约翰内斯·开普勒Johannes Kepler

约翰内斯·开普勒现在主要因发现以他名字命名的行星运动三定律而被铭记，这些定律分别发表于1609年和1619年。他还在光学方面做了重要工作（1604年、1611年），发现了两个新的正多面体（1619年），首次对等球密堆积进行了数学处理（导致了对蜂窝巢室形状的解释，1611年），首次证明了对数如何运作（1624年），并设计了一种求旋转体体积的方法，事后看来，这可以被视为对微积分发展的贡献（1615年、1616年）。此外，他计算了当时已知最精确的天文表，其持续的精确性大大有助于确立heliocentric天文学的真实性（Rudolphine Tables，乌尔姆，1627年）。

开普勒的大量通信保存了下来。他的许多信件几乎相当于一篇科学论文（当时还没有科学期刊），而收信人似乎因为觉得有趣而保留了这些信件。因此，我们对开普勒的生平，乃至他的性格，了解得相当多。部分正因如此，开普勒才多少有了一段作为半虚构人物的经历（见下文史学注记）。

童年

开普勒出生在斯瓦比亚的小镇安德烈·韦伊 der Stadt，并于1576年随父母搬到附近的Leonberg。他的父亲是一名雇佣兵，母亲是客栈老板的女儿。开普勒是他们的第一个孩子。他的父亲在开普勒五岁时最后一次离家，据信死于荷兰的战争中。小时候，开普勒与母亲住在祖父的客栈里。他告诉我们，他常常在客栈帮忙服务。可以想象，顾客有时会对这孩子非凡的算术能力感到困惑。

开普勒早年在当地一所学校接受教育，随后进入附近的一所神学院，之后他打算接受圣职，便进入图宾根大学就读，该校当时（如今依然）是路德宗正统派的堡垒。

开普勒的观点

开普勒一生都是一个极其虔诚的人。他的所有著作都大量提及上帝，他将自己的工作视为履行基督徒理解上帝作品之职责。正如开普勒所相信的，人是按上帝的形象造的，显然能够理解上帝所创造的宇宙。此外，开普勒确信上帝是按照一个数学方案创造宇宙的（这一信念见于柏拉图的著作，并与毕达哥拉斯相关联）。由于当时普遍接受数学提供了通向关于世界的真理的可靠方法（欧几里得的共同概念和公设被认为确实为真），我们在这里便有了一种理解宇宙的策略。既然有些作者给开普勒安上了非理性的名声，就值得指出，这种颇为乐观的认识论与神秘主义者的信念相去甚远，后者认为事物只能以依赖不受理性约束的洞见的不精确方式被理解。开普勒确实一再感谢上帝赐予他洞见，但这些洞见是以理性的方式呈现的。

大学教育

当时，大学里的所有学生通常都要修读“数学”课程。原则上，这包括四门数学科学：算术、几何、天文学和音乐。然而，具体教授什么似乎取决于各个大学。在图宾根，开普勒由当时一位领先的天文学家米夏埃尔·梅斯特林（1550 - 1631）教授天文学。课程中的天文学当然是地心天文学，即托勒密体系的现行版本，其中所有七颗行星——月亮、水星、金星、太阳、火星、木星和土星——都围绕地球运动，它们相对于恒星的位置通过组合圆周运动来计算。这个体系或多或少与当时的（亚里士多德）物理学观念一致，尽管存在某些困难，例如，是否可以将一个相对于自身中心而非相对于另一点（称为“等分点”）不均匀的圆周运动视为“均匀的”（因此显然可以接受为永恒的）。然而，总的来说，天文学家（他们自视为“数学家”）似乎满足于继续计算行星的位置，而让自然哲学家去担心数学模型是否与物理机制相对应。开普勒没有采取这种态度。他最早出版的著作（1596年）提议考虑行星的实际路径，而不是用来构建它们的圆。

开普勒不仅学习数学，还学习希腊语和希伯来语（两者都是阅读原始语言圣经所必需的）。教学使用拉丁语。在第一年结束时，开普勒除了数学外，所有科目都得了‘A’。很可能米夏埃尔·梅斯特林试图告诉他他可以做得更好，因为开普勒实际上是他选择教授更高级天文学的精选学生之一，通过向他们介绍尼古拉·哥白尼的新日心宇宙体系。正是从米夏埃尔·梅斯特林那里，开普勒得知On the revolutions的序言，解释这“只是数学”，并非出自尼古拉·哥白尼之手。开普勒似乎几乎立即接受了哥白尼体系在物理上是真实的；他接受它的理由将与他第一个宇宙模型联系起来讨论（见下文）。

似乎在开普勒的学生时代，就有迹象表明他的宗教信仰并不完全符合图宾根当前正统的路德教义，该教义在Confessio AugustanaⓉ（奥格斯堡信纲）中 formulated。开普勒与新教正统派的这些问题涉及圣体教义中物质与“精神”（一种非物质的实体）之间的所谓关系。这与开普勒的天文学有关，因为他显然在解释来自太阳的“力”[见关于开普勒行星定律的历史主题]如何影响行星时，发现了类似的智力困难。在他的著作中，开普勒习惯于直言不讳地表达自己的观点——这对历史学家来说非常方便。在现实生活中，类似的开放倾向似乎很可能导致图宾根的当局对他的宗教正统性产生了有充分根据的怀疑。这些可能解释了为什么米夏埃尔·梅斯特林说服开普勒放弃授圣职的计划，转而接受在格拉茨教授数学的职位。随后的几年里，宗教不容忍加剧。开普勒于1612年被逐出教会。这给他带来了很大的痛苦，但尽管他（那时）作为帝国数学家拥有相对较高的社会地位，他从未成功解除这一禁令。

开普勒的第一个宇宙模型（1596年）

在标准地心天文学中有七颗行星，而哥白尼体系只有六颗，月亮已成为天文学以前未知的一种天体，开普勒后来称之为“卫星”（他在1610年创造了这个名字来描述伽利略发现绕木星运行的卫星，字面意思是“侍从”）。为什么是六颗行星？

此外，在地心天文学中，无法利用观测来确定各行星轨道的相对大小；人们只是假定它们彼此接触。这似乎无需解释，因为它与自然哲学家的信念十分吻合：整个系统是由最外层天球的运动带动的，而这个天球位于“固定”恒星（其图案构成星座的那些恒星）天球之外一个（或许两个）天球处，又在土星天球之外。在哥白尼体系中，每个行星运动的周年分量都是地球周年运动的反映，这一事实使人们能够利用观测来计算每个行星轨道的尺寸，结果发现行星之间存在巨大的空间。为什么恰好是这些空间？
插图：Kepler_polyhedra.gif ↗
开普勒在Mysterium cosmographicumⓉ（《宇宙的奥秘》），图宾根，1596年中对这些问题的回答，在20世纪的读者看来颇为怪异（见右图）。他提出，如果画一个球面与土星轨道的内侧相切，并在该球内内接一个立方体，那么内接于该立方体的球面就是外接木星轨道的球面。然后，如果在木星轨道的内接球中画一个正四面体，该四面体的内切球就是外接火星轨道的球面，如此向内，在火星与地球之间放入正正十二面体，在地球与金星之间放入正二十面体，在金星与水星之间放入正八面体。这完美地解释了行星的数目：只有五种凸正多面体（正如欧几里得的Elements 第13卷所证明的）。它还与尼古拉·哥白尼推得的轨道大小令人信服地吻合，最大误差小于10%（即使现在，这对一个宇宙学模型来说也好得惊人）。开普勒并未用百分比误差来表述，而他的模型事实上是第一个数学宇宙学模型，但很容易看出他为何相信观测证据支持他的理论。

开普勒认为他的宇宙学理论为哥白尼理论提供了证据。在提出自己的理论之前，他给出了论证以确立哥白尼理论本身的可信性。开普勒断言，它相对于地心理论的优点在于其更强的解释力。例如，哥白尼理论可以解释为什么金星和水星从未被看到离太阳很远（它们位于地球与太阳之间），而在地心理论中这一事实得不到解释。开普勒在Mysterium cosmographicumⓉ（《宇宙的奥秘》）第一章中列出了九个这样的问题。

开普勒在格拉茨教书时进行了这项工作，但这本书由图宾根的米夏埃尔·梅斯特林监督出版。与从观测推导出的值并不完全一致，开普勒希望更好的观测能改善这种一致性，因此他将一本《宇宙的奥秘》寄给了当时最杰出的观测天文学家之一第谷·布拉赫（1546 - 1601）。第谷·布拉赫当时在布拉格工作（当时是神圣罗马帝国的首都），实际上已经写信给米夏埃尔·梅斯特林寻找数学助手。开普勒得到了这份工作。

‘与火星的战争’

自然，第谷·布拉赫的优先事项与开普勒的不同，开普勒很快发现自己正在研究火星轨道的棘手问题[见关于开普勒行星定律的历史主题]。在第谷·布拉赫去世（1601年）后，他继续这项工作，开普勒接替他成为帝国数学家。传统上，轨道由圆组成，只需要很少的观测值来确定圆的相对半径和位置。第谷·布拉赫进行了大量观测，开普勒决心尽可能最好地利用它们。本质上，他有如此多的观测数据，一旦他构建了一个可能的轨道，他就能用进一步的观测来检验它，直到达到令人满意的一致性。开普勒得出结论，火星的轨道是一个椭圆，太阳位于其一个焦点上（这一结果推广到所有行星后现在被称为“开普勒第一定律”），并且连接行星和太阳的线在行星描述其轨道时在相等时间内扫过相等面积（“开普勒第二定律”），即面积被用作时间的度量。在这项工作发表在Astronomia nova, ... Ⓣ（新天文学），海德堡，1609年之后，开普勒找到了其他行星的轨道，从而确立这两条定律也适用于它们。这两条定律都将行星的运动与太阳联系起来；开普勒的哥白尼主义对他的推理和演绎至关重要。

火星的实际计算过程极其费力——有近千张幸存的算术对开纸——开普勒自己将这项工作称为‘我与火星的战争’，但结果是一个与现代结果如此精确吻合的轨道，以至于比较时必须考虑到自开普勒时代以来轨道的长期变化。

观测误差

对开普勒来说，将他可能存在的轨道与观测进行核对的方法至关重要的是，他必须知道什么才算是足够吻合。由此产生了观测误差概念的首次明确使用。开普勒可能至少部分地将这一概念归功于第谷·布拉赫，后者曾对其仪器的性能做过详细检查（见第谷·布拉赫）。

光学与1604年的新星

关于火星的工作到1605年基本完成，但该书的出版有所延迟。与此同时，针对直接观测与使用camera obscura观测时月球视直径不同这一疑虑，开普勒做了一些光学方面的工作，提出了关于camera obscura的第一个正确的数学理论，并首次正确解释了人眼的工作方式，即在视网膜上形成倒立的图像。这些成果发表在Ad Vitellionem paralipomena, quibus astronomiae pars optica traditurⓉ（《对维泰洛的补充，论天文学的光学部分》），法兰克福，1604年。他还撰写了关于1604年新星的文章，该星现在通常被称为“开普勒超新星”，他否定了众多解释，并在某处指出，这颗星当然可能只是特殊的创造，“但在我们得出[那个结论]之前，我认为我们应该尝试其他一切”De stella novaⓉ（《论新星》），布拉格，1606年，第22章，KGW 1，第257页，第23行。

在伽利略使用望远镜发现木星卫星之后——该发现发表在他的Sidereal Messenger（威尼斯，1610年）中，开普勒对此写了一篇热情的回复（1610年）——开普勒写了一篇关于透镜性质的研究（这是光学方面的第一部此类著作），其中他提出了一种使用两片凸透镜的新型望远镜设计（Dioptrice，布拉格，1611年）。这种设计所成的最终图像是倒立的，它如此成功，以至于现在通常不被称为开普勒望远镜，而直接称为天文望远镜。

离开布拉格前往林茨

开普勒在布拉格的岁月相对平静，而且在科学上极其多产。事实上，即使事情进展不顺，他似乎也从未允许外部环境妨碍他继续工作。1611年末，情况开始变得非常糟糕。首先，他七岁的儿子去世了。开普勒写信给一位朋友说，这个孩子的死尤其难以承受，因为这个孩子让他想起了自己那个年纪的样子。接着开普勒的妻子去世了。然后，健康状况每况愈下的皇帝鲁道夫被迫退位，让位给他的兄弟马蒂亚斯，后者与鲁道夫一样是天主教徒，但（与鲁道夫不同）不相信对新教徒的宽容。开普勒不得不离开布拉格。离开之前，他将妻子的遗体移入儿子的墓中，并为他们写了一篇拉丁文墓志铭。他和剩下的孩子们搬到了林茨（今属奥地利）。

婚姻与酒桶
⟦N1⟧似乎是为爱情而娶了他的第一任妻子芭芭拉（尽管这桩婚事是通过中介安排的）。1613年的第二次婚姻则是出于实际需要；他需要有人照料孩子。⟦N2⟧的新妻子苏珊娜对⟦N3⟧的性格有了一次速成式的了解：由此产生的那本书的献词信解释说，在婚礼庆祝活动中，他注意到酒桶的容积是通过一根斜插进桶孔的杆来估算的，于是他开始琢磨这怎么可能行得通。其结果是一项关于旋转体体积的研究⟦E1⟧……⟦L2⟧（《酒桶的新立体测量学》），林茨，1615年，其中⟦N4⟧以⟦L3⟧的工作为基础，使用了分解为“不可分量”的方法。这一方法后来由⟦L4⟧（约1598—1647）加以发展，并且是⟦L5⟧微积分的前身之一。

开普勒似乎是为了爱情而娶了他的第一任妻子芭芭拉（尽管这桩婚姻是通过中介安排的）。1613年的第二次婚姻则是出于实际需要；他需要有人照顾孩子。开普勒的新妻子苏珊娜对开普勒的性格进行了一番速成了解：由此产生的著作的献词信解释说，在婚礼庆祝活动中，他注意到酒桶的体积是通过一根斜插进桶孔的杆来估算的，于是他开始思考这是如何做到的。结果是对旋转体体积的研究Nova stereometria doliorum ……Ⓣ（《酒桶的新立体几何》），林茨，1615年，其中开普勒以阿基米德的工作为基础，使用了分解为“不可分量”的方法。这种方法后来由博纳文图拉·卡瓦列里（约1598-1647）发展，并且是无穷小微积分渊源的一部分。

The Harmony of the World

开普勒作为帝国数学家的主要任务是编写基于第谷·布拉赫观测的天文表，但他真正想做的是写The Harmony of the World，自1599年起就计划作为他Mystery of the Cosmos的发展。这部关于宇宙论的第二部著作（Harmonices mundi libri V Ⓣ（世界的和谐，第5卷），林茨，1619年）提出了比早期更精细的数学模型，尽管多面体仍然存在。这部著作中的数学包括对镶嵌的首次系统处理，证明只有十三种凸均匀多面体（阿基米德立体），以及首次描述两种非凸正多面体（全部在第2卷中）。The Harmony of the World还包含现在被称为‘开普勒第三定律’的内容，即对于任意两颗行星，其周期的平方比将等于其轨道平均半径的立方比。从一开始，开普勒就寻求一个将轨道大小与周期联系起来的规则，但并没有像其他两条定律那样缓慢地逐步接近这条定律。事实上，尽管第三定律在Harmony of the World印刷版的某些最后部分中发挥了重要作用，但它实际上直到作品付印时才被发现。开普勒做了最后的修改。他自己讲述了最终成功的故事：-

……如果你想知道确切的时刻，它是在今年一千六百一十八年3月8日在心中构想出来的，但以一种不幸的方式提交计算，因此被当作错误而否定，最后在5月15日重新回来并采用新的进攻路线，攻占了我心灵的黑暗。我十七年来对第谷·布拉赫观测的研究与当前研究相结合，两者共同作用，提供了如此强有力的支持，以至于起初我相信自己在做梦，并把我的结论当作基本前提之一。但“任意两颗行星的周期时间之比恰好是它们平均距离的 sesquialterate 比……”这一点是绝对确定和精确的……
（《世界的和谐》第5卷，第3章，Aiton、Duncan和Field译，第411页）。

巫术审判

开普勒正在撰写他的Harmony of the World时，他的母亲被指控施行巫术。他寻求了图宾根法律系的帮助。Katharina 开普勒最终获释，至少部分原因是当局在使用酷刑时未遵循正确法律程序而引发的技术性异议。留存下来的文件令人不寒而栗。然而，开普勒继续工作。在前往符腾堡为母亲辩护的马车途中，他阅读了Vincenzo Galilei（约1520—1591，伽利略之父）的一部音乐理论著作，The Harmony of the World中多次提及此书。

天文表

计算表格是天文学家的日常事务，总是涉及繁重的算术。因此，当开普勒在1616年看到约翰·纳皮尔关于对数的工作（1614年出版）时，他非常高兴。然而，米夏埃尔·梅斯特林立刻告诉他，第一，一位严肃的数学家为仅仅是一种计算辅助工具而欣喜是不体面的；第二，信任对数是愚蠢的，因为没有人理解它们是如何运作的。（20世纪60年代初对计算机也有类似的评论。）开普勒对第二个反对意见的回答是发表一个关于对数如何运作的证明，其依据是一个无可挑剔的可靠来源：欧几里得的Elements第5卷。开普勒计算了八位对数的表格，这些表格与Rudolphine Tables一起出版（乌尔姆，1628年）。这些天文表格不仅使用了第谷·布拉赫的观测数据，还使用了开普勒的前两条定律。所有利用新观测数据的天文表格在出版后的最初几年都是准确的。Rudolphine Tables的非凡之处在于，它们在数十年间都被证明是准确的。随着年岁的增加，表格的持续准确性自然被视为开普勒定律正确性的论据，从而也是日心说天文学正确性的论据。开普勒履行其作为帝国数学家的枯燥官方任务，促成了他最殷切愿望的实现，即帮助确立哥白尼学说。

华伦斯坦

当Rudolphine Tables出版时，开普勒实际上已不再为皇帝工作（他于1626年离开林茨），而是为阿尔布雷希特·冯·华伦斯坦（1583 - 1632）工作，后者是三十年战争（1618 - 1648）中少数成功的军事领袖之一。

华伦斯坦像皇帝鲁道夫一样，期望开普勒根据占星术向他提供建议。开普勒自然不得不服从，但反复指出他不相信能做出精确的预测。像当时大多数人一样，开普勒接受占星术的原则，即天体可以影响地球上发生的事情（最明显的例子是太阳导致季节变化，月亮导致潮汐），但作为哥白尼主义者，他不相信星座的物理实在性。他的占星术仅基于天体位置之间的角度（“占星相位”）。他对传统占星术的复杂体系表示极度蔑视。

逝世

开普勒在雷根斯堡因病短暂卧床后去世。他当时正停留在该城，准备去收取与Rudolphine Tables有关的一些欠他的钱。他被安葬在当地教堂，但该教堂在三十年战争期间被毁，坟墓已无任何遗迹。

史学札记

人们有时对开普勒科学活动中所谓非理性因素大做文章。相信占星术的人常常声称，他的工作为他们的主张提供了在科学上体面的先例。已故的Arthur Koestler在其颇具影响力的Sleepwalkers中，把开普勒与火星的搏斗变成了现代科学内在非理性的论据。这两种论调都有许多心照不宣的追随者。然而，两者都建立在对开普勒著作极为片面的解读之上。尤其是，Koestler似乎并不具备理解开普勒方法的数学专业能力。更仔细的研究表明，Koestler的评判纯属错误。

开普勒工作中真正重要的非理性因素是他的基督教信仰。开普勒对数学广泛而成功的运用使他的工作看起来“现代”，但我们实际上面对的是一位基督教自然哲学家，对他而言，理解宇宙的本性包括理解其创造者的本性。

Johannes Kepler is now chiefly remembered for discovering the three laws of planetary motion that bear his name published in 1609 and 1619). He also did important work in optics (1604, 1611), discovered two new regular polyhedra (1619), gave the first mathematical treatment of close packing of equal spheres (leading to an explanation of the shape of the cells of a honeycomb, 1611), gave the first proof of how logarithms worked (1624), and devised a method of finding the volumes of solids of revolution that (with hindsight!) can be seen as contributing to the development of calculus (1615, 1616). Moreover, he calculated the most exact astronomical tables hitherto known, whose continued accuracy did much to establish the truth of heliocentric astronomy (Rudolphine Tables, Ulm, 1627).

A large quantity of Kepler's correspondence survives. Many of his letters are almost the equivalent of a scientific paper (there were as yet no scientific journals), and correspondents seem to have kept them because they were interesting. In consequence, we know rather a lot about Kepler's life, and indeed about his character. It is partly because of this that Kepler has had something of a career as a more or less fictional character (see historiographic note below).

Childhood

Kepler was born in the small town of Weil der Stadt in Swabia and moved to nearby Leonberg with his parents in 1576. His father was a mercenary soldier and his mother the daughter of an innkeeper. Johannes was their first child. His father left home for the last time when Johannes was five, and is believed to have died in the war in the Netherlands. As a child, Kepler lived with his mother in his grandfather's inn. He tells us that he used to help by serving in the inn. One imagines customers were sometimes bemused by the child's unusual competence at arithmetic.

Kepler's early education was in a local school and then at a nearby seminary, from which, intending to be ordained, he went on to enrol at the University of Tübingen, then (as now) a bastion of Lutheran orthodoxy.

Kepler's opinions

Throughout his life, Kepler was a profoundly religious man. All his writings contain numerous references to God, and he saw his work as a fulfilment of his Christian duty to understand the works of God. Man being, as Kepler believed, made in the image of God, was clearly capable of understanding the Universe that He had created. Moreover, Kepler was convinced that God had made the Universe according to a mathematical plan (a belief found in the works of Plato and associated with Pythagoras). Since it was generally accepted at the time that mathematics provided a secure method of arriving at truths about the world (Euclid's common notions and postulates being regarded as actually true), we have here a strategy for understanding the Universe. Since some authors have given Kepler a name for irrationality, it is worth noting that this rather hopeful epistemology is very far indeed from the mystic's conviction that things can only be understood in an imprecise way that relies upon insights that are not subject to reason. Kepler does indeed repeatedly thank God for granting him insights, but the insights are presented as rational.

University education

At this time, it was usual for all students at a university to attend courses on "mathematics". In principle this included the four mathematical sciences: arithmetic, geometry, astronomy and music. It seems, however, that what was taught depended on the particular university. At Tübingen Kepler was taught astronomy by one of the leading astronomers of the day, Michael Mästlin (1550 - 1631). The astronomy of the curriculum was, of course, geocentric astronomy, that is the current version of the Ptolemaic system, in which all seven planets - Moon, Mercury, Venus, Sun, Mars, Jupiter and Saturn - moved round the Earth, their positions against the fixed stars being calculated by combining circular motions. This system was more or less in accord with current (Aristotelian) notions of physics, though there were certain difficulties, such as whether one might consider as 'uniform' (and therefore acceptable as obviously eternal) a circular motion that was not uniform about its own centre but about another point (called an 'equant'). However, it seems that on the whole astronomers (who saw themselves as 'mathematicians') were content to carry on calculating positions of planets and leave it to natural philosophers to worry about whether the mathematical models corresponded to physical mechanisms. Kepler did not take this attitude. His earliest published work (1596) proposes to consider the actual paths of the planets, not the circles used to construct them.

At Tübingen, Kepler studied not only mathematics but also Greek and Hebrew (both necessary for reading the scriptures in their original languages). Teaching was in Latin. At the end of his first year Kepler got 'A's for everything except mathematics. Probably Mästlin was trying to tell him he could do better, because Kepler was in fact one of the select pupils to whom he chose to teach more advanced astronomy by introducing them to the new, heliocentric cosmological system of Copernicus. It was from Mästlin that Kepler learned that the preface to On the revolutions, explaining that this was 'only mathematics', was not by Copernicus. Kepler seems to have accepted almost instantly that the Copernican system was physically true; his reasons for accepting it will be discussed in connection with his first cosmological model (see below).

It seems that even in Kepler's student days there were indications that his religious beliefs were not entirely in accord with the orthodox Lutheranism current in Tübingen and formulated in the Confessio Augustana Ⓣ. Kepler's problems with this Protestant orthodoxy concerned the supposed relation between matter and 'spirit' (a non-material entity) in the doctrine of the Eucharist. This ties up with Kepler's astronomy to the extent that he apparently found somewhat similar intellectual difficulties in explaining how 'force' [See the History Topic on Kepler's planetary laws] from the Sun could affect the planets. In his writings, Kepler is given to laying his opinions on the line - which is very convenient for historians. In real life, it seems likely that a similar tendency to openness led the authorities at Tübingen to entertain well-founded doubts about his religious orthodoxy. These may explain why Mästlin persuaded Kepler to abandon plans for ordination and instead take up a post teaching mathematics in Graz. Religious intolerance sharpened in the following years. Kepler was excommunicated in 1612. This caused him much pain, but despite his (by then) relatively high social standing, as Imperial Mathematician, he never succeeded in getting the ban lifted.

Kepler's first cosmological model (1596)

Instead of the seven planets in standard geocentric astronomy the Copernican system had only six, the Moon having become a body of kind previously unknown to astronomy, which Kepler was later to call a 'satellite' (a name he coined in 1610 to describe the moons that Galileo had discovered were orbiting Jupiter, literally meaning 'attendant'). Why six planets?

Moreover, in geocentric astronomy there was no way of using observations to find the relative sizes of the planetary orbs; they were simply assumed to be in contact. This seemed to require no explanation, since it fitted nicely with natural philosophers' belief that the whole system was turned from the movement of the outermost sphere, one (or maybe two) beyond the sphere of the 'fixed' stars (the ones whose pattern made the constellations), beyond the sphere of Saturn. In the Copernican system, the fact that the annual component of each planetary motion was a reflection of the annual motion of the Earth allowed one to use observations to calculate the size of each planet's path, and it turned out that there were huge spaces between the planets. Why these particular spaces?
插图：Kepler_polyhedra.gif ↗
Kepler's answer to these questions, described in his Mysterium cosmographicum Ⓣ, Tübingen, 1596, looks bizarre to twentieth-century readers (see the figure on the right). He suggested that if a sphere were drawn to touch the inside of the path of Saturn, and a cube were inscribed in the sphere, then the sphere inscribed in that cube would be the sphere circumscribing the path of Jupiter. Then if a regular tetrahedron were drawn in the sphere inscribing the path of Jupiter, the insphere of the tetrahedron would be the sphere circumscribing the path of Mars, and so inwards, putting the regular dodecahedron between Mars and Earth, the regular icosahedron between Earth and Venus, and the regular octahedron between Venus and Mercury. This explains the number of planets perfectly: there are only five convex regular solids (as is proved in Euclid's Elements , Book 13). It also gives a convincing fit with the sizes of the paths as deduced by Copernicus, the greatest error being less than 10% (which is spectacularly good for a cosmological model even now). Kepler did not express himself in terms of percentage errors, and his is in fact the first mathematical cosmological model, but it is easy to see why he believed that the observational evidence supported his theory.

Kepler saw his cosmological theory as providing evidence for the Copernican theory. Before presenting his own theory he gave arguments to establish the plausibility of the Copernican theory itself. Kepler asserts that its advantages over the geocentric theory are in its greater explanatory power. For instance, the Copernican theory can explain why Venus and Mercury are never seen very far from the Sun (they lie between Earth and the Sun) whereas in the geocentric theory there is no explanation of this fact. Kepler lists nine such questions in the first chapter of the Mysterium cosmographicum Ⓣ.

Kepler carried out this work while he was teaching in Graz, but the book was seen through the press in Tübingen by Mästlin. The agreement with values deduced from observation was not exact, and Kepler hoped that better observations would improve the agreement, so he sent a copy of the Mysterium cosmographicum to one of the foremost observational astronomers of the time, Tycho Brahe (1546 - 1601). Tycho, then working in Prague (at that time the capital of the Holy Roman Empire), had in fact already written to Mästlin in search of a mathematical assistant. Kepler got the job.

The 'War with Mars'

Naturally enough, Tycho's priorities were not the same as Kepler's, and Kepler soon found himself working on the intractable problem of the orbit of Mars [See the History Topic on Kepler's planetary laws]. He continued to work on this after Tycho died (in 1601) and Kepler succeeded him as Imperial Mathematician. Conventionally, orbits were compounded of circles, and rather few observational values were required to fix the relative radii and positions of the circles. Tycho had made a huge number of observations and Kepler determined to make the best possible use of them. Essentially, he had so many observations available that once he had constructed a possible orbit he was able to check it against further observations until satisfactory agreement was reached. Kepler concluded that the orbit of Mars was an ellipse with the Sun in one of its foci (a result which when extended to all the planets is now called "Kepler's First Law"), and that a line joining the planet to the Sun swept out equal areas in equal times as the planet described its orbit ("Kepler's Second Law"), that is the area is used as a measure of time. After this work was published in Astronomia nova, ... Ⓣ, Heidelberg, 1609, Kepler found orbits for the other planets, thus establishing that the two laws held for them too. Both laws relate the motion of the planet to the Sun; Kepler's Copernicanism was crucial to his reasoning and to his deductions.

The actual process of calculation for Mars was immensely laborious - there are nearly a thousand surviving folio sheets of arithmetic - and Kepler himself refers to this work as 'my war with Mars', but the result was an orbit which agrees with modern results so exactly that the comparison has to make allowance for secular changes in the orbit since Kepler's time.

Observational error

It was crucial to Kepler's method of checking possible orbits against observations that he have an idea of what should be accepted as adequate agreement. From this arises the first explicit use of the concept of observational error. Kepler may have owed this notion at least partly to Tycho, who made detailed checks on the performance of his instruments (see the biography of Brahe).

Optics, and the New Star of 1604

The work on Mars was essentially completed by 1605, but there were delays in getting the book published. Meanwhile, in response to concerns about the different apparent diameter of the Moon when observed directly and when observed using a camera obscura, Kepler did some work on optics, and came up with the first correct mathematical theory of the camera obscura and the first correct explanation of the working of the human eye, with an upside-down picture formed on the retina. These results were published in Ad Vitellionem paralipomena, quibus astronomiae pars optica traditur Ⓣ, Frankfurt, 1604. He also wrote about the New Star of 1604, now usually called 'Kepler's supernova', rejecting numerous explanations, and remarking at one point that of course this star could just be a special creation 'but before we come to [that] I think we should try everything else' De stella nova Ⓣ, Prague, 1606, Chapter 22, KGW 1, p. 257, line 23.

Following Galileo's use of the telescope in discovering the moons of Jupiter, published in his Sidereal Messenger (Venice, 1610), to which Kepler had written an enthusiastic reply (1610), Kepler wrote a study of the properties of lenses (the first such work on optics) in which he presented a new design of telescope, using two convex lenses (Dioptrice, Prague, 1611). This design, in which the final image is inverted, was so successful that it is now usually known not as a Keplerian telescope but simply as the astronomical telescope.

Leaving Prague for Linz

Kepler's years in Prague were relatively peaceful, and scientifically extremely productive. In fact, even when things went badly, he seems never to have allowed external circumstances to prevent him from getting on with his work. Things began to go very badly in late 1611. First, his seven year old son died. Kepler wrote to a friend that this death was particularly hard to bear because the child reminded him so much of himself at that age. Then Kepler's wife died. Then the Emperor Rudolf, whose health was failing, was forced to abdicate in favour of his brother Matthias, who, like Rudolf, was a Catholic but (unlike Rudolf) did not believe in tolerance of Protestants. Kepler had to leave Prague. Before he departed he had his wife's body moved into the son's grave, and wrote a Latin epitaph for them. He and his remaining children moved to Linz (now in Austria).

Marriage and wine barrels

Kepler seems to have married his first wife, Barbara, for love (though the marriage was arranged through a broker). The second marriage, in 1613, was a matter of practical necessity; he needed someone to look after the children. Kepler's new wife, Susanna, had a crash course in Kepler's character: the dedicatory letter to the resultant book explains that at the wedding celebrations he noticed that the volumes of wine barrels were estimated by means of a rod slipped in diagonally through the bung-hole, and he began to wonder how that could work. The result was a study of the volumes of solids of revolution Nova stereometria doliorum ... Ⓣ, Linz, 1615, in which Kepler, basing himself on the work of Archimedes, used a resolution into 'indivisibles'. This method was later developed by Bonaventura Cavalieri (c. 1598 - 1647) and is part of the ancestry of the infinitesimal calculus.

The Harmony of the World

Kepler's main task as Imperial Mathematician was to write astronomical tables, based on Tycho's observations, but what he really wanted to do was write The Harmony of the World, planned since 1599 as a development of his Mystery of the Cosmos. This second work on cosmology (Harmonices mundi libri V Ⓣ, Linz, 1619) presents a more elaborate mathematical model than the earlier one, though the polyhedra are still there. The mathematics in this work includes the first systematic treatment of tessellations, a proof that there are only thirteen convex uniform polyhedra (the Archimedean solids) and the first account of two non-convex regular polyhedra (all in Book 2). The Harmony of the World also contains what is now known as 'Kepler's Third Law', that for any two planets the ratio of the squares of their periods will be the same as the ratio of the cubes of the mean radii of their orbits. From the first, Kepler had sought a rule relating the sizes of the orbits to the periods, but there was no slow series of steps towards this law as there had been towards the other two. In fact, although the Third Law plays an important part in some of the final sections of the printed version of the Harmony of the World, it was not actually discovered until the work was in press. Kepler made last-minute revisions. He himself tells the story of the eventual success:-

...and if you want the exact moment in time, it was conceived mentally on 8th March in this year one thousand six hundred and eighteen, but submitted to calculation in an unlucky way, and therefore rejected as false, and finally returning on the 15th of May and adopting a new line of attack, stormed the darkness of my mind. So strong was the support from the combination of my labour of seventeen years on the observations of Brahe and the present study, which conspired together, that at first I believed I was dreaming, and assuming my conclusion among my basic premises. But it is absolutely certain and exact that "the proportion between the periodic times of any two planets is precisely the sesquialterate proportion of their mean distances ..."
(Harmonice mundi Book 5, Chapter 3, trans. Aiton, Duncan and Field, p. 411).

Witchcraft trial

While Kepler was working on his Harmony of the World, his mother was charged with witchcraft. He enlisted the help of the legal faculty at Tübingen. Katharina Kepler was eventually released, at least partly as a result of technical objections arising from the authorities' failure to follow the correct legal procedures in the use of torture. The surviving documents are chilling. However, Kepler continued to work. In the coach, on his journey to Württemberg to defend his mother, he read a work on music theory by Vincenzo Galilei (c.1520 - 1591, Galileo's father), to which there are numerous references in The Harmony of the World.

Astronomical Tables

Calculating tables, the normal business for an astronomer, always involved heavy arithmetic. Kepler was accordingly delighted when in 1616 he came across Napier's work on logarithms (published in 1614). However, Mästlin promptly told him first that it was unseemly for a serious mathematician to rejoice over a mere aid to calculation and second that it was unwise to trust logarithms because no-one understood how they worked. (Similar comments were made about computers in the early 1960s.) Kepler's answer to the second objection was to publish a proof of how logarithms worked, based on an impeccably respectable source: Euclid's Elements Book 5. Kepler calculated tables of eight-figure logarithms, which were published with the Rudolphine Tables (Ulm, 1628). The astronomical tables used not only Tycho's observations, but also Kepler's first two laws. All astronomical tables that made use of new observations were accurate for the first few years after publication. What was remarkable about the Rudolphine Tables was that they proved to be accurate over decades. And as the years mounted up, the continued accuracy of the tables was, naturally, seen as an argument for the correctness of Kepler's laws, and thus for the correctness of the heliocentric astronomy. Kepler's fulfilment of his dull official task as Imperial Mathematician led to the fulfilment of his dearest wish, to help establish Copernicanism.

Wallenstein

By the time the Rudolphine Tables were published Kepler was, in fact, no longer working for the Emperor (he had left Linz in 1626), but for Albrecht von Wallenstein (1583 - 1632), one of the few successful military leaders in the Thirty Years' War (1618 - 1648).

Wallenstein, like the emperor Rudolf, expected Kepler to give him advice based on astrology. Kepler naturally had to obey, but repeatedly points out that he does not believe precise predictions can be made. Like most people of the time, Kepler accepted the principle of astrology, that heavenly bodies could influence what happened on Earth (the clearest examples being the Sun causing the seasons and the Moon the tides) but as a Copernican he did not believe in the physical reality of the constellations. His astrology was based only on the angles between the positions of heavenly bodies ('astrological aspects'). He expresses utter contempt for the complicated systems of conventional astrology.

Death

Kepler died in Regensburg, after a short illness. He was staying in the city on his way to collect some money owing to him in connection with the Rudolphine Tables. He was buried in the local church, but this was destroyed in the course of the Thirty Years' War and nothing remains of the tomb.

Historiographic note

Much has sometimes been made of supposedly non-rational elements in Kepler's scientific activity. Believing astrologers frequently claim his work provides a scientifically respectable antecedent to their own. In his influential Sleepwalkers the late Arthur Koestler made Kepler's battle with Mars into an argument for the inherent irrationality of modern science. There have been many tacit followers of these two persuasions. Both are, however, based on very partial reading of Kepler's work. In particular, Koestler seems not to have had the mathematical expertise to understand Kepler's procedures. Closer study shows Koestler was simply mistaken in his assessment.

The truly important non-rational element in Kepler's work is his Christianity. Kepler's extensive and successful use of mathematics makes his work look 'modern', but we are in fact dealing with a Christian Natural Philosopher, for whom understanding the nature of the Universe included understanding the nature of its Creator.