温伯格《解释世界》第四章 希腊化时代物理和技术

亚历山大去世后他的帝国分裂为几个国家。对科学史尤为重要的是埃及。埃及被希腊国王的继任者统治，从曾经是亚历山大手下将军的托勒密一世开始，一直到托勒密十五世—他是克里奥帕特拉与（或许是）凯撒大帝的儿子。安东尼与克里奥帕特拉于公元前31年在亚克提姆战役中被击败后不久，最后一位托勒密被暗杀，埃及被纳入罗马帝国。 从亚历山大到亚克提姆这段时期被称为希腊化时代（Hellenistic），该词（德文 Hellenismus）由约翰·古斯塔夫·德罗伊森19世纪30年代创造。我不知道德罗伊森是否有意为之，我觉得英文后缀“istic”具有贬义。比如仿古一词（archaistic），是用于描述对古代（archaic）的模仿，这个后缀似乎暗示希腊化时代文化不完全是希腊文化，而只是对公元前五世纪和四世纪古典时代成就的模仿。那些成就确实巨大，特别是在几何，戏剧，编史，建筑和雕刻方面，以及没有流传下来的其他古典艺术方面，如音乐和绘画。但事实上希腊化时代科学达到一个全新高度，其成就不只完全超越古典时代，而且只有十六和十七世纪发生的科学革命才可与之媲美。 希腊化时代最重要的科学中心是托勒密的首都亚历山大里亚，由亚历山大在尼罗河一个河口规划建造。亚历山大里亚成为希腊领域中最伟大的都城，在罗马帝国，其规模和财富仅次于罗马城。 公元前300年左右托勒密一世创建亚历山大博物馆作为他王宫的一部分。该博物馆原本是作为奉献给九位缪斯女神的文学和哲学研究中心。但是在公元前285年托勒密二世即位后，该博物馆也成为科学研究中心。虽然文学研究在该博物馆和亚历山大图书馆一直延续，但八位艺术女神的光芒完全被她们的科学姐妹—掌管天文的乌拉妮娅所掩盖。亚历山大博物馆和希腊科学远比托勒密王国更为长久，我们将会看到，古代科学中的一些最伟大的成就就是诞生于罗马帝国希腊领域，而且主要在亚历山大里亚。 希腊化时代埃及与希腊本土知识分子间的关系类似于二十世纪美国和欧洲间的联系。埃及的富饶以及至少托勒密前三世的慷慨支持，吸引了一大批早在雅典已经功成名就的学者到亚历山大里亚，就像二十世纪三十年代以后大批欧洲学者涌入美国。公元前300年前吕刻俄斯成员法勒鲁姆的德米特里厄斯成为博物馆的第一任主管，他把他在雅典的藏书也带了过来。大约在相同时间另一位吕刻俄斯成员兰萨库斯的斯特拉图被托勒密一世召到亚历山大里亚辅导他的儿子，可能是这个原因，他的儿子继承埃及王位后，博物馆转向科学研究。 在希腊化时代和罗马时代雅典与亚历山大里亚之间船的航行时间与二十世纪利物浦与纽约之间蒸汽船航行时间差不多，那时埃及与希腊之间人来人往。比如斯特拉图就没有在埃及留下，他后来又回到雅典成为吕刻俄斯的第三任校长。 斯特拉图是位极富洞察力的观测者。通过观察从屋檐下落的水滴如何在下落过程中越来越分离，开始连续的水体后来破裂为分离的水滴，他得出了落体向下加速的结论。这是由于下落最远的水滴下落时间也最长，由于加速，这些水滴比后面运行较短时间的水滴速度要快。 （见技术说明7）。斯特拉图也注意到当一个物体下落距离较短时，对地面的冲击很小，但是如果从很高处落下，对地面的冲击力巨大，从而说明落体速度下落过程中在增加。 像亚历山大里亚，米利都和雅典这样的希腊自然哲学中心本身也是商业中心，这点可能不是巧合。活跃的市场聚集了不同文化背景的人群，丰富了单一的农业。亚历山大里亚商业涉及范围甚广：海上货物从印度运输到地中海地区，期间穿越阿拉伯海，向上过红海，经陆地到尼罗河，从尼罗河往下一直运到亚历山大里亚。 但是亚历山大里亚与罗马理性氛围差异极大。首先博物馆学者不热衷于从泰勒斯到亚里士多德这些希腊学者一直关注的那种包罗万象的理论。正如弗洛里斯·科恩的评述：“雅典派思想博大，亚历山大派思想具体。”亚历山大派关注于去认识特定现象，从而能够获得实际进展。这其中包括光学和静水力学，尤其突出的是天文学，在第二部分我们将详细介绍。 希腊化时代希腊学者不再致力于研究一种包罗万象的理论，这不是一种退步。历史不断证实了解什么问题已经成熟，可以进行研究，什么问题还不够成熟，不是研究的时候，这对科学进展至关重要。比如二十世纪初包括亨德里克·洛伦兹和马克斯·亚伯拉罕在内的世界最优秀物理学家投身于研究新发现的电子的结构，结果一无所获。量子力学要等20年后才出现，在这之前无论谁去研究电子特性都不可能取得收获。阿尔伯特·爱因斯坦的狭义相对论之所以能够获得成功就在于他不去担心电子本身究竟是什么样的，而是关注于观测者的运动如何影响对物体（包括电子）的观察。爱因斯坦本人晚年试图解决自然力的统一，但没有取得进展，因为那时人们对这些力还知之不多。 希腊化时代科学家与他们的古典先驱另一个主要不同之处是希腊化时代不再囿于求知和实用（希腊语的episteme与techne, 或拉丁文的scientia与arts）间的区别。历史上许多哲学家对发明家的看法就像“仲夏夜之梦”中宫廷侍从菲劳斯特莱特对皮特·昆斯和他的演员的描述：“一些拥有粗糙的双手，现在在雅典做工的汉子，从来没有用过脑子。”作为一名从事像基本粒子和宇宙学这种不能马上转入实用学科研究的物理学家，我当然不会对纯粹的求知有不同看法，不过为了满足人类需要而去从事科学研究可以迫使科学家放弃诗意幻想，更加面对现实。 当然从早期人类学会如何用火烹食和如何用石头撞击来制造简单的工具开始，人们就对技术进步倍感兴趣。但是古典知识分子一直坚守的恃才傲物态度使柏拉图和亚里士多德这样的哲学家不去考虑将他们的理论付诸于技术应用。 虽然这种偏见在希腊化时代没有完全消失，但其影响力已然很弱。即使出生普通的人现在也可能以发明创造而一举成名。亚历山大里亚的克特西比乌是一个很好的例子。他是一位理发师的儿子，在公元前250年左右发明了压力泵和比早期水钟精确的多的新一代水钟，该水钟控制流出容器内的水位保持不变。克特西比乌声名远扬，两百年后罗马的维特鲁维奥在专著《关于建筑》中还对他做了记载。 希腊化时代一些技术是被从事系统性科学探索的学者所开发，这些探索有时也用于来引导技术，这点很重要。例如大约公元前250年生活在亚历山大里亚的拜占庭的斐罗（他是一位工程兵），在《机械设计》中介绍了避难所，堡垒，围攻，以及石弩（部分基于克特西比乌的设计）。但是在《气体力学》中，斐罗也给出了实验论证来支持阿那克西米尼，亚里士多德，以及斯特拉图关于空气真实存在的观点。比如如果一个空瓶开口向下浸入水中，水不会流入瓶中，因为瓶中空气无处可去。如果给瓶子开个小口可以让空气离开，水就会流入瓶子并将之充满。 一个实用科学学科时常受到希腊科学家的关注，甚至一直延续到罗马时代：即光的特性。这种对光特性的关注可以追溯到希腊化时代早期欧几里德的工作。 对欧几里德的一生人们知之不多。据说他生活在托勒密一世时期，可能创建了亚历山大博物馆的数学研究。他最为知名的成就是《几何原本》，该书从一系列几何定义，公理和假设开始，然后基本严格地从简到繁证明定理。但是欧几里德还写了《光学》，主要研究透视问题，研究镜面反射的《反射光学》一书也挂他的名字，但是现代历史学家怀疑他是否真是该书作者。 认真想起来，反射有些特别之处。当你观察平面镜中一个较小物体的反射时，你会看到物体图像位于确定位置，不会在镜面散开。然而从物体到镜面不同点然后回到眼睛可以有许多路径（注：古代世界通常认为当我们看某物体时，光线从眼睛运动到该物体，就好像视力是种触摸，我们需要伸出去才能看到我们想看的物体。后面的讨论我都采用现代解释，视觉是光线从物体运动到眼睛。不过好在分析反射和折射时，光的运动方向没有影响。）很明显光实际上只经过了一条路径，这样物体图像才呈现在这条路径与镜面接触点。但是镜面上这个点的位置由什么决定哪？在《反射光学》一书中有条基本原理回答了这个问题：射到平面镜的入射光与反射光与平面镜夹角相等。只有一条光线满足这个条件。 我们不知道希腊化时代的哪位科学家发现了这个原理。但我们确实知道，公元60年左右亚历山大里亚的希罗在他的《反射光学》书中给出了等角律的证明，该证明基于这样的假设：光线从物体到镜面然后返回到观测者眼睛的行径路线是最短的路线（见技术说明8）。希罗这样解释该原理：“众所周知大自然不会做无用功，也不会做没必要的努力。”可能他被亚里士多德的目的论所启发—一切发生皆有目的。但是希罗是对的，我们在第十四章会看到的，17世纪惠更斯从光的波动特征推导出最短距离原理（实际是最短时间）。这位探索光学基本原理的希罗应用他掌握的知识发明了实用测量仪器--经纬仪， 他也对虹吸作用做出了解释，并且设计了军用石弩和原始蒸汽机。 伟大的天文学家克罗蒂斯·托勒密（他不是国王的亲属）于公元150年左右在亚历山大里亚进一步促进了光学研究。他的著作《光学》拉丁文译文流传了下来，该拉丁文译本译自阿拉伯语译本，阿拉伯语译本又译自希腊原文（或许中间还有失传的古代叙利亚语），但阿拉伯语译本和希腊原文都已失传。在该书中托勒密描述了通过测量证实欧几里德和希罗的等角律。他把这个定律也应用到曲镜面的反射，像今天在游乐园里看到的那些。他对曲面镜反射的理解完全正确，这种反射与平面镜的反射一样，光线正切于反射点。 在《光学》最后一卷中托勒密研究了折射，当光线从一种透明物质比如空气到另一种透明物质比如水所产生的光线弯曲。他将一个周边刻有测量角的圆盘悬挂在盛满水的容器中部。通过对固定在圆盘上的管子内一个物体在水中下沉的观察，他可以测量入射光和反射光与垂直于表面直线的夹角，测量误差在几分之一到几度。在第十三章我们会看到，费马在17世纪得出关于这种夹角的正确定律，他把希罗应用到反射的原理做了简单的推广：折射时光线从物体到眼睛的行径路线不是距离最短，而是最快行径时间。对反射来说最短距离与最快时间没有差别，入射光与反射光穿过同一介质，距离与时间成正比。但对折射不然，光线从一种介质到另一种介质光速会发生变化。托勒密对此还不了解。正确的折射定律—斯涅尔定律（在法国称为笛卡尔定律）在17世纪早期才通过实验得以发现。 阿基米德可能是希腊化时代（或许任一时代）最成就斐然的科学家-技术专家。阿基米德公元前200多年生活在西西里希腊城市叙拉古，据说他至少到访过亚历山大里亚一次以上。他发明了多种滑轮和螺丝，以及各种打仗工具，比如基于对杠杆原理的掌握发明的“起重机”，可以将泊靠在岸边的战船抓起和掀翻。他发明的螺旋提水器在农业上应用了几个世纪，这种器械可以把溪水吸上来灌溉农田。至于阿基米德用曲面镜聚光点燃罗马战船保卫叙拉古的故事肯定是杜撰，不过这也反映出他在技术发明成就方面的名望。 在《论物体平衡》一书中阿基米德提出控制平衡法则：如果杠杆支点两端的物体重量反比于支点两边长度，则杠杆处于平衡状态。比如杠杆一端有5磅重，另一端一磅重，如果从支点到一磅重一端是从支点到5磅重一端的5倍长，那么杠杆平衡。 阿基米德最伟大的物理成就体现在他的专著《论浮体》一书中。阿基米德推断如果一部分流体被上部流体（或浮体，半潜体等）向下压的强度大于另一部分流体，流体将会发生流动直到流体各部位承担同样重量。他这样描述： 假设一种流体具有这样的特性，流体各部分均匀且连续，受力小的部分会受到受力大的部分的拉动。如果有任何物体沉在流体中以及流体受到压缩，那么流体的每一部分都会受到垂直向上流体的挤压。 阿基米德由此推断浮体将会下沉到其排出水的重量等于其自身重量的水位。（这就是为什么船的重量被称为“排水量”。）同样如果一个实体太重，不能浮在液面，如果从天枰臂用一根绳悬挂浸在液体中，则该实体“重量会轻于其真实重量，减轻重量等于排出液量。”（见技术说明9）。从物体真实重量与其悬挂在水中减轻重量的比值可以得出物体的“比重”，比重定义为物体重量与同体积水重量的比值。每种物质都有其特定比重，金的比重为19.32， 铅为11.34， 等等。阿基米德可以借助于从对静态流体的系统理论研究得出的方法来分辨王冠是否由纯金打造还是掺和了廉价金属。阿基米德是否实际应用了这种方法并不清楚，但是人们在好几个世纪都用此办法来鉴定物体成分。 阿基米德更加令人瞩目的成就在数学方面。基于一种积分计算雏形技术，他可以计算各种平面图形和实体的面积和体积。比如圆面积等于周长的一半乘以半径。（见技术说明10）。应用几何方法，他得出我们今天称之为PI（阿基米德没有用这个术语）--即周长与直径之比，其值在31/7和3 10/71之间。西塞罗说他曾经在阿基米德的墓碑上看到一个圆柱体外切一个球体，球体表面与与圆柱体的边和上下底相接触，类似于一个乒乓球正好放入一个易拉罐中。很明显阿基米德非常骄傲他证实了这种情形下球体体积等于圆柱体体积的三分之二。 罗马历史学家李维记录了有关阿基米德去世的一个轶闻。阿基米德死于公元前212年，在罗马士兵由马尔库斯·克劳狄乌斯·马塞拉斯率领下攻克叙拉古期间。（第二次步匿战争期间叙拉古被一亲迦太基派系掌管）。罗马士兵涌入叙拉古，据说阿基米德正在计算几何问题，一名罗马士兵发现了他，将他杀死。 除了无与伦比的阿基米德之外，另一位最伟大的希腊化时代数学家是比他年轻一些的同代人阿波罗尼奥斯。阿波罗尼奥斯约公元前262年出生于位于小亚细亚东南海岸的城市佩尔格，该城当时处于正在兴起的帕加马王国的控制之下。他在托勒密三世和四世统治期间（公元前247年到203年）到访过亚历山大里亚。他最伟大的成就是关于圆锥曲线：如椭圆，抛物线和双曲线。这些曲线可以由一个平面以不同角度切割圆锥体而形成。这些圆锥曲线理论后来对开普勒和牛顿至关重要，而在古代并没有得到物理应用。 不过这些辉煌的希腊数学过于注重几何，缺乏一些现代物理不可缺少的方法。希腊人从来没有学会书写和应用代数公式。像E = mc2和F = ma这样的公式是现代物理的核心。（大约在公元250年活跃在亚历山大里亚的丢番图将公式应用到纯数学，但是他方程中的符号只是代表整数和有理数，与今天物理上应用的公式有很大不同。）即使要用到几何，现代物理学家也一般会用代数来表示几何，这是17世纪勒内·笛卡尔和其他人发明的解析几何方法，在第十三章将对此作出介绍。可能是由于希腊数学当之无愧的声誉，这种几何风格一直沿用到17世纪的科学革命时期。当伽利略在他1623年创作的《试金者》一书中力图歌颂数学时，他讲到几何（注）：“这本包罗万象的作品里的哲学不时另我们眼界大开，这就是宇宙；但是除非一个人首先掌握了其所用的语言，了解书里的角色，否则无法明白书的内容。这里用的语言是数学语言，里面的角色是三角，圆，以及其他几何图形；没有这些知识人们不可能读懂书中任何内容，他会迷失在黑暗的迷宫中。“ （注：《试金者》是伽利略对耶稣会教士的反驳，以书信形式写给教皇侍从弗吉尼诺·切萨里尼。我们在第十一章会看到，伽利略在《试金者》中攻击了第谷·布拉赫和耶稣会教士有关彗星比月亮距地球更远的正确观点。（这里的引用来自莫里斯·菲诺切罗的译本《纯粹的伽利略》，2008年出版，第183页）。）伽利略重视几何甚于代数，这点落后于他所处的时代。他自己的作品用到了代数，但与他同代一些人相比他更侧重于几何，比现代物理杂志里涉及到的几何要多得多。 现代社会纯科学有其自身领地，纯科学追求知识本身而不去关注实用。古代社会在科学家认识到验证理论的必要性之前，科技应用具有特别重要意义，因为当一位科学家不是只夸夸其谈，而是真正将科学理论赋之于实用并取得成效的话，会得到额外重视。如果阿基米德应用他的测量比重方法将镀金铝皇冠误认为纯金皇冠，他在叙拉古就不会那么受欢迎。 我不想夸大科技在希腊化和罗马时代的重要程度。许多克特西比乌和希罗发明的设备不过是些玩具或舞台道具。历史学家推测在一种基于奴隶制的经济模式中没有对节省劳动力设备的需求，否则希罗玩具一样的蒸汽机可能会得以发展。军用和民用工程在古代非常重要，亚历山大里亚的国王一直支持对石弩和其他武器的研究，这些研究可能就在亚历山大博物馆进行，但是似乎并没有从当时的科学中得到帮助。 希腊科学有一个非常具有实用价值而且也最为发达的领域，这就是天文学。我们在第二部分将做介绍。 上面讲到科学的实际应用会为科学研究提供动力，但医疗确是个例外。在现代社会之前那些备受尊重的医生一直固守传统疗法，比如放血疗法，但其疗效从来没有经过试验验证，事实上其对人造成的伤害远大于疗效。当19世纪引入真正有效的抗菌术—一种具备科学基础的技术时，大多数医生起初极力抗拒使用。到二十世纪才要求新药在被批准使用前必须经过临床试验。早期医生确实学习识别不同疾病，而且对有些病还有有效疗法，比如用含奎宁的金鸡纳树皮治疗痢疾。他们知道如何准备止痛剂，麻醉剂，催吐剂，泻药，安眠药，以及毒药。但是常有人说二十世纪之前，普通病人不去看医生的话可能会过的更好。 那时的医疗并非没有理论基础。那时有体液学说，即存在四种体液：血液质，粘液质，黒胆质，以及黄胆质。这些体液分别控制我们乐观，冷漠，犹豫，和暴躁。体液学说是由古典希腊时期希泼克拉底，或他的同事以他的名义所创建。正如后来约翰·多恩在“早安”中所描述，该理论认为“凡是死亡皆为调和不当所致”。罗马时期帕加马的盖伦采纳了体液学说，他的著作在公元1000年后先在阿拉伯，然后在欧洲具有极大影响力。我没有看到任何对体液学说有效性的实验验证，虽然该理论被广泛接受（现在的印度传统医学--阿育吠陀医学仍然采用体液学说，不过只有三种体液：粘液质，胆汁质，和风质。） 除了体液学说，欧洲的医生还需要掌握另一门医疗理论：占星术，这种应用一直延续到现代。讽刺的是由于医药医生可以在大学学习这些理论，他们比外科医生拥有更高的声望，而实际上外科医生掌握真正有用的东西，比如处理骨折，但这些直到现代才在大学接受训练。 那么为什么这些教条医疗持续了如此长的时间而没有通过实践得以修正？我们可以肯定的是生物学的进展确实比天文学要艰难的多。我们在第八章会讨论太阳，月亮，以及其他行星的视运行如此有规律，这样人们非常容易发现早期理论的不足。历经几个世纪的不断认识，人们可以得出更加完善的理论。但是如果一个博学的医生虽然付诸最大努力仍然没能挽回患者的生命，谁又能理清原由。也许病人就医太晚，也许没有完全遵循医嘱。至少体液学说和占星术有些像科学，否则又能如何？难道回到给阿斯克勒皮俄斯供奉动物？ 另外一个因数可能是对病人而言最重要的是希望疾病可以得到治愈。这赋予了医生特有的权威。医生也会维护这种权威以便行使他们的治疗方案。不只在医疗方面，权威人士一般都会拒绝可能会使他们的权威受到损失的任何调查。

Following Alexander’s death his empire split into several successor states. Of these, the most important for the history of science was Egypt. Egypt was ruled by a succession of Greek kings, starting with Ptolemy I, who had been one of Alexander’s generals, and ending with Ptolemy XV, the son of Cleopatra and (perhaps) Julius Caesar. This last Ptolemy was murdered soon after the defeat of Antony and Cleopatra at Actium in 31 BC, when Egypt was absorbed into the Roman Empire. This age, from Alexander to Actium,1 is commonly known as the Hellenistic period, a term (in German, Hellenismus) coined in the 1830s by Johann Gustav Droysen. I don’t know if this was intended by Droysen, but to my ear there is something pejorative about the English suffix “istic.” Just as “archaistic,” for instance, is used to describe an imitation of the archaic, the suffix seems to imply that Hellenistic culture was not fully Hellenic, that it was a mere imitation of the achievements of the Classical age of the fifth and fourth centuries BC. Those achievements were very great, especially in geometry, drama, historiography, architecture, and sculpture, and perhaps in other arts whose Classical productions have not survived, such as music and painting. But in the Hellenistic age science was brought to heights that not only dwarfed the scientific accomplishments of the Classical era but were not matched until the scientific revolution of the sixteenth and seventeenth centuries. The vital center of Hellenistic science was Alexandria, the capital city of the Ptolemies, laid out by Alexander at one mouth of the Nile. Alexandria became the greatest city in the Greek world; and later, in the Roman Empire, it was second only to Rome in size and wealth. Around 300 BC Ptolemy I founded the Museum of Alexandria, as part of his royal palace. It was originally intended as a center of literary and philological studies, dedicated to the nine Muses. But after the accession of Ptolemy II in 285 BC the Museum also became a center of scientific research. Literary studies continued at the Museum and Library of Alexandria, but now at the Museum the eight artistic Muses were outshone by their one scientific sister—Urania, the Muse of astronomy. The Museum and Greek science outlasted the kingdom of the Ptolemies, and, as we shall see, some of the greatest achievements of ancient science occurred in the Greek half of the Roman Empire, and largely in Alexandria. The intellectual relations between Egypt and the Greek homeland in Hellenistic times were something like the connections between America and Europe in the twentieth century.2 The riches of Egypt and the generous support of at least the first three Ptolemies brought to Alexandria scholars who had made their names in Athens, just as European scholars flocked to America from the 1930s on. Starting around 300 BC, a former member of the Lyceum, Demetrius of Phaleron, became the first director of the Museum, bringing his library with him from Athens. At around the same time Strato of Lampsacus, another member of the Lyceum, was called to Alexandria by Ptolemy I to serve as tutor to his son, and may have been responsible for the turn of the Museum toward science when that son succeeded to the throne of Egypt. The sailing time between Athens and Alexandria during the Hellenistic and Roman periods was similar to the time it took for a steamship to go between Liverpool and New York in the twentieth century, and there was a great deal of coming and going between Egypt and Greece. For instance, Strato did not stay in Egypt; he returned to Athens to become the third director of the Lyceum. Strato was a perceptive observer. He was able to conclude that falling bodies accelerate downward, by observing how drops of water falling from a roof become farther apart as they fall, a continuous stream of water breaking up into separating drops. This is because the drops that have fallen farthest have also been falling longest, and since they are accelerating this means that they are traveling faster than drops following them, which have been falling for a shorter time. (See Technical Note 7.) Strato noted also that when a body falls a very short distance the impact on the ground is negligible, but if it falls from a great height it makes a powerful impact, showing that its speed increases as it falls.3 It is probably no coincidence that centers of Greek natural philosophy like Alexandria as well as Miletus and Athens were also centers of commerce. A lively market brings together people from different cultures, and relieves the monotony of agriculture. The commerce of Alexandria was far- ranging: seaborne cargoes being taken from India to the Mediterranean world would cross the Arabian Sea, go up the Red Sea, then go overland to the Nile and down the Nile to Alexandria. But there were great differences in the intellectual climates of Alexandria and Athens. For one thing, the scholars of the Museum generally did not pursue the kind of all-embracing theories that had preoccupied the Greeks from Thales to Aristotle. As Floris Cohen has remarked,4 “Athenian thought was comprehensive, Alexandrian piecemeal.” The Alexandrians concentrated on understanding specific phenomena, where real progress could be made. These topics included optics and hydrostatics, and above all astronomy, the subject of Part II. It was no failing of the Hellenistic Greeks that they retreated from the effort to formulate a general theory of everything. Again and again, it has been an essential feature of scientific progress to understand which problems are ripe for study and which are not. For instance, leading physicists at the turn of the twentieth century, including Hendrik Lorentz and Max Abraham, devoted themselves to understanding the structure of the recently discovered electron. It was hopeless; no one could have made progress in understanding the nature of the electron before the advent of quantum mechanics some two decades later. The development of the special theory of relativity by Albert Einstein was made possible by Einstein’s refusal to worry about what electrons are. Instead he worried about how observations of anything (including electrons) depend on the motion of the observer. Then Einstein himself in his later years addressed the problem of the unification of the forces of nature, and made no progress because no one at the time knew enough about these forces. Another important difference between Hellenistic scientists and their Classical predecessors is that the Hellenistic era was less afflicted by a snobbish distinction between knowledge for its own sake and knowledge for use—in Greek, episteme versus techne (or in Latin, scientia versus ars). Throughout history, many philosophers have viewed inventors in much the same way that the court chamberlain Philostrate in A Midsummer Night’s Dream described Peter Quince and his actors: “Hard-handed men, who work now in Athens, and never yet labor’d with their minds.” As a physicist whose research is on subjects like elementary particles and cosmology that have no immediate practical application, I am certainly not going to say anything against knowledge for its own sake, but doing scientific research to fill human needs has a wonderful way of forcing the scientist to stop versifying and to confront reality.5 Of course, people have been interested in technological improvement since early humans learned how to use fire to cook food and how to make simple tools by banging one stone on another. But the persistent intellectual snobbery of the Classical intelligentsia kept philosophers like Plato and Aristotle from directing their theories toward technological applications. Though this prejudice did not disappear in Hellenistic times, it became less influential. Indeed, people, even those of ordinary birth, could become famous as inventors. A good example is Ctesibius of Alexandria, a barber’s son, who around 250 BC invented suction and force pumps and a water clock that kept time more accurately than earlier water clocks by keeping a constant level of water in the vessel from which the water flowed. Ctesibius was famous enough to be remembered two centuries later by the Roman Vitruvius in his treatise On Architecture. It is important that some technology in the Hellenistic age was developed by scholars who were also concerned with systematic scientific inquiries, inquiries that were sometimes themselves used in aid of technology. For instance, Philo of Byzantium, who spent time in Alexandria around 250 BC, was a military engineer who in Mechanice syntaxism wrote about harbors, fortifications, sieges, and catapults (work based in part on that of Ctesibius). But in Pneumatics, Philo also gave experimental arguments supporting the view of Anaximenes, Aristotle, and Strato that air is real. For instance, if an empty bottle is submerged in water with its mouth open but facing downward, no water will flow into it, because there is nowhere for the air in the bottle to go; but if a hole is opened so that air is allowed to leave the bottle, then water will flow in and fill the bottle.6 There was one scientific subject of practical importance to which Greek scientists returned again and again, even into the Roman period: the behavior of light. This concern dates to the beginning of the Hellenistic era, with the work of Euclid. Little is known of the life of Euclid. He is believed to have lived in the time of Ptolemy I, and may have founded the study of mathematics at the Museum in Alexandria. His best-known work is the Elements,7 which begins with a number of geometric definitions, axioms, and postulates, and moves on to more or less rigorous proofs of increasingly sophisticated theorems. But Euclid also wrote the Optics, which deals with perspective, and his name is associated with the Catoptrics, which studies reflection by mirrors, though modern historians do not believe that he was its author. When one thinks of it, there is something peculiar about reflection. When you look at the reflection of a small object in a flat mirror, you see the image at a definite spot, not spread out over the mirror. Yet there are many paths one can draw from the object to various spots on the mirror and then to the eye.* Apparently there is just one path that is actually taken, so that the image appears at the one point where this path strikes the mirror. But what determines the location of that point on the mirror? In the Catoptrics there appears a fundamental principle that answers this question: the angles that a light ray makes with a flat mirror when it strikes the mirror and when it is reflected are equal. Only one light path can satisfy this condition. We don’t know who in the Hellenistic era actually discovered this principle. We do know, though, that sometime around AD 60 Hero of Alexandria in his own Catoptrics gave a mathematical proof of the equal-angles rule, based on the assumption that the path taken by a light ray in going from the object to the mirror and then to the eye of the observer is the path of shortest length. (See Technical Note 8.) By way of justification for this principle, Hero was content to say only, “It is agreed that Nature does nothing in vain, nor exerts herself needlessly.”8 Perhaps he was motivated by the teleology of Aristotle —everything happens for a purpose. But Hero was right; as we will see in Chapter 14, in the seventeenth century Huygens was able to deduce the principle of shortest distance (actually shortest time) from the wave nature of light. The same Hero who explored the fundamentals of optics used that knowledge to invent an instrument of practical surveying, the theodolite, and also explained the action of siphons and designed military catapults and a primitive steam engine. The study of optics was carried further about AD 150 in Alexandria by the great astronomer Claudius Ptolemy (no kin of the kings). His book Optics survives in a Latin translation of a lost Arabic version of the lost Greek original (or perhaps of a lost Syriac intermediary). In this book Ptolemy described measurements that verified the equal-angles rule of Euclid and Hero. He also applied this rule to reflection by curved mirrors, of the sort one finds today in amusement parks. He correctly understood that reflections in a curved mirror are just the same as if the mirror were a plane, tangent to the actual mirror at the point of reflection. In the final book of Optics Ptolemy also studied refraction, the bending of light rays when they pass from one transparent medium such as air to another transparent medium such as water. He suspended a disk, marked with measures of angle around its edge, halfway in a vessel of water. By sighting a submerged object along a tube mounted on the disk, he could measure the angles that the incident and refracted rays make with the normal to the surface, a line perpendicular to the surface, with an accuracy ranging from a fraction of a degree to a few degrees.9 As we will see in Chapter 13, the correct law relating these angles was worked out by Fermat in the seventeenth century by a simple extension of the principle that Hero had applied to reflection: in refraction, the path taken by a ray of light that goes from the object to the eye is not the shortest, but the one that takes the least time. The distinction between shortest distance and least time is irrelevant for reflection, where the reflected and incident ray are passing through the same medium, and distance is simply proportional to time; but it does matter for refraction, where the speed of light changes as the ray passes from one medium to another. This was not understood by Ptolemy; the correct law of refraction, known as Snell’s law (or in France, Descartes’ law), was not discovered experimentally until the early 1600s AD. The most impressive of the scientist-technologists of the Hellenistic era (or perhaps any era) was Archimedes. Archimedes lived in the 200s BC in the Greek city of Syracuse in Sicily, but is believed to have made at least one visit to Alexandria. He is credited with inventing varieties of pulleys and screws, and various instruments of war, such as a “claw,” based on his understanding of the lever, with which ships lying at anchor near shore could be seized and capsized. One invention used in agriculture for centuries was a large screw, by which water could be lifted from streams to irrigate fields. The story that Archimedes used curved mirrors that concentrated sunlight to defend Syracuse by setting Roman ships on fire is almost certainly a fable, but it illustrates his reputation for technological wizardry. In On the Equilibrium of Bodies, Archimedes worked out the rule that governs balances: a bar with weights at both ends is in equilibrium if the distances from the fulcrum on which the bar rests to both ends are inversely proportional to the weights. For instance, a bar with five pounds at one end and one pound at the other end is in equilibrium if the distance from the fulcrum to the one-pound weight is five times larger than the distance from the fulcrum to the five-pound weight. The greatest achievement of Archimedes in physics is contained in his book On Floating Bodies.10 Archimedes reasoned that if some part of a fluid was pressed down harder than another part by the weight of fluid or floating or submerged bodies above it, then the fluid would move until all parts were pressed down by the same weight. As he put it,

Let it be supposed that a fluid is of such a character that, the parts lying evenly and being continuous, that part which is thrust the less is driven along by that which is thrust the more; and that each of its parts is thrust by the fluid which is above it in a perpendicular direction if the fluid be sunk in anything and compressed by anything else.

From this Archimedes deduced that a floating body would sink to a level such that the weight of the water displaced would equal its own weight. (This is why the weight of a ship is called its “displacement.”) Also, a solid body that is too heavy to float and is immersed in the fluid, suspended by a cord from the arm of a balance, “will be lighter than its true weight by the weight of the fluid displaced.” (See Technical Note 9.) The ratio of the true weight of a body and the decrease in its weight when it is suspended in water thus gives the body’s “specific gravity,” the ratio of its weight to the weight of the same volume of water. Each material has a characteristic specific gravity: for gold it is 19.32, for lead 11.34, and so on. This method, deduced from a systematic theoretical study of fluid statics, allowed Archimedes to tell whether a crown was made of pure gold or gold alloyed with cheaper metals. It is not clear that Archimedes ever put this method into practice, but it was used for centuries to judge the composition of objects. Even more impressive were Archimedes’ achievements in mathematics. By a technique that anticipated the integral calculus, he was able to calculate the areas and volumes of various plane figures and solid bodies. For instance, the area of a circle is one-half the circumference times the radius (see Technical Note 10). Using geometric methods, he was able to show that what we call pi (Archimedes did not use this term), the ratio of the circumference of a circle to its diameter, is between 31/7 and 310/71. Cicero said that he had seen on the tombstone of Archimedes a cylinder circumscribed about a sphere, the surface of the sphere touching the sides and both bases of the cylinder, like a single tennis ball just fitting into a tin can. Apparently Archimedes was most proud of having proved that in this case the volume of the sphere is two-thirds the volume of the cylinder. There is an anecdote about the death of Archimedes, related by the Roman historian Livy. Archimedes died in 212 BC during the sack of Syracuse by Roman soldiers under Marcus Claudius Marcellus. (Syracuse had been taken over by a pro-Carthaginian faction during the Second Punic War.) As Roman soldiers swarmed over Syracuse, Archimedes was supposedly found by the soldier who killed him, while he was working out a problem in geometry. Aside from the incomparable Archimedes, the greatest Hellenistic mathematician was his younger contemporary Apollonius. Apollonius was born around 262 BC in Perga, a city on the southeast coast of Asia Minor, then under the control of the rising kingdom of Pergamon, but he visited Alexandria in the times of both Ptolemy III and Ptolemy IV, who between them ruled from 247 to 203 BC. His great work was on conic sections: the ellipse, parabola, and hyperbola. These are curves that can be formed by a plane slicing through a cone at various angles. Much later, the theory of conic sections was crucially important to Kepler and Newton, but it found no physical applications in the ancient world. Brilliant work, but with its emphasis on geometry, there were techniques missing from Greek mathematics that are essential in modern physical science. The Greeks never learned to write and manipulate algebraic formulas. Formulas like E = mc2 and F = ma are at the heart of modern physics. (Formulas were used in purely mathematical work by Diophantus, who flourished in Alexandria around AD 250, but the symbols in his equations were restricted to standing for whole or rational numbers, quite unlike the symbols in the formulas of physics.) Even where geometry is important, the modern physicist tends to derive what is needed by expressing geometric facts algebraically, using the techniques of analytic geometry invented in the seventeenth century by René Descartes and others, and described in Chapter 13. Perhaps because of the deserved prestige of Greek mathematics, the geometric style persisted until well into the scientific revolution of the seventeenth century. When Galileo in his 1623 book The Assayer wanted to sing the praises of mathematics, he spoke of geometry:* “Philosophy is written in this all-encompassing book that is constantly open to our eyes, that is the universe; but it cannot be understood unless one first learns to understand the language and knows the characters in which it is written. It is written in mathematical language, and its characters are triangles, circles, and other geometrical figures; without these it is humanly impossible to understand a word of it, and one wanders in a dark labyrinth.” Galileo was somewhat behind the times in emphasizing geometry over algebra. His own writing uses some algebra, but is more geometric than that of some of his contemporaries, and far more geometric than what one finds today in physics journals. In modern times a place has been made for pure science, science pursued for its own sake without regard to practical applications. In the ancient world, before scientists learned the necessity of verifying their theories, the technological applications of science had a special importance, for when one is going to use a scientific theory rather than just talk about it, there is a large premium on getting it right. If Archimedes by his measurements of specific gravity had identified a gilded lead crown as being made of solid gold, he would have become unpopular in Syracuse. I don’t want to exaggerate the extent to which science-based technology was important in Hellenistic or Roman times. Many of the devices of Ctesibius and Hero seem to have been no more than toys, or theatrical props. Historians have speculated that in an economy based on slavery there was no demand for laborsaving devices, such as might have been developed from Hero’s toy steam engine. Military and civil engineering were important in the ancient world, and the kings in Alexandria supported the study of catapults and other artillery, perhaps at the Museum, but this work does not seem to have gained much from the science of the time. The one area of Greek science that did have great practical value was also the one that was most highly developed. It was astronomy, to which we will turn in Part II.

There is a large exception to the remark above that the existence of practical applications of science provided a strong incentive to get the science right. It is the practice of medicine. Until modern times the most highly regarded physicians persisted in practices, like bleeding, whose value had never been established experimentally, and that in fact did more harm than good. When in the nineteenth century the really useful technique of antisepsis was introduced, a technique for which there was a scientific basis, it was at first actively resisted by most physicians. Not until well into the twentieth century were clinical trials required before medicines could be approved for use. Physicians did learn early on to recognize various diseases, and for some they had effective remedies, such as Peruvian bark—which contains quinine—for malaria. They knew how to prepare analgesics, opiates, emetics, laxatives, soporifics, and poisons. But it is often remarked that until sometime around the beginning of the twentieth century the average sick person would do better avoiding the care of physicians. It is not that there was no theory behind the practice of medicine. There was humorism, the theory of the four humors—blood, phlegm, black bile, and yellow bile, which (respectively) make us sanguine, phlegmatic, melancholy, or choleric. Humorism was introduced in classical Greek times by Hippocrates, or by colleagues of his whose writings were ascribed to him. As briefly stated much later by John Donne in “The Good Morrow,” the theory held that “whatever dies was not mixed equally.” The theory of humorism was adopted in Roman times by Galen of Pergamon, whose writings became enormously influential among the Arabs and then in Europe after about AD 1000. I am not aware of any effort while humorism was generally accepted ever to test its effectiveness experimentally. (Humorism survives today in Ayurveda, traditional Indian medicine, but with just three humors: phlegm, bile, and wind.) In addition to humorism, physicians in Europe until modern times were expected to understand another theory with supposed medical applications: astrology. Ironically, the opportunity for physicians to study these theories at universities gave medical doctors much higher prestige than surgeons, who knew how to do really useful things like setting broken bones but until modern times were not usually trained in universities. So why did the doctrines and practices of medicine continue so long without correction by empirical science? Of course, progress is harder in biology than in astronomy. As we will discuss in Chapter 8, the apparent motions of the Sun, Moon, and planets are so regular that it was not difficult to see that an early theory was not working very well; and this perception led, after a few centuries, to a better theory. But if a patient dies despite the best efforts of a learned physician, who can say what is the cause? Perhaps the patient waited too long to see the doctor. Perhaps he did not follow the doctor’s orders with sufficient care. At least humorism and astrology had an air of being scientific. What was the alternative? Going back to sacrificing animals to Aesculapius? Another factor may have been the extreme importance to patients of recovery from illness. This gave physicians authority over them, an authority that physicians had to maintain in order to impose their supposed remedies. It is not only in medicine that persons in authority will resist any investigation that might reduce their authority.