温伯格《解释世界》第八章 行星的运行

天空中中不只太阳和月亮每天环绕天北极与其他星星一起东升西落的同时也自西向东在黄道上穿行。古代几个文明社会都注意到有五个“星星”与太阳和月亮几乎在同样的轨道上在背景恒星中自西向东穿行。古希腊人称之为游星，或行星，并用神的名字将它们命名为：赫尔密斯，阿富罗底，阿瑞斯，宙斯和克洛诺斯，罗马人译为水星，金星，火星，木星和土星。另外他们采纳了巴比伦的方法，将太阳和月亮也作为行星（注：为了避免混淆，在本章中当我提到行星时，我只是指水星，金星，火星，木星和土星），这样一共有七个行星，每周七天就是源自于此。（注：在英文中可以看出一周每天与行星和神的对应。星期六（Saturday），星期日(Sunday)和星期一(Monday)很明显对应土星(Saturn)，太阳(Sun)和月亮（Moon）。星期二(Tuesday)，星期三(Wednesday)，星期四(Thursday)和星期五(Friday)是基于相应的与拉丁神相同的日耳曼神：蒂尼（Tyr）为火星，沃登（Wotan）为水星，托尔（Thor）为木星，弗丽嘉（Frigga）为金星）。 行星在天穹穿行速度不同。水星和金星用一年时间完成环绕黄道十二宫一周的穿行；火星用一年322天；木星用11年315天，土星用29年166天。这些是行星平均穿行周期，因为这些行星在黄道上并不是以匀速穿行，有时它们甚至会逆行一段时间，然后又回到向东的穿行方向。现代科学的诞生史很大程度上涉及如何解释行星的这种独特运行，期间跨越2000多年。 毕达哥拉斯学派早期曾提出了行星，太阳和月亮运行理论。他们想像五颗行星，太阳，月亮，以及地球都在环绕中心之火运行。为了解释我们在地球上为什么看不到中心之火，毕达哥拉斯学派假设我们生活在远离中心之火一面（像所有前苏格拉底学者一样，毕达哥拉斯学派也认为地球为扁平形状。他们认为地球像一个圆盘，其中一面总是面对中心之火，我们生活在另一面。他们用地球围绕中心之火的每日运行来解释我们看到的太阳，月亮，行星和恒星每日围绕地球的视运行）。据亚里士多德和埃提乌斯记载，公元前50世纪毕达哥拉斯学派的菲洛劳斯引入了一个反地球，该反地球位于地球和中心之火之间，或中心之火的另一面，我们看不到它。亚里士多德解释说反地球的引入是由于毕达哥拉斯学派对数字的迷恋。地球，太阳，月亮，五颗行星加恒星圈构成围绕中心之火的九个物体，但毕达哥拉斯学派认为应该有10个才对，因为10是完美的数字，10 = 1 + 2 + 3 + 4。亚里士多德略带嘲笑的描述到：毕达哥拉斯学派认为数字是一切的根本，整个天穹就是一个音阶和数字。如果他们可以证明一些数字和音阶的属性与天穹部分和整体的属性相匹配，他们就会收集起来融入他们的理论，如果其中有任何罅隙，他们会为之补缀，以便自圆其说。比如他们认为数字10是完美的数，天空中运行的物体应该有十个，但是只能看到9个运行的物体，为了满足他们的理论，他们引入了第十个—反地球。 显然毕达哥拉斯学派从来没有尝试证明他们的理论如何对太阳，月亮和行星相对背景恒星的视运行做出合理解释。人类解释这些视运行的努力持续了多个世纪，一直到开普勒时代才基本完成。 人们发明了各种设备来辅助研究，比如用圭表来研究太阳的运行，用其他一些仪器来测量恒星和行星的视角，或这些天体与地平线的夹角。当然这些观察都是肉眼天文学。令人不解的是托勒密虽然深入研究了折射和反射现象（包括恒星视位置受大气折射影响），但他从来没有意识到可以用透镜和曲面镜来放大天体，就像伽利略的折射望远镜或牛顿的反射望远镜一样。我们后面还会介绍托勒密在天文学史上起的关键作用。 希腊在天文学方面的伟大进展不只是依仗各种仪器，他们在数学领域的成就也起了重要作用。古代和中世纪天文学领域主要争议不是地心说和日心说之争，而是两种关于太阳，月亮和行星围绕静止地球运行的不同构想之争。后面我们会看到，这种争议与人们对数学在自然科学中作用的不同观点有关。 这个故事始于我称之为柏拉图的家庭作业。据新柏拉图派学者辛普里丘公元530年对亚里士多德《论天》评注中记载： 柏拉图建立天体匀速规则圆周运动原理。基于此他给数学家提出如下问题：采用什么样的匀速规则圆周运动假说可以保存行星的视运行。 “保存视运行”是传统译法。柏拉图的问题是如何组合行星（这里包括太阳和月亮）的匀速同向圆周运动可以与我们的实际观察结果一致。 与柏拉图同代的数学家尼多斯的欧多克索斯首先对此问题进行了研究。他建立了一个数学模型，在他的著作《论速率》中他详细描述了该模型。但《论速率》一书已失传，我们现在对该模型的了解来自于亚里士多德和辛普里丘的记载。欧多克索斯的模型是这样的：恒星由一个天球携带每日由东向西运行，太阳，月亮和行星则由不同天球携带，这些天球又由其他天球携带。最简单的模型是太阳有两个天球，外天球每天环绕地球由东向西运行一周，其轴及运行速度与恒星天球一致。太阳位于内天球赤道，内天球就像连接在外天球一样与外天球同步运行，同时内天球每年也围绕自身的轴由西向东运行一周。内天球轴相对外天球轴倾斜231/2度，这样即可以解释太阳每天的视运行，也可以解释太阳每年在黄道带的视运行。同样月亮可以认为由另两个相对运行的天球携带，不同之处在于携带月亮运行的内天球每月从西向东运行一周，而不是一年。据说欧多克索斯给太阳和月亮还分别加了第三个天球，原因不详。该理论被称为“同心圆”理论，因为携带行星，太阳和月亮的天球都有同一中心，即地球的中心。 行星的不规则运行给人们带来更大难度。欧多克索斯给每颗行星设四个天球。最外天球围绕地球每日由东向西运行一周，其旋转轴与恒星天球，太阳和月亮天球的外天球旋转轴一致。下一个天球类似于太阳和月亮的内天球，它们以不同速度缓慢环绕相对外天球轴倾斜231/2度的轴由西向东运行。两个内天球以同速反向围绕两个近平行轴旋转，这两个轴与两个外天球轴成很大交角。行星链接在最内天球。两个外天球带来行星与恒星相同的每日绕地运行以及在黄道带的长期均运行。两个内天球的轴如果完全平行，它们相向运行的效果将彼此抵消，但是因为它们的轴不完全平行，它们给每个行星在黄道带内的运行叠加出一个8字形，这用来解释行星的逆行。希腊人把这种轨迹称为马镣，因为它像拴马的缰绳。 欧多克索斯的模型与对太阳，月亮和行星的观察并不完全相符。比如他的模型无法解释季节的长短，我们在第六章介绍过，攸克特蒙利用圭表发现季节长短不同。另外该模型不能解释水星的运行，对金星和火星的运行解释也不理想。为了改进欧多克索斯的模型，塞西卡斯的卡利帕斯提出了一个新模型。他给太阳和月亮多加了两个天球，给水星，金星和火星各多家了一个天球，虽然卡利帕斯的模型给行星的视运行引入了一些虚构特征，但总体上比欧多克索斯的模型要好一些。 欧多克索斯和卡利帕斯的同心圆模型都给太阳，月亮和行星分别设置了一套天球，它们的外天球都与携带恒星的天球同步运行。这是现代物理学家称为“硬凑”的早期实例。如果不理解根本原因，只是调整一些参数来得出相符的结果，我们批评这种理论为硬凑理论。科学理论出现硬凑就像自然界在发出痛苦的哭泣，它在期待着一个更加合理的理论。 现代物理学家对硬凑的不满引发了基础物理领域一个重大的发现。二十世纪五十年代后期人们发现两种不稳定的粒子𝞃和𝝝以不同方式衰变—𝝝衰变为两个更轻的粒子，称为𝞹介子，而𝞃衰变为三个𝞹介子。𝞃和𝝝粒子不只有相同的质量，他们也有相同的平均寿命，虽然它们的衰变模式完全不同。物理学家那时认为𝞃和𝝝不可能是同一粒子，因为出于繁杂的原由自然界的左右对称（指自然定律保持镜像不变）不容许同一粒子有时衰变为两个𝞹介子，有时衰变为三个𝞹介子。以那时我们所掌握的知识，我们完全可以调整我们的理论常数来使得𝞃和𝝝粒子有相同的质量和寿命，但是人们很难接受这样的一个理论—这种硬凑没有出路。后来人们发现根本不需要硬凑，因为这两种粒子其实就是同一粒子。虽然将原子以及原子核聚集在一起的力遵循左右对称原理，但是各种衰变过程并不遵循此原理，包括𝞃和𝝝的衰变。发现这一现象的物理学家不相信𝞃和𝝝质量和寿命的相同只是巧合—那需要对理论进行过度的硬凑。 今天我们面临着一个硬凑问题更为令人苦恼。1998年天文学家发现宇宙的膨胀不但没有由于银河系间的相互引力作用而减慢，反而在加速膨胀。这种加速与空间自身能量有关，人们称之为暗能量。理论表明暗能量包括几部分。一些部分我们可以计算，但其他部分我们不能计算。我们可以计算的暗能量部分比天文学家观测结果大出56个数量级，即1后面56个0。我们可以假设暗能量可计算部分与不可计算部分几乎抵消，那样一来我们的计算结果并不离谱，但需要正好56个数量级才能去抵消。这种尺度的硬凑无法令人容忍。理论物理学家一直在辛苦的努力工作，寻求更好的方法来解释暗能量比我们计算结果小如何之多的根本原因。在第十一章我们会介绍其中一种可能的解释。 同时我们也必须承认有些看起来像是硬凑的例子其实只是巧合。比如地日与地月距离之比与日月直径之比刚好相等，这样从地球上观察日月大小一致，日全食时月亮正好遮盖全部太阳。这只是巧合，没有任何理由。 亚里士多德改进了欧多克索斯和卡利帕斯模型。在《形而上学》中他提出将所有天球连接起来成为一个单一衔接系统。与欧多克索斯和卡利帕斯给最外行星--土星设四个天球不同，他只给土星设三个内天球。土星由东向西的日视运行可由这三个天球与恒星天球的连接来解释。亚里士多德在土星三个天球之内又加了另外三个反向运行的天球，用来抵消土星三个天球的运行对相邻行星--木星的影响，木星的外天球与木星和土星之间三个额外天球的最内一个天球相连接。 通过额外增加三个反向运行天球，并且将土星外天球与恒星天球衔接，亚里士多德取得很好成效。人们不用再去问为什么土星的日视运行与恒星一致，因为土星与恒星天球是连接在一起的。但是接着亚里士多德给木星设了与欧多克索斯和卡利帕斯同样的四个天球，这完全破坏了整个模型。这样一来木星将每天绕土星和它自身最外天球一周，结果木星会每天绕地球两周。他是否忘记了土星天球内的三个反向运行天球只会抵消土星的独特运行，而不会抵消每日环绕地球的运行。 更为糟糕的是亚里士多德在木星四个天球之内加了三个反向运行天球来抵消木星的独特运行，但不会抵消其日视运行，然后他给下一个行星—火星设了与卡利帕斯模型一样的5个天球，结果导致火星每天环绕地球运行三周。同样亚里士多德的模型中金星，水星，太阳和月亮分别环绕地球运行4周，5周，6周和7周。 在我阅读亚里士多德《形而上学》著作看到这些明显差错时我倍感吃惊，后来我知道有几位作家，包括德雷尔，托马斯·希思和罗斯等，早已注意到这点。有些人将之归咎于文本差错。但如果亚里士多德确实像在标准《形而上学》版本介绍的那样呈现他的模型，那这不能理解为他的思考模式与我们不同，或他与我们关注点不同。我们可以断定在应用他的思考模式，研究他所关注问题方面他太粗心或愚蠢。 即使亚里士多德设了正确的反向运行天球数目可以使每个行星与恒星同步日环绕地球运行一周，他的模型也存在过度硬凑的问题。为了抵消土星独特运行对木星运行的影响，土星内的三个反向运行天球必须与土星的三个天球运行速度完全一致，只有这样才能抵消。距地球近的行星也是这样。另外与欧多克索斯和卡利帕斯模型一样，亚里士多德的模型中水星和金星第二个天球必须与太阳的第二个天球运行速度完全一致，这样才能解释水星，金星和太阳沿黄道的同步运行。我们看到的内行星在天空中永远不会离太阳很远，比如金星永远是晨星或晚星，永远不可能在午夜高空中看到金星。 古代天文学家中至少有一位以非常严肃的态度对待硬凑问题，他就是蓬杜斯的赫拉克利德斯--公元前四世纪柏拉图学院学生，后来柏拉图去西西里后可能由他负责该学院。辛普里丘和埃提乌斯都记载了赫拉克利德斯讲授地球绕自转轴自转设想，这一下就解决了恒星，行星，太阳和月亮围绕地球运行问题。（注：每年3651/4天，地球实际自转了3661/4次。太阳看起来一年围绕地球运行3651/4次，因为同时地球自转了3661/4次，地球以同向围绕太阳运行了一圈，看上去就像太阳围绕地球运行了3651/4次。由于地球相对恒星用365.25天（每天24小时）自转了366.25次，那地球自转一次的时间为（365.25×24小时）/366.25，即23小时，56分钟，4秒。这是恒星日。）赫拉克利德斯的想法在晚古和中世纪时期作品中不时被提及。没有任何资料证明阿里斯塔克斯在晚于赫拉克利德斯一个世纪后的作品中提及到地球不只围绕太阳公转，也围绕自转轴自转。 据公元四世纪将《蒂迈欧篇》从希腊文译为拉丁文的基督徒卡尔齐地乌斯记载，赫拉克利德斯也提出水星和金星在天空中的位置永远不会远离太阳，它们应该是围绕太阳运行，而不是地球，这样一来又可以去掉欧多克索斯，卡利帕斯和亚里士多德模型中的部分硬凑，太阳和内行星的第二个假想天球不再需要。但是太阳，月亮和三个外行星仍然围绕一个位置固定但自转的地球运行。该理论对内行星效果很好，因为它对内行星视运行的描述与哥白尼最简单的模型完全一致，哥白尼的模型中水星，金星和地球围绕太阳做匀速圆周运动。对内行星而言，赫拉克利德斯和哥白尼的唯一区别只是视角不同—一个是基于地球，一个是基于太阳。 欧多克索斯，卡利帕斯和亚里士多德的模型除了硬凑问题，还存在另外一个问题：这些同心圆模型与观察并不完全符合。那时人们认为行星自身发光，因为按照同心圆模型，由天球携带的行星与地球距离不变，行星的亮度不应该发生变化。然而很明显行星亮度变化很大。正如辛普里丘引述，大约公元200年左右逍遥派哲学家索西吉斯曾评述： 然而欧多克索斯的假想并没有与观察相符，只是符合人们早已知道的一些现象以及他们自己所接受的部分。这里需要的是去解释其他一些现象，塞西卡斯的卡利帕斯尝试去改进欧多克索斯不能够做到的部分，卡利帕斯是否成功？。。。。。。我指的是行星时常会看起来很近，有时又会看起来远离我们。有些行星看起来很明显。那个叫做金星的星星，以及那个叫做火星的星星，当它们逆行一半时看起来大出许多倍，无月之夜金星甚至会使物体投下阴影。 我们应该将辛普里丘和索西吉斯提到的行星大小理解为视亮度；肉眼无法看出任何行星的大小，但是光线越亮，看起来越大。 事实上索西吉斯的论据并不充分。行星反射太阳光（与月亮一样），即使按照欧多克索斯的模型，行星相的变化也会引起亮度的变化（类似月相变化）。但到伽利略时代人们才知道这点。不过即使把行星相变考虑进去，同心圆理论亮度变化与实际观察也不相符。 从职业天文学家（或许包括哲学家）角度来说，欧多克索斯，卡利帕斯和亚里士多德的同心圆理论在希腊化以及罗马时代被另一个新理论完全取代，该理论对太阳和行星视运行做出的解释要好得多。此理论基于三个数学概念—本轮，偏心圆，以及匀速点，后面将详细介绍。我们不知道是谁发明了本轮和偏心圆，但是我们确信希腊化时代数学家佩尔格的阿波罗尼奥斯和天文学家尼西亚的喜帕恰斯对此早有所知，在第六章和第七章我们曾介绍过这两位学者。我们对本轮和偏心圆理论的了解来自于克罗狄斯·托勒密的介绍，他自己又引入了匀速点，后来该理论就一直与托勒密联系在一起。 托勒密活跃于公元150年左右，那时正处于罗马帝国鼎盛时期的安敦宁王朝时代。他在亚历山大博物馆工作，去世于公元161年之后。在第四章我们介绍过他在折射和反射方面的研究。他在天文学领域的成就记载于《至大论》一书，阿拉伯人译为《天文学大成》--即后来欧洲人所知悉之名。《天文学大成》获得极大成功，后来的书吏甚至不再抄写像喜帕恰斯这样早期天文学家的成果，导致我们现在难以区分托勒密与早期天文学家的成就。 《天文学大成》改进了喜帕恰斯星表，书中一共记录了1028颗星，比喜帕恰斯星表多出几百颗星。书中不但列出星星在天空中的方位，也标出它们的亮度。（注：从托勒密时代一直到现代以来人们对星星视亮度的分类都用“星等”来表示，星星越暗，星等越高。天空中最亮的恒星—天狼星星等为-1.4，亮星织女星星等为0，肉眼刚刚可以看到的恒星星等为6。天文学家诺曼·波格森1856年对比了许多恒星实测视亮度与它们历史记录星等，基于此他定义如果恒星星等相差5个单位，亮度相差100倍。）托勒密关于行星，太阳和月亮的理论对后世科学更为重要。《天文学大成》记载的理论方法极富现代性。其中对行星的运动提出包含多个可调参数的数学模型，然后调整这些参数使数学模型预测结果与实际观察结果相符。后面我们会介绍一个这方面例子，也与偏心圆和匀速点有关。 托勒密理论最简单的模型是这样的，行星在被称为“本轮”的圆周上运行，行星不是围绕地球运行，而是围绕一个运动的点运行，该点围绕地球在另一个被称为“均轮”的圆周上运行。内行星水星和金星分别以每88天和225天在本轮上运行一周。模型要求本轮的中心在均轮上围绕地球运行一周正好一年，而且该中心一直位于地球和太阳的连线上。 我们很容易理解该模型为何有效。从行星的视运行中看不出行星离我们有多远，这样在托勒密理论中行星在天空中的视运行不取决于本轮和均轮的绝对大小，而是取决于它们大小之比。如果托勒密愿意，他可以扩大金星本轮和均轮的大小，但保持其比值不变，对水星也可做同样处理。这样可以让两个行星有相同的均轮，即太阳运行轨道，太阳就成为均轮上的点，内行星围绕太阳在自己的本轮上运行。喜帕恰斯和托勒密的模型不是这样，但是其得出的内行星视运行结果完全一样。因为这里只是运行轨道整体规模的不同，不影响视运行结果。这个本轮理论特例与前面介绍过的赫拉克利德斯理论一样，他的理论认为水星和金星围绕太阳运行，而太阳围绕地球运行。我们说过，赫拉克利德斯的理论之所以有效是因为其等同于地球和内行星围绕太阳运行，两个理论只是观察点不同。所以托勒密的本轮理论可以得出与赫拉克利德斯理论同样的水星和金星视运行结果，而且与实测非常相符，不是巧合。 托勒密完全可以把同样的本轮和均轮理论应用到外行星—火星，木星和土星，但是要使理论成立，行星在本轮上的运行需要比本轮中心在均轮上的运行慢得多。我不知道这有什么问题，但是不知出于什么原因，托勒密选择了另外一个途径。在他的最简单模型中，每颗外行星围绕均轮上的一个点在本轮上每年运行一周，均轮上的点以更长时间围绕地球运行：火星1.88年一周，木星11.9年一周，土星29.5年一周。这里还有另外的要求—本轮中心与行星连线一直保持与地球和太阳连线平行。该模型与实际观察外行星的视运行非常相符，主要原因与内行星一样，改变本轮和均轮大小，保持其比值不变，得到的行星视运行都一样。其中一个本轮和均轮特例大小值可以与最简单的哥白尼理论一致，区别只在于观察点不同：基于地球或太阳。对于外行星，这个特例是本轮的半径等于地日距离。（参见技术说明13）。 托勒密理论很好地解释了行星的逆行。比如火星在本轮上运行到最接近地球时看起来开始在黄道带上朝后运行，因为这时它在本轮上的运行方向与本轮在均轮上的运行方向相反，而且更快。其实这只是参考系的转换，按现代理论地球和火星都围绕太阳运行，当地球超越火星时看起来火星像在黄道带上朝后运行。这时候火星看起来最亮（如上面辛普里丘所引述），因为它离地球最近，我们看到火星的一面朝向太阳。 不要以为喜帕恰斯，阿波罗尼奥斯和托勒密的理论只是空想，运气好碰巧与观察相符，与现实无关。该模型最简单形式—即每颗行星只设一个本轮，不加其他复杂设计—得出的太阳和行星视运行结果与哥白尼理论最简单形式—即地球和其他行星以太阳为中心做匀速圆周运动—得出的预测结果完全一致。与前面对水星和金星运行所做出的解释一样（详细解释参见技术说明13），这是由于托勒密理论是一大类理论中的一个特例，这类理论都可以得出同样的太阳和行星视运行结果，其中之一（不是托勒密采用的那个）与哥白尼理论最简单形式得出的太阳和行星的相对运行结果完全一致。 希腊天文学故事本可完结与此。遗憾的是哥白尼理论最简单形式与托勒密理论最简单形式一样，其预测结果与实际观察并不完全相符，这点哥白尼自己也很清楚。从开普勒和牛顿时代以来，我们开始知道地球和其他行星运行轨道不是正圆，太阳并不位于运行轨道的中心，地球和其他行星也不是以匀速运行，希腊天文学家对这些一无所知。开普勒之前的天文史一大部分是修正托勒密和哥白尼最简单模型中的误差。 柏拉图要求匀速圆周运动，就我们所知古代没有人设想过天体非圆周运动，对于匀速，托勒密倒是愿意放弃。限于圆周运动，托勒密和他的先驱们采用了多种复杂的方法来使他们的理论与对太阳，月亮和行星的观察相符。（注：关于本轮的起源，托勒密在《天文学大成》第12卷开头暗示佩尔格的阿波罗尼奥斯应用本轮和偏心圆来解释太阳的视运行）。 其中一个复杂方法是再加更多的本轮。托勒密发现唯一一个需要加更多本轮的行星是水星，其运行轨道与其他行星相比更偏离正圆。另一个复杂方法是引入“偏心圆”—地球不是位于每颗行星均轮的中心，而是偏离一些。比如托勒密理论中金星均轮中心与地球之间偏差均轮半径的百分之二。（注：对于太阳的运行，可以把偏心圆当作一种本轮，本轮中心与太阳的连线一直与地球和太阳均轮中心连线平行，这样将太阳轨道中心偏离地球。月亮和行星也一样） 偏心圆可以与托勒密引入的另一个数学概念—“匀速点”结合起来。这是为了让行星除了由于其本轮造成的的速度改变，在自己轨道上也以变速度运行。人们可能以为从地球看上去每颗行星，或准确地说每颗行星本轮中心以匀速（比如每天多少度）围绕我们运行，但托勒密知道这与实际观察不符。引入偏心圆后人们又会以为行星本轮中心以匀速围绕行星均轮中心运行，而不是地球，这也不行。为此托勒密给每颗行星引入了后来被称为匀速点的概念， 该点与地球位于均轮中心两侧，距离均轮中心距离相等。（注：托勒密并没有使用“匀速点”一词。他采用的是“二等分偏心圆”，指均轮的中心位于匀速点与地球连线的中心。）行星本轮中心以匀角速围绕匀速点运行。设地球和匀速点与均轮中心等距并不是基于什么哲学的先入之见，而是多次尝试的结果，这一设定可以使理论预测与观察相符。 托勒密模型与观察仍然有不小偏差。正如我们在第11章介绍开普勒时会看到的，如果对每颗行星采用一个本轮，对太阳和每颗行星采用一个偏心圆和一个匀速点，只要保持一致的话，完全可以极好地模拟行星的运行，包括地球的椭圆形运行轨迹—其误差极小，不用天文望远镜的话完全观察不到误差。可是托勒密没有保持这种一致性。他没有采用匀速点来描述太阳围绕地球的视运行，这一疏忽也干扰了对行星运行的预测（因为行星的位置是相对于太阳的位置）。乔治·史密斯后来强调到，这体现出古代和中世纪天文学与现代科学的差距，托勒密后没有人认真分析托勒密模型预测的偏差进而提出更好的理论。 月球运行存在特别之处，那些可以很好地描述太阳和行星运行的理论对月球的描述并不非常适用。其原因到牛顿时代才完全被人所知。这是因为月球运行受到两个物体—太阳和地球重力的巨大影响，而行星运行几乎完全只受到一个物体—太阳重力的控制。喜帕恰斯曾经提出一个月球运行理论，他只用一个本轮，通过调整这个本轮来解释每次日月食的间隔。但是托勒密早已意识到，该模型对日月食间月球在黄道带上位置的预测并不理想。托勒密采用了更加复杂的模型修正了这一不足，但是他的模型自身存在问题：他的模型地月距离变化很大，这样造成月球视大小变化比实际观测变化大得多。 前面讲过，在托勒密和他的前辈理论体系中从对行星的观测中只能得知它们各自均轮和本轮大小比值，无法得知均轮和本轮的绝对大小。（注：即使加上偏心圆和匀速点也一样。通过观测只能对每个行星确定地球以及匀速点到均轮中心距离与均轮和本轮半径的比值大小。）托勒密在《天文学大成》之后写的《行星假说》一书中弥补了这一不足。在该著作中他援引了可能是来自于亚里士多德的一个先验原理—即宇宙体系不存在间隙。每颗行星，以及太阳和月亮都各占有一部分空间，每部分空间都在行星，太阳和月亮各自距地球最近到最远范围之间。所有这些空间衔接在一起，中间没有间隙。采用这种模式，一旦知道了行星，太阳和月亮距地球的远近次序，就可以确定行星，太阳和月亮的运行轨道相对大小。月球据地球很近，地月距离（以地球半径为单位）可以采用多种方法来估算，包括在第七章解释的喜帕恰斯方法。托勒密自己采用了视差法，通过天顶与月亮间的观察角以及计算假想从地心观察此角的角度大小，托勒密计算出地月距离与地球半径之比。（见技术说明14）这样按照托勒密的设想，只需要知道太阳和行星轨道环绕地球的次序就可以得出太阳和行星距地球的距离。 自古以来人们都一直认为月亮距地球最近，因为太阳和行星都不时会被月亮遮挡。另外人们自然认为距离地球最远的行星围绕地球运行速度一定最慢，这样火星，木星和土星距离地球一定应该由近到远。但是太阳，金星和火星看上去都平均用一年时间环绕地球运行一周，它们距离地球的远近次序颇有争议。托勒密猜想距离地球由近到远的次序是：月亮，水星，金星，太阳，然后是火星，木星和土星。托勒密计算的太阳，月亮和行星与地球距离与地球半径比值比实际值要小得多，他计算的太阳和月亮结果与第七章介绍的阿里斯塔克斯结果相近（这可能不是出于巧合）。 托勒密天文学因其本轮，匀速点，偏心圆的繁杂而备受诟病。但是不要以为托勒密只是为了弥补地心学说的错误而刻意为之。这种复杂设计（除了给每颗行星设一个本轮，太阳没有本轮）与地球围绕太阳运行还是太阳围绕地球运行没有关系。之所以需要这些复杂设计是由于行星实际运行轨道不是正圆，太阳并不是位于行星运行轨道中心，行星运行速度也不是匀速。这些到开普勒时代人们才真正了解。哥白尼最初始的理论其实也一样复杂，因为他也一样认为行星和地球在做匀速圆周运动。好在这个近似偏差不大，本轮理论最简单模型--即每个行星设一个本轮，太阳没有本轮—比欧多克索斯，卡利帕斯和亚里士多德的同心圆模型要好得多。如果托勒密给太阳和每颗行星都叫上匀速点和偏心圆，那其理论与观察结果将非常接近，其误差用那时的技术几乎无法发现。 但这并没有解决托勒密和亚里士多德间关于行星运行理论的不同（？）。托勒密理论虽然与实测结果更相符，但是确实违背了亚里士多德物理的假设前提—即所有天体的运行轨迹都是正圆，正圆中心是地球的中心。即使对那些没有先入为主的人来说，行星在本轮上这种怪异的运行方式也难以让人接受。 亚里士多德的支持者（常为物理学家或哲学家）与托勒密的支持者（一般为天文学家或数学家）间的争论持续了将近一千五百年。亚里士多德学派承认托勒密模型与实际观察结果更加符合，但是他们认为这只对数学家有意义，与认识世界没有关系。这种态度在罗兹的吉米纽斯的论述中表现的淋漓尽致。吉米纽斯活跃于公元前70年，阿弗罗狄西亚的亚历山大三个世纪之后引用了他的论述，接着辛普里丘在对亚里士多德《物理学》评述中又做了引用。该论述详细阐明了自然学家（有时翻译为“物理学家”）和天文学家间的争论： 探究自然在于认识天空和天体的特质，它们的力量以及它们出现和消失的本质。通过宙斯可以揭示出它们大小，形状以及位置的真相。天文学并不去回答这些问题，而是致力于揭示天体的有序性，展示宇宙真实有序，天文学也讨论地球，太阳和月亮的形状，大小和相对距离，日月食，天体之合，以及它们运行轨迹的特征。因为天文学涉及研究天体的尺度，形状，自然就需要用到算术和几何。天文学致力于研究这些问题，并应用算术和几何来得出其结果。天文学家和自然学家时常得出相同的认知，比如太阳体积巨大，地球为球形，但是他们采用不同的研究方法。自然学家从天体特性中证实自己的观点，包括从天体的力量，从它们自身益处，或从它们的出现和改变。而天文学家则从它们的形状，大小，运行特征以及运行时间角度来讨论。。。。。。总之天文学家并不关心本质上什么是静止，什么是运动；他宁可假设什么在静止，什么在运动，然后研究天体运行服从于那个假设。他需要从自然学家那里得到第一基本原理，即天体进行简单，规则，有序运行；从这个原理出发，他指出所有天体都做圆形运行，包括那些彼此平行运行以及延斜圆运行。 吉米纽斯提到的“自然学家”有些类似于今天的理论物理学家，但差别很大。吉米纽斯延续了亚里士多德的思想，认为自然学家依据第一原理，包括目的性原理：自然学家认为天体存在“有益”。吉米纽斯认为只有天文学家应用数学，作为观测的辅助。吉米纽斯想象不到理论与观察间发展出的相互作用。现代理论物理学家确实从基本原理出发进行推理，但是他们在工作中应用数学，他们的原理也是源于观察，并以数学形式表述，根本不会考虑什么是“有益”。 从吉米纽斯提到的行星“彼此平行运行以及延斜圆运行”，人们自然会想到欧多克索斯，卡利帕斯和亚里士多德提出的同心圆沿倾斜轴运行的模型。作为亚里士多德学派中杰出一员，吉米纽斯当然会忠诚于亚里士多德理论。另一方面，公元100年左右创作《蒂迈欧篇》评述的阿弗罗狄西亚的阿德拉斯陀斯，以及后一代数学家土麦那的西昂信服阿波罗尼奥斯和喜帕恰斯的理论，他们致力于通过将本轮和均轮解释为透明圆形实体（类型亚里士多德的同心圆，但不是同心圆）来推广这一理论。 有些学者面对两种对立行星运行理论间的冲突无所适从，他们干脆宣称人类根本无法认识天体现象。公元五世纪中叶，新柏拉图派教徒普罗克鲁斯在评述《蒂迈欧篇》时宣称： 我们在探讨月球轨道下面的地面物体时，我们容易达成一致，因为组成它们的材料不稳定，多数情况下我们都可以掌握。但当我们开始认识天体时，我们应用感性，采用各种完全不同的手段。。。。。。从对天体的一些发现中清晰体现出这点—--对同一天体从不同假设出发得出相同的结论。有些采用本轮假设，有些采用偏心圆假设，还有采用没有行星的反向运转球体，这些都可以保存视运行。无疑上帝的判断是确定的。但是对于我们人类，我们也只能满足于“接近”事实，只因我们是人类，只依据类似来发表看法，更像寓言故事。 这里普罗克鲁斯有三点错误。他没有抓住问题的关键，托勒密的本轮和均轮理论比亚里士多德同心圆“反向运行”理论在“保存视运行”方面要好的多。他也有一小点技术上的差错：在提到“采用本轮假设，采用偏心圆假设来保存视运行”时，普罗克鲁斯似乎没有意识到本轮其实可以当作偏心圆（见第92 页注释），它们不是不同的假设，只是用不同方法表达数学上的同一假设。更离谱的是普罗克鲁斯认为认识天体比认识月球轨道下面的地面物体要难得多。正好相反，我们可以精确计算太阳系星体的运行，但是我们仍然不能预测地震和台风。不只普罗克鲁斯一个人有这种想法，后面我们会看到几个世纪后摩西·迈蒙尼德又重复了这种对认识行星运行规律的悲观情绪。 早期为物理学家，后来转变为哲学家的皮埃尔·迪昂在其创作于二十世纪头十年的作品中强力支持托勒密学派，因为他们的模型与观测数据更相符，但是他反对西昂和阿德拉斯托斯试图将该模型与真实现实相联系的想法。也许由于迪昂是位虔诚的宗教徒，他寻求将科学作用限于构建与观察相符的数学理论，而不是致力于做出解释。我不赞同这种观点，我这一代物理学家显然认为我们在做出解释，我们也常常用解释这一词，我们不只是在进行简单描述。确实描述和解释之间无法明确界定。我要说的是我们通过展示自然规律如何遵循更基本的自然规律来对世界做出解释。什么是我们说的基本规律哪？在我们表述牛顿重力和运动定律比开普勒的三大行星运行定律更为基本时，我想我们知道我们在说什么。牛顿成功之处在于解释行星的运行，而不只是简单描述。牛顿并没有解释重力，他也知道他没有做出解释，这就是解释的本质—总会有一些留下来等待后世做出解释。 行星的这种奇异运行让我们无法用行星计时，计日或定方向。希腊化时期以来人们给行星赋予了一种不同的应用—用于星相，一种源于巴比伦的伪科学。（注：星相源于巴比伦一说在贺拉斯作品中有所体现，“不要问上帝给你我设了什么结局（我们不容许知道），里奥克农，不要干预巴比伦的占星术。无论那是什么，又有什么更好的忍受办法。”）现代天文学和星相学间的明确界限在古代和中世纪并不清晰，那时人们还不知道人类情感与支配恒星和行星运行的定律间没有关联。托勒密时代以来各个政府都支持对天文学的研究，主要是希望能够揭示未来，这样一来天文学家自然会在星相学方面花费大量时间。事实上托勒密不只创作了最伟大的天文学宏著《天文学大成》，他也创造了星相学教科书《星象四书》。 我不能就这样乏味地结束希腊天文学。我下面引用托勒密从天文研究中得到的乐趣来给本书第二部分一个愉快的结尾： 我知道我是一个凡人；但当我探寻天上群星时，我的双脚不再立足于大地，此刻我与宙斯同行，分享众神的仙肴。

The Sun and Moon are not alone in moving from west to east through the zodiac while they share the quicker daily revolution of the stars from east to west around the north celestial pole. In several ancient civilizations it was noticed that over many days five “stars” travel from west to east through the fixed stars along pretty much the same path as the Sun and Moon. The Greeks called them wandering stars, or planets, and gave them the names of gods: Hermes, Aphrodite, Ares, Zeus, and Cronos, translated by the Romans into Mercury, Venus, Mars, Jupiter, and Saturn. Following the lead of the Babylonians, they also included the Sun and Moon as planets,* making seven in all, and on this based the week of seven days.* The planets move through the sky at different speeds: Mercury and Venus take 1 year to complete one circuit of the zodiac; Mars takes 1 year and 322 days; Jupiter 11 years and 315 days; and Saturn 29 years and 166 days. All these are average periods, because the planets do not move at constant speed through the zodiac—they even occasionally reverse the direction of their motion for a while, before resuming their eastward motion. Much of the story of the emergence of modern science deals with the effort, extending over two millennia, to explain the peculiar motions of the planets. An early attempt at a theory of the planets and Sun and Moon was made by the Pythagoreans. They imagined that the five planets, together with the Sun and Moon and also the Earth, all revolve around a central fire. To explain why we on Earth do not see the central fire, the Pythagoreans supposed that we live on the side of the Earth that faces outward, away from the fire. (Like almost all the pre-Socratics, the Pythagoreans believed the Earth to be flat; they thought of it as a disk always presenting the same side to the central fire, with us on the other side. The daily motion of the Earth around the central fire was supposed to explain the apparent daily motion of the more slowly moving Sun, Moon, planets, and stars around the Earth.)1 According to Aristotle and Aëtius, the Pythagorean Philolaus of the fifth century BC invented a counter-Earth, orbiting where on our side of the Earth we can’t see it, either between the Earth and the central fire or on the other side of the central fire from the Earth. Aristotle explained the introduction of the counter-Earth as a result of the Pythagoreans’ obsession with numbers. The Earth, Sun, Moon, and five planets together with the sphere of the fixed stars made nine objects about the central fire, but the Pythagoreans supposed that the number of these objects must be 10, a perfect number in the sense that 10 = 1 + 2 + 3 + 4. As described somewhat scornfully by Aristotle,2 the Pythagoreans

supposed the elements of numbers to be the elements of all things, and the whole heaven to be a musical scale and a number. And all the properties of numbers and scales which they could show to agree with the attributes and parts and the whole arrangement of the heavens, they collected and fitted into their scheme, and if there was a gap anywhere, they readily made additions so as to make their whole theory coherent. For example, as the number 10 is thought to be perfect and to comprise the whole nature of numbers, they say that the bodies which move through the heavens are ten, but as the visible bodies are only nine, to meet this they invent a tenth—the “counter- Earth.”

Apparently the Pythagoreans never tried to show that their theory explained in detail the apparent motions in the sky of the Sun, Moon, and planets against the background of fixed stars. The explanation of these apparent motions was a task for the following centuries, not completed until the time of Kepler. This work was aided by the introduction of devices like the gnomon, for studying the motions of the Sun, and other instruments that allowed the measurement of angles between the lines of sight to various stars and planets, or between such astronomical objects and the horizon. Of course, all this was naked- eye astronomy. It is ironic that Claudius Ptolemy, who had deeply studied the phenomena of refraction and reflection (including the effects of refraction in the atmosphere on the apparent positions of stars) and who as we will see played a crucial role in the history of astronomy, never realized that lenses and curved mirrors could be used to magnify the images of astronomical bodies, as in Galileo Galilei’s refracting telescope and the reflecting telescope invented by Isaac Newton. It was not just physical instruments that furthered the great advances of scientific astronomy among the Greeks. These advances were made possible also by improvements in the discipline of mathematics. As matters worked out, the great debate in ancient and medieval astronomy was not between those who thought that the Earth or the Sun was in motion, but between two different conceptions of how the Sun and Moon and planets revolve around a stationary Earth. As we will see, much of this debate had to do with different conceptions of the role of mathematics in the natural sciences. This story begins with what I like to call Plato’s homework problem. According to the Neoplatonist Simplicius, writing around AD 530 in his commentary on Aristotle’s On the Heavens,

Plato lays down the principle that the heavenly bodies’ motion is circular, uniform, and constantly regular. Therefore he sets the mathematicians the following problem: What circular motions, uniform and perfectly regular, are to be admitted as hypotheses so that it might be possible to save the appearances presented by the planets?3

“Save (or preserve) the appearances” is the traditional translation; Plato is asking what combinations of motion of the planets (here including the Sun and Moon) in circles at constant speed, always in the same direction, would present an appearance just like what we actually observe. This question was first addressed by Plato’s contemporary, the mathematician Eudoxus of Cnidus.4 He constructed a mathematical model, described in a lost book, On Speeds, whose contents are known to us from descriptions by Aristotle5 and Simplicius.6 According to this model, the stars are carried around the Earth on a sphere that revolves once a day from east to west, while the Sun and Moon and planets are carried around the Earth on spheres that are themselves carried by other spheres. The simplest model would have two spheres for the Sun. The outer sphere revolves around the Earth once a day from east to west, with the same axis and speed of rotation as the sphere of the stars; but the Sun is on the equator of an inner sphere, which shares the rotation of the outer sphere as if it were attached to it, but that also revolves around its own axis from west to east once a year. The axis of the inner sphere is tilted by 23½° to the axis of the outer sphere. This would account both for the Sun’s daily apparent motion, and for its annual apparent motion through the zodiac. Likewise the Moon could be supposed to be carried around the Earth by two other counter-rotating spheres, with the difference that the inner sphere on which the Moon rides makes a full rotation from west to east once a month, rather than once a year. For reasons that are not clear, Eudoxus is supposed to have added a third sphere each for the Sun and Moon. Such theories are called “homocentric,” because the spheres associated with the planets as well as the Sun and the Moon all have the same center, the center of the Earth. The irregular motions of the planets posed a more difficult problem. Eudoxus gave each planet four spheres: the outer sphere rotating once a day around the Earth from east to west, with the same axis of rotation as the sphere of the fixed stars and the outer spheres of the Sun and Moon; the next sphere like the inner spheres of the Sun and Moon revolving more slowly at various speeds from west to east around an axis tilted by about 23½° to the axis of the outer sphere; and the two innermost spheres rotating, at exactly the same rates, in opposite directions around two nearly parallel axes tilted at large angles to the axes of the two outer spheres. The planet is attached to the innermost sphere. The two outer spheres give each planet its daily revolution following the stars around the Earth and its average motion over longer periods through the zodiac. The effects of the two oppositely rotating inner spheres would cancel if their axes were precisely parallel, but because these axes are supposed to be not quite parallel, they superimpose a figure eight motion on the average motion of each planet through the zodiac, accounting for the occasional reversals of direction of the planet. The Greeks called this path a hippopede because it resembled the tethers used to keep horses from straying. The model of Eudoxus did not quite agree with observations of the Sun, Moon, and planets. For instance, its picture of the Sun’s motion did not account for the differences in the lengths of the seasons that, as we saw in Chapter 6, had been found with the use of the gnomon by Euctemon. It quite failed for Mercury, and did not do well for Venus or Mars. To improve things, a new model was proposed by Callippus of Cyzicus. He added two more spheres to the Sun and Moon, and one more each to Mercury, Venus, and Mars. The model of Callippus generally worked better than that of Eudoxus, though it introduced some new fictitious peculiarities to the apparent motions of the planets. In the homocentric models of Eudoxus and Callippus, the Sun, Moon, and planets were each given a separate suite of spheres, all with outer spheres rotating in perfect unison with a separate sphere carrying the fixed stars. This is an early example of what modern physicists call “fine-tuning.” We criticize a proposed theory as fine-tuned when its features are adjusted to make some things equal, without any understanding of why they should be equal. The appearance of fine-tuning in a scientific theory is like a cry of distress from nature, complaining that something needs to be better explained. A distaste for fine-tuning led modern physicists to make a discovery of fundamental importance. In the late 1950s two types of unstable particle called tau and theta had been identified that decay in different ways—the theta into two lighter particles called pions, and the tau into three pions. Not only did the tau and theta particles have the same mass—they had the same average lifetime, even though their decay modes were entirely different! Physicists assumed that the tau and the theta could not be the same particle, because for complicated reasons the symmetry of nature between right and left (which dictates that the laws of nature must appear the same when the world is viewed in a mirror as when it is viewed directly) would forbid the same particle from decaying sometimes into two pions and sometimes into three. With what we knew at the time, it would have been possible to adjust the constants in our theories to make the masses and lifetimes of the tau and theta equal, but one could hardly stomach such a theory—it seemed hopelessly fine-tuned. In the end, it was found that no fine-tuning was necessary, because the two particles are in fact the same particle. The symmetry between right and left, though obeyed by the forces that hold atoms and their nuclei together, is simply not obeyed in various decay processes, including the decay of the tau and theta.7 The physicists who realized this were right to distrust the idea that the tau and the theta particles just happened to have the same mass and lifetime— that would take too much fine-tuning. Today we face an even more distressing sort of fine-tuning. In 1998 astronomers discovered that the expansion of the universe is not slowing down, as would be expected from the gravitational attraction of galaxies for each other, but is instead speeding up. This acceleration is attributed to an energy associated with space itself, known as dark energy. Theory indicates that there are several different contributions to dark energy. Some contributions we can calculate, and others we can’t. The contributions to dark energy that we can calculate turn out to be larger than the value of the dark energy observed by astronomers by about 56 orders of magnitude—that is, 1 followed by 56 zeroes. It’s not a paradox, because we can suppose that these calculable contributions to dark energy are nearly canceled by contributions we can’t calculate, but the cancellation would have to be precise to 56 decimal places. This level of fine-tuning is intolerable, and theorists have been working hard to find a better way to explain why the amount of dark energy is so much smaller than that suggested by our calculations. One possible explanation is mentioned in Chapter 11. At the same time, it must be acknowledged that some apparent examples of fine-tuning are just accidents. For instance, the distances of the Sun and Moon from the Earth are in just about the same ratio as their diameters, so that seen from Earth, the Sun and Moon appear about the same size, as shown by the fact that the Moon just covers the Sun during a total solar eclipse. There is no reason to suppose that this is anything but a coincidence. Aristotle took a step to reduce the fine-tuning of the models of Eudoxus and Callippus. In Metaphysics8 he proposed to tie all the spheres together in a single connected system. Instead of giving the outermost planet, Saturn, four spheres like Eudoxus and Callippus, he gave it only their three inner spheres; the daily motion of Saturn from east to west was explained by tying these three spheres to the sphere of the fixed stars. Aristotle also added, inside the three of Saturn, three extra spheres that rotated in opposite directions, so as to cancel the effect of the motion of the three spheres of Saturn on the spheres of the next planet, Jupiter, whose outer sphere was attached to the innermost of the three extra spheres between Jupiter and Saturn. At the cost of adding these three extra counter-rotating spheres, by tying the outer sphere of Saturn to the sphere of the fixed stars Aristotle had accomplished something rather nice. It was no longer necessary to wonder why the daily motion of Saturn should precisely follow that of the stars—Saturn was physically tied to the sphere of the stars. But then Aristotle spoiled it all: he gave Jupiter all four spheres that had been given to it by Eudoxus and Callippus. The trouble with this was that Jupiter would then get a daily motion from that of Saturn and also from the outermost of its own four spheres, so that it would go around the Earth twice a day. Did he forget that the three counter-rotating spheres inside the spheres of Saturn would cancel only the special motions of Saturn, not its daily revolution around the Earth? Worse yet, Aristotle added only three counter-rotating spheres inside the four spheres of Jupiter, to cancel its own special motions but not its daily motion, and then gave Mars, the next planet, the full five spheres given to it by Callippus, so that Mars would go around the Earth three times a day. Continuing in this way, in Aristotle’s scheme Venus, Mercury, the Sun, and the Moon would in a day respectively go around the Earth four, five, six, and seven times. I was struck by this apparent failure when I read Aristotle’s Metaphysics, and then I learned that it had already been noticed by several authors, including J. L. E. Dreyer, Thomas Heath, and W. D. Ross.9 Some of them blamed it on a corrupt text. But if Aristotle really did present the scheme described in the standard version of Metaphysics, then this cannot be explained as a matter of his thinking in different terms from ours, or being interested in different problems from ours. We would have to conclude that on his own terms, in working on a problem that interested him, he was being careless or stupid. Even if Aristotle had put in the right number of counter-rotating spheres, so that each planet would follow the stars around the Earth just once each day, his scheme still relied on a great deal of fine- tuning. The counter-rotating spheres introduced inside the spheres of Saturn to cancel the effect of Saturn’s special motions on the motions of Jupiter would have to revolve at precisely the same speed as the three spheres of Saturn for the cancellation to work, and likewise for the planets closer to the Earth. And, just as for Eudoxus and Callippus, in Aristotle’s scheme the second spheres of Mercury and Venus would have to revolve at precisely the same speed as the second sphere of the Sun, in order to account for the fact that Mercury, Venus, and the Sun move together through the zodiac, so that the inner planets are never seen far in the sky from the Sun. Venus, for instance, is always the morning star or the evening star, never seen high in the sky at midnight. At least one ancient astronomer seems to have taken the problem of fine-tuning very seriously. This was Heraclides of Pontus. He was a student at Plato’s Academy in the fourth century BC, and may have been left in charge of it when Plato went to Sicily. Both Simplicius10 and Aëtius say that Heraclides taught that the Earth rotates on its axis,* eliminating at one blow the supposed simultaneous daily revolution of the stars, planets, Sun, and Moon around the Earth. This proposal of Heraclides was occasionally mentioned by writers in late antiquity and the Middle Ages, but it did not became popular until the time of Copernicus, again presumably because we do not feel the Earth’s rotation. There is no indication that Aristarchus, writing a century after Heraclides, suspected that the Earth not only moves around the Sun but also rotates on its own axis. According to Chalcidius (or Calcidius), a Christian who translated the Timaeus from Greek to Latin in the fourth century, Heraclides also proposed that since Mercury and Venus are never seen far in the sky from the Sun, they revolve about the Sun rather than about the Earth, thus removing another bit of fine-tuning from the schemes of Eudoxus, Callippus, and Aristotle: the artificial coordination of the revolutions of the second spheres of the Sun and inner planets. But the Sun and Moon and three outer planets were still supposed to revolve about a stationary, though rotating, Earth. This theory works very well for the inner planets, because it gives them precisely the same apparent motions as the simplest version of the Copernican theory, in which Mercury, Venus, and the Earth all go at constant speed on circles around the Sun. As far as the inner planets are concerned, the only difference between Heraclides and Copernicus is point of view—either based on the Earth or based on the Sun. Besides the fine-tuning inherent in the schemes of Eudoxus, Callippus, and Aristotle, there was another problem: these homocentric schemes did not agree very well with observation. It was believed then that the planets shine by their own light, and since in these schemes the spheres on which the planets ride always remain at the same distance from the Earth’s surface, the planets’ brightness should never change. It was obvious however that their brightness changed very much. As quoted by Simplicius,11 around AD 200 the philosopher Sosigenes the Peripatetic had commented:

However the [hypotheses] of the associates of Eudoxus do not preserve the phenomena, and just those which had been known previously and were accepted by themselves. And what necessity is there to speak about other things, some of which Callippus of Cyzicus also tried to preserve when Eudoxus had not been able to do so, whether or not Callippus did preserve them? . . . What I mean is that there are many times when the planets appear near, and there are times when they appear to have moved away from us. And in the case of some [planets] this is apparent to sight. For the star which is called Venus and also the one which is called Mars appear many times larger when they are in the middle of their retrogressions so that in moonless nights Venus causes shadows to be cast by bodies.

Where Simplicius or Sosigenes refers to the size of planets, we presumably should understand their apparent luminosity; with the naked eye we can’t actually see the disk of any planet, but the brighter a point of light is, the larger it seems to be. Actually, this argument is not as conclusive as Simplicius thought. The planets (like the Moon) shine by the reflected light of the Sun, so their brightness would change even in the schemes of Eudoxus et al. as they go through different phases (like the phases of the Moon). This was not understood until the work of Galileo. But even if the phases of the planets had been taken into account, the changes in brightness that would be expected in homocentric theories would not have agreed with what is actually seen. For professional astronomers (if not for philosophers) the homocentric theory of Eudoxus, Callippus, and Aristotle was supplanted in the Hellenistic and Roman eras by a theory that did much better at accounting for the apparent motions of the Sun and planets. This theory is based on three mathematical devices—the epicycle, the eccentric, and the equant—to be described below. We do not know who invented the epicycle and eccentric, but they were definitely known to the Hellenistic mathematician Apollonius of Perga and to the astronomer Hipparchus of Nicaea, whom we met in Chapters 6 and 7.12 We know about the theory of epicycles and eccentrics through the writings of Claudius Ptolemy, who invented the equant, and with whose name the theory has ever after been associated. Ptolemy flourished around AD 150, in the age of the Antonine emperors at the height of the Roman Empire. He worked at the Museum of Alexandria, and died sometime after AD 161. We have already discussed his study of reflection and refraction in Chapter 4. His astronomical work is described in Megale Syntaxis, a title transformed by the Arabs to Almagest, by which name it became generally known in Europe. The Almagest was so successful that scribes stopped copying the works of earlier astronomers like Hipparchus; as a result, it is difficult now to distinguish Ptolemy’s own work from theirs. The Almagest improved on the star catalog of Hipparchus, listing 1,028 stars, hundreds more than Hipparchus, and giving some indication of their brightness as well as their position in the sky.* Ptolemy’s theory of the planets and the Sun and Moon was much more important for the future of science. In one respect the work on this theory described in the Almagest is strikingly modern in its methods. Mathematical models are proposed for planetary motions containing various free numerical parameters, which are then found by constraining the predictions of the models to agree with observation. We will see an example of this below, in connection with the eccentric and equant. In its simplest version, the Ptolemaic theory has each planet revolving in a circle known as an “epicycle,” not about the Earth, but about a moving point that goes around the Earth on another circle known as a “deferent.” The inner planets, Mercury and Venus, go around the epicycle in 88 and 225 days, respectively, while the model is fine-tuned so that the center of the epicycle goes around the Earth on the deferent in precisely one year, always remaining on the line between the Earth and the Sun. We can see why this works. Nothing in the apparent motion of planets tells us how far away they are. Hence in the theory of Ptolemy, the apparent motion of any planet in the sky does not depend on the absolute sizes of the epicycle and deferent; it depends only on the ratio of their sizes. If Ptolemy had wanted to he could have expanded the sizes of both the epicycle and the deferent of Venus, keeping their ratio fixed, and likewise of Mercury, so that both planets would have the same deferent, namely, the orbit of the Sun. The Sun would then be the point on the deferent about which the inner planets travel on their epicycles. This is not the theory proposed by Hipparchus or Ptolemy, but it gives the motion of the inner planets the same appearance, because it differs only in the overall scale of the orbits, which does not affect apparent motions. This special case of the epicycle theory is just the same as the theory attributed to Heraclides discussed above, in which Mercury and Venus go around the Sun while the Sun goes around the Earth. As already mentioned, Heraclides’ theory works well because it is equivalent to one in which the Earth and inner planets go around the Sun, the two theories differing only in the point of view of the astronomer. So it is no accident that the epicycle theory of Ptolemy, which gives Mercury and Venus the same apparent motions as the theory of Heraclides, also works pretty well in comparison with observation. Ptolemy could have applied the same theory of epicycles and deferents to the outer planets—Mars, Jupiter, and Saturn—but to make the theory work it would have been necessary to make the planets’ motion around the epicycles much slower than the motion of the epicycles’ centers around the deferents. I don’t know what would have been wrong with this, but for one reason or another Ptolemy chose a different path. In the simplest version of his scheme, each outer planet goes on its epicycle around a point on the deferent once a year, and that point on the deferent goes around the Earth in a longer time: 1.88 years for Mars, 11.9 years for Jupiter, and 29.5 years for Saturn. Here there is a different sort of fine-tuning—the line from the center of the epicycle to the planet is always parallel to the line from the Earth to the Sun. This scheme agrees fairly well with the observed apparent motions of the outer planets because here, as for the inner planets, the different special cases of this theory that differ only in the scale of the epicycle and deferent (keeping their ratio fixed) all give the same apparent motions, and there is one special value of this scale that makes this model the same as the simplest Copernican theory, differing only in point of view: Earth or Sun. For the outer planets, this special choice of scale is the one for which the radius of the epicycle equals the distance of the Sun from the Earth. (See Technical Note 13.) Ptolemy’s theory nicely accounted for the apparent reversal in direction of planetary motions. For instance, Mars seems to go backward in its motion through the zodiac when it is on a point in its epicycle closest to the Earth, because then its supposed motion around the epicycle is in the direction opposite to the supposed motion of the epicycle around the deferent, and faster. This is just a transcription into a frame of reference based on the Earth of the modern statement that Mars seems to go backward in the zodiac when the Earth is passing it as they both go around the Sun. This is also the time when it is brightest (as noted in the above quotation from Simplicius), because at this time it is closest to the Earth, and the side of Mars that we see faces the Sun. The theory developed by Hipparchus, Apollonius, and Ptolemy was not a fantasy that, by good luck, just happened to agree fairly well with observation but had no relation to reality. As far as the apparent motions of the Sun and planets are concerned, in its simplest version, with just one epicycle for each planet and no other complications, this theory gives precisely the same predictions as the simplest version of the theory of Copernicus—that is, a theory in which the Earth and the other planets go in circles at constant speed with the Sun at the center. As already explained in connection with Mercury and Venus (and further explained in Technical Note 13), this is because the Ptolemaic theory is in a class of theories that all give the same apparent motions of the Sun and planets, and one member of that class (though not the one adopted by Ptolemy) gives precisely the same actual motions of the Sun and planets relative to one another as given by the simplest version of the Copernican theory. It would be nice to end the story of Greek astronomy here. Unfortunately, as Copernicus himself well understood, the predictions of the simplest version of the Copernican theory for the apparent motions of the planets do not quite agree with observation, and so neither do the predictions of the simplest version of the Ptolemaic theory, which are identical. We have known since the time of Kepler and Newton that the orbits of the Earth and the other planets are not exactly circular, the Sun is not exactly at the center of these orbits, and the Earth and the other planets do not travel around their orbits at exactly constant speed. Of course, none of that was understood in modern terms by the Greek astronomers. Much of the history of astronomy until Kepler was taken up with trying to accommodate the small inaccuracies in the simplest versions of both the Ptolemaic and the Copernican theories. Plato had called for circles and uniform motion, and as far as is known no one in antiquity conceived that astronomical bodies could have any motion other than one compounded of circular motions, though Ptolemy was willing to compromise on the issue of uniform motion. Working under the limitation to orbits composed of circles, Ptolemy and his forerunners invented various complications to make their theories agree more accurately with observation, for the Sun and Moon as well as for the planets.* One complication was just to add more epicycles. The only planet for which Ptolemy found this necessary was Mercury, whose orbit differs from a circle more than that of any other planet. Another complication was the “eccentric”; the Earth was taken to be, not at the center of the deferent for each planet, but at some distance from it. For instance, in Ptolemy’s theory the center of the deferent of Venus was displaced from the Earth by 2 percent of the radius of the deferent.* The eccentric could be combined with another mathematical device introduced by Ptolemy, the “equant.” This is a prescription for giving a planet a varying speed in its orbit, apart from the variation due to the planet’s epicycle. One might imagine that, sitting on the Earth, we should see each planet, or more precisely the center of each planet’s epicycle, going around us at a constant rate (say, in degrees of arc per day), but Ptolemy knew that this did not quite agree with actual observation. Once an eccentric was introduced, one might instead imagine that we should see the centers of the planets’ epicycles go at a constant rate, not around the Earth, but around the centers of the planets’ deferents. Alas, that didn’t work either. Instead, for each planet Ptolemy introduced what came to be called an equant,* a point on the opposite side of the center of the deferent from the Earth, but at an equal distance from this center; and he supposed that the centers of the planets’ epicycles go at a constant angular rate about the equant. The fact that the Earth and the equant are at an equal distance from the center of the deferent was not assumed on the basis of philosophical preconceptions, but found by leaving these distances as free parameters, and finding the values of the distances for which the predictions of the theory would agree with observation. There were still sizable discrepancies between Ptolemy’s model and observation. As we will see when we come to Kepler in Chapter 11, if consistently used the combination of a single epicycle for each planet and an eccentric and equant for the Sun and for each planet can do a good job of imitating the actual motion of planets including the Earth in elliptical orbits—good enough to agree with almost any observation that could be made without telescopes. But Ptolemy was not consistent. He did not use the equant in describing the supposed motion of the Sun around the Earth; and this omission—since the locations of planets are referred to the position of the Sun—also messed up the predictions of planetary motions. As George Smith has emphasized,13 it is a sign of the distance between ancient or medieval astronomy and modern science that no one after Ptolemy appears to have taken these discrepancies seriously as a guide to a better theory. The Moon presented special difficulties: the sort of theory that worked pretty well for the apparent motions of the Sun and planets did not work well for the Moon. It was not understood until the work of Isaac Newton that this is because the Moon’s motion is significantly affected by the gravitation of two bodies—the Sun as well as the Earth—while the planets’ motion is almost entirely governed by the gravitation of a single body: the Sun. Hipparchus had proposed a theory of the Moon’s motion with a single epicycle, which was adjusted to account for the length of time between eclipses; but as Ptolemy recognized, this model did not do well in predicting the location of the Moon on the zodiac between eclipses. Ptolemy was able to fix this flaw with a more complicated model, but his theory had its own problems: the distance between the Moon and the Earth would vary a good deal, leading to a much larger change in the apparent size of the Moon than is observed. As already mentioned, in the system of Ptolemy and his predecessors there is no way that observation of the planets could have indicated the sizes of their deferents and epicycles; observation could have fixed only the ratio of these sizes for each planet.* Ptolemy filled this gap in Planetary Hypotheses, a follow-up to the Almagest. In this work he invoked an a priori principle, perhaps taken from Aristotle, that there should be no gaps in the system of the world. Each planet as well as the Sun and Moon was supposed to occupy a shell, extending from the minimum to the maximum distance of the planet or Sun or Moon from the Earth, and these shells were supposed to fit together with no gaps. In this scheme the relative sizes of the orbits of the planets and Sun and Moon were all fixed, once one decided on their order going outward from the Earth. Also, the Moon is close enough to the Earth so that its absolute distance (in units of the radius of the Earth) could be estimated in various ways, including the method of Hipparchus discussed in Chapter 7. Ptolemy himself developed the method of parallax: the ratio of the distance to the Moon and the radius of the Earth can be calculated from the observed angle between the zenith and the direction to the Moon and the calculated value that this angle would have if the Moon were observed from the center of the Earth.14 (See Technical Note 14.) Hence, according to Ptolemy’s assumptions, to find the distances of the Sun and planets all that was necessary was to know the order of their orbits around the Earth. The innermost orbit was always taken to be that of the Moon, because the Sun and the planets are each occasionally eclipsed by the Moon. Also, it was natural to suppose that the farthest planets are those that appear to take the longest to go around the Earth, so Mars, Jupiter, and Saturn were generally taken in that order going away from the Earth. But the Sun, Venus, and Mercury all on average appear to take a year to go around the Earth, so their order remained a subject of controversy. Ptolemy guessed that the order going out from the Earth is the Moon, Mercury, Venus, the Sun, and then Mars, Jupiter, and Saturn. Ptolemy’s results for the distances of the Sun, Moon, and planets as multiples of the diameter of the Earth were much smaller than their actual values, and for the Sun and Moon similar (perhaps not coincidentally) to the results of Aristarchus discussed in Chapter 7. The complications of epicycles, equants, and eccentrics have given Ptolemaic astronomy a bad name. But it should not be thought that Ptolemy was stubbornly introducing these complications in order to make up for the mistake of taking the Earth as the unmoving center of the solar system. The complications, beyond just a single epicycle for each planet (and none for the Sun), had nothing to do with whether the Earth goes around the Sun or the Sun around the Earth. They were made necessary by the fact, not understood until Kepler’s time, that the orbits are not circles, the Sun is not at the center of the orbits, and the velocities are not constant. The same complications also affected the original theory of Copernicus, who assumed that the orbits of planets and the Earth had to be circles and the speeds constant. Fortunately, this is a pretty good approximation, and the simplest version of the epicycle theory, with just one epicycle for each planet and none for the Sun, worked far better than the homocentric spheres of Eudoxus, Callippus, and Aristotle. If Ptolemy had included an equant along with an eccentric for the Sun as well as for each planet, the discrepancies between theory and observation would have been too small to be detected with the methods then available. But this did not settle the issue between the Ptolemaic and Aristotelian theories of planetary motions. The Ptolemaic theory agreed better with observation, but it did violence to the assumption of Aristotelian physics that all celestial motions must be composed of circles whose center is the center of the Earth. Indeed, the queer looping motion of planets moving on epicycles would have been hard to swallow even for someone who had no stake in any other theory. For fifteen hundred years the debate continued between the defenders of Aristotle, often called physicists or philosophers, and the supporters of Ptolemy, generally referred to as astronomers or mathematicians. The Aristotelians often acknowledged that the model of Ptolemy fitted the data better, but they regarded this as just the sort of thing that might interest mathematicians, not relevant for understanding reality. Their attitude was expressed in a statement by Geminus of Rhodes, who flourished around 70 BC, quoted about three centuries later by Alexander of Aphrodisias, who in turn was quoted by Simplicius,15 in a commentary on Aristotle’s Physics. This statement lays out the great debate between natural scientists (sometimes translated “physicists”) and astronomers:

It is the concern of physical inquiry to enquire into the substance of the heavens and the heavenly bodies, their powers and the nature of their coming-to-be and passing away; by Zeus, it can reveal the truth about their size, shape, and positioning. Astronomy does not attempt to pronounce on any of these questions, but reveals the ordered nature of the phenomena in the heavens, showing that the heavens are indeed an ordered cosmos, and it also discusses the shapes, sizes, and relative distances of the Earth, Sun, and Moon, as well as eclipses, the conjunctions of the heavenly bodies, and qualities and quantities inherent in their paths. Since astronomy touches on the study of the quantity, magnitude, and quality of their shapes, it understandably has recourse to arithmetic and geometry in this respect. And about these questions, which are the only ones it promised to give an account of, it has the power to reach results through the use of arithmetic and geometry. The astronomer and the natural scientist will accordingly on many occasions set out to achieve the same objective, for example, that the Sun is a sizeable body, that the Earth is spherical, but they do not use the same methodology. For the natural scientist will prove each of his points from the substance of the heavenly bodies, either from their powers, or from the fact that they are better as they are, or from their coming-to-be and change, while the astronomer argues from the properties of their shapes and sizes, or from quantity of movement and the time that corresponds to it. . . . In general it is not the concern of the astronomer to know what by nature is at rest and what by nature is in motion; he must rather make assumptions about what stays at rest and what moves, and consider with which assumptions the appearances in the heavens are consistent. He must get his first basic principles from the natural scientist, namely that the dance of the heavenly bodies is simple, regular, and ordered; from these principles he will be able to show that the movement of all the heavenly bodies is circular, both those that revolve in parallel courses and those that wind along oblique circles.

The “natural scientists” of Geminus share some characteristics of today’s theoretical physicists, but with huge differences. Following Aristotle, Geminus sees the natural scientists as relying on first principles, including teleological principles: the natural scientist supposes that the heavenly bodies “are better as they are.” For Geminus it is only the astronomer who uses mathematics, as an adjunct to his observations. What Geminus does not imagine is the give-and-take that has developed between theory and observation. The modern theoretical physicist does make deductions from basic principles, but he uses mathematics in this work, and the principles themselves are expressed mathematically and are learned from observation, certainly not by considering what is “better.” In the reference by Geminus to the motions of planets “that revolve in parallel courses and those that wind along oblique circles” one can recognize the homocentric spheres rotating on tilted axes of the schemes of Eudoxus, Callippus, and Aristotle, to which Geminus as a good Aristotelian would naturally be loyal. On the other hand, Adrastus of Aphrodisias, who around AD 100 wrote a commentary on the Timaeus, and a generation later the mathematician Theon of Smyrna were sufficiently convinced by the theory of Apollonius and Hipparchus that they tried to make it respectable, by interpreting the epicycles and deferents as solid transparent spheres, like the homocentric spheres of Aristotle, but now not homocentric. Some writers, facing the conflict between the rival theories of the planets, threw up their hands, and declared that humans were not meant to understand celestial phenomena. Thus, in the mid–fifth century AD, in his commentary on the Timaeus, the Neoplatonist pagan Proclus proclaimed:16

When we are dealing with sublunary things, we are content, because of the instability of the material which goes to constitute them, to grasp what happens in most instances. But when we want to know heavenly things, we use sensibility and call upon all sorts of contrivances quite removed from likelihood. . . . That this is the way things stand is plainly shown by the discoveries made about these heavenly things—from different hypotheses we draw the same conclusions relative to the same objects. Among these hypotheses are some which save the phenomena by means of epicycles, others which do so by means of eccentrics, still others which save the phenomena by means of counterturning spheres devoid of planets. Surely the god’s judgement is more certain. But as for us, we must be satisfied to “come close” to those things, for we are men, who speak according to what is likely, and whose lectures resemble fables.

Proclus was wrong on three counts. He missed the point that the Ptolemaic theories that used epicycles and eccentrics did a far better job of “saving the phenomena” than the Aristotelian theory using the hypothesis of homocentric “counterturning spheres.” There is also a minor technical point: in referring to hypotheses “which save the phenomena by means of epicycles, others which do so by means of eccentrics” Proclus seems not to realize that in the case where an epicycle can play the role of an eccentric (discussed in footnote *), these are not different hypotheses but different ways of describing what is mathematically the same hypothesis. Above all, Proclus was wrong in supposing that it is harder to understand heavenly motions than those here on Earth, below the orbit of the Moon. Just the reverse is true. We know how to calculate the motions of bodies in the solar system with exquisite precision, but we still can’t predict earthquakes or hurricanes. But Proclus was not alone. We will see his unwarranted pessimism regarding the possibility of understanding the motion of the planets repeated centuries later, by Moses Maimonides. Writing in the first decade of the twentieth century, the physicist turned philosopher Pierre Duhem17 took the side of the Ptolemaics because their model fitted the data better, but he disapproved of Theon and Adrastus for trying to lend reality to the model. Perhaps because he was deeply religious, Duhem sought to restrict the role of science merely to the construction of mathematical theories that agree with observation, rather than encompassing efforts to explain anything. I am not sympathetic to this view, because the work of my generation of physicists certainly feels like explanation as we ordinarily use the word, not like mere description.18 True, it is not so easy to draw a precise distinction between description and explanation. I would say that we explain some generalization about the world by showing how it follows from some more fundamental generalization, but what do we mean by fundamental? Still, I think we know what we mean when we say that Newton’s laws of gravitation and motion are more fundamental than Kepler’s three laws of planetary motion. The great success of Newton was in explaining the motions of the planets, not merely describing them. Newton did not explain gravitation, and he knew that he had not, but that is the way it always is with explanation— something is always left for future explanation.

Because of their odd motions, the planets were useless as clocks or calendars or compasses. They were put to a different sort of use in Hellenistic times and afterward for purposes of astrology, a false science learned from the Babylonians.* The sharp modern distinction between astronomy and astrology was less clear in the ancient and medieval worlds, because the lesson had not yet been learned that human concerns were irrelevant to the laws governing the stars and planets. Governments from the Ptolemies on supported the study of astronomy largely in the hope that it would reveal the future, and so naturally astronomers spent much of their time on astrology. Indeed, Claudius Ptolemy was the author not only of the greatest astronomical work of antiquity, the Almagest, but also of a textbook of astrology, the Tetrabiblos. But I can’t leave Greek astronomy on this sour note. For a happier ending to Part II of this book, I’ll quote Ptolemy on his pleasure in astronomy:19

I know that I am mortal and the creature of a day; but when I search out the massed wheeling circles of the stars, my feet no longer touch the Earth, but, side by side with Zeus himself, I take my fill of ambrosia, the food of the gods.