温伯格《解释世界》第十一章 太阳系问题得以解决

无论如何评价科学革命，可以确定它始于哥白尼。尼古拉·哥白尼1473年出生于波兰一个早先从西里西亚移民过来的家庭。尼古拉10岁丧父，幸运的是他的舅舅抚养了他，他的舅舅从效力教会中致富，几年后成为波兰东北部瓦尔米亚（也叫厄米兰得）的主教。哥白尼在克拉科夫大学完成学业之后（可能也学了天文学课程），1496年进入博洛尼亚大学攻读教会法，在这里作为天文学家多梅尼科·玛丽亚·诺瓦拉（雷格蒙塔努斯的学生）的助手开始进行天文观测。在博洛尼亚哥白尼获悉在他的舅舅的资助下他被任命为伐米亚弗龙堡（也叫费劳恩译）座堂圣职团的16名教士之一，他此后一直从此职务获得不菲的收入，而甚少承担宗教职责。哥白尼从来没有成为一名神父。在帕多瓦大学短暂攻读一段医学之后，他于1503年在费拉拉大学获宗教法博士学位，不久之后他就返回波兰。1510年他定居于弗龙堡，在这里他建立了一座小型天文观测站，他一直在这里生活到1543年去世。 哥白尼回到弗龙堡不久就创作了一部匿名短文，后来题为《试论天体运行的假设》，通常称为《小注解》。《小注解》在作者去世很久之后才发表，所以没有他其他作品那样具有影响力，但记述了指导他工作的观点。 在对早期行星运行理论作为了简短批评之后，哥白尼在《小注解》中阐述了他新理论的七条原理。下面是这七条原理的改述以及注释。 1. 不存在天体运行轨道的共同中心。（历史学家对哥白尼是否认同亚里士多德提出的天体由实体球体携带有不同意见。） 2. 地球中心不是宇宙中心，其中心只是月球运行轨道中心，以及地面物体受重力吸引朝向的中心。 3. 除月球之外所有天体都围绕太阳运行，因而太阳是宇宙中心。（但是正如下面要讲到的，哥白尼认为地球和其他行星的运行轨道中心不是太阳，而是靠近太阳的一点。） 4. 日地之间距离与地球恒星间的距离相比可以忽略不计。（可能哥白尼用这个假设来解释我们为什么观察不到周年视差—即由于地球围绕太阳运行而引起的恒星视周年移动。但是在《小注解》中没有提及视差问题。） 5. 恒星每日围绕地球的视运行完全是由于地球自转。 6. 太阳的视运行是由于地球自转和地球（与其他行星一样）围绕太阳公转。 7. 行星的视逆行是由于地球的运行，当地球超越火星，木星或土星，或被水星或金星超越。 哥白尼不能在《小注解》中说他的理论比托勒密理论更加符合观测结果。因为该理论确实没有更加符合。事实上也不可能，这是由于哥白尼理论大部分基于他从托勒密《天文学大成》中推导出的数据，而不是他自己的观测。哥白尼没有去寻求新的观测数据，而是特别指出他的理论中一系列审美优势。 其中一个优势是可以用地球的转动来解释太阳，恒星和其他行星各种视运行。这样哥白尼可以去除托勒密理论中的“硬凑”—即水星和金星本轮中心一直位于地球和太阳的连线之上，火星，木星和土星与它们各自本轮中心的连线一直平行于地球和太阳连线。结果每个内行星本轮中心围绕地球运转一周以及每个外行星在本轮上运转一周都需要硬凑到刚好整一年。哥白尼知道这种非自然要求其实只是我们从围绕太阳运行的平台上观察太阳系的镜像。 哥白尼理论另一个审美优势是其对行星运转轨道大小的确定。前面讲过托勒密体系中行星的视运行不取决于本轮和均轮的大小，而是取决于每个行星本轮和均轮半径之比。如果愿意，甚至可以将水星的均轮定为比土星均轮都大，只要相应调整水星本轮大小就可以。受托勒密《行星假说》影响，天文学家通常按照一颗行星距地球最大距离等于远离地球方向的下一颗行星距离地球最小距离的假设来设定行星运行轨道大小。这可以按照行星远离地球的顺序选择来设定行星运行轨道的相对大小，但这种顺序选择是很随意的。无论如何《行星假说》中的设想既不是基于观测，也没有被观测所证实。 相反若想让哥白尼体系与观测相符，每颗行星轨道半径就需要与地球轨道半径有确定比例关系。（注：第八章讲过，托勒密理论最简单模型中（每个行星有一个本轮，太阳没有本轮）只有一种特例与哥白尼理论最简便模型一致，差别只在视角不同：该特例是内行星均轮与太阳围绕地球运转轨道一致，外行星本轮半径都等于地日距离，而且内行星的本轮半径以及外行星的均轮半径与哥白尼理论中行星轨道半径一致。）特别是由于托勒密给内行星和外行星引入了不同的本轮（先不提轨道椭圆带来的复杂性），内行星本轮与均轮的半径之比必须等于它们与太阳以及日地距离之比，外行星本轮与均轮的半径之比必须反比于它们与太阳以及日地距离之比。（见技术说明13）哥白尼并没有这样呈现他的结果；他用了复杂的“三角法则”，给人一直错觉好像他做出新的预测且被观测所验证。但是他确实得出了正确的行星运行轨道半径大小。他发现行星距离太阳的次序为水星，金星，地球，火星，木星，和土星；这与它们公转周期次序完全一致，哥白尼估算其公转周期依次为3个月，9个月，1年，2年半，12年和30年。虽然那时还没有计算行星在其轨道运转速度的理论，不过哥白尼肯定认为行星运行轨道越大，其环绕太阳运行越慢。 哥白尼理论是这样的一个经典例子--一个理论虽然没有实验证实支持其优于其他理论，但基于审美准则可以认定该理论。《小注解》中的哥白尼理论用地球的自转和公转一举解决了托勒密理论中的一些令人费解问题，而且哥白尼理论在有关行星次序以及轨道大小方面比托勒密理论更加明确。哥白尼承认毕达哥拉斯学派早就提出过地动观，但是他也（正确地）指出他们只是“平白主张”该观点，缺乏他所提出的论据。 除了对托勒密理论中的硬凑以及行星运行轨道大小和次序的不确定之外，哥白尼对其他一些部分也不满意。哥白尼忠实于柏拉图行星匀速圆周运动的声明，他不接受托勒密使用匀速点的方法来处理从常速圆周运动的实际偏离。哥白尼像伊本·阿尔·沙提尔一样引入了更多本轮：给水星引入六个；给月球引入三个；金星，火星，木星和土星各四个。这里哥白尼没有比《天文学大成》有所提高。 哥白尼的成就又一次展现出物理科学史上时常发生的一幕：一个简单漂亮与观测结果比较相符的理论常常比一个复杂丑陋但与观测结果非常相符的理论更加接近真相。哥白尼理论最简单的形式是设定包括地球在内的行星都以太阳为中心做匀速圆周运动，完全不用本轮。这与托勒密天文学最简单形式—每个行星设一个本轮，太阳和月亮不设本轮，不用偏心圆和匀速点—完全一致。这种形式与观测结果并不完全相符，因为学校运行轨迹不是正圆，而是接近正圆的椭圆；它们的速度只是接近匀速；太阳不在每个椭圆的中心，而是稍许偏离中心，位于椭圆一个焦点上。（见技术说明18）哥白尼完全可以仿效托勒密给包括地球在内的每个行星运行轨道加偏心圆和匀速点，这样的话其与观测间的误差极小，用那时天文学家的观察方法完全不会察觉到这么小的误差。 量子力学发展史上也有这么一幕呈现不用过度担忧与观测结果间微小误差的重要性。1925年埃尔温·薛定谔摸索出计算最简单原子—氢原子能级的方法。他的计算结果与这些能级总体上符合的很好，但是他的计算结果精度上—把狭义相对论与牛顿力学间的差别考虑进去—与测量能级不相符。薛定谔思考了一阵，后来他意识到能够计算总体能级已经是一个显著成就，值得发表，对相对论效应的处理可以等以后解决。几年之后保罗·狄拉克做出了计算。 哥白尼除了用了许多本轮，他也引入其他复杂机制，类似托勒密天文学的偏心圆。地球轨道中心不在太阳，而在稍许偏离太阳的一点上。这种复杂机制可以用来大致解释各种现象，比如攸克特蒙发现的季节长度不同，实际上这是由于太阳不再地球椭圆形轨道的中心，而在其中一个焦点上，以及地球运行速度不是匀速。 哥白尼引入的另一个复杂机制其实是误解。哥白尼似乎认为地球围绕太阳公转一周会引起地球自转轴围绕垂直于地球公转轨道平面旋转360度，就像一名舞者在进行单脚尖旋转时伸展开手臂的手指会围绕垂直方向每次旋转360度。（哥白尼可能受到行星乘坚固透明球体运行旧观念的影响.）当然地轴事实上并没有每年发生明显改变，这样哥白尼不得不给地球另加除公转和自转外的第三种运行来抵消地轴的摇摆。哥白尼假设这种摇摆不会完全被抵消掉，每隔许多年地轴会摇摆一周，造成喜帕恰斯所发现的二分点缓慢的进动。牛顿之后人们才明白地球围绕太阳的公转对地轴方向没有影响，除了太阳和月亮对地球赤道隆起带的引力作用。所以（正如开普勒提出的）哥白尼设置的相互抵消实际上没有必要。 虽然加了这些复杂机制，哥白尼理论仍然比托勒密理论简单，但不是特别显著。哥白尼当时没有意识到，实际上如果他不管本轮，将理论的一些细小误差留给后世解决，他的理论会更接近真相。 《小注解》并没有介绍技术细节。这些细节在哥白尼的巨著《天体运行论》--即人们熟知的《运行论》中做了完整介绍，该书1543年出版，那时哥白尼已处于弥留的时刻。该书以对教皇保罗三世亚历山大·法尔内塞的献词开头。书中哥白尼再次提起亚里士多德同心圆与托勒密偏心圆和本轮间的争论，他指出前者不符合观测结果，而后者“与规则运行的第一原理冲突。”为了支持他提出的地球运转的大胆设想，哥白尼引用了普鲁塔克一段话： 有人认为地球不动。毕达哥拉斯学派的菲洛劳斯坚信地球像太阳和月亮一样围绕中心之火以斜圆运行。蓬杜斯的赫拉克利德斯和毕达哥拉斯学派的艾克方杜斯主张地球在运动，但不是在行进，而是像轮子一样从西向东围绕自己中心转动。 （ 在《运行论》标准版本中哥白尼没有提及阿里斯塔克斯，但是在原文中出现过，后来删除了。）哥白尼接着解释到既然前人曾经有过地动设想，也应该容许他测试此观点。然后他描述他的结论： 假定地球具有我在本书后面所赋予的那些运动，我经过长期、认真的研究终于发现如果把其他行星的运动与地球的轨道运行联系在一起，并按每颗行星的运转来计算，那么不仅可以对所有的行星和球体得出它们的观测现象，还可以使它们的顺序和大小以及苍穹本身全都联系在一起了，以至不能移动某一部分的任何东西而不在其他部分和整个宇宙中引起混乱。 与在《小注解》中一样，哥白尼强调了他的理论比托勒密理论预测精确的事实；按他的理论可以得出满足观测结果的唯一的行星次序以及轨道大小，而托勒密理论不能确定这些。当然如果哥白尼不假定他的理论正确的话，他就无法证实他的轨道半径是正确的；这只有等到伽利略对行星相位的观测之后才得以解决。 《运行论》大部分内容都非常专业，充实了《小注解》中的观点。值得特别指出的一点是在卷一中哥白尼阐述了运行由圆组成的先验主张。卷一第一章如此开头：

首先，我们应当指出，宇宙是球形的。这要么是因为在一切形状中球形是最完美的，不需要衔接，而且是不能增减的整体（这里哥白尼听起来像柏拉图）；要么是因为它是一切形状中容积最大的，最宜于包罗一切事物（也就是说同等表面积下圆的体积最大）；甚至还因为宇宙的个别部分--我指的是太阳、月球、行星和恒星--看起来都呈这种图形（他怎么可能知道恒星形状？）；乃至于万物都趋向于由这种边界所包围，就像水滴和其他液滴所呈现出的样子(这是表面张力，与行星没有关系)。因此没有人怀疑神圣物体拥有这种形状。 接着在第四章他解释到天体的运行是“匀速，永恒，圆周或由圆周运行组成。” 在卷一后面哥白尼指出了他的日心体系中最漂亮的一面：它解释了为什么水星和金星看起来从来不会在天空中离太阳太远。比如之所以从来不会看到金星远离太阳45度以上是因为它围绕太阳运行轨道大约是地球轨道的百分之70。（见技术说明19。）我们在第11章讲过，托勒密理论需要硬凑水星和金星的运行，让它们本轮中心一直保持在地球和太阳连线上。哥白尼体系也不再需要托勒密对外行星做出的硬凑—即要求保持每个外行星与其本轮中心连线平行于地日连线。 哥白尼体系甚至在《运行论》出版之前就遭遇宗教领袖的反对。十九世纪康奈尔首任校长安德鲁·迪克森·怀特创作的名著《基督教世界中科学与神学作战史》夸大了这种冲突，该书引用的路德，梅兰希顿，加尔文，和卫斯理许多名言都不可靠。但是冲突确实存在。《桌边谈话录》中记载了马丁·路德在维滕堡与他的信徒的对话，1539年6月4日记载了： 某位新星相家试图证实地球在运转，而不是天空，太阳，与月亮 … （路德评述说）“是这样的，那谢想表现自己聪明的人拒绝接受他人认可的东西。他一定要自己做出些什么。这就是那个傻瓜所做的，他想完全改写天文学。即使陷入这些混乱，我仍然相信神圣经文，因为约书亚命令太阳静止，而非地球。” 《运行论》出版之后几年，路德的同事菲利普·梅兰希通（1497-1560）加入了对哥白尼的攻击，他引用了《传道书》1：5—“太阳也会升起，然后太阳下落，很快回到升起之处。” 与《圣经》描述的冲突自然会引起那些已经用经文替代了教皇权威的新教徒的不满。除此之外，有一点可能引起所有宗教人士的不满：人类的家园—地球，现在被贬为只是与另外五个行星一样的一颗行星。 即使在《运行论》印刷时就出现了问题。哥白尼将他的手稿交给纽伦堡的出版社，出版社任命路德教一名牧师安德烈亚斯·奥西安德为编辑，奥西安德的业余爱好是天文学。也许是为了表达自己的见解，奥西安德在书中加入了一段前言，人们原以为这是哥白尼所写，直到后一个世纪才由开普勒揭示出真相。在此段前言中奥西安德借哥白尼之口否认试图去呈现行星真实运行特征，其中说道： 天文学家的职责是通过仔细且专业的研究来构建天体（视）运行过程。这样他需要设想并设计这些运行的原因或做出假设。因为他根本无法获得真实原由，他只能采用各种假设，只要能够从几何原理正确计算以后以及过去的运行即可。 奥西安德的前言结尾处写道： 只要是在谈假设，谁也不要指望从天文学得到任何确定的东西，而天文学也提供不出这样的东西。如果不了解这一点，他就会把为另一个目的提出的想法认为是真理，于是读完此书后比他刚开始读时更为愚蠢。 这与大约公元前70年吉米纽斯的观点一致（在第八章做了引述），但是完全有悖于哥白尼在《小注解》与《运行论》中所呈现的对我们现在称之为太阳系做出真实描述的意图。 不管个别牧师如何看待日心说，基督教新教并没有压制哥白尼理论。天主教在十七世纪之前也没有有组织地反对哥白尼。著名的乔尔丹诺·布鲁诺1600年被罗马宗教裁判所处死事件并不是由于他捍卫哥白尼，而是由于他的异端学说，（按那时的法规）他确实有罪。但是后面我们会看到，到十七世纪天主教会开始大规模禁止哥白尼观点。 对后世科学真正重要的是哥白尼学说受到同仁天文学界的接受。第一位信服哥白尼的是他唯一的学生--雷提卡斯，1540年雷提卡斯发表了对哥白尼理论的介绍，1543年他帮助将《运行论》交到纽伦堡出版社手里。（一开始应该由雷提卡斯写《运行论》的前言，但是他去了莱比锡履职，这个任务不幸交给了奥西安德。）雷提卡斯早期协助梅兰希顿将维滕堡大学建立成数学和天文学研究中心。 伊拉斯谟·雷茵霍尔德1551年在普鲁士公爵资助下应用哥白尼理论编制了新的天文表--普鲁士星表，人们可以使用该表计算行星在黄道任何时间的位置，这给哥白尼理论带来巨大声誉。这比1275年阿方索10世统治时期在卡斯蒂利亚编制的阿方索星表有显著改进。事实上这种改进不是由于哥白尼理论的优势，而是由于1275年与1551年之间积累的大量新的观察数据，或许也是由于日心说要简单的多，计算起来极其容易。当然坚守地球静止的人会辩称《运行论》只是为计算提供了方便，并不是世界的真实图像。事实上虽然耶稣会天文学家和数学家克里斯托弗卢斯·克拉维斯在教皇格里高利十三世领导下应用普鲁士星表制定了1582年日历改革，带来了我们现在使用的格里高利历，但克拉维斯从来没有放弃地球静止的观点。 有一位数学家试图协调地球静止与哥白尼理论间的矛盾。1568年梅兰希通的女婿，维滕堡数学教授卡斯帕·波瑟在《天体假说》中声称通过数学转换可以将哥白尼理论改写为地球静止而非太阳静止的形式。这正是后来波瑟的一名学生第谷·布拉赫得出的结果。 第谷·布拉赫是望远镜出现之前历史上最为高超的天文观测者，他提出的天体运行理论仅次于哥白尼。第谷1546年出生于斯坎尼亚省基（现在的瑞典南部，1658年前属于丹麦）一个丹麦贵族家庭。他在哥本哈根大学接受教育，1560年他成功预测了日偏食，为此他喜出望外。后来他转到德国和瑞士，在莱比锡大学，维滕堡大学，罗斯托克大学，巴塞尔大学和奥格斯堡大学学习。这期间他研究了普鲁士星表，此星表预测的1563年土星和木星的合只差几天，而较早的阿方索星表则差了几个月，这给他留下深刻印象。 回到丹麦以后，第谷先在他的舅舅位于斯坎尼亚省荷瑞维德的家中生活了一段时间。1572年他在仙后座观察到一颗他称之为的“新星”（现在知道那是一颗恒星发生了热核爆炸，叫Ia超新星。1952年射电天文学家发现此爆炸残留，他们发现其距离为9千光年，该恒星距离地球太远，在爆炸之前不用望远镜根本看不到。）第谷使用他自己建造的六分仪对这颗新星进行了长达数月的观测，他发现该新星没有任何周日视差（即每日在恒星间的位置变化），如果该新星像月球一样离地球近，或者比月球还近，那由于地球的自转（或者其他天体每日围绕地球的转动），就应该有周日视差。（见技术手册20）他断言：“这颗新星并不位于空气之上，月球之下，它不会离地球很近…应该远远在月球之外，位于天球中。”这与亚里士多德月球之外的天穹不变的理论直接相冲突，第谷因此一举成名。 1576年丹麦国王腓特烈二世将位于斯坎尼亚与丹麦西兰岛之间海峡内的汶岛赠与第谷，并许诺他一笔费用来支持他在汶岛建造维护居所和科学研究场地。第谷建造了乌拉妮娅城堡，其中包括天文观测台，图书馆，实验室和印刷厂。内部装饰有过去天文学家喜帕恰斯，托勒密，阿尔·巴塔尼和哥白尼的画像，以及科学资助人海塞-卡塞尔伯爵威廉姆四世的画像。第谷在汶岛培训助手，立即着手天文观测。 第谷1577年就已经观测到一颗彗星，他发现该彗星观测不到周日视差。这不只再次异于亚里士多德理论，证实月球之外的天穹会发生变化。而且现在第谷也可以断定彗星运行路径穿过了亚里士多德提出的同心圆球体或托勒密提出的球体。（如果这些球体是坚固实体，那这就是一个问题。在第八章我们讲过，亚里士多德就是这样教授的，后来又由希腊化时代天文学家在托勒密理论中得以继承。坚固实球体观点在现代早期曾复苏，很快被第谷排除。）彗星比超新星爆炸要频繁的多，在接下来几年第谷可以对其他彗星重复他的观测。 第谷从1583年起开始致力于一种全新的行星运行理论研究，该理论基本观点是地球静止，太阳和月亮围绕地球运行，而五颗行星围绕太阳运行。在第谷1588年出版的关于1577年出现的彗星著作第八章中介绍了该理论。按照该理论，地球既不自转也不公转，太阳，月亮，行星和恒星除了有慢速运转，它们也每天从东到西围绕地球运行一周。有些天文学家采纳了一种“半第谷”理论，即行星围绕着太阳，太阳又围绕着地球运行，但地球会自转，恒星静止不动。（最早倡导半第谷理论的是尼古拉斯·赖默斯·巴尔，虽然他并没有称之为半第谷理论，因为他认为第谷从他那里偷窃了原始第谷系统。） 前面多次讲过，第谷理论与托勒密理论的一个特例（托勒密从来没有这样考虑过）完全一致，即内行星的均轮与太阳围绕地球运行轨道吻合，外行星本轮半径等于太阳围绕地球轨道半径。如果只考虑天体的相对运行以及运行速度，该理论与哥白尼理论也等同，区别只是视角不同：哥白尼理论太阳静止，第谷理论地球既不自转也不公转。至于在观测方面，第谷理论自动会预测没有年恒星视差，不需要去假定恒星比太阳和行星距离地球要远的多（当然我们现在知道确实如此）。该理论也不再需要奥里斯姆对曾经误导托勒密和布里丹古老问题的解答：即向上抛出的物体由于地球的自转或公转会落到后面。 第谷对后世天文学家最为重要的贡献不是他的理论，而是他前无古人的观测精度。二十世纪七十年代我访问汶岛时没有看到第谷的建筑，但是第谷曾经放置他的观测设备的巨大岩石地基仍然在那里。（我访问之后那里建立了一个博物馆和花园）。使用这些设备，第谷观测天体位置的精度误差只有1/15o。在乌拉妮娅城堡旧址立着伊瓦尔·约翰森1936年雕刻的第谷花岗岩雕像—雕像仰望天穹，非常适合天文学家。 资助第谷的腓特烈二世1588年去世。克里斯蒂安四世即位，丹麦人现在仍然认为他是丹麦最伟大的帝王之一，可惜他对支持天文研究没有兴趣。第谷在汶岛的天文观测结束于1597年，后来他周游汉堡，德累斯顿，维滕堡和布拉格。在布拉格他成为圣罗马皇帝鲁道夫二世的皇家数学家，他开始制作新的天文表--鲁道夫天文表。第谷1601年去世后，开普勒承接了这个工作。 约翰尼斯·开普勒是第一位认识到非匀速圆周运动的天文学家，该问题曾一直困惑着自柏拉图以来的天文学家。开普勒五岁时看到1577年出现的彗星（第谷从他在汶岛建立的新天文台观测的第一颗彗星），他备受激励。开普勒在图宾根大学求学，该大学在梅兰希顿领导下在神学和数学领域成就斐然。开普勒在图宾根大学学习这两门学科，但他逐渐对数学更感兴趣。他从图宾根数学教授迈克尔·麦斯特林那里学到了开普勒理论，很快他就接受了该理论。 1594年开普勒受聘于奥地利南部城市格拉茨一所路德学校教授数学。在那里他出版了第一部原创作品--《宇宙的奥秘》。我们前面讲过，哥白尼理论的一大优点是容许人们通过天文观测确定行星远离太阳的次序以及运行轨道大小。开普勒在这部作品中设想这些轨道为半透明球体携带的行星勾画出的正圆（那时人们都这样认为），它们依照哥白尼理论围绕太阳运行。这些球体不是二维曲面，而是薄壳层，内外半径等于行星距离太阳最大和最小距离。开普勒设想这些球体半径受一个先决条件约束，即每个球体（除了最外一个球体—土星球体）正好内切五个正多面体中的一个，而且每个球体（除了最内球体—水星球体）正好外切五个正多面体中的一个。具体地说，开普勒放置的远离太阳的次序为：（1）水星球，（2）正八面体，（3）金星球，（4）正二十面体，（5）地球球体，（6）正十二面体，（7）火星球体，（8）正四面体，（9）木星球体。（10）立方体，最后（11）土星球体，依次紧密契合。 该方案确定了所有行星轨道相对大小，除了可以通过选择五种正多面体占据行星之间空间的次序，没有其他方法可以改变结果。正多面体次序一共有30种组合方法（注：5个不同事物的次序存在120种组合；五个中的任何一个都可以是第一个，剩下4个中的任何一个都可以是第二个，剩下3个中的任何一个都可以是第三个，剩下2个中的任何一个都可以是第四个，最后只剩下一个是第五个，这样五个事物的组合总数是5×4×3×2×1=120。但是把外接球与内切球的比值考虑进去，五个正多边形并不完全不同，该比值对立方体和正八面体是一样的，正二十面体和正十二面体的该比值也一样。因而五个正多边形的两组组合中交换立方体和正八面体，或交换正二十面体和正十二面体，得到的太阳系模型是一致的。不同模型的组合共有120/（2×2）=30.），所以并不奇怪开普勒可以找到其中的次序来使得行星轨道大小预测结果与哥白尼结果相吻合。 事实上开普勒最初设想对水星效果很差，开普勒做了回避,对其他行星也一般。但是与科学复兴时代的多数人一样，开普勒受柏拉图哲学影响极深，而且他同柏拉图一样，对正多面体只有五种可能甚感兴趣，这样只可能有六颗行星，包括地球。他骄傲的宣称：“现在你知道行星个数的理由了吧。” 即使开普勒的设想做的再好，现在也不会被人看重，这不是因为我们已经摒弃了过去那种像对正多面体数学可能性数量的柏拉图式幻想。现在仍然有其他各种数字让物理学家着迷。比如人们知道可以实施算术计算（包括除法）的只有四类“数”：实数，复数（包含负一的平方根），以及更奇异的四元数和八元数。一些物理学家费尽心思想把四元数和八元数与实数和复数一起结合到基本物理定律之中。我们现代之所以对开普勒的思路如此陌生不是他致力于为正多面体赋予基本物理意义，而是他用于解决行星轨道问题，其实这只是一种偶然。无论自然界基本定律是什么，我们今天可以确信它们不会涉及行星轨道半径。 这不是开普勒糊涂。在他那时代没有人知道（开普勒根本不相信）恒星其实也都是太阳，拥有各自的行星系统，并不仅仅是位于土星球体外面球体上的光点。太阳系被认为是整个宇宙，在时间的开端被创造出来。那时很自然地认为太阳系的完整结构与自然界其他物体一样基础。（？） 今天的理论物理处于同样境地。我们以为我们所谓的宇宙膨胀—我们观测到的巨大星系云向各方均匀远去—就是整个宇宙。我们认为我们测量的自然界常数--比如不同基本粒子的质量—最终会从目前未知的自然界基本定律中推导出来。但是有可能我们所谓膨胀的宇宙只是巨大“多重宇宙”的一小部分，此“多重宇宙”其中包含众多像我们观测到的膨胀，在多重宇宙不同地方自然界常数值也不同。这样的话这些常数就是环境参数，无法从基本原理中推导出来，正如无法从基本原理中推导出行星与太阳的距离。我们最多也就是指望人为估算（？）。我们所处的银河中有几十亿行星，其中只有极少数具有适合产生生命的温度及化学组分，但是很明显当生命出现并演化出天文学家后，他们会发现他们生活在这极少数行星上。因此上其实毫不奇怪我们所生活的行星距离太阳不会是现在地球距离太阳的两倍远，也不会只有一半。同样在多重宇宙中只有个别次宇宙具有可以容许生命演化的物理常数，但是当然科学家们会发现他们生活在这些个别的次宇宙中。这曾经用于解释第八章介绍过的在暗能量发现之前暗能量的数量级。当然所有这些都只是推测，但这会警示我们在试图解开自然常数之谜时我们可能会经历与开普勒试图解释太阳系大小所经历的同样的失望。 一些顶级物理学家强烈反对多重宇宙观，因为他们难以接受有些自然界常数无法计算的可能性。多重宇宙观可能完全是错误的，所以不能过早就放弃计算我们所知物理常数的努力。但是不能把做不了这些计算会使我们不快作为反对多重宇宙观的论证。无论自然界最终定律是什么，没有理由认为它们是为了让物理学家快乐而设计。 开普勒在格拉茨开始与第谷·布拉赫通信联系，第谷读了开普勒的《宇宙的奥秘》。第谷邀请开普勒造访乌拉妮娅城堡，但是开普勒觉得太远。1600年2月开普勒接受第谷的邀请到布拉格（从1583年成为圣罗马帝国首都）拜访第谷。开普勒在这里开始研究第谷的观测数据，特别是火星运行数据，他发现这些数据与托勒密理论间存在0.13o差别。（注：火星的运行用于测试行星理论最为理想。与水星和金星不同，火星在夜晚会升的很高，观测起来极为容易。在给定年度时间段，火星沿运行轨道公转次数比木星和土星要多许多。而且火星运行轨道偏离正圆比除水星之外的其他行星都大（水星永远离太阳很近，难以观测）。因此上火星偏离匀速圆周运动比其他行星都要明显。） 开普勒与第谷相处的并不愉快，于是开普勒回到了格拉茨。那时正好赶上基督新教徒被驱逐出格拉茨，开普勒在1600年8月不得不带着他的家人离开。回到布拉格后，开普勒开始与第谷合作制作鲁道夫天文表—用于替代雷茵霍尔德的普鲁士星表。第谷1601年去世后，开普勒的事业暂时得以解决，他被任命为鲁道夫二世的宫廷数学家，继任第谷的职位。 这位皇帝热衷于星相学，所以开普勒作为宫廷数学家的职责也包括占星术。他在图宾根做学生时就曾受雇从事占星术，尽管他自己很怀疑占星术做出的预测。幸运的是他还有时间投身真正的科学。1604年他在蛇夫座观察到一颗新的星星，这是1987年之前在我们银河系或接近我们银河系观察到的最后一次超新星爆炸。同年他发表了《天文学的光学部分》，一部关于光学理论和对天文学应用的著作，其中包括大气折射对观察行星的影响。 开普勒继续行星运行方面的研究工作，他多次尝试通过增加偏心，本轮，以及匀速点来让哥白尼理论与第谷精确的观察数据相符，但多次失败。1605年开普勒完成了这项工作，但是由于与第谷后嗣的争执，他没能发表。最后在1609年开普勒在《新天文学》（？）。 《新天文学》第三部为地球引入了匀速点和偏心圆，对哥白尼理论做了重大改进，这样在地球运行轨道中心的另一端（远离太阳）引入了一个新点，该点与地球的连线以匀速运行。这样去除了从托勒密时代以来一直干扰行星运行理论的大部分偏差，但是第谷的测量数据如此精确，开普勒仍然发现他的理论与观测有一些不符。 开普勒对此甚感无望，他觉得需要放弃柏拉图，亚里士多德，托勒密，哥白尼和第谷所坚信的行星轨道由圆形组成的假设。他转而断定行星轨道为椭圆形。最终在《新天文学》第58章（共70章）他给出了精确结论。他断定行星（包括地球）运行轨道为椭圆，太阳在其中一个焦点，而不在中心，人们后来称之为开普勒第一定律。正如圆可以用一个数来完全描述（其大小）--即圆的半径，椭圆可以用两个数来完全描述（其大小和方位）--长轴和短轴的长度，或等效于长轴长度与一个被称为“偏心率”的数值，从偏心率可以得知长轴和短轴有多大差别。（见技术说明18。）椭圆两个焦点是位于长轴上的两个点，等间距地位于中心两侧，两点距离等于偏心率乘以椭圆长轴长度。如果偏心率为0，椭圆两轴长度相等，两个焦点合并为一个圆点，椭圆成为一个正圆。 事实上开普勒所知悉的行星都有小的偏心率，如下表所列其现代值（回算到1900年数值）： 行星 偏心率 水星 0.205615 金星 0.006820 地球 0.016750 火星 0.093312 木星 0.048332 土星 0.055890

这就是为什么哥白尼与托勒密理论最简单的模型（哥白尼理论中不加本轮，托勒密理论中五个行星每个只设一个本轮）计算效果很好的原由。（注：行星的椭圆形轨道主要效果不在椭圆本身，而是太阳位于一个焦点，不是中心。准确的说，焦点与中心的距离正比于偏心率，而椭圆上任何一点到焦点的距离改变值正比于偏心率的平方，如果偏心率较小，该值或很小。比如如果偏心率为0.1（接近于火星轨道），行星与太阳的最小距离只比最大距离小百分之0.5，而太阳与轨道中心的距离为轨道半径均值的百分之10。） 用椭圆替代正圆还有另一个深远影响。球体的转动可以产生正圆，可是没有任何实体的转动可以产生椭圆。这样再结合第谷从1577年观测彗星得出的结论可以完全否决行星被旋转球体携带的陈旧观点，开普勒在他的《宇宙的奥秘》中也曾持此观点。现在开普勒和他的继任者设想行星在真空空间各自轨道上穿行。 《新天文学》的计算也用到了人们后来称之为的开普勒第二定律，不过一直到1621年开普勒才在《哥白尼天文学概要》中做了清晰阐明。开普勒第二定律指出行星在其轨道运行时速度如何变化。该定律表述为行星运行时相等时间太阳和行星的连线所扫过的面积相等。当行星靠近太阳时应该比远离太阳时在轨道上运行更长距离，这样开普勒第二定律结果是每颗行星接近太阳时运行速度一定加快。除了正比于偏心率平方的细微校正，开普勒第二定律与行星和另一个焦点（非太阳所在的焦点）连线以常数运转—即每秒转动相同角度—的表述相同。（参见技术说明21）。这样大体上开普勒第二定律与过去的匀速点—位于圆心远离太阳（或对托勒密理论远离地球）并且到圆心距离相等，行星和匀速点连线以匀速围绕匀速点运行理论得出的行星运行速度相同。这表明匀速点不过是椭圆的另一个空焦点。只有第谷无与伦比的数值精度才让开普勒认识到采用偏心圆和匀速点不够，必须用椭圆替代正圆。 第二定律也有深远应用价值，至少是对开普勒。在《宇宙的奥秘》中开普勒设想行星被“动力幽灵”所推动。现在开普勒发现每颗行星与太阳间的距离增大时速度减慢，他继而推断行星是被太阳所发射出的力驱动： 如果你用“力”一词替代“幽灵”一词，你就会知道《火星评述》（新天文学）中天体物理所基于的原理。因为受到所教授的斯卡里格（注：这里指朱利亚斯·凯撒·斯卡里格，他竭力维护亚里士多德，反对哥白尼）有关动力智能的影响，我以前完全相信驱动行星的本因是幽灵。但是当我认识到行星到太阳间距离增大时驱动动力减弱，正如太阳光衰弱一样，我断定这个力一定好像是有形的。 当然行星持续运行不是由于太阳发射出的力，而是因为它们的动量没被消耗。但是它们固守在自己轨道运行而没有飞到天际确是受太阳发射出的力—重力的作用，所以开普勒并没有完全错误。 超距力的观点这时已经流行，部分是由于皇家外科医学院院长，伊丽莎白一世的宫廷御医威廉姆·吉尔伯特在磁性方面的研究工作，开普勒参考了他的研究。如果开普勒提到的“幽灵”确是指该词的本意，那么从基于幽灵过渡到基于力的“物理”是终结古代将宗教与自然科学混杂在一起的关键一步。 《新天文学》的创作没有为了回避冲突。通过在全书名中采用“物理”一词，开普勒向那些亚里士多德追随者中盛行的旧观念发起挑战，天文学应该关注于用数学方法描述现象，若要真正掌握人们必须依靠物理—即亚里士多德的物理。开普勒宣称只有像他这样的天文学家在从事真正的物理。不过事实上开普勒的众多见解都受到一个错误物理观点的指引，他以为太阳用类似于磁力这样的力驱动行星在各自轨道运行。 开普勒同样挑战所有反对哥白尼的人。《新天文学》引言中含有这样一段： 对傻瓜的忠告。如果有人过于愚蠢，不能理解天文学，或胆子太小，不能坚守他的信仰同时接受哥白尼，我对他的忠告是他既然已经拒绝接受天文研究，随心所欲谴责那些哲学研究，他就应该少管闲事，会到家里清理自己的污垢。 开普勒前两条定律没有提及不同行星运行轨道的对比。1619年出版的《宇宙谐和论》填补了这一空缺，人们后来称之为开普勒第三定律，即“任何两颗行星运行周期之比等于平均距离的（3/2）次方之比。”（注：后面的讨论中会说明开普勒提到的平均距离不是指时间上的平均距离，而是指与太阳最小和最大距离的平均值。正如技术说明18所介绍，行星与太阳最小和最大距离为（1-e）a和(1+e)a，这里e是偏心率，a是椭圆长轴一半（即半长轴）,所以平均值正好为a。技术说明18还会介绍这也是行星在其轨道上运行时和太阳距离的平均。）也就是说每颗行星恒星周期（在其轨道上完成一整周运行需要的时间）的平方正比于椭圆长轴的立方。这样如果T是单位为年的恒星周期，a是以天文单位（AU）度量的半长轴，1个天文单位定义为地球的半长轴，那么开普勒第三定律是说T2/a3对每颗行星相等。因为我们定义地球T为1年，a为1个天文单位，那以此单位T2/a3等于1，按照开普勒第三定律每颗行星都应该遵守T2/a3=1。下表给出准确的现代值： 行星 a（AU） T(年) T2/a3 水星 0.38710 0.24085 1.0001 金星 0.72333 0.61521 0.9999 地球 1.00000 1.00000 1.0000 火星 1.52369 1.88809 1.0079 木星 5.2028 11.8622 1.001 土星 9.540 29.4577 1.001

（每颗行星T2/a3并不完全相等，只是由于行星间重力场的相互影响） 开普勒从来没有完全摒弃柏拉图思想，他尝试重新用他早期在《宇宙的奥秘》中采用的正多面体来解释轨道大小。他也曾考虑毕达哥拉斯学派行星运行周期形成一种音阶的观点。与那个时代其他科学家一样，开普勒也只是部分拥有刚刚出现的新科学主张，他部分上仍然具有古老的哲学和诗意传统。 《鲁道夫星表》年终于完成。该星表基于开普勒的第一和第二定律，比以前的普鲁士星表精度有显著提高。新星表预测年将发生水星凌日（即会观测到水星穿越太阳）。开普勒没能看到。开普勒由于基督新教徒的身份再次被逐出天主教奥地利，他与年去世于雷根斯堡。 哥白尼和开普勒日心说方面的成就是数学上的简易和自恰，而不是与观测相符更好。我们前面讲过，哥白尼与托勒密理论最简单形式对太阳和行星的预测一致，与观测也非常相符，如果托勒密像对行星一样给太阳也设了匀速点和偏心圆，他再加上一些本轮，那他完全可以实现开普勒所实现的对哥白尼理论的改进效果。第一个决定性地支持日心说优于托勒密系统的观察证据由伽利略·伽利雷做出。 伽利略是与牛顿，达尔文和爱因斯坦比肩的历史上最伟大的科学家之一。他引入望远镜彻底改变了天文观测手段，他对运动的研究为现代实验物理提供了范例。另外他的科学事业极富戏剧性，我们这里只能做简要叙述。 伽利略1564年出生于比萨，一个并不富裕的贵族托斯卡纳人，父亲是音乐理论家芬琴齐奥·伽利雷。在佛罗伦萨修道院学习之后，他1581年进入比萨大学学医。此阶段他追随亚里士多德，对一位医学学术，这不奇怪。后来伽利略的兴趣由医学转向数学，他还在托斯卡纳首都佛罗伦萨教授了一段数学。1589年伽利略被召回比萨担任数学教授。 在比萨大学伽利略开始他的落体研究。在《论运动》一书中介绍了他的部分工作，但他从来没有将该书发表。与亚里士多德相反，伽利略推断重落体速度并不显著取决于其重量。传说他测试从比萨斜塔抛下不同重量的物体，不过这没有证据。伽利略在比萨没有发表任何他在落体方面的研究工作。 1591年伽利略搬到帕多瓦任帕多瓦大学数学教授，当时该大学属于威尼斯共和国，是欧洲最富学术盛名的大学。从1597年开始除了大学工资，他从制作和出售数学仪器中阿获得部分收入，这些仪器被用于商业和战争。 1597年伽利略获得两册开普勒的《宇宙的奥秘》.他给开普勒写信，承认他与开普勒一样都是哥白尼派，虽然他还没有公开。开普勒回信说伽利略应该明确支持哥白尼，他敦促：“伽利略，站出来！” 不久伽利略就开始与亚里士多德学派产生了冲突。当是在帕多瓦与在意大利其他地方一样，亚里士多德学派主导着哲学教学。1604年他做了关于开普勒观测到的“新恒星”的演讲。与第谷和开普勒一样，他推断月球轨道之外的太空确实会发生变化。他一度的朋友，帕多瓦哲学教授塞萨尔·克里蒙尼尼为此而攻击他。伽利略创作了两个农民间以质朴的帕多瓦方言的对话来回击克里蒙尼尼。克里蒙尼尼农民辩称一般的测量规则不能应用到天体。伽利略的农民回应说哲学家对测量一无所知：无论是测量天体或测量玉米粥，人们必须依赖于数学。 1609年开启了一场天文学革命，这年伽利略第一次听说了一种新出现的叫做小望远镜的荷兰装置。人们早就知道充满水的玻璃球体具有放大特性，比如罗马政治家和哲学家塞内加就曾对此有所提及。阿尔·海什木研究过放大效果，1267年罗杰·培根在《大著作》中描述了玻璃的放大作用。随着玻璃制作技术的改善，到十四世纪阅读眼镜已很普及。但若要放大远方物体则需要一对镜片，一个镜片将物体任何一点发出的平行光线聚焦使其交汇，另一个镜片收集这些光线（如果光线仍然交汇的话采用凹面镜，如果光线开始重新发散则采用凸面镜）水平方向发送到S 眼睛。（放松眼球晶状体将平行光线聚焦到视网膜上一点；该点位置取决于平行光线方向。）十七世纪早期荷兰开始制作采用这种设计的眼镜，1608年即位眼镜制作者为他们的眼镜申请专利。他们的申请遭到拒绝，理由是这种设备也广为人知。很快法国和意大利都有了这种小望远镜，但其放大能力只有3到4倍。（即如果到远方两点视线夹角为某一小角度，那用这种小望远镜看的话这两点是原角度的3或4倍。） 伽利略在1609年听说了这种小望远镜，不久他就改进了制作，让第一个镜片凸面向前，平面朝后，用长焦距（焦距是体现镜片光学特性的长度。凸透镜焦距是平行光入射聚焦点到镜片的距离。凹透镜使聚光束折射为平行光，其焦距为如果没有凹透镜的话光线聚焦点到镜片的距离。焦距取决于镜片曲率半径以及空气和玻璃的光速比。见技术说明22。），第二个镜片凹面面相第一个镜片，平面朝后，用较短焦距。采用这种设计的话若要将极远距离光源以平行光束发送到眼睛，镜片距离需要等于焦距差，这样放大倍数第一个镜片与第二个镜片焦距之比。（见技术说明23。）伽利略很快就可以做到8倍或9倍放大。1609年8月23号他向威尼斯执政官和显贵们展示了他的小望远镜，他还示范用他的小望远镜海里的船可以比肉眼提前两小时看到。这种设备的价值对像威尼斯这样的海洋强权显而易见。伽利略将他的小望远镜捐献给威尼斯共和国后，他的工资涨了三倍，他获得终身教职。到11月伽利略将他的小望远镜放大倍数提高到20倍，他开始将其用于天文观测。 应用后来被称为天文望远镜的小望远镜，伽利略做出了天文史上极其重要的六个发现。1610年3月出版的《星空信使》记述了其中前四个发现： 1. 1609年11月20日，伽利略首次用天文望远镜观测新月。在亮面他观察到月亮表面崎岖不平： 通过多次重复(对月球表面)的观察，我们确信我们看到的月球表面并不是光滑，平坦，呈完美球形，虽然多数哲学家都认为月球和其他天体应该这样，然而相反，其表面参差不齐，凹凸不平，满是大坑和突起。它与地球表面很相似，各地都有连绵的山脉和深谷。 他在靠近明暗交界的暗面可以看到许多亮点，他对此的解释是这是太阳即将从月球地平线升起时所照亮的山顶。从这些亮点到明暗交界的距离他甚至估算出有些大山至少4英里高。（见技术说明24。）伽利略也对月球暗面微弱的亮光做出了解释。他不认同伊拉斯谟·雷茵霍尔德和第谷·布拉赫提出的各种解释，他们设想这是月球自身发的光或来自金星或恒星的光，他正确地指出“这神奇的光”是地球反射的太阳光，正如地球夜晚会被月亮反射的太阳光隐约照亮一样。如此一来像月球这样的天体与地球其实没有太大区别。 2. 伽利略用他的小望远镜可以观察到比六等星暗的多的“几乎不可思议的星群”，这些恒星根本无法肉眼观察。在肉眼可见的昴星团六颗星星中他又发现了40多颗其他星星，在猎户座他可以看到500多颗以前从来没有看到过的星星。将他的天文望远镜对向银河，他可以看到银河其实包含众多星星，与大阿尔伯特猜测的一样。 3. 伽利略宣称从他的天文望远镜看到的行星是“完整的圆形球体，像一个一个小的月亮，”但是他观察到的星星不是这样。相反他发现虽然所有的星星从天文望远镜观察都要明亮很多，但它们并没有明显大很多。伽利略的解释有点混乱。他不知道星星的视尺寸是由于地球大气波动使得光线发生各向弯曲造成的，而不是被星星附近固有的任何东西的影响。也正是这种大气波动造成了星星的闪烁。（注：行星的视角足够大，其表面不同点的光线穿过地球大气层后距离变化比大气波动幅度大；这样大气波动对不同光线的影响不同，它们一般会彼此抵消而不会叠加。这就是为什么我们看不到行星闪烁的原因。）伽利略推断既然从他的望远镜看不出恒星的形状，它们比行星距离我们一定远得多。正如伽利略后来所记述，这有助于解释如果地球围绕太阳运行，为什么我们观测不到周年恒星视差。 4. 《星空信使》记述的最为重要和令人印象深刻的发现是1610年1月7日做出的。伽利略用他的天文望远镜观察木星，他发现|“三颗小星星位于它附近，很小但非常亮。”起初伽利略认为这只是三颗恒星，只不过太暗，以前看不到，另他倍感惊奇的是它们沿黄道排列，两颗在木星东边，一颗在西边。但第二个晚上所有三颗“恒星”都位于木星西边，1月10日只能看到两颗，都位于东边。最后在1月13日，他可以看到四颗“恒星”，依然差不多沿黄道排列。伽利略断定木星轨道有四个卫星陪伴，类似于地球的月亮，它们与月亮一样在差不多同一行星轨道平面运行，靠近于黄道—即地球环绕太阳轨道平面。（我们现在知道这些是木星最大的四颗卫星：盖尼米得，伊奥, 卡里斯托和欧罗巴，它们被以丘比特神的男性和女性爱人命名。）(注：伽利略如果知道这些命名一直沿用下来一定很不快。 这是1614年西蒙·迈尔为木星卫星所命名，西蒙·迈尔是德国天文学家，他与伽利略争辩究竟谁是第一个发现这些卫星的人。) 这个发现是对哥白尼理论的有力支持。该木星与其卫星系统像是哥白尼所设想的太阳与其行星系统的微缩模型，明确表明天体可以围绕其他物体运行，而不是围绕地球。另外土星的卫星也平息了对哥白尼理论的另一个反对声音—即如果地球在运行，那为什么月球没有被落在后面？ 伽利略1611年末测量了他所发现的四颗土星卫星的公转周期，但是来不及在《星空信使》中发表。1612年他将这些结果在另一部关于其他方面的作品首页发表。下表给出了伽利略以及现代结果，以天，小时，和分钟为单位： 土星卫星 周期（伽利略） 周期（现代） 伊奥 1天18小时30分钟 1天18小时29分钟 欧罗巴 3天13小时20分钟 3天13小时18分钟 盖尼米得 7天4小时0分钟 7天4小时0分钟 卡里斯托 16天18小时0分钟 16天18小时5分钟

伽利略测量之精确证实他的观测有多么仔细，计时有多么精准。（注：伽利略应该没有使用时钟，而是观察恒星的视运行。每恒星日恒星围绕地球约24小时运行360度，这样恒星位置改变1度意味着1/360乘以24小时时间的变化，即4分钟。） 伽利略将《星空信使》奉献给他的前学生科西莫二世迪·梅迪奇，现任托斯卡纳大公，他将土星的四颗伴星命名为“梅迪奇星星”。这种恭维颇有用意。伽利略在帕多瓦薪金丰厚，但是他得知他的收入不会增加。而且此薪金要求他承担教学任务，影响了他的研究工作。他与科西莫达成一个协议，他被任命为宫廷数学家和哲学家，在比萨任教授，但不需要承担教学任务。伽利略坚持“宫廷哲学家”头衔，因为尽管开普勒等人通过数学大大推进了天文学，尽管有克莱文思的据理力争，数学家地位仍然低于哲学家。而且伽利略也希望人们把他的工作作为哲学家所说的“物理”那样严肃对待，这是研究太阳，月亮和行星的特性，而不只是用数学计算视运行。 1610年夏伽利略离开帕多瓦去了佛罗伦萨，最终证明这是个灾难性决定。帕多瓦位于威尼斯共和国领域，那时受梵蒂冈的控制比意大利其他地方要弱，伽利略离开之前几年帕多瓦曾成功地拒绝了一个教皇的禁令。到了佛罗伦萨伽利略容易受到教会的制约。现代大学的系主任可能会觉得这个风险是对伽利略逃避教学任务的公正惩罚。但是这个惩罚没有立即发生。 5. 1610年6月伽利略做出了第五个重大天文发现。他观察金星，发现金星也存在相，与月亮的相一样。他给开普勒发了一个编码消息：“爱神（金星）之母仿效月亮女神辛西娅（月球）。”托勒密与哥白尼理论都容许相的存在，但两种理论的相不同。托勒密理论中金星一直位于太阳和地球之间，这样金星不会超过半圆。然而按照哥白尼理论，当金星位于远离地球的轨道另一边时会被完全照亮。 这是证明托勒密理论错误的第一个直接证据。前面说过托勒密理论得出的从地球观察太阳与行星的视运行与哥白尼理论得出的一致，无论我们如何选择每颗行星均轮大小。但是从行星上观察，托勒密理论得出的太阳和行星视运行与哥白尼理论不同。当然伽利略不可能去其他任何行星那里观察太阳和行星如何运行。但是从金星的相他可以知道从金星上观察到的太阳光方向—亮面朝向太阳。托勒密理论只有一个特例可以给出正确结果，即金星和水星的均轮与太阳轨道一致，前面说过这正是第谷的理论。托勒密以及其追随者从来没有采用过这个说法。 6. 伽利略到了佛罗伦萨以后发现了一种巧妙的方法来研究太阳表面，他用天文望远镜将其投影到一个屏幕。采用这种方法他做出第六项发现：黑点在太阳表面移动。1613年他将观察结果发表在《有关太阳黑子的信件》，后面会详细介绍。 历史上多次出现新技术扩展了纯科学的研究。十九世纪真空泵的改进使人们得以在真空管内做放电现象实验，这导致电子的发现。伊尔福公司开发的感光乳剂带来二战后10年间许多新基本粒子的发现。二战中微波雷达的研发使人们可以用微波来探测原子，这提供了1947年对量子电动力学至关重要的测试。而且我们也不要忘记圭表。但是这些新技术带来的科学成果都无法与伽利略用天文望远镜产生的成功相媲美。 对伽利略发现的反应有的谨慎，有的热烈。伽利略帕多瓦的老对手塞萨尔·克里蒙尼尼拒绝用天文望远镜观测，比萨大学的哲学教授朱利欧·来博瑞也一样。另一方面伽利略被选为几年前刚刚成立的欧洲第一所科学院林嗣科学院的院士。开普勒从伽利略寄给他的天文望远镜证实了伽利略的发现。（开普勒弄清了天文望远镜的原理，不久他就发明了具有双凸镜的天文望远镜。） 伽利略起初与教会没有冲突，这可能是他还没有明确支持哥白尼。《星空信使》书中只在末尾当讲到如果地球运行，为什么月球没有被落到后面时提到一次哥白尼。那时与罗马宗教裁判所有冲突的不是伽利略，而是像克里蒙尼尼这样的亚里士多德学派成员，其理由也是与1277年对亚里士多德多种教义的谴责一样。伽利略即与亚里士多德学派也与耶稣会有争执，从长远来看对他没有好处。 1611年7月。伽利略刚在佛罗伦萨担任新职不久就与哲学家们进行了一场争论，这些哲学家坚信他们所谓的亚里士多德信条，认为固态冰比液态水密度（单位体积重量）大。曾经是决定处死布鲁诺的罗马宗教裁判所一员的红衣主教罗伯特·贝拉明支持伽利略，他们坚称冰浮在水上，所以一定并水轻。在1612年出版的《水中浮体对话集》中伽利略公开了他有关浮体的结论。 1613年伽利略在辩论一个不太重要的天文问题时惹恼了耶稣会，也包括克里斯托夫·席耐尔：太阳黑子是否是与太阳相关的一部分—像伽利略认为的也许是太阳表面之上的云，这样就会给天体并不完美提供一个例证（就像月面山峰一样）？或者它们只是比水星更近围绕太阳运行的行星？如果可以证实这些只是云，那么那些声称太阳围绕地球运行的人就不能同样声称如果地球围绕太阳运行的化地球上的云会被落在后面。在1613年出版的《有关太阳黑子的信件》中伽利略提出太阳黑子接近太阳边部时看起来会变窄，说明靠近边部它们处于倾斜，因此上它们被太阳表面所携带，随表面的旋转而旋转。关于是谁第一个发现太阳黑子也是争论的一部分。这只是与耶稣会冲突加剧中的一幕，不能说其中哪一方总是不公。对后来产生更大影响的是在《有关太阳黑子的信件》中伽利略终于明确表达了对哥白尼的支持。 随着《试金者》的出版伽利略与教会的冲突于1623年开始激化。这部作品是对耶稣会数学家奥拉齐奥·格拉西的反驳，格拉西提出与第谷同样的一个其实是完全正确的结论—即之所以观测不到彗星周日视差是因为彗星位于月球之外。伽利略反而提出一个奇特的理论，认为彗星是太阳光受大气线性扰动产生的反射，之所以不会有周日视差是因为这种扰动随着地球的转动而转动。也许伽利略真正的敌人不是奥拉齐奥·格拉西，而是第谷·布拉赫，第谷提出的地心说行星运行理论那时的观测结果无以反驳。 那些年教会还能够容忍哥白尼体系可以作为计算行星视运行的纯数学工具，而不是真实描述行星以及行星运行的理论。比如贝拉明1615年写给那不勒斯修道士保罗·安东尼奥·佛斯卡里尼的信中对佛斯卡里尼支持哥白尼系统即给以安慰也提出警告： 在我看来尊敬的伽利略先生会小心翼翼地用假设而非绝对来向你告知，正如我一直以来都相信哥白尼也是这样说的。（贝拉明是不是相信了奥西安德的前言？伽利略肯定不会）说假设地球运动而太阳静止比偏心和本轮学说得出更好的视运行结果值得赞扬。（贝拉明明显没有意识到哥白尼与托勒密一样用了本轮，只不过没有那么多。）这没有危险，也满足数学要求。但是如果一定坚持太阳确实位于世界中心静止不动，太阳只会自转而不会由东向西运行，地球位于第三层天球飞速绕太阳旋转，这就很危险。这不只会激怒所有的哲学家和神学家，也会危害信仰，背叛圣经。 意识到哥白尼理论渐渐聚集起来的麻烦，伽利略1615年给托斯卡纳公爵夫人洛林的克里斯汀娜（伽利略参加了她与已故的费迪南多大公的婚礼）写了一封有名的关于科学与宗教的信件。与哥白尼在《运行论》所述一样，伽利略提到拉克坦提乌斯不相信地球为球形是用经文反驳科学发现的一个可怕事例。他也驳斥了早先路德用对约书亚书的字面解释来反对哥白尼，证明太阳在运行。伽利略解释说圣经不是天文学课本，对五颗行星，圣经之提到金星，而且只有几次。给克里斯汀娜的信中最有名的一句话是：“下面我要说我听到的一位极富盛名的牧师所言：’圣灵是要教授我们如何上天堂，而不是天体如何运行。’” （伽利略旁注这位极富盛名的牧师是梵蒂冈图书馆长，学者恺撒·巴洛尼奥主教。）伽利略对约书亚书提到的太阳曾经停止不动做出解释：这是太阳的自转（伽利略通过观测太阳黑子运动了解了自转）发生了停止，这有又接着导致地球和其他行星停止了公转和自转，正如圣经所描述的延长了战争时间。我们不知道伽利略是真的相信这个荒谬的说法还只是寻求政治庇护。 伽利略1615年没有听从朋友们的建议，亲自前去罗马反对对哥白尼观点的压制。教皇保罗五世不想有争执，他听从贝拉明的建议，决定将哥白尼理论交给一组神学家讨论。他们裁定哥白尼体系“哲学上愚昧，荒唐，多处违背圣经的表述，完全是异端邪说。” 1616年2月伽利略被召集到宗教裁判所接受两条密令。一份签署文件要求他不许坚持或维护哥白尼学说。另一份未签署文件更严格，要求他不许坚持，维护，或以任何方式教授哥白尼学说。1616年3月宗教裁判所发布正式命令，虽然没有提及伽利略，但是查禁佛斯卡里尼的著作，要求删除对哥白尼的记述。《运行论》被列在天主教禁书目录之中。不过一些天主教天文学家并没有回到托勒密或亚里士多德体系，比如耶稣会牧师乔万尼·巴特斯达·里奇奥利在他1651年出版的《新天文学大成》中支持第谷体系，那时的观测还不能否认该体系。一直到1835年《运行论》都在禁书目录中，这严重影响了科学在一些天主教国家的传授，比如在西班牙。 1624年马费奥·巴尔贝里尼成为教皇乌班八世，伽利略燃起了希望。巴尔贝里尼是佛罗伦萨人，仰慕伽利略。他欢迎伽利略到罗马，并且召集6名听众。在这些对话中伽利略解释了他从1616年以来一直研究的潮汐理论。 伽利略的理论主要基于地球的运行。其观点是但地球围绕太阳公转时地球也在自转，海水会随着前后晃动，在这过程中地面一点沿地球公转轨道转动方向的净速度不断发生增减变化。这就有了一天一个周期的海水波动，与其他振动一样，也有半天周期，三分之一天周期等等的泛频周期。这里还没有涉及到月球的影响，但是人们自古以来就知道“大潮”发生在满月和新月，而“小潮”发生在半月。伽利对月球影响做出的解释是说地球在新月时—月亮位于地球和太阳之间—公转速度增加，在满月时—月亮位于远离太阳的地球另一边—公转速度减慢。 这当然不是伽利略的最佳成果。不用重力理论伽利略根本无法去正确认识潮汐。但是伽利略应该知道不能靠这个没有明确实证支持的潮汐假想理论来证实地球的运转。 教皇声明如果伽利略只把地球运转作为数学假说，而不是实际情况，那就容许他发表他的潮汐理论。乌班解释说他并不同意宗教裁判所1616年做出的公开法令，但是他还不准备废除该法令。对话中伽利略没有向教皇提及宗教裁判所给他的私密法令。 1632年伽利略做好了发表他的潮汐理论的准备，这已经走向对哥白尼理论的完整支持。现在教会还没有对伽利略公开谴责，所以当他向本地主教申请出版一本新书时很快获得许可。这就是他的《对话》（关于托勒密和哥白尼两大世界体系对话）。 书名很特别。那时不只两个，而是四个世界体系：除了托勒密和哥白尼，还有亚里士多德的围绕地球的同心圆体系，以及第谷的太阳和月亮围绕静止地球其他行星围绕太阳的体系。伽利略为什么没有考虑亚里士多德和第谷体系哪？ 关于亚里士多德体系，人们可以说其与观测不符，但是两千年来人们一直知道其与观测不符，但也没有失去其追随者。看看第10章引用的法兰卡斯特罗在十六世纪初期给出的理由。伽利略在一个世纪之后明显认为这些理由不值一驳，但是不清楚这是任何发生的。 另一方面第谷体系效果极好，无法不予理会。伽利略肯定知道第谷体系。也许伽利略认为他的潮汐理论表明地球确实在运转，但是他的理论并没有任何成功实例支撑。也或许伽利略不想让哥白尼与强大的第谷来竞争。 《对话》采用三个人物的对话形式：萨尔维亚蒂—代表伽利略，名字来自于伽利略的朋友佛罗伦萨贵族菲利普·萨尔维亚蒂，辛普利西奥—一名亚里士多德学派学者，名字可以来自于辛普里丘（也可能用来代表头脑简单的人）；萨格莱多--名字来源于伽利略的朋友梵蒂冈数学家乔瓦尼·弗朗西斯科·萨格莱多，在前面两人间做出明智地判断。前三天的对话中萨尔维亚蒂驳倒了辛普利西奥，潮汐问题只在第四天才提及。这显然违背了宗教裁判所给伽利略未签署文件中下达的指令，而且也可以说违背了相对宽松的签署文件中的指令（不许坚持或维护哥白尼学说）。更糟糕的是《对话》用意大利为写作，而非拉丁文，任何识字的意大利人都可以阅读，不限于学者。 这时候有人给教皇乌班展示了1616年宗教裁判所给伽利略的未签署法令，这可能是伽利略早期辩论太阳黑子和彗星时得罪的一些人所为。乌班怀疑他本人就是辛普利西奥的原型，这更使他大为震怒。虽然辛普利西奥说了一些教皇还是巴尔贝里尼主教时说过的话，但也完全无助。（？）宗教裁判所下令禁止《对话》的销售，但为时已晚--该书已全部售完。 1633年伽利略受到审判。主要是他违背1616年宗教裁判所的法令。伽利略被严刑威胁，要求他接受认罪协议，承认个人虚荣心作祟做出过分行为。但是后来他被宣布“强烈怀疑是异教徒”，判处终身监禁，强迫宣誓放弃地球围绕太阳运行观点。（一个杜撰的故事说伽利略离开法庭后嘴里念念有词，“Eppur si muove”, 意思是“它确实在动。”） 幸运的是伽利略并没有被非常粗暴对待。他被容许作为锡耶纳大主教的客人在家软禁，后来又软禁到自己佛罗伦萨附近阿切特里的住宅，这里靠近他的女儿，妹妹玛丽亚·西莉斯特和妹妹阿坎格拉的修道院居所。我们在第十二章会看到，伽利略这些年又重新回到半世纪前在比萨进行的关于运动问题的研究。 伽利略1642年去世时任被监禁在阿切特里居所。直到1835年类似伽利略支持哥白尼体系这样的著作才被从天主教会禁书目录中删除，但哥白尼天文学在多说天主教国家一级基督新教国家早已广为人知。二十世纪教会为伽利略恢复了名誉。1979年教皇约翰·保罗二世指伽利略给克里斯汀娜的信件为“构想出重要的认识论模式，对圣经与科学的融合不可或缺。”成立了一个委员会调查伽利略案，调查报告说伽利略时代的教会犯了错误。教皇回应说：“那时的神学家坚持地球中心说，他们的错误是认为我们对物理世界结构的理解受圣经字义所限定。” 我觉得这还不够。教会当然无法回避现在众所周知的在地动方面的错误。但是即使教会是正确的，伽利略天文学不对，那判罚伽利略监禁，拒绝他出版的权利也是错误的，就像即使乔尔丹诺·布鲁诺是异教徒，将他用火烧死也完全是错误的。幸好虽然我不知道教会是否明确承认，但今天不再会发生这种事情。除了一些伊斯兰国家会惩罚对上帝的亵渎或叛教，整个世界已经吸取了教训，政府和宗教机构无权对宗教意见实施刑事处罚，无论这意见是对或错。 由哥白尼，第谷·布拉赫，开普勒，以及伽利略的计算和观测应运而生了对太阳系正确的描述，这体现在开普勒三大定律之中。但对行星为什么遵循这些定律的解释还需要再等到下一代--牛顿的诞生。

Whatever the scientific revolution was or was not, it began with Copernicus. Nicolaus Copernicus was born in 1473 in Poland of a family that in an earlier generation had emigrated from Silesia. Nicolaus lost his father at the age of ten, but was fortunate to be supported by his uncle, who had become rich in the service of the church and a few years later became bishop of Varmia (or Ermeland) in northeast Poland. After an education at the University of Cracow, probably including courses in astronomy, in 1496 Copernicus enrolled as a student of canon law at the University of Bologna and began astronomical observations as an assistant to the astronomer Domenico Maria Novara, who had been a student of Regiomontanus. While at Bologna Copernicus learned that, with the help of his uncle’s patronage, he had been confirmed as one of the 16 canons of the cathedral chapter of Frombork (or Frauenburg), in Varmia, from which for the rest of his life he derived a good income with little in the way of ecclesiastical duties. Copernicus never became a priest. After studying medicine briefly at the University of Padua, in 1503 he picked up a degree of doctor of laws at the University of Ferrara and soon afterward returned to Poland. In 1510 he settled in Frombork, where he constructed a small observatory, and where he remained until his death in 1543. Soon after he came to Frombork, Copernicus wrote a short anonymous work, later titled De hypothesibus motuum coelestium a se constitutis commentariolus, and generally known as the Commentariolus, or Little Commentary.1 The Commentariolus was not published until long after its author’s death, and so was not as influential as his later writings, but it gives a good account of the ideas that guided his work. After a brief criticism of earlier theories of the planets, Copernicus in the Commentariolus states seven principles of his new theory. Here is a paraphrase, with some added comments:

1. There is no one center of the orbits of the celestial bodies. (There is disagreement among historians whether Copernicus thought that these bodies are carried on material spheres,2 as supposed by Aristotle.) 2. The center of the Earth is not the center of the universe, but only the center of the Moon’s orbit, and the center of gravity toward which bodies on Earth are attracted. 3. All the heavenly bodies except the Moon revolve about the Sun, which is therefore the center of the universe. (But as discussed below, Copernicus took the center of the orbits of the Earth and other planets to be, not the Sun, but rather a point near the Sun.) 4. The distance between the Earth and the Sun is negligible compared with the distance of the fixed stars. (Presumably Copernicus made this assumption to explain why we do not see annual parallax, the apparent annual motion of the stars caused by the Earth’s motion around the Sun. But the problem of parallax is nowhere mentioned in the Commentariolus.) 5. The apparent daily motion of the stars around the Earth arises entirely from the Earth’s rotation on its axis. 6. The apparent motion of the Sun arises jointly from the rotation of the Earth on its axis and the Earth’s revolution (like that of the other planets) around the Sun. 7. The apparent retrograde motion of the planets arises from the Earth’s motion, occurring when the Earth passes Mars, Jupiter, or Saturn, or is passed in its orbit by Mercury or Venus. Copernicus could not claim in the Commentariolus that his scheme fitted observation better than that of Ptolemy. For one thing, it didn’t. Indeed, it couldn’t, since for the most part Copernicus based his theory on data he inferred from Ptolemy’s Almagest, rather than on his own observations.3 Instead of appealing to new observations, Copernicus pointed out a number of his theory’s aesthetic advantages. One advantage is that the motion of the Earth accounted for a wide variety of apparent motions of the Sun, stars, and the other planets. In this way, Copernicus was able to eliminate the “fine-tuning” assumed in the Ptolemaic theory, that the center of the epicycles of Mercury and Venus had to remain always on the line between the Earth and the Sun, and that the lines between Mars, Jupiter, and Saturn and the centers of their respective epicycles had to remain always parallel to the line between the Earth and the Sun. In consequence the revolution of the center of the epicycle of each inner planet around the Earth and the revolution of each outer planet by a full turn on its epicycle all had to be fine-tuned to take precisely one year. Copernicus saw that these unnatural requirements simply mirrored the fact that we view the solar system from a platform revolving about the Sun. Another aesthetic advantage of the Copernican theory had to do with its greater definiteness regarding the sizes of planetary orbits. Recall that the apparent motion of the planets in Ptolemaic astronomy depends, not on the sizes of the epicycles and deferents, but only on the ratio of the radii of the epicycle and deferent for each planet. If one liked, one could even take the deferent of Mercury to be larger than the deferent of Saturn, as long as the size of Mercury’s epicycle was adjusted accordingly. Following the lead of Ptolemy in Planetary Hypotheses, astronomers customarily assigned sizes to the orbits, on the assumption that the maximum distance of one planet from the Earth equals the minimum distance from the Earth of the next planet outward. This fixed the relative sizes of planetary orbits for any given choice of the order of the planets going out from the Earth, but that choice was still quite arbitrary. In any case, the assumptions of Planetary Hypotheses were neither based on observation nor confirmed by observation. In contrast, for the scheme of Copernicus to agree with observation, the radius of every planet’s orbit had to have a definite ratio to the radius of the Earth’s orbit.* Specifically, because of the difference in the way that Ptolemy had introduced epicycles for the inner and outer planets (and leaving aside complications associated with the ellipticity of the orbits), the ratio of the radii of the epicycles and deferents must equal the ratio of the distances from the Sun of the planets and Earth for the inner planets, and equal the inverse of this ratio for the outer planets. (See Technical Note 13.) Copernicus did not present his results this way; he gave them in terms of a complicated “triangulation scheme,” which conveyed a false impression that he was making new predictions confirmed by observation. But he did in fact get the right radii of planetary orbits. He found that going out from the Sun, the planets are in order Mercury, Venus, Earth, Mars, Jupiter, and Saturn; this is precisely the same as the order of their periods, which Copernicus estimated to be respectively 3 months, 9 months, 1 year, 2½ years, 12 years, and 30 years. Though there was as yet no theory that dictated the speeds of the planets in their orbits, it must have seemed to Copernicus evidence of cosmic order that the larger the orbit of a planet, the more slowly it moves around the Sun.4 The theory of Copernicus provides a classic example of how a theory can be selected on aesthetic criteria, with no experimental evidence that favors it over other theories. The case for the Copernican theory in the Commentariolus was simply that a great deal of what was peculiar about the Ptolemaic theory was explained at one blow by the revolution and rotation of the Earth, and that the Copernican theory was much more definite than the Ptolemaic theory about the order of the planets and the sizes of their orbits. Copernicus acknowledged that the idea of a moving Earth had long before been proposed by the Pythagoreans, but he also noted (correctly) that this idea had been “gratuitously asserted” by them, without arguments of the sort he himself was able to advance. There was something else about the Ptolemaic theory that Copernicus did not like, besides its fine- tuning and its uncertainty regarding the sizes and order of planetary orbits. True to Plato’s dictum that planets move on circles at constant speed, Copernicus rejected Ptolemy’s use of devices like the equant to deal with the actual departures from circular motion at fixed speed. As had been done by Ibn al- Shatir, Copernicus instead introduced more epicycles: six for Mercury; three for the Moon; and four each for Venus, Mars, Jupiter, and Saturn. Here he made no improvement over the Almagest. This work of Copernicus illustrates another recurrent theme in the history of physical science: a simple and beautiful theory that agrees pretty well with observation is often closer to the truth than a complicated ugly theory that agrees better with observation. The simplest realization of the general ideas of Copernicus would have been to give each planet including the Earth a circular orbit at constant speed with the Sun at the center of all orbits, and no epicycles anywhere. This would have agreed with the simplest version of Ptolemaic astronomy, with just one epicycle for each planet, none for the Sun and Moon, and no eccentrics or equants. It would not have precisely agreed with all observations, because planets move not on circles but on nearly circular ellipses; their speed is only approximately constant; and the Sun is not at the center of each ellipse but at a point a little off-center, known as the focus. (See Technical Note 18.) Copernicus could have done even better by following Ptolemy and introducing an eccentric and equant for each planetary orbit, but now also including the orbit of the Earth; the discrepancy with observation would then have been almost too small for astronomers of the time to measure. There is an episode in the development of quantum mechanics that shows the importance of not worrying too much about small conflicts with observation. In 1925 Erwin Schrödinger worked out a method for calculating the energies of the states of the simplest atom, that of hydrogen. His results were in good agreement with the gross pattern of these energies, but the fine details of his result, which took into account the departures of the mechanics of special relativity from Newtonian mechanics, did not agree with the fine details of the measured energies. Schrödinger sat on his results for a while, until he wisely realized that getting the gross pattern of the energy levels was a significant accomplishment, well worth publishing, and that the correct treatment of relativistic effects could wait. It was provided a few years later by Paul Dirac. In addition to numerous epicycles, there was another complication adopted by Copernicus, one similar to the eccentric of Ptolemaic astronomy. The center of the Earth’s orbit was taken to be, not the Sun, but a point at a relatively small distance from the Sun. These complications approximately accounted for various phenomena, such as the inequality of the seasons discovered by Euctemon, which are really consequences of the fact that the Sun is at the focus rather than the center of the Earth’s elliptical orbit, and the Earth’s speed in its orbit is not constant. Another of the complications introduced by Copernicus was made necessary only by a misunderstanding. Copernicus seems to have thought that the revolution of the Earth around the Sun would give the axis of the Earth’s rotation each year a 360° turn around the direction perpendicular to the plane of the Earth’s orbit, somewhat as a finger at the end of the outstretched arm of a dancer executing a pirouette would undergo a 360° turn around the vertical direction for each rotation of the dancer. (He may have been influenced by the old idea that the planets ride on solid transparent spheres.) Of course, the direction of the Earth’s axis does not in fact change appreciably in the course of a year, so Copernicus was forced to give the Earth a third motion, in addition to its revolution around the Sun and its rotation around its axis, which would almost cancel this swiveling of its axis. Copernicus assumed that the cancellation would not be perfect, so that the Earth’s axis would swivel around over very many years, producing the slow precession of the equinoxes that had been discovered by Hipparchus. After Newton’s work it became clear that the revolution of the Earth around the Sun in fact has no influence on the direction of the Earth’s axis, aside from tiny effects due to the action of the gravity of the Sun and Moon on the Earth’s equatorial bulge, and so (as Kepler argued) no cancellation of the sort arranged by Copernicus is actually necessary. With all these complications, the theory of Copernicus was still simpler than that of Ptolemy, but not dramatically so. Though Copernicus could not have known it, his theory would have been closer to the truth if he had not bothered with epicycles, and had left the small inaccuracies of the theory to be dealt with in the future. The Commentariolus did not give much in the way of technical details. These were supplied in his great work De Revolutionibus Orbium Coelestium,5 commonly known as De Revolutionibus, finished in 1543 when Copernicus was on his deathbed. The book starts with a dedication to Alessandro Farnese, Pope Paul III. In it Copernicus raised again the old argument between the homocentric spheres of Aristotle and the eccentrics and epicycles of Ptolemy, pointing out that the former do not account for observations, while the latter “contradict the first principles of regularity of motion.” In support of his daring in suggesting a moving Earth, Copernicus quoted a paragraph of Plutarch:

Some think that the Earth remains at rest. But Philolaus the Pythagorean believes that, like the Sun and Moon, it revolves around the fire in an oblique circle. Heraclides of Pontus and Ecphantus the Pythagorean make the Earth move, not in a progressive motion, but like a wheel in a rotation from west to east about its own center.

(In the standard edition of De Revolutionibus Copernicus makes no mention of Aristarchus, but his name had appeared originally, and had then been struck out.) Copernicus went on to explain that since others had considered a moving Earth, he too should be permitted to test the idea. He then described his conclusion:

Having thus assumed the motions which I ascribe to the Earth later in the volume, by long and intense study I finally found that if the motions of the other planets are correlated with the orbiting of the Earth, and are computed for the revolution of each planet, not only do their phenomena follow therefrom but also the order and size of all the planets and spheres, and heaven itself is so linked together that in no portion of it can anything be shifted without disrupting the remaining parts and the universe as a whole.

As in the Commentariolus, Copernicus was appealing to the fact that his theory was more predictive than Ptolemy’s; it dictated a unique order of planets and the sizes of their orbits required to account for observation, while Ptolemy’s theory left these undetermined. Of course, Copernicus had no way of confirming that his orbital radii were correct without assuming the truth of his theory; this had to wait for Galileo’s observations of planetary phases. Most of De Revolutionibus is extremely technical, fleshing out the general ideas of the Commentariolus. One point worth special mention is that in Book 1 Copernicus states an a priori commitment to motion composed of circles. Thus Chapter 1 of Book I begins:

First of all, we must note that the universe is spherical. The reason is either that, of all forms, the sphere is the most perfect, needing no joint and being a complete whole, which can neither be increased nor diminished [here Copernicus sounds like Plato]; or that it is the most capacious of figures, best suited to enclose and retain all things [that is, it has the greatest volume for a given surface area]; or even that all the separate parts of the universe, I mean the Sun, Moon, planets and stars are seen to be of this shape [how could he know anything about the shape of the stars?]; or that wholes strive to be circumscribed by this boundary, as is apparent in drops of water and other fluid bodies when they seek to be self-contained [this is an effect of surface tension, which is irrelevant on the scale of planets]. Hence no one will question that the attribution of this form belongs to the divine bodies.

He then went on to explain in Chapter 4 that in consequence the movement of the heavenly bodies is “uniform, eternal, and circular, or compounded of circular motions.” Later in Book 1, Copernicus pointed out one of the prettiest aspects of his heliocentric system: it explained why Mercury and Venus are never seen far in the sky from the Sun. For instance, the fact that Venus is never seen more than about 45° from the Sun is explained by the fact that its orbit around the Sun is about 70 percent the size of the orbit of the Earth. (See Technical Note 19.) As we saw in Chapter 11, in Ptolemy’s theory this had required fine-tuning the motion of Mercury and Venus so that the centers of their epicycles are always on the line between the Earth and the Sun. The system of Copernicus also made unnecessary Ptolemy’s fine-tuning of the motion of the outer planets, which kept the line between each planet and the center of its epicycle parallel to the line between the Earth and the Sun. The Copernican system ran into opposition from religious leaders, beginning even before publication of De Revolutionibus. This conflict was exaggerated in a famous nineteenth-century polemic, A History of the Warfare of Science with Theology in Christendom by Cornell’s first president, Andrew Dickson White,6 which offers a number of unreliable quotations from Luther, Melanchthon, Calvin, and Wesley. But a conflict did exist. There is a record of Martin Luther’s conversations with his disciples at Wittenberg, known as Tischreden, or Table Talk.7 The entry for June 4, 1539, reads in part:

There was mention of a certain new astrologer who wanted to prove that the Earth moves and not the sky, the Sun, and the Moon. . . . [Luther remarked,] “So it goes now. Whoever wants to be clever must agree with nothing that others esteem. He must do something of his own. This is what that fool does who wishes to turn the whole of astronomy upside down. Even in these things that are thrown into disorder I believe in the Holy Scriptures, for Joshua commanded the Sun to stand still and not the Earth.”8

A few years after the publication of De Revolutionibus, Luther’s colleague Philipp Melanchthon (1497– 1560) joined the attack on Copernicus, now citing Ecclesiastes 1:5—“The Sun also rises, and the Sun goes down, and hastens to his place where he rose.” Conflicts with the literal text of the Bible would naturally raise problems for Protestantism, which had replaced the authority of the pope with that of Scripture. Beyond this, there was a potential problem for all religions: man’s home, the Earth, had been demoted to just one more planet among the other five. Problems arose even with the printing of De Revolutionibus. Copernicus had sent his manuscript to a publisher in Nuremberg, and the publisher appointed as editor a Lutheran clergyman, Andreas Osiander, whose hobby was astronomy. Probably expressing his own views, Osiander added a preface that was thought to be by Copernicus until the substitution was unmasked in the following century by Kepler. In this preface Osiander had Copernicus disclaiming any intention to present the true nature of planetary orbits, as follows:9

For it is the duty of an astronomer to compose the history of the [apparent] celestial motions through careful and expert study. Then he must conceive and devise the causes of these motions or hypotheses about them. Since he cannot in any way attain to the true cause, he will adopt whatever suppositions enable the motions to be computed correctly from the principles of geometry for the future as well as for the past. Osiander’s preface concludes:

So far as hypotheses are concerned, let no one expect anything certain from astronomy, which cannot furnish it, lest he accept as the truth ideas conceived for another purpose, and depart from this study a greater fool than when he entered it.

This was in line with the views of Geminus around 70 BC (quoted here in Chapter 8), but it was quite contrary to the evident intention of Copernicus, in both the Commentariolus and De Revolutionibus, to describe the actual constitution of what is now called the solar system. Whatever individual clergymen may have thought about a heliocentric theory, there was no general Protestant effort to suppress the works of Copernicus. Nor did Catholic opposition to Copernicus become organized until the 1600s. The famous execution of Giordano Bruno by the Roman Inquisition in 1600 was not for his defense of Copernicus, but for heresy, of which (by the standards of the time) he was surely guilty. But as we will see, the Catholic church did in the seventeenth century put in place a very serious suppression of Copernican ideas. What was really important for the future of science was the reception of Copernicus among his fellow astronomers. The first to be convinced by Copernicus was his sole pupil, Rheticus, who in 1540 published an account of the Copernican theory, and who in 1543 helped to get De Revolutionibus into the hands of the Nuremberg publisher. (Rheticus was initially supposed to supply the preface to De Revolutionibus, but when he left to take a position in Leipzig the task unfortunately fell to Osiander.) Rheticus had earlier assisted Melanchthon in making the University of Wittenberg a center of mathematical and astronomical studies. The theory of Copernicus gained prestige from its use in 1551 by Erasmus Reinhold, under the sponsorship of the duke of Prussia, to compile a new set of astronomical tables, the Prutenic Tables, which allow one to calculate the location of planets in the zodiac at any given date. These were a distinct improvement over the previously used Alfonsine Tables, which had been constructed in Castile in 1275 at the court of Alfonso X. The improvement was in fact due, not to the superiority of the theory of Copernicus, but rather to the accumulation of new observations in the centuries between 1275 and 1551, and perhaps also to the fact that the greater simplicity of heliocentric theories makes calculations easier. Of course, adherents of a stationary Earth could argue that De Revolutionibus provided only a convenient scheme for calculation, not a true picture of the world. Indeed, the Prutenic Tables were used by the Jesuit astronomer and mathematician Christoph Clavius in the 1582 calendar reform under Pope Gregory XIII that gave us our modern Gregorian calendar, but Clavius never gave up his belief in a stationary Earth. One mathematician tried to reconcile this belief with the Copernican theory. In 1568, Melanchthon’s son-in-law Caspar Peucer, professor of mathematics at Wittenberg, argued in Hypotyposes orbium coelestium that it should be possible by a mathematical transformation to rewrite the theory of Copernicus in a form in which the Earth rather than the Sun is stationary. This is precisely the result achieved later by one of Peucer’s students, Tycho Brahe. Tycho Brahe was the most proficient astronomical observer in history before the introduction of the telescope, and the author of the most plausible alternative to the theory of Copernicus. Born in 1546 in the province of Skåne, now in southern Sweden but until 1658 part of Denmark, Tycho was a son of a Danish nobleman. He was educated at the University of Copenhagen, where in 1560 he became excited by the successful prediction of a partial solar eclipse. He moved on to universities in Germany and Switzerland, at Leipzig, Wittenberg, Rostock, Basel, and Augsburg. During these years he studied the Prutenic Tables and was impressed by the fact that these tables predicted the date of the 1563 conjunction of Saturn and Jupiter to within a few days, while the older Alfonsine Tables were off by several months. Back in Denmark, Tycho settled for a while in his uncle’s house at Herrevad in Skåne. There in 1572 he observed in the constellation Cassiopeia what he called a “new star.” (It is now recognized as the thermonuclear explosion, known as a type Ia supernova, of a preexisting star. The remnant of this explosion was discovered by radio astronomers in 1952 and found to be at a distance of about 9,000 light-years, too far for the star to have been seen without a telescope before the explosion.) Tycho observed the new star for months, using a sextant of his own construction, and found that it did not exhibit any diurnal parallax, the daily shift in position among the stars of the sort that would be expected to be caused by the rotation of the Earth (or the daily revolution around the Earth of everything else) if the new star were as close as the Moon, or closer. (See Technical Note 20.) He concluded, “This new star is not located in the upper regions of the air just under the lunar orb, nor in any place closer to Earth . . . but far above the sphere of the Moon in the very heavens.”10 This was a direct contradiction of the principle of Aristotle that the heavens beyond the orbit of the Moon can undergo no change, and it made Tycho famous. In 1576 the Danish king Frederick II gave Tycho the lordship of the small island of Hven, in the strait between Skåne and the large Danish island of Zealand, along with a pension to support the building and maintenance of a residence and scientific establishment on Hven. There Tycho built Uraniborg, which included an observatory, library, chemical laboratory, and printing press. It was decorated with portraits of past astronomers—Hipparchus, Ptolemy, al-Battani, and Copernicus—and of a patron of the sciences: William IV, landgrave of Hesse-Cassel. On Hven Tycho trained assistants, and immediately began observations. Already in 1577 Tycho observed a comet, and found that it had no observable diurnal parallax. Not only did this show, again contra Aristotle, that there was change in the heavens beyond the orbit of the Moon. Now Tycho could also conclude that the path of the comet would have taken it right through either Aristotle’s supposed homocentric spheres or the spheres of the Ptolemaic theory. (This, of course, would be a problem only if the spheres were conceived as hard solids. This was the teaching of Aristotle, which we saw in Chapter 8 had been carried over to the Ptolemaic theory by the Hellenistic astronomers Adrastus and Theon. The idea of hard spheres was revived in early modern times,11 not long before Tycho ruled it out.) Comets occur more frequently than supernovas, and Tycho was able to repeat these observations on other comets in the following years. From 1583 on, Tycho worked on a new theory of the planets, based on the idea that the Earth is at rest, the Sun and Moon go around the Earth, and the five known planets go around the Sun. It was published in 1588 as Chapter 8 of Tycho’s book on the comet of 1577. In this theory the Earth is not supposed to be moving or rotating, so in addition to having slower motions, the Sun, Moon, planets, and stars all revolve around the Earth from east to west once a day. Some astronomers adopted instead a “semi-Tychonic” theory, in which the planets revolve around the Sun, the Sun revolves around the Earth, but the Earth rotates and the stars are at rest. (The first advocate of a semi-Tychonic theory was Nicolas Reymers Bär, although he would not have called it a semi-Tychonic system, for he claimed Tycho had stolen the original Tychonic system from him.)12 As mentioned several times above, the Tychonic theory is identical to the version of Ptolemy’s theory (never considered by Ptolemy) in which the deferents of the inner planets are taken to coincide with the orbit of the Sun around the Earth, and the epicycles of the outer planets have the same radius as the Sun’s orbit around the Earth. As far as the relative separations and velocities of the heavenly bodies are concerned, it is also equivalent to the theory of Copernicus, differing only in the point of view: a stationary Sun for Copernicus, or a stationary and nonrotating Earth for Tycho. Regarding observations, Tycho’s theory had the advantage that it automatically predicted no annual stellar parallax, without having to assume that the stars are very much farther from Earth than the Sun or planets (which, of course, we now know they are). It also made unnecessary Oresme’s answer to the classic problem that had misled Ptolemy and Buridan: that objects thrown upward would seemingly be left behind by a rotating or moving Earth. For the future of astronomy, the most important contribution of Tycho was not his theory, but the unprecedented accuracy of his observations. When I visited Hven in the 1970s, I found no sign of Tycho’s buildings, but there, still in the ground, were the massive stone foundations to which Tycho had anchored his instruments. (A museum and formal gardens have been put in place since my visit.) With these instruments, Tycho was able to locate objects in the sky with an uncertainty of only 1/15°. Also at the site of Uraniborg stands a granite statue, carved in 1936 by Ivar Johnsson, showing Tycho in a posture appropriate for an astronomer, facing up into the sky.13 Tycho’s patron Frederick II died in 1588. He was succeeded by Christian IV, whom Danes today regard as one of their greatest kings, but who unfortunately had little interest in supporting work on astronomy. Tycho’s last observations from Hven were made in 1597; he then left on a journey that took him to Hamburg, Dresden, Wittenberg, and Prague. In Prague, he became the imperial mathematician to the Holy Roman Emperor Rudolph II and started work on a new set of astronomical tables, the Rudolphine Tables. After Tycho’s death in 1601, this work was continued by Kepler. Johannes Kepler was the first to understand the nature of the departures from uniform circular motion that had puzzled astronomers since the time of Plato. As a five-year-old he was inspired by the sight of the comet of 1577, the first comet that Tycho had studied from his new observatory on Hven. Kepler attended the University of Tübingen, which under the leadership of Melanchthon had become eminent in theology and mathematics. At Tübingen Kepler studied both of these subjects, but became more interested in mathematics. He learned about the theory of Copernicus from the Tübingen mathematics professor Michael Mästlin and became convinced of its truth. In 1594 Kepler was hired to teach mathematics at a Lutheran school in Graz, in southern Austria. It was there that he published his first original work, the Mysterium Cosmographicum (Mystery of the Description of the Cosmos). As we have seen, one advantage of the theory of Copernicus was that it allowed astronomical observations to be used to find unique results for the order of planets outward from the Sun and for the sizes of their orbits. As was still common at the time, Kepler in this work conceived these orbits to be the circles traced out by planets carried on transparent spheres, revolving in the Copernican theory around the Sun. These spheres were not strictly two-dimensional surfaces, but thin shells whose inner and outer radii were taken to be the minimum and maximum distance of the planet from the Sun. Kepler conjectured that the radii of these spheres are constrained by an a priori condition, that each sphere (other than the outermost sphere, of Saturn) just fits inside one of the five regular polyhedrons, and each sphere (other than the innermost sphere, of Mercury) just fits outside one of these regular polyhedrons. Specifically, in order outward from the Sun, Kepler placed (1) the sphere of Mercury, (2) then an octahedron, (3) the sphere of Venus, (4) an icosahedron, (5) the sphere of Earth, (6) a dodecahedron, (7) the sphere of Mars, (8) a tetrahedron, (9) the sphere of Jupiter, (10) a cube, and finally (11) the sphere of Saturn, all fitting together tightly. This scheme dictated the relative sizes of the orbits of all the planets, with no freedom to adjust the results, except by choosing the order of the five regular polyhedrons that fit into the spaces between the planets. There are 30 different ways of choosing the order of the regular polyhedrons,* so it is not surprising that Kepler could find one way of choosing their order so that the predicted sizes of planetary orbits would roughly fit the results of Copernicus. In fact, Kepler’s original scheme worked badly for Mercury, requiring Kepler to do some fudging, and only moderately well for the other planets.* But like many others at the time of the Renaissance, Kepler was deeply influenced by Platonic philosophy, and like Plato he was intrigued by the theorem that regular polyhedrons exist in only five possible shapes, leaving room for only six planets, including the Earth. He proudly proclaimed, “Now you have the reason for the number of planets!” No one today would take seriously a scheme like Kepler’s, even if it had worked better. This is not because we have gotten over the old Platonic fascination with short lists of mathematically possible objects, like regular polyhedrons. There are other such short lists that continue to intrigue physicists. For instance, it is known that there are just four kinds of “numbers” for which a version of arithmetic including division is possible: the real numbers, complex numbers (involving the square root of −1), and more exotic quantities known as quaternions and octonions. Some physicists have expended much effort trying to incorporate quaternions and octonions as well as real and complex numbers in the fundamental laws of physics. What makes Kepler’s scheme so foreign to us today is not his attempt to find some fundamental physical significance for the regular polyhedrons, but that he did this in the context of planetary orbits, which are just historical accidents. Whatever the fundamental laws of nature may be, we can be pretty sure now that they do not refer to the radii of planetary orbits. This was not just stupidity on Kepler’s part. In his time no one knew (and Kepler did not believe) that the stars were suns with their own systems of planets, rather than just lights on a sphere somewhere outside the sphere of Saturn. The solar system was generally thought to be pretty much the whole universe, and to have been created at the beginning of time. It was perfectly natural then to suppose that the detailed structure of the solar system is as fundamental as anything else in nature. We may be in a similar position in today’s theoretical physics. It is generally supposed that what we call the expanding universe, the enormous cloud of galaxies that we observe rushing apart uniformly in all directions, is the whole universe. We think that the constants of nature we measure, such as the masses of the various elementary particles, will eventually all be deduced from the yet unknown fundamental laws of nature. But it may be that what we call the expanding universe is just a small part of a much larger “multiverse,” containing many expanding parts like the one we observe, and that the constants of nature take different values in different parts of the multiverse. In this case, these constants are environmental parameters that will never be deduced from fundamental principles any more than we can deduce the distances of the planets from the Sun from fundamental principles. The best we could hope for would be an anthropic estimate. Of the billions of planets in our own galaxy, only a tiny minority have the right temperature and chemical composition to be suitable for life, but it is obvious that when life does begin and evolves into astronomers, they will find themselves on a planet belonging to this minority. So it is not really surprising that the planet on which we live is not twice or half as far from the Sun as the Earth actually is. In the same way, it seems likely that only a tiny minority of the subuniverses in the multiverse would have physical constants that allow the evolution of life, but of course any scientists will find themselves in a subuniverse belonging to this minority. This had been offered as an explanation of the order of magnitude of the dark energy mentioned in Chapter 8, before dark energy was discovered.14 All this, of course, is highly speculative, but it serves as a warning that in trying to understand the constants of nature we may face the same sort of disappointment Kepler faced in trying to explain the dimensions of the solar system. Some distinguished physicists deplore the idea of a multiverse, because they cannot reconcile themselves to the possibility that there are constants of nature that can never be calculated. It is true that the multiverse idea may be all wrong, and so it would certainly be premature to give up the effort to calculate all the physical constants we know about. But it is no argument against the multiverse idea that it would make us unhappy not to be able to do these calculations. Whatever the final laws of nature may be, there is no reason to suppose that they are designed to make physicists happy. At Graz Kepler began a correspondence with Tycho Brahe, who had read the Mysterium Cosmographicum. Tycho invited Kepler to visit him in Uraniborg, but Kepler thought it would be too far to go. Then in February 1600 Kepler accepted Tycho’s invitation to visit him in Prague, the capital since 1583 of the Holy Roman Empire. There Kepler began to study Tycho’s data, especially on the motions of Mars, and found a discrepancy of 0.13° between these data and the theory of Ptolemy.* Kepler and Tycho did not get along well, and Kepler returned to Graz. At just that time Protestants were being expelled from Graz, and in August 1600 Kepler and his family were forced to leave. Back in Prague, Kepler began a collaboration with Tycho, working on the Rudolphine Tables, the new set of astronomical tables intended to replace Reinhold’s Prutenic Tables. After Tycho died in 1601, Kepler’s career problems were solved for a while by his appointment as Tycho’s successor as court mathematician to the emperor Rudolph II. The emperor was enthusiastic about astrology, so Kepler’s duties as court mathematician included the casting of horoscopes. This was an activity in which he had been employed since his student days at Tübingen, despite his own skepticism about astrological prediction. Fortunately, he also had time to pursue real science. In 1604 he observed a new star in the constellation Ophiuchus, the last supernova seen in or near our galaxy until 1987. In the same year he published Astronomiae Pars Optica (The Optical Part of Astronomy), a work on optical theory and its applications to astronomy, including the effect of refraction in the atmosphere on observations of the planets. Kepler continued work on the motions of planets, trying and failing to reconcile Tycho’s precise data with Copernican theory by adding eccentrics, epicycles, and equants. Kepler had finished this work by 1605, but publication was held up by a squabble with the heirs of Tycho. Finally in 1609 Kepler published his results in Astronomia Nova (New Astronomy Founded on Causes, or Celestial Physics Expounded in a Commentary on the Movements of Mars). Part III of Astronomia Nova made a major improvement in the Copernican theory by introducing an equant and eccentric for the Earth, so that there is a point on the other side of the center of the Earth’s orbit from the Sun around which the line to the Earth rotates at a constant rate. This removed most of the discrepancies that had bedeviled planetary theories since the time of Ptolemy, but Tycho’s data were good enough so that Kepler could see that there were still some conflicts between theory and observation. At some point Kepler became convinced that the task was hopeless, and that he had to abandon the assumption, common to Plato, Aristotle, Ptolemy, Copernicus, and Tycho, that planets move on orbits composed of circles. Instead, he concluded that planetary orbits have an oval shape. Finally, in Chapter 58 (of 70 chapters) of Astronomia Nova, Kepler made this precise. In what later became known as Kepler’s first law, he concluded that planets (including the Earth) move on ellipses, with the Sun at a focus, not at the center. Just as a circle is completely described (apart from its location) by a single number, its radius, any ellipse can be completely described (aside from its location and orientation) by two numbers, which can be taken as the lengths of its longer and shorter axes, or equivalently as the length of the longer axis and a number known as the “eccentricity,” which tells us how different the major and minor axes are. (See Technical Note 18.) The two foci of an ellipse are points on the longer axis, evenly spaced around the center, with a separation from each other equal to the eccentricity times the length of the longer axis of the ellipse. For zero eccentricity, the two axes of the ellipse have equal length, the two foci merge to a single central point, and the ellipse degenerates into a circle. In fact, the orbits of all the planets known to Kepler have small eccentricities, as shown in the following table of modern values (projected back to the year 1900): Planet Eccentricity Mercury 0.205615 Venus 0.006820 Earth 0.016750

Mars 0.093312 Jupiter 0.048332 Saturn 0.055890

This is why simplified versions of the Copernican and Ptolemaic theories (with no epicycles in the Copernican theory and only one epicyle for each of the five planets in the Ptolemaic theory) would have worked pretty well.* The replacement of circles with ellipses had another far-reaching implication. Circles can be generated by the rotation of spheres, but there is no solid body whose rotation can produce an ellipse. This, together with Tycho’s conclusions from the comet of 1577, went far to discredit the old idea that planets are carried on revolving spheres, an idea that Kepler himself had assumed in the Mysterium Cosmographicum. Instead, Kepler and his successors now conceived of planets as traveling on freestanding orbits in empty space. The calculations reported in Astronomia Nova also used what later became known as Kepler’s second law, though this law was not clearly stated until 1621, in his Epitome of Copernican Astronomy. The second law tells how the speed of a planet changes as the planet moves around its orbit. It states that as the planet moves, the line between the Sun and the planet sweeps out equal areas in equal times. A planet has to move farther along its orbit to sweep out a given area when it is near the Sun than when it is far from the Sun, so Kepler’s second law has the consequence that each planet must move faster the closer it comes to the Sun. Aside from tiny corrections proportional to the square of the eccentricity, Kepler’s second law is the same as the statement that the line to the planet from the other focus (the one where the Sun isn’t) turns at a constant rate—that is, it turns by the same angle in every second. (See Technical Note 21.) Thus to a good approximation, Kepler’s second law gives the same planetary velocities as the old idea of an equant, a point on the opposite side of the center of the circle from the Sun (or, for Ptolemy, from the Earth), and at the same distance from the center, around which the line to the planet turns at a constant rate. The equant was thus revealed as nothing but the empty focus of the ellipse. Only Tycho’s superb data for Mars allowed Kepler to conclude that eccentrics and equants are not enough; circular orbits had to be replaced with ellipses.15 The second law also had profound applications, at least for Kepler. In Mysterium Cosmographicum Kepler had conceived of the planets as being moved by a “motive soul.” But now, with the speed of each planet found to decrease as its distance from the Sun increases, Kepler instead concluded that the planets are impelled in their orbits by some sort of force radiating from the Sun:

If you substitute the word “force” [vis] for the word “soul” [anima], you have the very principle on which the celestial physics in the Commentary on Mars [Astronomia Nova] is based. For I formerly believed completely that the cause moving the planets is a soul, having indeed been imbued with the teaching of J. C. Scaliger* on motive intelligences. But when I recognized that this motive cause grows weaker as the distance from the Sun increases, just as the light of the Sun is attenuated, I concluded that this force must be as it were corporeal.16

Of course, the planets continue in their motion not because of a force radiating from the Sun, but rather because there is nothing to drain their momentum. But they are held in their orbits rather than flying off into interstellar space by a force radiating from the Sun, the force of gravitation, so Kepler was not entirely wrong. The idea of force at a distance was becoming popular at this time, partly owing to the work on magnetism by the president of the Royal College of Surgeons and court physician to Elizabeth I, William Gilbert, to whom Kepler referred. If by “soul” Kepler had meant anything like its usual meaning, then the transition from a “physics” based on souls to one based on forces was an essential step in ending the ancient mingling of religion with natural science. Astronomia Nova was not written with the aim of avoiding controversy. By using the word “physics” in the full title, Kepler was throwing out a challenge to the old idea, popular among followers of Aristotle, that astronomy should concern itself only with the mathematical description of appearances, while for true understanding one must turn to physics—that is, to the physics of Aristotle. Kepler was staking out a claim that it is astronomers like himself who do true physics. In fact, much of Kepler’s thinking was inspired by a mistaken physical idea, that the Sun drives the planets around their orbits, by a force similar to magnetism. Kepler also challenged all opponents of Copernicanism. The introduction to Astronomia Nova contains the paragraph:

Advice for idiots. But whoever is too stupid to understand astronomical science, or too weak to believe Copernicus without [it] affecting his faith, I would advise him that, having dismissed astronomical studies, and having damned whatever philosophical studies he pleases, he mind his own business and betake himself home to scratch in his own dirt patch.17

Kepler’s first two laws had nothing to say about the comparison of the orbits of different planets. This gap was filled in 1619 in Harmonices mundi, by what became known as Kepler’s third law:18 “the ratio which exists between the periodic times of any two planets is precisely the ratio of the 3/2-power of the mean distances.”* That is, the square of the sidereal period of each planet (the time it takes to complete a full circuit of its orbit) is proportional to the cube of the longer axis of the ellipse. Thus if T is the sidereal period in years, and a is half the length of the longer axis of the ellipse in astronomical units (AU), with 1 AU defined as half the longer axis of the Earth’s orbit, then Kepler’s third law says that T2 / a3 is the same for all planets. Since the Earth by definition has T equal to 1 year and a equal to 1 AU, in these units it has T2 / a3 equal to 1, so according to Kepler’s third law each planet should also have T2 / a3 = 1. The accuracy with which modern values follow this rule is shown in the following table:

(The departures from perfect equality of T2 / a3 for the different planets are due to tiny effects of the gravitational fields of the planets themselves acting on each other.) Never entirely emancipated from Platonism, Kepler tried to make sense of the sizes of the orbits, resurrecting his earlier use of regular polyhedrons in Mysterium Cosmographicum. He also played with the Pythagorean idea that the different planetary periods form a sort of musical scale. Like other scientists of the time, Kepler belonged only in part to the new world of science that was just coming into being, and in part also to an older philosophical and poetic tradition. The Rudolphine Tables were finally completed in 1627. Based on Kepler’s first and second laws, they represented a real improvement in accuracy over the previous Prutenic Tables. The new tables predicted that there would be a transit of Mercury (that is, that Mercury would be seen to pass across the face of the Sun) in 1631. Kepler did not see it. Forced once again as a Protestant to leave Catholic Austria, Kepler died in 1630 in Regensburg. The work of Copernicus and Kepler made a case for a heliocentric solar system based on mathematical simplicity and coherence, not on its better agreement with observation. As we have seen, the simplest versions of the Copernican and Ptolemaic theories make the same predictions for the apparent motions of the Sun and planets, in pretty good agreement with observation, while the improvements in the Copernican theory introduced by Kepler were the sort that could have been matched by Ptolemy if he had used an equant and eccentric for the Sun as well as for the planets, and if he had added a few more epicycles. The first observational evidence that decisively favored heliocentrism over the old Ptolemaic system was provided by Galileo Galilei. With Galileo, we come to one of the greatest scientists of history, in a class with Newton, Darwin, and Einstein. He revolutionized observational astronomy with his introduction and use of the telescope, and his study of motion provided a paradigm for modern experimental physics. Further, to an extent that is unique, his scientific career was attended by high drama, of which we can here give only a condensed account. Galileo was a patrician though not wealthy Tuscan, born in Pisa in 1564, the son of the musical theorist Vincenzo Galilei. After studies at a Florentine monastery, he enrolled as a medical student at the University of Pisa in 1581. Unsurprisingly for a medical student, at this point in his life he was a follower of Aristotle. Galileo’s interests then shifted from medicine to mathematics, and for a while he gave mathematics lessons in Florence, the capital of Tuscany. In 1589 Galileo was called back to Pisa to take the chair of mathematics. While at the University of Pisa Galileo started his study of falling bodies. Some of his work is described in a book, De Motu (On Motion), which he never published. Galileo concluded, contrary to Aristotle, that the speed of a heavy falling body does not depend appreciably on its weight. It’s a nice story that he tested this by dropping various weights from Pisa’s Leaning Tower, but there is no evidence for this. While in Pisa Galileo published nothing about his work on falling bodies. In 1591 Galileo moved to Padua to take the chair of mathematics at its university, which was then the university of the republic of Venice and the most intellectually distinguished university in Europe. From 1597 on he was able to supplement his university salary with the manufacture and sale of mathematical instruments, used in business and war. In 1597 Galileo received two copies of Kepler’s Mysterium Cosmographicum. He wrote to Kepler, acknowledging that he, like Kepler, was a Copernican, though as yet he had not made his views public. Kepler replied that Galileo should come out for Copernicus, urging, “Stand forth, O Galileo!”19 Soon Galileo came into conflict with the Aristotelians who dominated the teaching of philosophy at Padua, as elsewhere in Italy. In 1604 he lectured on the “new star” observed that year by Kepler. Like Tycho and Kepler he drew the conclusion that change does occur in the heavens, above the orbit of the Moon. He was attacked for this by his sometime friend Cesare Cremonini, professor of philosophy at Padua. Galileo replied with an attack on Cremonini, written in a rustic Paduan dialect as a dialogue between two peasants. The Cremonini peasant argued that the ordinary rules of measurement do not apply in the heavens; and Galileo’s peasant replied that philosophers know nothing about measurement: for this one must trust mathematicians, whether for measurement of the heavens or of polenta. A revolution in astronomy began in 1609, when Galileo first heard of a new Dutch device known as a spyglass. The magnifying property of glass spheres filled with water was known in antiquity, mentioned for instance by the Roman statesman and philosopher Seneca. Magnification had been studied by al- Haitam, and in 1267 Roger Bacon had written about magnifying glasses in Opus Maius. With improvements in the manufacture of glass, reading glasses had become common in the fourteenth century. But to magnify distant objects, it is necessary to combine a pair of lenses, one to focus the parallel rays of light from any point on the object so that they converge, and the second to gather these light rays, either with a concave lens while they are still converging or with a convex lens after they begin to diverge again, in either case sending them on parallel directions into the eye. (When relaxed the lens of the eye focuses parallel rays of light to a single point on the retina; the location of that point depends on the direction of the parallel rays.) Spyglasses with such an arrangement of lenses were being produced in the Netherlands by the beginning of the 1600s, and in 1608 several Dutch makers of spectacles applied for patents on their spyglasses. Their applications were rejected, on the ground that the device was already widely known. Spyglasses were soon available in France and Italy, but capable of magnification by only three or four times. (That is, if the lines of sight to two distant points are separated by a certain small angle, then with these spyglasses they seemed to be separated by three or four times that angle.) Sometime in 1609 Galileo heard of the spyglass, and soon made an improved version, with the first lens convex on the side facing forward and planar on the back, and with long focal length,* while the second was concave on the side facing the first lens and planar on the back side, and with shorter focal length. With this arrangement, to send the light from a point source at very large distances on parallel rays into the eye, the distance between the lenses must be taken as the difference of the focal lengths, and the magnification achieved is the focal length of the first lens divided by the focal length of the second lens. (See Technical Note 23.) Galileo was soon able to achieve a magnification by eight or nine times. On August 23, 1609, he showed his spyglass to the doge and notables of Venice and demonstrated that with it ships could be seen at sea two hours before they became visible to the naked eye. The value of such a device to a maritime power like Venice was obvious. After Galileo donated his spyglass to the Venetian republic, his professorial salary was tripled, and his tenure was guaranteed. By November Galileo had improved the magnification of his spyglass to 20 times, and he began to use it for astronomy. With his spyglass, later known as a telescope, Galileo made six astronomical discoveries of historic importance. The first four of these he described in Siderius Nuncius (The Starry Messenger),20 published in Venice in March 1610.

1. On November 20, 1609, Galileo first turned his telescope on the crescent Moon. On the bright side, he could see that its surface is rough:

By oft-repeated observations of [lunar markings] we have been led to the conclusion that we certainly see the surface of the Moon to be not smooth, even, and perfectly spherical, as the great crowd of philosophers have believed about this and other heavenly bodies, but on the contrary, to be uneven, rough, and crowded with depressions and bulges. And it is like the face of the Earth itself, which is marked here and there with chains of mountains and depths of valleys.

On the dark side, near the terminator, the boundary with the bright side, he could see spots of light, which he interpreted as mountaintops illuminated by the Sun when it was just about to come over the lunar horizon. From the distance of these bright spots from the terminator he could even estimate that some of these mountains were at least four miles high. (See Technical Note 24.) Galileo also interpreted the observed faint illumination of the dark side of the Moon. He rejected various suggestions of Erasmus Reinhold and of Tycho Brahe that the light comes from the Moon itself or from Venus or the stars, and correctly argued that “this marvelous brightness” is due to the reflection of sunlight from the Earth, just as the Earth at night is faintly illuminated by sunlight reflected from the Moon. So a heavenly body like the Moon was seen to be not so very different from the Earth.

2. The spyglass allowed Galileo to observe “an almost inconceivable crowd” of stars much dimmer than stars of the sixth magnitude, and hence too dim to have been seen with the naked eye. The six visible stars of the Pleiades were found to be accompanied with more than 40 other stars, and in the constellation Orion he could see over 500 stars never seen before. Turning his telescope on the Milky Way, he could see that it is composed of many stars, as had been guessed by Albertus Magnus.

3. Galileo reported seeing the planets through his telescope as “exactly circular globes that appear as little moons,” but he could not discern any such image of the stars. Instead, he found that, although all stars seemed much brighter when viewed with his telescope, they did not seem appreciably larger. His explanation was confused. Galileo did not know that the apparent size of stars is caused by the bending of light rays in various directions by random fluctuations in the Earth’s atmosphere, rather than by anything intrinsic to the neighborhood of the stars. It is these fluctuations that cause stars to appear to twinkle.* Galileo concluded that, since it was not possible to make out the images of stars with his telescope, they must be much farther from us than are the planets. As Galileo noted later, this helped to explain why, if the Earth revolves around the Sun, we do not observe an annual stellar parallax.

4. The most dramatic and important discovery reported in Siderius Nuncius was made on January 7, 1610. Training his telescope on Jupiter, Galileo saw that “three little stars were positioned near him, small but very bright.” At first Galileo thought that these were just another three fixed stars, too dim to have been seen before, though he was surprised that they seemed to be lined up along the ecliptic, two to the east of Jupiter and one to the west. But on the next night all three of these “stars” were to the west of Jupiter, and on January 10 only two could be seen, both to the east. Finally on January 13, he saw that four of these “stars” were now visible, still more or less lined up along the ecliptic. Galileo concluded that Jupiter is accompanied in its orbit with four satellites, similar to Earth’s Moon, and like our Moon revolving in roughly the same plane as planetary orbits, which are close to the ecliptic, the plane of the Earth’s orbit around the Sun. (They are now known as the four largest moons of Jupiter: Ganymede, Io, Callisto, and Europa, named after the god Jupiter’s male and female lovers.)* This discovery gave important support to the Copernican theory. For one thing, the system of Jupiter and its moons provided a miniature example of what Copernicus had conceived to be the system of the Sun and its surrounding planets, with celestial bodies evidently in motion about a body other than the Earth. Also, the example of Jupiter’s moons put to rest the objection to Copernicus that, if the Earth is moving, why is the Moon not left behind? Everyone agreed that Jupiter is moving, and yet its moons were evidently not being left behind. Though the results were too late to be included in Siderius Nuncius, Galileo by the end of 1611 had measured the periods of revolution of the four Jovian satellites that he had discovered, and in 1612 he published these results on the first page of a work on other matters.21 Galileo’s results are given along with modern values in days (d), hours (h), and minutes (m) in the table below: Jovian satellite Period (Galileo) Period (modern) Io 1d 18h 30m 1d 18h 29m Europa 3d 13h 20m 3d 13h 18m Ganymede 7d 4h 0m 7d 4h 0m Callisto 16d 18h 0m 16d 18h 5m

The accuracy of Galileo’s measurements testifies to his careful observations and precise timekeeping.* Galileo dedicated Siderius Nuncius to his former pupil Cosimo II di Medici, now the grand duke of Tuscany, and he named the four companions of Jupiter the “Medicean stars.” This was a calculated compliment. Galileo had a good salary at Padua, but he had been told that it would not again be increased. Also, for this salary he had to teach, taking time away from his research. He was able to strike an agreement with Cosimo, who named him court mathematician and philosopher, with a professorship at Pisa that carried no teaching duties. Galileo insisted on the title “court philosopher” because despite the exciting progress made in astronomy by mathematicians such as Kepler, and despite the arguments of professors like Clavius, mathematicians continued to have a lower status than that enjoyed by philosophers. Also, Galileo wanted his work to be taken seriously as what philosophers called “physics,” an explanation of the nature of the Sun and Moon and planets, not just a mathematical account of appearances. In the summer of 1610 Galileo left Padua for Florence, a decision that turned out eventually to be disastrous. Padua was in the territory of the republic of Venice, which at the time was less under Vatican influence than any other state in Italy, having successfully resisted a papal interdict a few years before Galileo’s departure. Moving to Florence made Galileo much more vulnerable to control by the church. A modern university dean might feel that this danger was a just punishment for Galileo’s evasion of teaching duties. But for a while the punishment was deferred.

5. In September 1610 Galileo made the fifth of his great astronomical discoveries. He turned his telescope on Venus, and found that it has phases, like those of the Moon. He sent Kepler a coded message: “The Mother of Loves [Venus] emulates the shapes of Cynthia [the Moon].” The existence of phases would be expected in both the Ptolemaic and the Copernican theories, but the phases would be different. In the Ptolemaic theory, Venus is always more or less between the Earth and the Sun, so it can never be as much as half full. In the Copernican theory, on the other hand, Venus is fully illuminated when it is on the other side of its orbit from the Earth. This was the first direct evidence that the Ptolemaic theory is wrong. Recall that the Ptolemaic theory gives the same appearance of solar and planetary motions seen from the Earth as the Copernican theory, whatever we choose for the size of each planet’s deferent. But it does not give the same appearance as the Copernican theory of solar and planetary motions as seen from the planets. Of course, Galileo could not go to any planet to see how the motions of the Sun and other planets appear from there. But the phases of Venus did tell him the direction of the Sun as seen from Venus—the bright side is the side facing the Sun. Only one special case of Ptolemy’s theory could give that correctly, the case in which the deferents of Venus and Mercury are identical with the orbit of the Sun, which as already remarked is just the theory of Tycho. That version had never been adopted by Ptolemy, or by any of his followers.

6. At some time after coming to Florence, Galileo found an ingenious way to study the face of the Sun, by using a telescope to project its image on a screen. With this he made his sixth discovery: dark spots were seen to move across the Sun. His results were published in 1613 in his Sunspot Letters, about which more later.

There are moments in history when a new technology opens up large possibilities for pure science. The improvement of vacuum pumps in the nineteenth century made possible experiments on electrical discharges in evacuated tubes that led to the discovery of the electron. The Ilford Corporation’s development of photographic emulsions allowed the discovery of a host of new elementary particles in the decade following World War II. The development of microwave radar during that war allowed microwaves to be used as a probe of atoms, providing a crucial test of quantum electrodynamics in 1947. And we should not forget the gnomon. But none of these new technologies led to scientific results as impressive as those that flowed from the telescope in the hands of Galileo. The reactions to Galileo’s discoveries ranged from caution to enthusiasm. Galileo’s old adversary at Padua, Cesare Cremonini, refused to look through the telescope, as did Giulio Libri, professor of philosophy at Pisa. On the other hand, Galileo was elected a member of the Lincean Academy, founded a few years earlier as Europe’s first scientific academy. Kepler used a telescope sent to him by Galileo, and confirmed Galileo’s discoveries. (Kepler worked out the theory of the telescope and soon invented his own version, with two convex lenses.) At first, Galileo had no trouble with the church, perhaps because his support for Copernicus was still not explicit. Copernicus is mentioned only once in Siderius Nuncias, near the end, in connection with the question why, if the Earth is moving, it does not leave the Moon behind. At the time, it was not Galileo but Aristotelians like Cremonini who were in trouble with the Roman Inquisition, on much the same grounds that had led to the 1277 condemnation of various tenets of Aristotle. But Galileo managed to get into squabbles with both Aristotelian philosophers and Jesuits, which in the long run did him no good. In July 1611, shortly after taking up his new position in Florence, Galileo entered into a debate with philosophers who, following what they supposed to be a doctrine of Aristotle, argued that solid ice had a greater density (weight per volume) than liquid water. The Jesuit cardinal Roberto Bellarmine, who had been on the panel of the Roman Inquisition that sentenced Bruno to death, took Galileo’s side, arguing that since ice floats, it must be less dense than water. In 1612 Galileo made his conclusions about floating bodies public in his Discourse on Bodies in Water.22 In 1613 Galileo antagonized the Jesuits, including Christoph Scheiner, in an argument over a peripheral astronomical issue: Are sunspots associated with the Sun itself—perhaps as clouds immediately above its surface, as Galileo thought, which would provide an example (like lunar mountains) of the imperfections of heavenly bodies? Or are they little planets orbiting the Sun more closely than Mercury? If it could be established that they are clouds, then those who claimed that the Sun goes around the Earth could not also claim that the Earth’s clouds would be left behind if the Earth went around the Sun. In his Sunspot Letters of 1613, Galileo argued that sunspots seemed to narrow as they approach the edge of the Sun’s disk, showing that near the disk’s edge they were being seen at a slant, and hence were being carried around with the Sun’s surface as it rotates. There was also an argument over who had first discovered sunspots. This was only one episode in an increasing conflict with the Jesuits, in which unfairness was not all on one side.23 Most important for the future, in Sunspot Letters Galileo at last came out explicitly for Copernicus. Galileo’s conflict with the Jesuits heated up in 1623 with the publication of The Assayer. This was an attack on the Jesuit mathematician Orazio Grassi for Grassi’s perfectly correct conclusion, in agreement with Tycho, that the lack of diurnal parallax shows that comets are beyond the orbit of the Moon. Galileo instead offered a peculiar theory, that comets are reflections of the sun’s light from linear disturbances of the atmosphere, and do not show diurnal parallax because the disturbances move with the Earth as it rotates. Perhaps the real enemy for Galileo was not Orazio Grassi but Tycho Brahe, who had presented a geocentric theory of the planets that observation could not then refute. In these years it was still possible for the church to tolerate the Copernican system as a purely mathematical device for calculating apparent motions of planets, though not as a theory of the real nature of the planets and their motions. For instance, in 1615 Bellarmine wrote to the Neapolitan monk Paolo Antonio Foscarini with both a reassurance and a warning about Foscarini’s advocacy of the Copernican system:

It seems to me that Your Reverence and Signor Galileo would act prudently by contenting yourselves with speaking hypothetically and not absolutely, as I have always believed Copernicus to have spoken. [Was Bellarmine taken in by Osiander’s preface? Galileo certainly was not.] To say that by assuming the Earth in motion and the Sun immobile saves all the appearances better than the eccentrics and epicycles ever did is to speak well indeed. [Bellarmine apparently did not realize that Copernicus like Ptolemy had employed epicycles, only not so many.] This holds no danger and it suffices for the mathematician. But to want to affirm that the Sun really remains at rest at the world’s center, that it turns only on itself without running from East to West, and that the Earth is situated in the third heaven and turns very swiftly around the Sun, that is a very dangerous thing. Not only may it irritate all the philosophers and scholastic theologians, it may also injure the faith and render Holy Scripture false.24

Sensing the trouble that was gathering over Copernicanism, Galileo in 1615 wrote a celebrated letter about the relation of science and religion to Christina of Lorraine, grand duchess of Tuscany, whose wedding to the late grand duke Ferdinando I Galileo had attended.25 As Copernicus had in De Revolutionibus, Galileo mentioned the rejection of the spherical shape of the Earth by Lactantius as a horrible example of the use of Scripture to contradict the discoveries of science. He also argued against a literal interpretation of the text from the Book of Joshua that Luther had earlier invoked against Copernicus to show the motion of the Sun. Galileo reasoned that the Bible was hardly intended as a text on astronomy, since of the five planets it mentions only Venus, and that just a few times. The most famous line in the letter to Christina reads, “I would say here something that was heard from an ecclesiastic of the most eminent degree: ‘That the intention of the Holy Ghost is to teach us how one goes to heaven, not how heaven goes.’ ” (A marginal note by Galileo indicated that the eminent ecclesiastic was the scholar Cardinal Caesar Baronius, head of the Vatican library.) Galileo also offered an interpretation of the statement in Joshua that the Sun had stood still: it was the rotation of the Sun, revealed to Galileo by the motion of sunspots, that had stopped, and this in turn stopped the orbital motion and rotation of the Earth and other planets, which as described in the Bible extended the day of battle. It is not clear whether Galileo actually believed this nonsense or was merely seeking political cover. Against the advice of friends, Galileo in 1615 went to Rome to argue against the suppression of Copernicanism. Pope Paul V was anxious to avoid controversy and, on the advice of Bellarmine, decided to submit the Copernican theory to a panel of theologians. Their verdict was that the Copernican system is “foolish and absurd in Philosophy, and formally heretical inasmuch as it contradicts the express position of Holy Scripture in many places.”26 In February 1616 Galileo was summoned to the Inquisition and received two confidential orders. A signed document ordered him not to hold or defend Copernicanism. An unsigned document went further, ordering him not to hold, defend, or teach Copernicanism in any way. In March 1616 the Inquisition issued a public formal order, not mentioning Galileo but banning Foscarini’s book, and calling for the writings of Copernicus to be expurgated. De Revolutionibus was put on the Index of books forbidden to Catholics. Instead of returning to Ptolemy or Aristotle, some Catholic astronomers, such as the Jesuit Giovanni Battista Riccioli in his 1651 Almagestum Novum, argued for Tycho’s system, which could not then be refuted by observation. De Revolutionibus remained on the Index until 1835, blighting the teaching of science in some Catholic countries, such as Spain. Galileo hoped for better things after 1624, when Maffeo Barberini became Pope Urban VIII. Barberini was a Florentine and an admirer of Galileo. He welcomed Galileo to Rome and granted him half a dozen audiences. In these conversations Galileo explained his theory of the tides, on which he had been working since before 1616. Galileo’s theory depended crucially on the motion of the Earth. In effect, the idea was that the waters of the oceans slosh back and forth as the Earth rotates while it goes around the Sun, during which movement the net speed of a spot on the Earth’s surface along the direction of the Earth’s motion in its orbit is continually increasing and decreasing. This sets up a periodic ocean wave with a one-day period, and as with any other oscillation, there are overtones, with periods of half a day, a third of a day, and so on. So far, this leaves out any influence of the Moon, but it had been known since antiquity that the higher “spring” tides occur at full and new moon, while the lower “neap” tides are at the times of half-moon. Galileo tried to explain the influence of the Moon by supposing that for some reason the Earth’s orbital speed is increased at new moon, when the Moon is between the Earth and the Sun, and decreased at full moon, when the Moon is on the other side of the Earth from the Sun. This was not Galileo at his best. It’s not so much that his theory was wrong. Without a theory of gravitation there was no way that Galileo could have correctly understood the tides. But Galileo should have known that a speculative theory of tides that had no significant empirical support could not be counted as a verification of the Earth’s motion. The pope said that he would permit publication of this theory of tides if Galileo would treat the motion of the Earth as a mathematical hypothesis, not as something likely to be true. Urban explained that he did not approve of the Inquisition’s public order of 1616, but he was not ready to rescind it. In these conversations Galileo did not mention to the pope the Inquisition’s private orders to him. In 1632 Galileo was ready to publish his theory of the tides, which had grown into a comprehensive defense of Copernicanism. As yet, the church had made no public criticism of Galileo, so when he applied to the local bishop for permission to publish a new book it was granted. This was his Dialogo (Dialogue Concerning the Two Chief Systems of the World—Ptolemaic and Copernican). The title of Galileo’s book is peculiar. There were at the time not two but four chief systems of the world: not just the Ptolemaic and Copernican, but also the Aristotelian, based on homocentric spheres revolving around the Earth, and the Tychonic, with the Sun and Moon going around a stationary Earth but all other planets going around the Sun. Why did Galileo not consider the Aristotelian and Tychonic systems? About the Aristotelian system, one can say that it did not agree with observation, but it had been known to disagree with observation for two thousand years without losing all its adherents. Just look back at the argument made by Fracastoro at the beginning of the sixteenth century, quoted in Chapter 10. Galileo a century later evidently thought such arguments not worth answering, but it is not clear how that came about. On the other hand, the Tychonic system worked too well for it to be justly dismissed. Galileo certainly knew about Tycho’s system. Galileo may have thought his own theory of the tides showed that the Earth does move, but this theory was not supported by any quantitative successes. Or perhaps Galileo just did not want to expose Copernicus to competition with the formidable Tycho. The Dialogo took the form of a conversation among three characters: Salviati, a stand-in for Galileo named for Galileo’s friend the Florentine nobleman Filippo Salviati; Simplicio, an Aristotelian, perhaps named for Simplicius (and perhaps intended to represent a simpleton); and Sagredo, named for Galileo’s Venetian friend the mathematician Giovanni Francesco Sagredo, to judge wisely between them. The first three days of the conversation showed Salviati demolishing Simplicio, with the tides brought in only on the fourth day. This certainly violated the Inquisition’s unsigned order to Galileo, and arguably the less stringent signed order (not to hold or defend Copernicanism) as well. To make matters worse, the Dialogo was in Italian rather than Latin, so that it could be read by any literate Italian, not just by scholars. At this point, Pope Urban was shown the unsigned 1616 order of the Inquisition to Galileo, perhaps by enemies that Galileo had made in the earlier arguments over sunspots and comets. Urban’s anger may have been amplified by a suspicion that he was the model for Simplicio. It didn’t help that some of the pope’s words when he was Cardinal Barberini showed up in the mouth of Simplicio. The Inquisition ordered sales of the Dialogo to be banned, but it was too late—the book was already sold out. Galileo was put on trial in April 1633. The case against him hinged on his violation of the Inquisition’s orders of 1616. Galileo was shown the instruments of torture and tried a plea bargain, admitting that personal vanity had led him to go too far. But he was nevertheless declared under “vehement suspicion of heresy,” condemned to eternal imprisonment, and forced to abjure his view that the Earth moves around the Sun. (An apocryphal story has it that as Galileo left the court, he muttered under his breath, “Eppur si muove,” that is, “But it does move.”) Fortunately Galileo was not treated as roughly as he might have been. He was allowed to begin his imprisonment as a guest of the archbishop of Siena, and then to continue it in his own villa at Arcetri, near Florence, and near the convent residence of his daughters, Sister Maria Celeste and Sister Arcangela.27 As we will see in Chapter 12, Galileo was able during these years to return to his work on the problem of motion, begun a half century earlier at Pisa. Galileo died in 1642 while still under house arrest in Arcetri. It was not until 1835 that books like Galileo’s that advocated the Copernican system were removed from the Index of books banned by the Catholic church, though long before that Copernican astronomy had become widely accepted in most Catholic as well as Protestant countries. Galileo was rehabilitated by the church in the twentieth century.28 In 1979 Pope John Paul II referred to Galileo’s Letter to Christina as having “formulated important norms of an epistemological character, which are indispensable to reconcile Holy Scripture and science.”29 A commission was convened to look into the case of Galileo, and reported that the church in Galileo’s time had been mistaken. The pope responded, “The error of the theologians of the time, when they maintained the centrality of the Earth, was to think that our understanding of the physical world’s structure was, in some way, imposed by the literal sense of the Sacred Scripture.”30 My own view is that this is quite inadequate. The church of course cannot avoid the knowledge, now shared by everyone, that it had been wrong about the motion of the Earth. But suppose the church had been correct and Galileo mistaken about astronomy. The church would still have been wrong to sentence Galileo to imprisonment and to deny his right to publish, just as it had been wrong to burn Giordano Bruno, heretic as he was.31 Fortunately, although I don’t know if this has been explicitly acknowledged by the church, it would not today dream of such actions. With the exception of those Islamic countries that punish blasphemy or apostasy, the world has generally learned the lesson that governments and religious authorities have no business imposing criminal penalties on religious opinions, whether true or false. From the calculations and observations of Copernicus, Tycho Brahe, Kepler, and Galileo there had emerged a correct description of the solar system, encoded in Kepler’s three laws. An explanation of why the planets obey these laws had to wait a generation, until the advent of Newton.