other topics which were not studied again in this depth until modern times.

Having reached a position of great power in the government, pushing through effective reforms with great success, he was accused of dishonest practice in the autumn of 1077. These charges were completely false, but brought by political opponents of Wang's reform programme. Shen was a brilliant man, but his skill was in accomplishing tasks and not in political manoeuvring. He had suddenly become exposed since the politically skilful Wang had become frustrated and had retired in the autumn of 1076. It was Wang's replacement who brought the false charges against Shen.

Shen was sent to Shenxi province as Commissioner for Prefectural and Military Affairs. There, in 1081, he organised an advance against the Tanguts winning major victories and extending the control of the Sung dynasty over new regions. The emperor was delighted and Shen made proposals to him to fortify the region. However there were still those who were working against him and his plans were overturned. Instead plans for fortifications which he considered useless were acted on instead. An attack by the Tanguts saw the Sung troops defeated and sustain heavy losses. Although he had been in no way responsible for the defeat he was blamed by Wang's successor and was relieved of his command and banished.

This seems to have been fortunate as far as science is concerned since after he was banished he wrote his scientific works. He was banned from taking part in official events and made to live essentially under house arrest. He was free to undertake scientific work, however, and he worked on a project he had begun at the Emperor's request in 1076, namely to produce maps of all Chinese territory. He produced 23 maps, all drawn to the scale of 1:900,000. Shen knew that maps tended not to survive for long, whereas books had a better chance of long term survival. He therefore encoded his maps in terms of directional coordinates and distances, writing [1]:-

Thus although in later generations the maps may be lost, given my book the territorial divisions may be laid out according to the twenty-four directions, and the maps speedily reconstructed without the least discrepancy.

Maps are of great military significance, as well as scientific interest, so his project found favour with the government. After six years of being subjected to what was close to house arrest, Shen was allowed to live in a place of his own choosing. He already had purchased a property when he had been a leading figure ten years earlier but he had never seen his property on the outskirts of Jiangsu (Zhenjiang). In 1086 he was allowed to visit it and found a place of great beauty which he had dreamt of for many years. He named it Dream Brook and in 1086 he received a pardon, a state pension, and permission to live where he chose. He spent the last seven years of his life in isolation at Dream Brook.

He composed many works, but the most famous was Meng ch'i pi t'an (Brush talks from Dream Brook). He wrote [1]:-

Because I had only my writing brush and ink slab to converse with, I call it Brush Talks.

It is a remarkable scientific document which contains his work on mathematics, music, astronomy, calendars, cartography, geology, optics and medicine. There is also a reference to a magnetic compass and its use in cartography.

Shen Kua claimed that fossilised plants were evidence for changes in climate. He recognised fossils of certain sea creatures in rock far from the sea and understood what this meant. Observing seashells in strata of the T'ai-hang Shan mountains, he deduced that these mountains, though now far from the sea, must once have been a sea shore [1]:-

I saw bivalve shells and ovoid rocks running horizontally through a cliff like a belt. this was once a seashore, although the sea is now hundreds of miles east. What we call our continent is an inundation of silt ...

Let us look briefly at some of his achievements. First let us see what he achieved in mathematics. The first thing to note is that Shen took a much more abstract view of mathematics than the Chinese generally did. Most Chinese mathematics was motivated by practical problems, but Shen was happy to look at mathematical problems for their own sake. Let us look at three particular problems which he studied.

He considered the problem of the go board which had 19 × 19 lines giving 361 intersections. On each intersection there was one of three possibilities: no counter, a white counter, or a black counter. Shen asked how any possibilities there were. It is an easy calculation for us, namely $3^{361}$. This, of course, is an exceedingly large number and Shen's problem was to express such a number within the Chinese number system. The answer is approximately $1.74 \times 10^{172}$ and Shen writes this as wan wan wan ... wan. Here wan = $10^{4}$ and he is writing a product of, presumably, 43 wan. We say presumably since copyists made errors in reproducing how many wan there were, possibly not understanding the significance of what Shen was doing.

The second example that we shall give to illustrate Shen's mathematics is that of a pile of unit sized items. If the top of the pile is a rectangle $a \times b$ and the base of the pile is a rectangle $c \times d$, and there are $h$ layers of items in the pile then he showed that the number of items in the pile is $((2b + d)a + (2d + b)c + (c - a))h/6$. Needham [5] sees this as a step towards a volume integral and, if this is stretching things a bit too far, one certainly can say that there is an abstraction here. Shen does not specify what the items are, but leaves the formula general, he takes them to be of unit volume and also neglects (deliberately) volumes left between the items.

In his astronomical work Shen had to convert from coordinates given in equatorial form to those given in ecliptic form. This requires at least some understanding of spherical geometry and trigonometry. In fact Shen demonstrated a remarkable ability to view spatial arrangements and he gave an approximate formula for the length of a circular arc in terms of the cord subtending the arc. There is evidence in this work that he was coming close to trigonometry, effectively having the sine function for his use.

Shen's work in astronomy is remarkable in many ways. He attempted to explain why the motion of the planets across the sky was as observed with periods of retrograde motion. Other Chinese astronomers had only looked at the positions of the planets without ever trying to explain what was observed. Unlike Greek astronomers who tried to explain all motions as circular, Shen proposed that planets moved in a willow leaf motion composed with circular motion round the earth. He is also said to have constructed a armillary sphere, a water clock, and a bronze gnomon, a pointer whose shadow gives the time of mid-day.

In addition to Brush talks from Dream Brook Shen wrote Supplement to Brush Talks and Sequel to Brush Talks. He also wrote a medical work Good prescriptions and Record of longings forgotten which was observations on living in the mountains but it has not survived. His treatise on calendar reform is also lost as is about half of a posthumous compilation of his unpublished writings.

Sivin writes in [1]:-

Breadth of interest alone does not account for Shen's importance for the study of the Chinese scientific intellect. Another aspect is his profound technical curiosity. ... Above all, one is aware in Shen, as in other great scientific figures, of a special directness. A member of a society in which the weight of the past always lay heavily on work of the mind, he nevertheless often cut past deeply ingrained structures and assumptions.