温伯格《解释世界》第七章 测量太阳，月亮和地球

希腊天文学最突出的成就是对地球，太阳和月亮大小的测量以及测量日月之间和地日之间的距离。其成就不是体现在计算结果有多么精确（由于所依据的观测数据过于原始，因此那时不可能计算出准确的大小和距离），而是数学第一次被正确的应用到对自然世界的定量计算中。 做出这些计算首先必须掌握日食和月食特性，而且需要认识地球是个球体。基督教殉道者希波吕托斯和时常被人引用的哲学家埃提乌斯（时间不详）都把最早对日食和月食的认识归功于阿那克萨哥拉。阿那克萨哥拉是爱奥尼亚希腊人，约公元前500年出生于克拉佐美纳伊（靠近士麦那），在雅典执教。也许是基于巴门尼德所观察到的月亮明亮一面总是面向太阳，阿那克萨哥拉断言：“太阳赋予月球光芒”。由此可以自然推定月食是当月球穿过地影时发生的。据说他也明白日食发生在月影落在地球时刻。 亚里士多德应用推理加观察正确认识了地球的形状。戴奥真尼斯·拉尔修与希腊地理学家斯特雷波都认为巴门尼德在早于亚里士多德多年以前已经知道地球为球形。但是我们不了解巴门尼德是如何得出他的结论的（如果他真的知道）。亚里士多德在《论天》中介绍了地球是球体的理论和经验理由。如我们在第三章所述，按照亚里士多德的物质先验论，重元素土和水（稍弱）会自然趋向宇宙中心，而气和火（更强）趋向远离宇宙中心。地球是球形，其中心与宇宙重合，这样可使最多的元素趋向中心。亚里士多德并没有局囿于理论论证。他也介绍了地球是球体的经验证明。月食时地球落在月球上的影子成弧形（注：有人主张亚里士多德关于地影在月球上的形状理由不能让人信服（见纽格鲍尔， 《古代数学天文史》，纽约， 1975， 1093-94页），因为陆地的多种变化以及月球形状也或产生同样的弧形），另外当我们由北向南旅行时会发现天上星星的位置像在发生变化： 月食时轮廓总呈弧形，因为月食是由于地球遮挡造成的，这样轮廓的形状取决于地球表面的形状，因此上地球一定是球形。另外我们通过对天上星星的观察可以明显看出不仅地球为圆形，而且其周长一定不会很长。因为我们向南或北行进一小距离就会感觉到地平面明显的变化。我是说我们头顶的星星变化很大，当我们向北或南行进。我们看到的星星不同。确实在埃及和塞浦路斯附近看到的一些星星在北部地区看不到。有些北方星星永远不会离开观察范围，一直在那一带升起和落下。 亚里士多德对数学的态度很典型，他没有尝试应用这些对星星的观察来定量计算地球的大小。除此之外，我也很困惑亚里士多德为什么没有引用水手们都非常熟悉的一个现象。远方的船只在天色晴朗的日子首先映入眼帘的是船桅—地球的弧线遮挡了船的其他部位—当船更加接近时，可以看到船的全貌。（注：塞缪尔·艾略特·莫里森在哥伦布传记中引用了这个论据来证实与广为流传的猜想相反，早在哥伦布航行之前人们已经知道地球是球形。在卡斯提尔宫廷关于是否支持哥伦布提出的远航辩论中关注的不是地球的形状，而是大小。哥伦布认为地球不大，他完全拥有足够的食物和水去完成从西班牙到亚洲东海岸的航行。他对地球大小的认知完全错误，当然后来当时未知的位于欧洲和亚洲间的美洲大陆救了他。） 不可小觑亚里士多德对地球球形形态的认识。阿那克西曼德曾认为地球为圆柱形，我们生活在圆柱的平面上。阿那克西米尼认为地球为扁平形，太阳，月亮和星星漂浮在空气中，当它们位于地球高部位后面时会被挡住，因而我们看不见。色诺芬尼曾写道：“我们眼前看到的只是地球的上部，在我们脚下地球向下延伸无限远。”后来的德谟克里特和阿那克萨哥拉都像阿那克西米尼一样认为地球为扁平形。 我怀疑很多人之所以坚信地球为扁平是因为认为球形地球存在一个明显的问题：如果地球为球形，那为什么旅行者不会掉下去？其实亚里士多德物质理论给出了很好的解释。亚里士多德知道并不存在普世的“向下”方向—即任何地方的物体都会朝向这个方向下落。实际上地球上任何地方由重元素土和水所组成的物质都会倾向于落向宇宙中心，这与观察一致。在这方面亚里士多德中元素的自然位置在宇宙中心的理论有些类似于现代的重力理论，主要区别在于对亚里士多德来说只存在一个宇宙中心，然而今天我们知道超大物体在自身重力的作用下都会倾向于收缩为球形，并对其他物体产生向心引力。亚里士多德的理论无法解释为什么除了地球，其他所有星体也应该是球体，但是他知道至少月亮是球体，其理由是月相从满月到新月又回到满月的逐渐变化。 亚里士多德之后天文学家和哲学家（像拉克坦提乌斯等几个除外）中主流共识都认为地球为球形。阿基米德甚至应用心灵之眼在一杯水中看到了地球球形形态；在《论浮体》命题二，他提出：“静止流体表面都是球面，其中心为地球中心。”（这只有不存在表面张力的条件下才成立，阿基米德忽略了这点。） 现在让我开始介绍萨摩斯的阿里斯塔克斯的成就，从某些方面可以说这是古代世界将数学应用到自然科学方面的典范。阿里斯塔克斯公元前310年左右出生于萨摩斯的爱奥尼亚岛。他是雅典吕刻俄斯第三任校长兰萨库斯的斯特拉图的学生。后来在亚历山大一直工作到公元前230年左右去世。幸运的是他的巨著《论日月大小和距离》流传下来。在其中阿里斯塔克斯把四个天文观测作为假设： 1. 半月时日月距离比四分之一圆小四分之一圆的三十分之一。（也就是说当月亮半圆时，视线与月亮和太阳的交角比90度小3度，所以是87度） 2. 日食时月亮刚好覆盖太阳 3. 地影宽度相当于两个月亮。（最简单的解释是指在月球位置，月食时两个月亮直径的球体可以占满地影。这可能是通过测量月亮一角刚开始被地影遮挡到完全遮挡的时间，以及从完全遮挡到月食消失的时间发现的。） 4. 月亮占黄道带的15分之一。（整个黄道带为完整的360度，但是阿里斯塔克斯这里显然是指黄道带的一个星座。黄道带包含12个星座，这样一个星座占3600/12=300， 其15分之一为2度。） 从以上假设阿里斯塔克斯推导出： 1. 地日距离比地月距离远19倍到20倍。 2. 太阳直径比月亮直径大19倍到20倍。 3. 地球直径比月亮直径大108/43倍到60/19倍。 4. 地月距离比月亮直径大30倍到45/2倍。 阿里斯塔克斯计算那时还没有三角学，他需要通过繁琐的几何架构来得出这些上限和下限值。今天我们应用三角学可以得出更精确的结果。比如从第一条假设我们可以得知地日距离比地月距离远87度的正割函数倍（余割函数的倒数），即19.1倍，确实在19倍与20倍之间。（阿里斯塔克斯的这个以及其他几个结论在技术说明11中用现代方法做了重新推导）。 阿里斯塔克斯的数学推理没有瑕疵，但是他的计算结果不对，因为他四个假设中作为起点的数据错的很离谱。当月亮为半圆时视线与月亮和太阳的实际交角不是87度，而是89.853度，这样太阳距地球比月亮距地球远390倍，比阿里斯塔克斯想的要远很多。采用肉眼观察无法得出这么准确的测量数据，尽管如此阿里斯塔克斯应该说明半月时视线与月亮和太阳的实际交角不会小于87度。另外月亮视角为0.519度，不是2度，这样地月距离是月亮直径的111倍。阿里斯塔克斯应该做的更好，阿基米德在《数沙者》中暗示他在后期工作中确有改进。（注：阿基米德《数沙者》中有个令人关注的论述，阿里斯塔克斯发现“太阳看上去是黄道带的1/720（阿基米德著作，翻译汉斯，剑桥大学出版社，1897年，223页）。也就是说从地球上看日盘视角是360度的1/720 , 即0.5度，这与正确数值0.519度相差无几。阿基米德甚至宣称他用自己的观察证实了这点。但是我们前面看到，在阿里斯塔克斯流传下来的作品中他认为月盘视角为2度，而且他也记录日盘与月盘视尺寸一样大。那阿基米德是否引用了阿里斯塔克斯后期没有流传下来的测量数据？又或许他只是引用了他自己的测量数据，只是归于阿里斯塔克斯？我听说学者们认为产生这个误差的原因在于拷贝错误或翻译错误。但是看起来不可能。前面说过，阿里斯塔克斯从测量月亮视角得出地月距离比月亮直径大30到45/2倍，这与0.5度视角很不匹配。应用现代三角学我们知道如果月亮视角为2度，那地月距离为月球直径的28.6倍，此值在30与45/2之间。（《数沙者》不是一本严肃的天文学著作，它只是阿基米德用来证实他可以做出超大数据计算，比如充满恒星球体需要的沙粒数量）。阿里斯塔克斯与我们现代科学的差距不在其观察数据的错误。现代天文观测与实验物理仍然偶尔会出现严重误差。比如十九世纪三十年代所计算的宇宙膨胀速度比我们现代正确掌握的速度快7倍。阿里斯塔克斯与今天天文学家和物理学家的真正不同不在于他的数据误差。而是他从来没有试图分析数据的不确定性，或承认这些数据可能是不够完善的。 今天的物理学家和天文学家都要接受非常严格地评估实验误差的训练。虽然我在读大学本科时已经知道我要做一名不需要做实验的理论物理学家，但在康奈尔大学我必须与其他物理系同学一样上实验课。课上我们大部分时间都花在估算我们所做实验的测量误差。但是历史上历经数哉才得以认识到不确定性的重要。就我所知，在古代和中世纪没有人曾尝试严肃地评估测量误差，我们在第14章会看到，即使牛顿对实验误差也漫不经心。 在阿里斯塔克斯身上我们还看到数学威力的一个负面效应。他的书读起来像欧几里德的《几何原本》：第一点到第四点的数据作为公设，由此他应用严格的数学计算推导出结论。但是他的观测误差远远大于他通过计算给出的尺寸和距离的狭窄区间。也许阿里斯塔克斯并不是真的认为半圆时目视日月视线交角为87度，他只不过把它作为一个例子来说明由此可以推导出什么结果。阿里斯塔克斯的同代人对阿里斯塔克斯作为一个数学家一无所知。不像他的老师斯特拉图以“物理学家”闻名。 阿里斯塔克斯确实取得一个重要的定性成果：太阳比地球大很多。为了强调这点，阿里斯塔克斯特别指出太阳体积是地球体积的（361/30）3（大约218）倍。当然我们现在知道实际上比这个值要大出许多。 阿基米德和普鲁塔克都指出阿里斯塔克斯从太阳的巨大体积推断不是太阳环绕地球运转，而是地球环绕太阳运转。阿基米德在《数沙者》中记载阿里斯塔克斯推断不只地球绕日运行，而且其运行轨道半径与地球和恒星间的距离相比微不足道。可能阿里斯塔克斯是为了解决地动观引发的一个问题。正如从旋转木马上观察一个地面物体会发现它像在前后移动，那从旋转的地球上观察恒星也应该在一年内看起来像在前后移动。亚里士多德可能意识到这点，如果地球在运转，他评述，那么“应该看到恒星的交错和转向，但这并没有被观察到。同样的恒星总是在地球上同一地起落。”具体地说，如果地球绕日运行，那每颗星星都应该看起来在天空中勾画出一条封闭曲线，其大小取决于地球绕日轨道直径与地球到恒星距离之比。 如果地球绕日运行，为什么古代天文学家没有观察到这种被称为周年视差的恒星年视运动。要使得这种视差小到人们根本无法观察得到，就需要假定恒星距地球必须远过一定的距离。很遗憾阿基米德在《数沙者》中没有明确提及视差，我们不知道古代是否有人应用这个论据来设定地球到恒星距离的一个下限值。 亚里士多德也提出几个其他论据来否认地动观。其中一些是基于在第三章介绍过的他的向宇宙中心自然运动理论，但是有一个论证是基于观察。亚里士多德解释道，如果地球在运转，那向上抛出的物体由于地球转动而会落到后面，而不会落回到原地。然而实际上正如他所述，“向上直线抛出的重物总是落回到原地，即使被抛出无限远。”该论证历史上被不断重复，比如克罗蒂斯·托勒密（我们在第四章介绍过）在公元150年，以及让·布里丹在中世纪，尼克尔·奥里斯姆最终给出针对该观点的答案（我们在第十章会介绍）。 如果我们有古代太阳系机械模型—太阳系仪的记载的话，我们可以推断出地动观传播有多远。西塞罗在《论共和国》中讲述了一场关于太阳系仪的对话，该对话发生在早于他出生之前23年的公元前129年。（注：1901年采海绵潜水员在地中海克利特岛与希腊大陆之间的安提凯特拉岛发现了一个古代装置，即著名的安提凯特拉机械装置。 据说该装置丢失于公元前150年到100年间的一次沉船事故。虽然安提凯特拉机械装置现在不过是一块遭受腐蚀的青铜，但一些学者应用X光对其内部机构进行了研究，推断出其工作原理。显然这不是太阳系仪，而是历法设备，它可以显示出太阳以及其他行星在黄道带任何日期的视位置。最重要的一点是其复杂精密的齿轮结构，充分体现了希腊化时代的高超技术。） 对话中一位叫卢修斯·福利乌斯·夫利阿斯的据说讲述了阿基米德制作的一个太阳系仪在叙拉古沦陷后被占领者马塞拉斯拿走，后来在马塞拉斯孙子房间看到过。从这种第三手资料无法了解太阳系仪的工作原理。（《论共和国》这部分的一些页码丢失了），但是故事中有一处西塞罗引述夫利阿斯的话说在这个太阳系仪上“刻画着太阳，月亮，以及其他五个被称为游走星(行星)的运行，”这里明显表明在太阳系仪上运行的是太阳，而不是地球。 在第八章我们会看到早在阿里斯塔克斯之前毕达哥拉斯学派已经提出地球和太阳都在围绕中心之火运行的观点。虽然他们没有给出任何证据，但是不知何故他们的观点却流传下来，而阿里斯塔克斯的观点几乎被遗忘。现代所知只有一位古代天文学家采纳了阿里斯塔克斯的日心说—即鲜为人知的塞琉西亚的栖来克斯，他活跃在公元前150年左右。在哥白尼和伽利略时代当天文学家或传教士提及地动说，他们会将之称作毕达哥拉斯学说，而不是阿里斯塔克斯学说。我2005年参观萨摩斯岛时发现许多酒吧和餐厅以毕达哥拉斯命名，没有一个以萨摩斯的阿里斯塔克斯命名。 我们很容易理解为什么古代社会难以接受地动学说。人们感受不到这种运动，而且在十四世纪之前没有人知道其实人们没有什么理由一定应该感受到这种运动。另外阿基米德或其他任何人都没有说明阿里斯塔克斯如何解决从运转的地球观察行星的视运动应该是什么样子的。 古代社会最伟大的天文观测家喜帕恰斯大大提高了地球到月球距离测量精度。喜帕恰斯公元前161年到146年间在亚历山大进行天文观测，后来也许在罗德岛从事天文观测到公元前127年。他的所有著作几乎全部失传，我们主要从三个世纪之后的克罗蒂斯·托勒密的介绍中了解他的天文工作。他的一个计算是基于对日食的观察，我们现在知道该日食发生在公元前189年3月14号。此次日食在亚历山大里亚可以观察到日全食，而在赫勒斯滂（现代达达尼尔海峡，位于亚洲和欧洲之间）太阳只被遮盖4/5。因为月球和太阳的视直径几乎相等（喜帕恰斯测量值为33弧秒，或0.55度），喜帕恰斯推断从赫勒斯滂和从亚历山大里亚看到的月亮方向相差0.55度的1/5，即0.11度。从对太阳的观察中喜帕恰斯知道赫勒斯滂以及亚历山大里亚的纬度，他也知道日食发生时月亮在天空的位置，这样他可以算出地球到月球距离与地球半径的比值。考虑到月亮视大小每月的不同，喜帕恰斯推断地月距离在地球半径的71倍到83倍间变化。地月平均距离实际是地球半径的60倍。 这里我需要插入介绍一下喜帕恰斯另一项伟大的成就，虽然这项成就与测量尺寸和距离没有直接关系。喜帕恰斯制作了包含800颗恒星的星表，每颗星都标有天文位置。我们现代最好的星表标有由一颗人造地球卫星所测定的118000颗恒星的位置，该地球卫星被命名为喜帕恰斯卫星，这个命名显然非常恰当。 喜帕恰斯在测量恒星位置过程中发现了一个异常的现象，该现象一直到牛顿的伟大成就之后才得以解释。要理解该发现，首先需要介绍一下如何描述天体位置。喜帕恰斯星表没有流传下来，我们不知道他如何描述这些位置。从罗马时代以来有两种可能常用的方法，一种方法将恒星作为球体上的点（托勒密用此方法制作他的星表），球赤道为黄道，即太阳一年在背景星星中的视运行轨迹。在此球体上标记星星的黄经和黄纬的方法与在地球上标记点位的经纬度方法一样。（注：黄纬是恒星与黄道间的角距。地球上我们从格林威治子午线开始测量经度，而黄经是恒星在一个固定黄纬圈内与太阳在春分点的天子午线的角距离。）另一种方法也把恒星作为球体上的点，但是这个球体对着地轴，而不是黄道（喜帕恰斯可能就用此方法）；球北极即是天北极，星星每晚看起来围绕该点运行。在这个球体上的坐标不用黄纬或黄经，而是用赤纬和赤经。 据托勒密记载，喜帕恰斯的测量非常精确，他注意到他观察的角宿一的赤经与许久之前天文学家提摩卡里斯在亚历山大观察结果差2度。这并不是因为角宿一相对其他恒星的位置发生了变化，而是秋分时太阳在天球上的位置发生了变化，赤经是从此点开始测量的。 现在很难准确确定这个变化用了多长时间。提摩卡里斯出生于公元前320年，早于喜帕恰斯130年。据信他在公元前280年英年早逝，比喜帕恰斯早逝160年。如果我们设想他们俩观察角宿一时间差150年，那么他们的观察说明太阳在秋分的位置每75年改变1度（注：托勒密基于他自己对轩辕十四的观察在《天文学大成》中给出约每100年1度的变化值）。以这个速度的话二分点完成绕黄道12宫360度一周进动需要360乘以75年，即27000年。 今天我们知道岁差是由于地轴围绕一个垂直于公转轨道方向的旋转（就像陀螺轴的旋转一样），该方向与地轴的交角依然近乎固定为23.5度。二分点是地日连线垂直于地轴时的日期，所以地轴的旋转造成二分点的进动。我们在第14章会讲到牛顿首先对地轴旋转做出了解释，这是由于太阳和月亮对地球赤道隆起的引力导致的。实际上地轴用25727年旋转360度。喜帕恰斯的成果能够如此精确的预测出岁差周期令人赞叹。（另外正是由于岁差的原因古代航海家不用北极星，而用天北极方向的星座来判断北方。北极星相对其他恒星并没有发生变化，但在古代地轴并不像现在这样指向北极星，未来北极星又会偏离天北极。） 现在回到天体测量，阿里斯塔克斯和喜帕恰斯计算结果都表述为日月大小和距离与地球大小的比值。地球大小的测量是由晚于阿里斯塔克斯几十年的埃拉托色尼做出的。埃拉托色尼出生于公元前273年的昔兰尼--位于今天利比亚的地中海沿岸希腊都城，昔兰尼建城于公元前630年，后来成为托勒密王国的一部分。埃拉托色尼在雅典求学，部分在吕刻俄斯，大约公元前245年被托勒密三世召回到亚历山大里亚任博物馆研究员，并给后来的托勒密四世做导师。约公元前234年他被任命为图书馆第五任馆长。他的主要著作《测量地球》，《地理学概论》，和《赫尔墨斯》等很不幸全部失传,但在古代被人广泛引用。 约公元前50年之后的斯多葛派哲学家克利奥米德在《论天体》中描述了埃拉托色尼对地球的测量。埃拉托色尼观察到夏至正午太阳位于亚历山大里亚正南的埃及西恩纳的头顶，而在亚历山大里亚圭表显示夏至正午太阳偏离头顶一圈的五十分之一，即7.2度。由此他推断地球的周长是亚历山大里亚与西恩纳之间距离的50倍。（参见技术说明12）。当时测得的亚历山大里亚与西恩纳之间距离为5000希腊里（可能是步行者测出的，他们受训每步都迈一样步长），这样地球的周长为250000希腊里。 这个估算是否准确？我们不知道埃拉托色尼使用的希腊里有多长，克利奥米德可能也不知道。因为该单位从来没有一个标准的定义（不像我们用的英里或公里）。但是即使不知道希腊里的长度，我们也可以判断埃拉托色尼计算的精度。地球周长实际上是亚历山大里亚与西恩纳（现代阿斯旺）距离的47.9倍，这样埃拉托色尼得出的地球的周长是亚历山大里亚与西恩纳之间距离50倍的结果非常靠谱，不管希腊里的长度是多少。（注：埃拉托色尼有运气成分。西恩纳不在亚历山大里亚的正南（其经度为32.9度，而亚历山大里亚的经度为29.9度），另外夏至正午太阳不在西恩纳的正头顶，而是偏离垂向0.4度。这两个误差部分抵消。埃拉托色尼真正测量的是地球周长与亚历山大到北回归线（克利奥米德称之为夏季热带圈）间距离的比值，夏至午时太阳正好在北回归线头顶。 亚历山大里亚纬度为31.2度，北回归线纬度为23.5度，与亚历山大里亚差7.7度，这样地球周长是亚历山大里亚与北回归线距离的3600/7.70=46.75倍，与埃拉托色尼给出的50倍相差不大。）。无论他的地理学如何，至少在天文学应用方面，埃拉托色尼做的非常出色。

One of the most remarkable achievements of Greek astronomy was the measurement of the sizes of the Earth, Sun, and Moon, and the distances of the Sun and Moon from the Earth. It is not that the results obtained were numerically accurate. The observations on which these calculations were based were too crude to yield accurate sizes and distances. But for the first time mathematics was being correctly used to draw quantitative conclusions about the nature of the world. In this work, it was essential first to understand the nature of eclipses of the Sun and Moon, and to discover that the Earth is a sphere. Both the Christian martyr Hippolytus and Aëtius, a much-quoted philosopher of uncertain date, credit the earliest understanding of eclipses to Anaxagoras, an Ionian Greek born around 500 BC at Clazomenae (near Smyrna), who taught in Athens.1 Perhaps relying on the observation of Parmenides that the bright side of the Moon always faces the Sun, Anaxagoras concluded, “It is the Sun that endows the Moon with its brilliance.”2 From this, it was natural to infer that eclipses of the Moon occur when the Moon passes through the Earth’s shadow. He is also supposed to have understood that eclipses of the Sun occur when the Moon’s shadow falls on the Earth. On the shape of the Earth, the combination of reason and observation served Aristotle very well. Diogenes Laertius and the Greek geographer Strabo credit Parmenides with knowing, long before Aristotle, that the Earth is a sphere, but we have no idea how (if at all) Parmenides reached this conclusion. In On the Heavens Aristotle gave both theoretical and empirical arguments for the spherical shape of the Earth. As we saw in Chapter 3, according to Aristotle’s a priori theory of matter the heavy elements earth and (less so) water seek to approach the center of the cosmos, while air and (more so) fire tend to recede from it. The Earth is a sphere, whose center coincides with the center of the cosmos, because this allows the greatest amount of the element earth to approach this center. Aristotle did not rest on this theoretical argument, but added empirical evidence for the spherical shape of the Earth. The Earth’s shadow on the Moon during a lunar eclipse is curved,* and the position of stars in the sky seems to change as we travel north or south:

In eclipses the outline is always curved, and, since it is the interposition of the Earth that makes the eclipse, the form of the line will be caused by the form of the Earth’s surface, which is therefore spherical. Again, our observation of the stars make[s] it evident, not only that the Earth is circular, but also that it is a circle of no great size. For quite a small change of position on our part to south or north causes a manifest alteration of the horizon. There is much change, I mean, in the stars which are overhead, and the stars seen are different, as one moves northward or southward. Indeed there are some stars seen in Egypt and in the neighborhood of Cyprus that are not seen in the northerly regions; and stars, which in the north are never beyond the range of observation, in those regions rise and set.3

It is characteristic of Aristotle’s attitude toward mathematics that he made no attempt to use these observations of stars to give a quantitative estimate of the size of the Earth. Apart from this, I find it puzzling that Aristotle did not also cite a phenomenon that must have been familiar to every sailor. When a ship at sea is first seen on a clear day at a great distance it is “hull down on the horizon”—the curve of the Earth hides all but the tops of its masts—but then, as it approaches, the rest of the ship becomes visible.* Aristotle’s understanding of the spherical shape of the Earth was no small achievement. Anaximander had thought that the Earth is a cylinder, on whose flat face we live. According to Anaximenes, the Earth is flat, while the Sun, Moon, and stars float on the air, being hidden from us when they go behind high parts of the Earth. Xenophanes had written, “This is the upper limit of the Earth that we see at our feet; but the part beneath goes down to infinity.”4 Later, both Democritus and Anaxagoras had thought like Anaximenes that the Earth is flat. I suspect that the persistent belief in the flatness of the Earth may have been due to an obvious problem with a spherical Earth: if the Earth is a sphere, then why do travelers not fall off? This was nicely answered by Aristotle’s theory of matter. Aristotle understood that there is no universal direction “down,” along which objects placed anywhere tend to fall. Rather, everywhere on Earth things made of the heavy elements earth and water tend to fall toward the center of the world, in agreement with observation. In this respect, Aristotle’s theory that the natural place of the heavier elements is in the center of the cosmos worked much like the modern theory of gravitation, with the important difference that for Aristotle there was just one center of the cosmos, while today we understand that any large mass will tend to contract to a sphere under the influence of its own gravitation, and then will attract other bodies toward its own center. Aristotle’s theory did not explain why any body other than the Earth should be a sphere, and yet he knew that at least the Moon is a sphere, reasoning from the gradual change of its phases, from full to new and back again.5 After Aristotle, the overwhelming consensus among astronomers and philosophers (aside from a few like Lactantius) was that the Earth is a sphere. With the mind’s eye, Archimedes even saw the spherical shape of the Earth in a glass of water; in Proposition 2 of On Floating Bodies, he demonstrates, “The surface of any fluid at rest is the surface of a sphere whose center is the Earth.”6 (This would be true only in the absence of surface tension, which Archimedes neglected.) Now I come to what in some respects is the most impressive example of the application of mathematics to natural science in the ancient world: the work of Aristarchus of Samos. Aristarchus was born around 310 BC on the Ionian island of Samos; studied as a pupil of Strato of Lampsacus, the third head of the Lyceum in Athens; and then worked at Alexandria until his death around 230 BC. Fortunately, his masterwork On the Sizes and Distances of the Sun and Moon has survived.7 In it, Aristarchus takes four astronomical observations as postulates:

1. “At the time of Half Moon, the Moon’s distance from the Sun is less than a quadrant by one- thirtieth of a quadrant.” (That is, when the Moon is just half full, the angle between the lines of sight to the Moon and to the Sun is less than 90° by 3°, so it is 87°.) 2. The Moon just covers the visible disk of the Sun during a solar eclipse. 3. “The breadth of the Earth’s shadow is that of two Moons.” (The simplest interpretation is that at the position of the Moon, a sphere with twice the diameter of the Moon would just fill the Earth’s shadow during a lunar eclipse. This was presumably found by measuring the time from when one edge of the Moon began to be obscured by the Earth’s shadow to when it became entirely obscured, the time during which it was entirely obscured, and the time from then until the eclipse was completely over.) 4. “The Moon subtends one fifteenth part of the zodiac.” (The complete zodiac is a full 360° circle, but Aristarchus here evidently means one sign of the zodiac; the zodiac consists of 12 constellations, so one sign occupies an angle of 360°/12 = 30°, and one fifteenth part of that is 2°.)

From these assumptions, Aristarchus deduced in turn that: 1. The distance from the Earth to the Sun is between 19 and 20 times larger than the distance of the Earth to the Moon. 2. The diameter of the Sun is between 19 and 20 times larger than the diameter of the Moon. 3. The diameter of the Earth is between 108/43 and 60/19 times larger than the diameter of the Moon. 4. The distance from the Earth to the Moon is between 30 and 45/2 times larger than the diameter of the Moon.

At the time of his work, trigonometry was not known, so Aristarchus had to go through elaborate geometric constructions to get these upper and lower limits. Today, using trigonometry, we would get more precise results; for instance, we would conclude from point 1 that the distance from the Earth to the Sun is larger than the distance from the Earth to the Moon by the secant (the reciprocal of the cosine) of 87°, or 19.1, which is indeed between 19 and 20. (This and the other conclusions of Aristarchus are re-derived in modern terms in Technical Note 11.) From these conclusions, Aristarchus could work out the sizes of the Sun and Moon and their distances from the Earth, all in terms of the diameter of the Earth. In particular, by combining points 2 and 3, Aristarchus could conclude that the diameter of the Sun is between 361/60 and 215/27 times larger than the diameter of the Earth. The reasoning of Aristarchus was mathematically impeccable, but his results were quantitatively way off, because points 1 and 4 in the data he used as a starting point were badly in error. When the Moon is half full, the actual angle between the lines of sight to the Sun and to the Moon is not 87° but 89.853°, which makes the Sun 390 times farther away from the Earth than the Moon, and hence much larger than Aristarchus thought. This measurement could not possibly have been made by naked-eye astronomy, though Aristarchus could have correctly reported that when the Moon is half full the angle between the lines of sight to the Sun and Moon is not less than 87°. Also, the visible disk of the Moon subtends an angle of 0.519°, not 2°, which makes the distance of the Earth to the Moon more like 111 times the diameter the Moon. Aristarchus certainly could have done better than this, and there is a hint in Archimedes’ The Sand Reckoner that in later work he did so.* It is not the errors in his observations that mark the distance between the science of Aristarchus and our own. Occasional serious errors continue to plague observational astronomy and experimental physics. For instance, in the 1930s the rate at which the universe is expanding was thought to be about seven times faster than we now know it actually is. The real difference between Aristarchus and today’s astronomers and physicists is not that his observational data were in error, but that he never tried to judge the uncertainty in them, or even acknowledged that they might be imperfect. Physicists and astronomers today are trained to take experimental uncertainty very seriously. Even though as an undergraduate I knew that I wanted to be a theoretical physicist who would never do experiments, I was required along with all other physics students at Cornell to take a laboratory course. Most of our time in the course was spent estimating the uncertainty in the measurements we made. But historically, this attention to uncertainty was a long time in coming. As far as I know, no one in ancient or medieval times ever tried seriously to estimate the uncertainty in a measurement, and as we will see in Chapter 14, even Newton could be cavalier about experimental uncertainties. We see in Aristarchus a pernicious effect of the prestige of mathematics. His book reads like Euclid’s Elements: the data in points 1 through 4 are taken as postulates, from which his results are deduced with mathematical rigor. The observational error in his results was very much greater than the narrow ranges that he rigorously demonstrated for the various sizes and distances. Perhaps Aristarchus did not mean to say that the angle between the lines of sight to the Sun and the Moon when half full is really 87°, but only took that as an example, to illustrate what could be deduced. Not for nothing was Aristarchus known to his contemporaries as “the Mathematician,” in contrast to his teacher Strato, who was known as “the Physicist.” But Aristarchus did get one important point qualitatively correct: the Sun is much bigger than the Earth. To emphasize the point, Aristarchus noted that the volume of the Sun is at least (361/60)3 (about 218) times larger than the volume of the Earth. Of course, we now know that it is much bigger than that. There are tantalizing statements by both Archimedes and Plutarch that Aristarchus had concluded from the great size of the Sun that it is not the Sun that goes around the Earth, but the Earth that goes around the Sun. According to Archimedes in The Sand Reckoner,8 Aristarchus had concluded not only that the Earth goes around the Sun, but also that the Earth’s orbit is tiny compared with the distance to the fixed stars. It is likely that Aristarchus was dealing with a problem raised by any theory of the Earth’s motion. Just as objects on the ground seem to be moving back and forth when viewed from a carousel, so the stars ought to seem to move back and forth during the year when viewed from the moving Earth. Aristotle had seemed to realize this, when he commented9 that if the Earth moved, then “there would have to be passings and turnings of the fixed stars. Yet no such thing is observed. The same stars always rise and set in the same parts of the Earth.” To be specific, if the Earth goes around the Sun, then each star should seem to trace out in the sky a closed curve, whose size would depend on the ratio of the diameter of the Earth’s orbit around the Sun to the distance to the star. So if the Earth goes around the Sun, why didn’t ancient astronomers see this apparent annual motion of the stars, known as annual parallax? To make the parallax small enough to have escaped observation, it was necessary to assume that the stars are at least a certain distance away. Unfortunately, Archimedes in The Sand Reckoner made no explicit mention of parallax, and we don’t know if anyone in the ancient world used this argument to put a lower bound on the distance to the stars. Aristotle had given other arguments against a moving Earth. Some were based on his theory of natural motion toward the center of the universe, mentioned in Chapter 3, but one other argument was based on observation. Aristotle reasoned that if the Earth were moving, then bodies thrown straight upward would be left behind by the moving Earth, and hence would fall to a place different from where they were thrown. Instead, as he remarks,10 “heavy bodies forcibly thrown quite straight upward return to the point from which they started, even if they are thrown to an unlimited distance.” This argument was repeated many times, for instance by Claudius Ptolemy (whom we met in Chapter 4) around AD 150, and by Jean Buridan in the Middle Ages, until (as we will see in Chapter 10) an answer to this argument was given by Nicole Oresme. It might be possible to judge how far the idea of a moving Earth spread in the ancient world if we had a good description of an ancient orrery, a mechanical model of the solar system.* Cicero in On the Republic tells of a conversation about an orrery in 129 BC, twenty-three years before he himself was born. In this conversation, one Lucius Furius Philus was supposed to have told about an orrery made by Archimedes that had been taken after the fall of Syracuse by its conqueror Marcellus, and that was later seen in the house of Marcellus’ grandson. It is not easy to tell from this thirdhand account how the orrery worked (and some pages from this part of De Re Publica are missing), but at one point in the story Cicero quotes Philus as saying that on this orrery “were delineated the motion of the Sun and Moon and of those five stars that are called wanderers [planets],” which certainly suggests that the orrery had a moving Sun, rather than a moving Earth.11 As we will see in Chapter 8, long before Aristarchus the Pythagoreans had the idea that both the Earth and the Sun move around a central fire. For this they had no evidence, but somehow their speculations were remembered, while that of Aristarchus was almost forgotten. Just one ancient astronomer is known to have adopted the heliocentric ideas of Aristarchus: the obscure Seleucus of Seleucia, who flourished around 150 BC. In the time of Copernicus and Galileo, when astronomers and churchmen wanted to refer to the idea that the Earth moves, they called it Pythagorean, not Aristarchean. When I visited the island of Samos in 2005, I found plenty of bars and restaurants named for Pythagoras, but none for Aristarchus of Samos. It is easy to see why the idea of the Earth’s motion did not take hold in the ancient world. We do not feel this motion, and no one before the fourteenth century understood that there is no reason why we should feel it. Also, neither Archimedes nor anyone else gave any indication that Aristarchus had worked out how the motion of the planets would appear from a moving Earth. The measurement of the distance from the Earth to the Moon was much improved by Hipparchus, generally regarded as the greatest astronomical observer of the ancient world.12 Hipparchus made astronomical observations in Alexandria from 161 BC to 146 BC, and then continued until 127 BC, perhaps on the island of Rhodes. Almost all his writings have been lost; we know about his astronomical work chiefly from the testimony of Claudius Ptolemy, three centuries later. One of his calculations was based on the observation of an eclipse of the Sun, now known to have occurred on March 14, 189 BC. In this eclipse the disk of the Sun was totally hidden at Alexandria, but only four- fifths hidden on the Hellespont (the modern Dardanelles, between Asia and Europe). Since the apparent diameters of the Moon and Sun are very nearly equal, and were measured by Hipparchus to be about 33' (minutes of arc) or 0.55°, Hipparchus could conclude that the direction to the Moon as seen from the Hellespont and from Alexandria differed by one-fifth of 0.55°, or 0.11°. From observations of the Sun Hipparchus knew the latitudes of the Hellespont and Alexandria, and he knew the location of the Moon in the sky at the time of the eclipse, so he was able to work out the distance to the Moon as a multiple of the radius of the Earth. Considering the changes during a lunar month of the apparent size of the Moon, Hipparchus concluded that the distance from the Earth to the Moon varies from 71 to 83 Earth radii. The average distance is actually about 60 Earth radii. I should pause to say something about another great achievement of Hipparchus, even though it is not directly relevant to the measurement of sizes and distances. Hipparchus prepared a star catalog, a list of about 800 stars, with the celestial position given for each star. It is fitting that our best modern star catalog, which gives the positions of 118,000 stars, was made by observations from an artificial satellite named in honor of Hipparchus. The measurements of star positions by Hipparchus led him to the discovery of a remarkable phenomenon, which was not understood until the work of Newton. To explain this discovery, it is necessary to say something about how celestial positions are described. The catalog of Hipparchus has not survived, and we don’t know just how he described these positions. There are two possibilities commonly used from Roman times on. One method, used later in the star catalog of Ptolemy,13 pictures the fixed stars as points on a sphere, whose equator is the ecliptic, the path through the stars apparently traced in a year by the Sun. Celestial latitude and longitude locate stars on this sphere in the same way that ordinary latitude and longitude give the location of points on the Earth’s surface.* In a different method, which may have been used by Hipparchus,14 the stars are again taken as points on a sphere, but this sphere is oriented with the Earth’s axis rather than the ecliptic; the north pole of this sphere is the north celestial pole, about which the stars seem to revolve every night. Instead of latitude and longitude, the coordinates on this sphere are known as declination and right ascension. According to Ptolemy,15 the measurements of Hipparchus were sufficiently accurate for him to notice that the celestial longitude (or right ascension) of the star Spica had changed by 2° from what had been observed long before at Alexandria by the astronomer Timocharis. It was not that Spica had changed its position relative to the other stars; rather, the location of the Sun on the celestial sphere at the autumnal equinox, the point from which celestial longitude was then measured, had changed. It is difficult to be precise about how long this change took. Timocharis was born around 320 BC, about 130 years before Hipparchus; but it is believed that he died young around 280 BC, about 160 years before Hipparchus. If we guess that about 150 years separated their observations of Spica, then these observations indicate that the position of the Sun at the autumnal equinox changes by about 1° every 75 years.* At that rate, this equinoctal point would precess through the whole 360° circle of the zodiac in 360 times 75 years, or 27,000 years. Today we understand that the precession of the equinoxes is caused by a wobble of the Earth’s axis (like the wobble of the axis of a spinning top) around a direction perpendicular to the plane of its orbit, with the angle between this direction and the Earth’s axis remaining nearly fixed at 23.5°. The equinoxes are the dates when the line separating the Earth and the Sun is perpendicular to the Earth’s axis, so a wobble of the Earth’s axis causes the equinoxes to precess. We will see in Chapter 14 that this wobble was first explained by Isaac Newton, as an effect of the gravitational attraction of the Sun and Moon for the equatorial bulge of the Earth. It actually takes 25,727 years for the Earth’s axis to wobble by a full 360°. It is remarkable how accurately the work of Hipparchus predicted this great span of time. (By the way, it is the precession of the equinoxes that explains why ancient navigators had to judge the direction of north from the position in the sky of constellations near the north celestial pole, rather from the position of the North Star, Polaris. Polaris has not moved relative to the other stars, but in ancient times the Earth’s axis did not point at Polaris as it does now, and in the future Polaris will again not be at the north celestial pole.) Returning now to celestial measurement, all of the estimates by Aristarchus and Hipparchus expressed the size and distances of the Moon and Sun as multiples of the size of the Earth. The size of the Earth was measured a few decades after the work of Aristarchus by Eratosthenes. Eratosthenes was born in 273 BC at Cyrene, a Greek city on the Mediterranean coast of today’s Libya, founded around 630 BC, that had become part of the kingdom of the Ptolemies. He was educated in Athens, partly at the Lyceum, and then around 245 BC was called by Ptolemy III to Alexandria, where he became a fellow of the Museum and tutor to the future Ptolemy IV. He was made the fifth head of the Library around 234 BC. His main works—On the Measurement of the Earth, Geographic Memoirs, and Hermes—have all unfortunately disappeared, but were widely quoted in antiquity. The measurement of the size of the Earth by Eratosthenes was described by the Stoic philosopher Cleomedes in On the Heavens,16 sometime after 50 BC. Eratosthenes started with the observations that at noon at the summer solstice the Sun is directly overhead at Syene, an Egyptian city that Eratosthenes supposed to be due south of Alexandria, while measurements with a gnomon at Alexandria showed the noon Sun at the solstice to be one-fiftieth of a full circle, or 7.2°, away from the vertical. From this he could conclude that the Earth’s circumference is 50 times the distance from Alexandria to Syene. (See Technical Note 12.) The distance from Alexandria to Syene had been measured (probably by walkers, trained to make each step the same length) as 5,000 stadia, so the circumference of the Earth must be 250,000 stadia. How good was this estimate? We don’t know the length of the stadion as used by Eratosthenes, and Cleomedes probably didn’t know it either, since (unlike our mile or kilometer) it had never been given a standard definition. But without knowing the length of the stadion, we can judge the accuracy of Eratosthenes’ use of astronomy. The Earth’s circumference is actually 47.9 times the distance from Alexandria to Syene (modern Aswan), so the conclusion of Eratosthenes that the Earth’s circumference is 50 times the distance from Alexandria to Syene was actually quite accurate, whatever the length of the stadion.* In his use of astronomy, if not of geography, Eratosthenes had done quite well.