温伯格《解释世界》第十章 中世纪欧洲

随着罗马帝国在西方的衰落，拜占庭领地之外的欧洲变的贫穷，荒漠，基本上目不识丁。虽然一些文字保留了下来，但也只集中于教会，而且只是拉丁文。在中世纪的西欧基本上没有人认识希腊文。 一些希腊知识片段在修道院图书馆以拉丁文译本保存了下来，包括一部分柏拉图的《蒂迈欧篇》以及罗马贵族波伊提乌公元500年翻译的亚里士多德逻辑方面著作和一本算术课本。罗马人也用拉丁文写了一些对希腊科学的记载，其中最著名的是马提亚努斯·卡佩拉公元五世纪创作的百科全书，该书有个奇怪的书名--《水星与语言学的结合》，书中介绍了人文七科（作为语言学的基础）：文法，逻辑，修辞学，地理学，算术，天文学和音乐。在讨论天文学时马提亚努斯描写了赫拉克利德斯关于水星和金星围绕太阳运行，而太阳围绕地球运行的古老理论。千年之后哥白尼对此表达了赞美之意。即使有这些知识的碎片，中世纪早期的欧洲对希腊伟大的科学成就几乎一无所知。西欧持续不断受到入侵者哥特人，汪达尔人，胡人，阿瓦尔人，阿拉伯人，马扎尔人，以及北欧人的攻击，西欧人有太多其他事情去操心。 欧洲在十世纪和十一世纪开始重振。外族的入侵接近尾声，新技术提高了农业生产力。不过一直到十三世纪末欧洲才出现有影响力的科学研究，而真正的科学成果直到十六世纪才出现。然而这期间科学复兴的制度和知识基础已经打下。 公元十世纪和十一世纪—这段宗教时代—大量欧洲新财富自然流向教会，而不是农民。法国编年史家拉乌尔·格拉贝1030年做了极好地描述：“就像世界在晃动，丢掉旧衣，换上教会白袍。”隶属于教会的学校对后来教育尤为重要，比如在奥尔良，兰斯，拉昂，科隆，乌德勒支，桑斯，托莱多，沙特尔，以及巴黎的学校。 这些学校不只对牧师进行宗教培训，而且也培训他们罗马时代流传下来的世俗人文知识，这其中部分是基于波伊提乌和马提亚努斯的著作：包括文法，逻辑和修辞学构成的语文三科，以及特别是在沙特尔教授的由算术，几何，天文和音乐构成的数学四科。一些学校的历史可以追溯到查理大帝时期，但是在十一世纪他们开始吸引到一些智力超群的大师，有些学校还重新兴起对基督教教义和自然世界知识的融合。正如历史学家彼特·德尔所评论：“通过学习上帝所创造的事物，认识上帝创造这些事物的缘由，由此来了解上帝，许多人认为这是一种最为虔诚的精神。”比如沙特尔的蒂埃里，他在巴黎和沙特尔教学，1142年成为沙特尔学校校长，他用从《蒂迈欧篇》学到的四元素理论来解释《创世纪》所描述的世界起源。 另一项进展比教会学校的繁荣--虽然与其相关--更为重要。这就是重新兴起的对早期科学著作的翻译。初开始时这些翻译多数不是直接译自希腊文，而是译自阿拉伯文：包括阿拉伯科学家的成就，以及已被从希腊文译为阿拉伯文或先被从希腊文译为叙利亚文后又译为阿拉伯文的著作。 这场大翻译始于十世纪中期，例如在靠近基督教欧洲与倭马亚西班牙边界比利牛斯的圣玛利亚里波利修道院开展的翻译工作。我们可以从欧里亚克的吉尔伯特的职业生涯看出新知识如何在中世纪欧洲传播。欧里亚克的吉尔伯特公元945年出生于阿基坦的一个普通家庭，他在加泰罗尼亚学了一些阿拉伯数学和天文知识，在罗马生活了一段时间后去了兰斯，在那里教授阿拉伯数字和算盘，他还重整了教会学校；任兰斯的修道院院长，后又升为大主教；辅助法国新朝代创建者国王于格·卡佩的加冕典礼；追随德国皇帝奥托三世到意大利和马格德堡；任拉文纳大主教；公元999年被选为教皇--西尔维斯特二世。他的学生沙特尔的富尔伯特在兰斯教会学校学习，1006年成为沙特尔主教，掌管重建这座辉煌教堂的工作。 到十二世纪翻译的节奏开始加快。十二世纪初，英国人巴斯的阿德拉德遍访阿拉伯国家。他翻译了阿尔·花剌子模的著作， 在《自然问题》一书中他介绍了阿拉伯学术。沙特尔的蒂里学会了阿拉伯数学零的应用，他将之引入到欧洲。十二世纪最重要的翻译家可能是克雷莫纳的杰勒德。他在托莱多工作，阿拉伯人入侵之前托莱多是基督教西班牙的首都，虽然卡斯蒂利亚人1085年夺回了托莱多，但该城仍然是阿拉伯和犹太文明中心。他从阿拉伯文译为拉丁文的托勒密著作《天文学大成》把希腊天文学引入到中世纪欧洲。杰勒德也翻译了欧几里德的《几何原本》，以及阿基米德，阿尔·拉奇，阿尔·法甘尼，盖伦，伊本·西纳和阿尔·花剌子模的著作。诺曼人1091年占领阿拉伯西西里之后，希腊文被直接翻译为拉丁文，不再中间借助于阿拉伯文。 最具有直接影响力的是亚里士多德的翻译作品。多数亚里士多德著作是在托莱多从阿拉伯文翻译过来，例如杰勒德翻译了《论天》，《物理学》和《形而上学》。 亚里士多德的作品并不被教会普遍接受。中世纪基督教徒受柏拉图主义和新柏拉图主义影响更深，其中部分是圣·奥古斯丁的作用。亚里士多德的写作更富自然主义风格，这点与柏拉图不同。他认为宇宙遵循自然定律，虽然他的自然定律不尽完善，但仍然呈现出对上帝万能的约束，该情形也曾经深深困扰阿尔·安萨里。两个新托钵修会派别间的冲突至少部分是对亚里士多德不同认知间的冲突：方济会，即灰衣修士，成立于1209年，他们反对教授亚里士多德；布道兄弟会，即黑衣修士，1216年成立，他们支持“哲学家”。 这种冲突主要发生于欧洲新型高等教育机构—大学。巴黎一所教会学校与1200年获得开办大学的皇家特许。（博洛尼亚有更早一些的大学，但专攻法律和医学，对中世纪物理学的作用不大）但很快在1210年巴黎大学学者就被禁止教授亚里士多德自然哲学方面的内容。教皇格里高利九世1231年下令对亚里士多德作品进行修订，只保留教授那些安全的部分。 不是所有的地方都严禁亚里士多德。图卢兹大学从1229年建校起就开始教授亚里士多德作品。1234年巴黎也解禁了亚里士多德著作。后几十年学习亚里士多德成为那里的教学核心。这其中主要是两位十三世纪教士的贡献：大阿尔伯特和托马斯·阿奎纳。在他们辉煌时期他们被授予大博士头衔：大阿尔伯特被授予“全能博士”，托马斯被授予“天使博士”。 大阿尔伯特在帕多瓦和科隆学习，加入布道兄弟会，1241年去巴黎，从1245年到1248年他在巴黎担任给外籍专家的教授职务。后来他移居到科隆，在那里创建科隆大学。大阿尔伯特属于温和亚里士多德学派，他更倾向于托勒密体系，而非亚里士多德同心圆体系。但是他对托勒密体系与亚里士多德物理学间的冲突不安。他推测天上银河是由许多恒星组成。（与亚里士多德相反）他认为月球表面的痕迹是月球自身的瑕疵。大阿尔伯特开创的先例不久由另一位德国布道兄弟会会员弗赖堡的迪特里希所继承。他独自重复了阿尔·法里斯早先有关彩虹方面的部分工作。1941年梵蒂冈宣布大阿尔伯特为所有科学家的圣师。 托马斯·阿奎纳出生于意大利南部的一个次贵族家庭。在蒙特卡西诺修道院和那不勒斯大学完成学业后，他没有像他的家庭期望的那样成为一个富裕修道院的院长，而是像大阿尔伯特一样成为了一名布道兄弟会会员。托马斯移居到巴黎和科隆师从大阿尔伯特学习。后来他又回到巴黎，从1256年到1259年，以及从1269年到1272年任大学教授。 阿奎纳最伟大的著作是《神学大全》，一部系统融合亚里士多德哲学与基督教神学的作品。书中他持中立立场，介于极端亚里士多德学派—该派随伊本·路世德被称为阿威罗伊派，以及极端反亚里士多德学派—比如新成立的奥古斯丁会成员—之间。阿奎纳极力反对普遍认为（可能并不公平）是十三世纪阿威罗伊派的布拉班特的西基尔和达契亚的柏修斯的一个教义。该教义认为哲学可以给出答案（？），比如物质可以不朽以及死者不能复生，而宗教对此存在着错误认知。对阿奎纳而言，只存在一个真相。天文学方面阿奎纳倾向于亚里士多德同心圆理论，他认为该理论是基于理性思维，而托勒密的理论只是与观测相符，其他假想也可能与观测数据相符。另一方面阿奎纳不认同亚里士多德运动理论；他认为即使在真空任何运动也需要一定时间。阿奎纳鼓励他的同代人—英国人布道兄弟会会员穆尔贝克的威廉姆直接从希腊文将亚里士多德，阿基米德，以及其他一些学者作品译为拉丁文。到1255年巴黎学生都要考核他们对阿奎纳作品的掌握情况。 但是亚里士多德的麻烦还没结束。从十二世纪五十年代开始，方济会圣博纳旺蒂尔主导了在巴黎反对亚里士多德理论。1245年教皇因诺森特四世在图卢兹全面禁止亚里士多德的著作。1270年巴黎主教唐皮耶禁止教授亚里士多德的13个命题。教皇约翰二十一世下令唐皮耶进一步加强禁令，1277年唐皮耶谴责了亚里士多德或阿奎纳的219个教义。坎特伯雷大主教罗伯特·科瓦达将对亚里士多德的谴责扩展到英国。他的继任者约翰·佩坎1284年持续了这种谴责。 1277年所谴责的命题可以按照被谴责原由细分。有些命题直接与经文冲突—比如认为世界永恒的命题： 9 不存在第一个人，也不会有最后一个人。相反，人类一直是而且还将会一代一代繁衍下去。 87 世界永生，包括其中所有物种。时间永恒，运动，物质，动因和受体一样永恒。 还有些被谴责的命题讲述了认识真相的方法，挑战了宗教权威，比如： 38 除非不证自明或可以从不证自明的事物中断定，否则不要轻信。 150 不要对权威深信不疑 151 不要认为掌握了神学就掌握了真相。 最后，一些被谴责的命题曾经同样困惑阿尔·安萨里，这些哲学和科学理论制约了上帝的自由，例如： 34 第一推动力不能创建多个世界 49 上帝不能让天体呈直线运动，因为这样会留下一个真空 141 上帝不能无中生有，也不能创造多于(三维)的空间。 对亚里士多德和阿奎纳命题的谴责并没有持续很久。新教皇约翰二十二世曾接受布道兄弟会培养，在他的主导下，托马斯·阿奎纳于1323年被封为圣徒。1325年巴黎主教废除了谴责，他颁布法令：“我们全部废除上述由于教授，或认为教授了神圣的托马斯作品而受到谴责或被逐出教会的法令，为此我们对这些作品不做判断，留给学者自由讨论。”1341年巴黎大学要求人文学者发誓他们会教授“亚里士多德体系，以及他的评注者阿威罗伊，或其他评注者，除非其内容背离信仰。” 历史学家不认同这一幕谴责及后面的复兴对后世科学的影响。这里有两个问题：如果那些谴责没被废除，会对科学有什么影响？如果从来没有发生过对讲授亚里士多德和阿奎纳的谴责，对科学有会有什么影响？ 在我看来如果那些谴责没有被废除，对科学将是场灾难。这到不是因为亚里士多德关于自然世界的结论有多重要，实际上其中大多数都是错误的：比如与亚里士多德理论恰恰相反，人类并不是自古就存在的；肯定存在多行星系统，也可能有多次宇宙大爆炸；许多天体沿直线运行；真空不是没有可能；现代弦理论认为存在三维以上的空间，其他几维空间之所以无法观察得到是因为这些空间发生了卷缩。这种谴责的危害在于命题被谴责的理由，而非命题本身。 虽然亚里士多德自然定律不正确，但重要的是要相信自然界存在定律。如果容许基于上帝万能而谴责像命题34，49和141这样对自然的归纳，那基督教欧洲将会陷落到阿尔·安萨里主张的伊斯兰那种情形。 另外对那些质疑宗教权威命题（如上面引用的38，150和153条） 的谴责部分也是中世纪大学人文和神学教员间冲突中的一幕。神学地位明显更高，学习神学可以获得神学博士学位，而人文学最多只能授予硕士学位。（神学博士在大学排在领先位置，其次为法律博士，医学博士，然后才是人文硕士。）废除那些谴责不会给人文带来与神学同等地位，但有助于将人文教员从神学同事的脑力约束中解放出来。 现在很难判断如果那些谴责从来没有发生的话会有什么后果。我们后面会看到在十四世纪亚里士多德物理和天文方面的权威在巴黎和牛津不断受到挑战，虽然一些新的观点需要以只是异想而非事实来掩饰。如果在十三世纪亚里士多德的权威没有由于受谴责而被削弱，是否还有可能来挑战亚里士多德？戴维·林德伯格引述尼克尔·奥里斯姆（后面会有详细介绍）的例子，他在1377年争辩说可以容许去想象地球在无限空间直线运动，因为“如果不这样就维持了在巴黎所进行的谴责。”也许可以把十三世纪发生的事件总结为这种谴责把科学从亚里士多德的教条中解放了出来，而废除谴责把科学从基督教教条中解放了出来。 大翻译时代以及对于是否接受亚里士多德理论的冲突之后，具有开创性的科学工作终于在十四世纪欧洲开花结果。其中的领军人物是1296年出生于阿拉斯的法国人让·布里丹，他的一生大部分在巴黎渡过。布里丹是位世俗牧师，他不属于任何宗教派别。哲学方面他是唯名论者，认为个体才是真实存在的，而不是共相。布里丹于1328年和1340年两次被选为巴黎大学校长。 布里丹是经验主义者，他不认为科学原理一定服从逻辑：“这些原理不是一目了然的，我们可能长时间对它们产生怀疑。但它们之所以被称为原理是因为它们是无法证明的，它们不能由其他前提演绎出来，也不能有任何方法来证明。我们接受这些原理，因为我们已经多次观察到其真实性，而从来没有观察到其错误，” 了解这点对科学的未来至关重要，但并不容易。陈旧且无法实现的柏拉图自然科学纯粹演绎目标只能基于周密观察之上的小心分析，这阻碍了科学的进步。即使在今天人们也会不时遇到这种困惑。比如心理学家让·皮亚杰就认为他发现儿童可能天生拥有理解相对论的能力，但在后来他们失去了这种能力，这就像相对论只需要逻辑或哲学，而不是最终基于对以光速或接近光速运动物体观察而得出的结论。 布里丹虽然是经验主义者，但却不是实验主义者。与亚里士多德一样，他的推理基于日常观察，但他比亚里士多德更加小心翼翼去地达成宽泛结论。比如布里丹直面亚里士多德的一个古老问题：为什么沿水平或向上抛出的物体离开手后不会立即开始其所谓的自然运动：即向下直线运动。布里丹提出几条理由否定亚里士多德所给出的抛射物被空气携带的解释。首先空气只会阻止运动，而不会辅助运动，因为空气必须分开才能让实物穿过。其次怎么可能当抛出物体的手停止运动后，空气确会运动？再者尾端宽大的长矛受到空气推力，但尖头朝后的长矛在空气中的运行与其同样，甚至更快。（？） 布里丹不认为是空气在维持抛物的运行，他认为这是手赋予抛物的一种称为“原动力”的作用。我们前面讲过，菲洛波努斯的约翰也曾提出相似的观点，布里丹所称的原动力后来牛顿称之为“运动之量”，现代术语称为动量，但并不完全一样。布里丹认可亚里士多德的观点，处于运动状态的物体需要事物来维持其运动，他设想原动力在起这种作用，而并不只是运动的属性，比如动量。他从来没有把物体所携带的原动力理解为物体质量乘以物体运动速度，这是牛顿物理中动量的定义。但是他的认识是有意义的。让一个运动物体在一定时间之内停下来所需要的力正比于物体的动量，在这点上动量与布里丹的原动力起同样作用。 布里丹将原动力的想法扩展到圆周运动，他设想行星在原动力的作用下保持运动，上帝赋予行星原动力。布里丹采用这种方法寻求科学与宗教间的妥协，这在几个世纪之后非常盛行。上帝建立宇宙运行机制，后面所发生的一切受自然定律约束。虽然动量守恒确实可以维持行星运行，但不能像布里丹所设想的原动力让行星做圆周运动，那需要额外的力，后来认识到是重力。 布里丹也曾动过源自赫拉克利德斯的一个念头，即地球每天从西向东自转一周。他意识到这与设想天体每天由东向西围绕静止地球一周观察到的视运行是一致的。他也承认这个理论更为自然，因为地球与太阳，月亮，行星和恒星构成的苍穹相比小的太多。但是他最终还是否定了地球自转，理由是如果地球在转动，那向上射出的箭将落回到弓箭手的西边，因为地球在箭飞的过程中已经发生了转动。如果布里丹能够意识到地球的自转会给箭一个原动力，将携带箭与地球一起向东运行，那他本可以避免这个错误。相反他被原动力的概念所误导；他只考虑了弓给予箭的垂直原动力，而没有考虑从地球自转得到的水平原动力。 布里丹原动力概念持续了几个世纪。哥白尼十六世纪早期学习医学时帕多瓦大学还在教授该理论。伽利略在十六世纪末期在比萨大学也学了该理论。 布里丹在另一个问题—真空的不可能上也支持亚里士多德，但不同的是他的结论基于观察：当空气被从吸管中吸出，液体被吸入吸管，阻止真空的产生。拉动风箱把柄，空气进入风箱，不会生成真空。可以自然得出大自然讨厌真空的结论。我们在第12章会介绍，直到十七世纪人们才正确地用气压解释了这些现象。 布里丹的两个学生萨克森的阿尔伯特和尼克尔·奥里斯姆进一步深化了的他的思想。阿尔伯特哲学作品广位流传，但对科学做出重大贡献的是奥里斯姆。 奥里斯姆1325年出生于诺曼底，十四世纪四十年代到巴黎师从布里丹学习。他强烈反对通过“星相，泥土占卜，通灵术或其他类似的艺术（如果这些可以成为艺术的话）”来预测未来。1377年奥里斯姆被任命为诺曼底利泽尔地区的主教，1382年去世于此。 奥里斯姆的著作《论天与地》（为了让法国皇帝读起来方便采用本国语写作）扩大了对亚里士多德的评注，书中他多次挑战哲学家。奥里斯姆重新思索了不是天体围绕地球由东向西运转，而是地球由西向东沿自转轴自转的观点。布里丹和奥里斯姆都认同我们只能观察到相对运动，因而看到天体运行不能排除地球自转的可能性。奥里斯姆整理了各种反对意见，一一做了反驳。托勒密在《天文学大成》中辩称如果地球自转，那天上云朵以及抛出去的物体都会被甩到后面；我们上面也介绍过，布里丹否定地球自转的理由是如果地球从西向东自转，那向上射出的箭会由于地球的转动而落在后边，而实际上箭总是落回到一开始垂直向上射出时的原地。奥里斯姆认为地球携带了箭与地球一起自转，而且也携带了弓箭手，空气以及地面上的一切，奥里斯姆这里应用了布里丹原动力理论，反而其原创者并没有理解到这种应用， 奥里斯姆也回应了另外一个否定地球自转的理由—一个完全不同的理由，圣经（比如约书亚圣经）中提到太阳每日围绕地球运行。奥里斯姆回应说这只是迁就人们日常说法的传统，就像里面也写到上帝生气或懊悔—这些不能只从字面上去理解。奥里斯姆这里仿效托马斯·阿奎纳的做法，阿奎纳曾经反驳《创世纪》中的段落，其中上帝声称：“要有一穹窿于诸水当中，让其分开诸水。”阿奎纳解释说摩西是为了让他的听众容易理解而调整了说法，不应该从字面上理解内容。如果不是教会中有许多像阿奎纳和奥里斯姆这样具有开明观点的人的存在，圣经直译主义会阻碍科学的发展。 尽管奥里斯姆有这么多论证，他最终还是转向大众普遍接受的静止地球观。他说： 后来结果表明不能通过论证得出天体转动的结论 …，然而所有人都坚信，我也同样认为是天体而不是地球在转动：因为上帝创造了这个世界，其不可能发生运动，尽管存在反面理由，因为很明显这些没有最终说服力。然而如果全面考虑，人们可能会相信地球在动，而非天体，因为反过来不能不辩自明。然而初看起来这个观点违背自然也违背我们的信仰，我所说的通过思索可以以这种方式来否认或验证那些用辩论去怀疑我们信仰的人。 我们不知道奥里斯姆是否真的不愿意最终承认地球自转，还是只为了口头应对正统宗教观。 奥里斯姆也预见到牛顿重力定律的一面。他认为重物并不一定落向地球中心，如果它们靠近其他世界的话就不会这样。认为存在类似地球的多个世界从神学上来说是个极其大胆的设想。上帝是否在其他世界也创作人类？基督是否也去其他世界救赎那些人类？这类问题层出不穷，极具颠覆性。 与布里丹不同，奥里斯姆是位数学家，他的主要数学贡献是改进了牛津大学早期的一项数学成就，所以我们现在需要从法国转向英国，回到更早一个时期。然后我们再回到奥里斯姆。 到十二世纪牛津已经成为坐落于泰晤士河上游的一个繁华市镇，吸引了众多学生和教师。十二世纪早期牛津的多所学校组成大学。牛津通常将1224年校长罗伯特·格罗斯特（后来成为林肯郡主教）列为第一任校长，他开创了中世纪牛津对自然哲学的关注（？）。格罗斯特直接阅读亚里士多德希腊文作品，他创作了关于光学，历法以及亚里士多德方面的著作。他在牛津大学的继任者常常引用到他的作品。 在《罗伯特·格罗斯特与实验科学的起源》中克罗姆比走的更远，他将格罗斯特列为带来现代科学诞生的实验科学进展过程中的核心人物。（？）这似乎夸大了格罗斯特的重要性。从克罗姆比的记述中可以清晰看出格罗斯特的“实验”只是对自然的被动观察，与亚里士多德方法相差无几。无论格罗斯特还是他中世纪继任者都没有寻求通过现代意义上的实验方法—即人为制造自然现象—来得出普适原理。格罗斯特的理论也受到不少赞誉，但是他的成就完全无法与希罗，托勒密和海什木光学方面，或喜帕恰斯，托勒密和阿尔·比鲁尼等人有关行星运行方面的成功理论相媲美。 格罗斯特对罗杰·培根影响很深，培根的智能和纯粹科学思想是他那个时代的精神代表。牛津求学之后，培根1240年到巴黎讲授亚里士多德，后来徘徊在巴黎和牛津之间，1257年成为方济会修士。与柏拉图一样，他热衷于数学，但很少用到数学。他创作了大量光学和地理学著作，但并没有在早期希腊和阿拉伯成就之上添加任何有价值的贡献。培根是技术上的乐观主义者，在他那年代很突出： 可以制造不用动物拉动而急速行走的车 … 飞行机器也可以制造出来，人座在机器之内，机器借助人工翅膀煽动空气而飞行，像鸟一样。 培根被称为“奇异的博士”，这很恰当。 1264年曾任英格兰大法官，后成为罗切斯特主教的沃尔特·墨顿创建了牛津大学第一所寄宿学院。牛津大学墨顿学院在十四世纪开展了深入地数学研究。其关键人物是四位学院院士：托马斯·布拉德沃丁(大约1295 – 1349)，威廉姆·海茨伯里（活跃于1335年），理查德·斯温斯黑德（活跃于 1340 – 1355），以及杜布莱顿的约翰（活跃于 1338 – 1348）。他们最引人注目的成就是墨顿学院平均速度定理，该定理第一次用数学方法描述了非匀速运动—即运动速度不是常量。 现在流传下来关于该定律最早记录出于威廉姆·海茨伯里（牛津大学1371年校长）的《求解诡辩规则》。他把物体非匀速运动某一时刻速度定义为假设在该时刻物体以匀速运动时行进距离与行进时间之比。这种定义是循环定义，没有意义。比较现代的定义（也许正是海茨伯里想要表达的）是物体非匀速运动时某一时刻速度等于此时刻极短时间间隔内运行距离与运行时间之比，此时间间隔极小，速度变化可以忽略不计。海茨伯里然后定义匀加速运动为相同时间间隔内速度增加相等的非匀速运动。然后他接着陈述该定理： 当运动物体从静止匀加速到某一（速度），在此时间段其行进距离等于假如该物体以此末速度匀速运行相同时间运行距离的一半。这种运动总体上相当于速度增加值的平均值，正好等于末速度的一半。 也就是说，当物体以匀加速运动一段时间其行进距离等于假如该物体在此时间段内以速度为实际速度的平均值匀速运行的距离。如果一个物体从静止匀加速到某一终速，那么该物体此时段平均速度等于终速的一半，其运行距离等于终速的一半乘以运行时间。 海茨伯里，杜布莱顿的约翰以及尼克尔·奥里斯姆给出了该定理多种证明办法。其中奥里斯姆的证明最为有趣。他引入了用图形表达代数关系技术。采用这种技术，他可以将计算一个物体从静止匀加速到某一终速的运行距离简化为计算由边长分别等于运行时间和终速组成的直角三角形的面积（见技术说明17）。这意味着速度定理等效于基础几何定理，即直角三角形的面积等于由直角三角形两边组成的长方形面积的一半。 无论墨顿学院导师还是奥里斯姆都没有将平均速度定理应用到相关的一个最重要问题—即自由落体运动。对于各位院士或奥里斯姆本人来说，此定理只是一个智力训练，显示他们有能力用数学方法处理非匀速运动。如果平均速度定理展现了应用数学能力的提高，那同时也表明数学与自然科学间的契合有多么不易，即使现在也一样。 我们需要承认虽然落体的加速显而易见（正如斯特拉图曾描述的），但落体速度增加值正比于时间这点并不明显，这是匀加速特征，落体速度并不正比于下落距离。如果下落距离的改变率（即下落速度）正比于下落距离，那一旦落体开始下落，其下落距离将随时间呈指数增长（参考第十二章注释二），就像利息收入正比于存款额的银行账户金额会随时间呈指数增长（虽然如果利息低的话需要很长时期才能见效）。第一位猜测落体加速度正比于下落时间的学者是十六世纪的布道兄弟会修士多明戈·德索托，比奥里斯姆晚了两个世纪。 十四世纪中叶到十五世纪中叶是欧洲多难之秋。英法百年战争既耗尽了英国，也重创了法国。教会处于分裂状态，一个教皇在罗马，另一个教皇在亚维农。黑死病导致各地大量人口死亡。 也许是百年战争的结果，此阶段科学中心向东迁移，从法国和英国移到德国和意大利。尼古拉斯·库萨的一生就活跃于此两地。他1401年左右出生于德国摩泽尔河流域 库萨镇，1464年在意大利翁布里亚去世。尼古拉斯在海德堡和帕多瓦接受教育，后来成为一名教会律师, 使节,1448年后成为一名红衣主教.他的著作明显体现出中世纪时期仍然难以将自然科学与神学和哲学区分开来。尼古拉斯闪烁其词地描写了地球的运转和宇宙的无限，但他没有用到数学。虽然后来开普勒和笛卡尔引用了他，但很难知道他们从他那里能够学到什么。 中世纪后期仍然延续着阿拉伯时代就存在的职业数理天文学家(他们应用托勒密体系）与医学家和哲学家（他们是亚里士多德的追随者）间的不同。在众多十五世纪天文学家中（多数在德国），有两位是波伊巴赫和他的学生柯尼斯堡的约翰·穆勒（雷格蒙塔努斯），他们一起继承并扩展了托勒密的本轮理论（注：后代作家乔治·哈特曼（1489-1564）声称他见过雷格蒙塔努斯的一封信中有这样一句话：“由于地球的转动，恒星的转动会发生细小变化”（科学传记字典，斯克里布纳出版社，纽约，1975， 第二卷，351页）。如果这是真的，那么雷格蒙塔努斯可能早已预见到哥白尼学说，虽然该说法与毕达哥拉斯学派的地球和太阳围绕世界中心运转的教义也相符）。哥白尼后来大量运用了雷格蒙塔努斯的《天文学大成概论》。医学家包括博洛尼亚的亚历桑德罗·阿基利尼（1463-1512）和维洛那的吉罗拉摩·法兰卡斯特罗（1478-1553），他们俩都在帕多瓦接受教育，是亚里士多德的坚定维护者。 法兰卡斯特罗带有偏见地描述了此冲突： 你知道那些职业天文学家常常发现非常难以解释行星的视运行。他们有两种解释理论：一种采用同心圆方法，另一种采用被称为偏心圆（本轮）的方法。这两种方法各有危害，也各有不足。那些采用同心圆理论的人从来无法对现象做出解释，而那些采用偏心圆的人确实可以更加准确地解释现象，但他们关于神圣天体的构想充满错误，可以说在亵渎神灵。因为他们赋予天体的位置和形状不符合上天。我们知道在古代欧多克索斯和卡利帕斯多次被这些困难所误导。喜帕恰斯属于最早一批人宁肯认同偏心圆，而不去发现其不足。托勒密追随他，不久几乎所有天文学家都被托勒密所征服。但是整个哲学界不断对这些天文学家，或至少是对这种偏心假想提出抗议。我在说什么？哲学家？大自然与天体本身也在不断抗议。至今没有一位哲学家允许这些怪物球体在神圣完美的天体中存在。 公平地说不是所有观测结果都站在托勒密一边，而不支持亚里士多德理论。亚里士多德同心圆体系中的一个缺陷是正如我们前面讲过的，公元200年索西吉斯提出该体系将行星与地球间的距离固定不变，这与行星看起来在围绕地球运行时亮度发生明暗变化的事实不符。但是托勒密体系朝相反方向走的太远。比如按照托勒密理论，金星与地球最大距离是最小距离的6.5倍。如果金星靠自身发光，那么（因为视亮度与距离的平方成反比）它的最大亮度将比最小亮度亮6.5的平方，即42倍，这当然不对。维也纳大学黑森的亨利(1325-1397)对托勒密的这点提出了批评。该问题的解决办法当然是行星不是自身发光。而是反射太阳光，所以它们的视亮度不只取决于它们与地球间的距离，而是也像月亮一样，取决于它们的相。当金星离地球最远时，它处于太阳远离地球一边，所以它完全被太阳所照亮。而当它离地球最近时，它大体上处于地球与太阳之间，我们只能看到它的暗面。金星相的效果与距离部分抵消，这样其亮度变化不大。直到伽利略发现金星的相之后这些才被人们所理解。 不久托勒密与亚里士多德天文学间的冲突就让步于一个更为深刻的冲突，即那些追随托勒密和亚里士多德地心说与复兴的阿里斯塔克斯日心说间的冲突。

As the Roman Empire decayed in the West, Europe outside the realm of Byzantium became poor, rural, and largely illiterate. Where some literacy did survive, it was concentrated in the church, and there only in Latin. In Western Europe in the early Middle Ages virtually no one could read Greek. Some fragments of Greek learning had survived in monastery libraries as Latin translations, including parts of Plato’s Timaeus and translations around AD 500 by the Roman aristocrat Boethius of Aristotle’s work on logic and of a textbook of arithmetic. There were also works written in Latin by Romans, describing Greek science. Most notable was a fifth-century encyclopedia oddly titled The Marriage of Mercury and Philology by Martianus Capella, which treated (as handmaidens of philology) seven liberal arts: grammar, logic, rhetoric, geography, arithmetic, astronomy, and music. In his discussion of astronomy Martianus described the old theory of Heraclides that Mercury and Venus go around the Sun while the Sun goes around the Earth, a description praised a millennium later by Copernicus. But even with these shreds of ancient learning, Europeans in the early Middle Ages knew almost nothing of the great scientific achievements of the Greeks. Battered by repeated invasions of Goths, Vandals, Huns, Avars, Arabs, Magyars, and Northmen, the people of Western Europe had other concerns. Europe began to revive in the tenth and eleventh centuries. The invasions were winding down, and new techniques improved the productivity of agriculture.1 It was not until the late thirteenth century that significant scientific work would begin again, and not much would be accomplished until the sixteenth century, but in the interval an institutional and intellectual foundation was laid for the rebirth of science. In the tenth and eleventh centuries—a religious age—much of the new wealth of Europe naturally went not to the peasantry but to the church. As wonderfully described around AD 1030 by the French chronicler Raoul (or Radulfus) Glaber, “It was as if the world, shaking itself and putting off the old things, were putting on the white robe of churches.” For the future of learning, most important were the schools attached to cathedrals, such as those at Orléans, Reims, Laon, Cologne, Utrecht, Sens, Toledo, Chartres, and Paris. These schools trained the clergy not only in religion but also in a secular liberal arts curriculum left over from Roman times, based in part on the writings of Boethius and Martianus: the trivium of grammar, logic, and rhetoric; and, especially at Chartres, the quadrivium of arithmetic, geometry, astronomy, and music. Some of these schools went back to the time of Charlemagne, but in the eleventh century they began to attract schoolmasters of intellectual distinction, and at some schools there was a renewed interest in reconciling Christianity with knowledge of the natural world. As remarked by the historian Peter Dear,2 “Learning about God by learning what He had made, and understanding the whys and wherefores of its fabric, was seen by many as an eminently pious enterprise.” For instance, Thierry of Chartres, who taught at Paris and Chartres and became chancellor of the school at Chartres in 1142, explained the origin of the world as described in Genesis in terms of the theory of the four elements he learned from the Timaeus. Another development was even more important than the flowering of the cathedral schools, though not unrelated to it. This was a new wave of translations of the works of earlier scientists. Translations were at first not so much directly from Greek as from Arabic: either the works of Arab scientists, or works that had earlier been translated from Greek to Arabic or Greek to Syriac to Arabic. The enterprise of translation began early, in the middle of the tenth century, for instance at the monastery of Santa Maria de Ripoli in the Pyrenees, near the border between Christian Europe and Ummayad Spain. For an illustration of how this new knowledge could spread in medieval Europe, and its influence on the cathedral schools, consider the career of Gerbert d’Aurillac. Born in 945 in Aquitaine of obscure parents, he learned some Arab mathematics and astronomy in Catalonia; spent time in Rome; went to Reims, where he lectured on Arabic numbers and the abacus and reorganized the cathedral school; became abbot and then archbishop of Reims; assisted in the coronation of the founder of a new dynasty of French kings, Hugh Capet; followed the German emperor Otto III to Italy and Magdeburg; became archbishop of Ravenna; and in 999 was elected pope, as Sylvester II. His student Fulbert of Chartres studied at the cathedral school of Reims and then became bishop of Chartres in 1006, presiding over the rebuilding of its magnificent cathedral. The pace of translation accelerated in the twelfth century. At the century’s start, an Englishman, Adelard of Bath, traveled extensively in Arab countries; translated works of al-Khwarizmi; and, in Natural Questions, reported on Arab learning. Somehow Thierry of Chartres learned of the use of zero in Arab mathematics, and introduced it into Europe. Probably the most important twelfth-century translator was Gerard of Cremona. He worked in Toledo, which had been the capital of Christian Spain before the Arab conquests, and though reconquered by Castilians in 1085 remained a center of Arab and Jewish culture. His Latin translation from Arabic of Ptolemy’s Almagest made Greek astronomy available to medieval Europe. Gerard also translated Euclid’s Elements and works by Archimedes, al- Razi, al-Ferghani, Galen, Ibn Sina, and al-Khwarizmi. After Arab Sicily fell to the Normans in 1091, translations were also made directly from Greek to Latin, with no reliance on Arabic intermediaries. The translations that had the greatest immediate impact were of Aristotle. It was in Toledo that the bulk of Aristotle’s work was translated from Arabic sources; for instance, there Gerard translated On the Heavens, Physics, and Meteorology. Aristotle’s works were not universally welcomed in the church. Medieval Christianity had been far more influenced by Platonism and Neoplatonism, partly through the example of Saint Augustine. Aristotle’s writings were naturalistic in a way that Plato’s were not, and his vision of a cosmos governed by laws, even laws as ill-developed as his were, presented an image of God’s hands in chains, the same image that had so disturbed al-Ghazali. The conflict over Aristotle was at least in part a conflict between two new mendicant orders: the Franciscans, or gray friars, founded in 1209, who opposed the teaching of Aristotle; and the Dominicans, or black friars, founded around 1216, who embraced “The Philosopher.” This conflict was chiefly carried out in new European institutions of higher learning, the universities. One of the cathedral schools, at Paris, received a royal charter as a university in 1200. (There was a slightly older university at Bologna, but it specialized in law and medicine, and did not play an important role in medieval physical science.) Almost immediately, in 1210, scholars at the University of Paris were forbidden to teach the books of Aristotle on natural philosophy. Pope Gregory IX in 1231 called for Aristotle’s works to be expurgated, so that the useful parts could be safely taught. The ban on Aristotle was not universal. His works were taught at the University of Toulouse from its founding in 1229. At Paris the total ban on Aristotle was lifted in 1234, and in subsequent decades the study of Aristotle became the center of education there. This was largely the work of two thirteenth- century clerics: Albertus Magnus and Thomas Aquinas. In the fashion of the times, they were given grand doctoral titles: Albertus was the “Universal Doctor,” and Thomas the “Angelic Doctor.” Albertus Magnus studied in Padua and Cologne, became a Dominican friar, and in 1241 went to Paris, where from 1245 to 1248 he occupied a professorial chair for foreign savants. Later he moved to Cologne, where he founded its university. Albertus was a moderate Aristotelian who favored the Ptolemaic system over Aristotle’s homocentric spheres but was concerned about its conflict with Aristotle’s physics. He speculated that the Milky Way consists of many stars and (contrary to Aristotle) that the markings on the Moon are intrinsic imperfections. The example of Albertus was followed a little later by another German Dominican, Dietrich of Freiburg, who independently duplicated some of al-Farisi’s work on the rainbow. In 1941 the Vatican declared Albertus the patron saint of all scientists. Thomas Aquinas was born a member of the minor nobility of southern Italy. After his education at the monastery of Monte Cassino and the University of Naples, he disappointed his family’s hopes that he would become the abbot of a rich monastery; instead, like Albertus Magnus, he became a Dominican friar. Thomas went to Paris and Cologne, where he studied under Albertus. He then returned to Paris, and served as professor at the university in 1256–1259 and 1269–1272. The great work of Aquinas was the Summa Theologica, a comprehensive fusion of Aristotelian philosophy and Christian theology. In it, he took a middle ground between extreme Aristotelians, known as Averroists after Ibn Rushd; and the extreme anti-Aristotelians, such as members of the newly founded Augustinian order of friars. Aquinas strenuously opposed a doctrine that was widely (but probably unjustly) attributed to thirteenth-century Averroists like Siger of Brabant and Boethius of Dacia. According to this doctrine, it is possible to hold opinions true in philosophy, such as the eternity of matter or the impossibility of the resurrection of the dead, while acknowledging that they are false in religion. For Aquinas, there could be only one truth. In astronomy, Aquinas leaned toward Aristotle’s homocentric theory of the planets, arguing that this theory was founded on reason while Ptolemaic theory merely agreed with observation, and another hypothesis might also fit the data. On the other hand, Aquinas disagreed with Aristotle on the theory of motion; he argued that even in a vacuum any motion would take a finite time. It is thought that Aquinas encouraged the Latin translation of Aristotle, Archimedes, and others directly from Greek sources by his contemporary, the English Dominican William of Moerbeke. By 1255 students at Paris were being examined on their knowledge of the works of Aquinas. But Aristotle’s troubles were not over. Starting in the 1250s, the opposition to Aristotle at Paris was forcefully led by the Franciscan Saint Bonaventure. Aristotle’s works were banned at Toulouse in 1245 by Pope Innocent IV. In 1270 the bishop of Paris, Étienne Tempier, banned the teaching of 13 Aristotelian propositions. Pope John XXI ordered Tempier to look further into the matter, and in 1277 Tempier condemned 219 doctrines of Aristotle or Aquinas.3 The condemnation was extended to England by Robert Kilwardy, the archbishop of Canterbury, and renewed in 1284 by his successor, John Pecham. The propositions condemned in 1277 can be divided according to the reasons for their condemnation. Some presented a direct conflict with scripture—for instance, propositions that state the eternity of the world:

9. That there was no first man, nor will there be a last; on the contrary, there always was and always will be the generation of man from man. 87. That the world is eternal as to all the species contained in it; and that time is eternal, as are motion, matter, agent, and recipient.

Some of the condemned doctrines described methods of learning truth that challenged religious authority, for instance:

38. That nothing should be believed unless it is self-evident or could be asserted from things that are self-evident. 150. That on any question, a man ought not to be satisfied with certitude based upon authority. 153. That nothing is known better because of knowing theology. Finally, some of the condemned propositions had raised the same issue that had concerned al-Ghazali, that philosophical and scientific reasoning seems to limit the freedom of God, for example:

34. That the first cause could not make several worlds. 49. That God could not move the heavens with rectilinear motion, and the reason is that a vacuum would remain. 141. That God cannot make an accident exist without a subject nor make more [than three] dimensions exist simultaneously.

The condemnation of propositions of Aristotle and Aquinas did not last. Under the authority of a new pope who had been educated by Dominicans, John XXII, Thomas Aquinas was canonized in 1323. In 1325 the condemnation was rescinded by the bishop of Paris, who decreed: “We wholly annul the aforementioned condemnation of articles and judgments of excommunication as they touch, or are said to touch, the teaching of blessed Thomas, mentioned above, and because of this we neither approve nor disapprove of these articles, but leave them for free scholastic discussion.”4 In 1341 masters of arts at the University of Paris were required to swear they would teach “the system of Aristotle and his commentator Averroes, and of the other ancient commentators and expositors of the said Aristotle, except in those cases that are contrary to the faith.”5 Historians disagree about the importance for the future of science of this episode of condemnation and rehabilitation. There are two questions here: What would have been the effect on science if the condemnation had not been rescinded? And what would have been the effect on science if there had never been any condemnation of the teachings of Aristotle and Aquinas? It seems to me that the effect on science of the condemnation if not rescinded would have been disastrous. This is not because of the importance of Aristotle’s conclusions about nature. Most of them were wrong, anyway. Contrary to Aristotle, there was a time before there were any men; there certainly are many planetary systems, and there may be many big bangs; things in the heavens can and often do move in straight lines; there is nothing impossible about a vacuum; and in modern string theories there are more than three dimensions, with the extra dimensions unobserved because they are tightly curled up. The danger in the condemnation came from the reasons why propositions were condemned, not from the denial of the propositions themselves. Even though Aristotle was wrong about the laws of nature, it was important to believe that there are laws of nature. If the condemnation of generalizations about nature like propositions 34, 49, and 141, on the ground that God can do anything, had been allowed to stand, then Christian Europe might have lapsed into the sort of occasionalism urged on Islam by al-Ghazali. Also, the condemnation of articles that questioned religious authority (such as articles 38, 150, and 153 quoted above) was in part an episode in the conflict between the faculties of liberal arts and theology in medieval universities. Theology had a distinctly higher status; its study led to a degree of doctor of theology, while liberal arts faculties could confer no degree higher than master of arts. (Academic processions were headed by doctors of theology, law, and medicine in that order, followed by the masters of arts.) Lifting the condemnation did not give the liberal arts equal status with theology, but it helped to free the liberal arts faculties from intellectual control by their theological colleagues. It is harder to judge what would have been the effect if the condemnations had never occurred. As we will see, the authority of Aristotle on matters of physics and astronomy was increasingly challenged at Paris and Oxford in the fourteenth century, though sometimes new ideas had to be camouflaged as being merely secundum imaginationem—that is, something imagined, rather than asserted. Would challenges to Aristotle have been possible if his authority had not been weakened by the condemnations of the thirteenth century? David Lindberg6 cites the example of Nicole Oresme (about whom more later), who in 1377 argued that it is permissible to imagine that the Earth moves in a straight line through infinite space, because “To say the contrary is to maintain an article condemned in Paris.”7 Perhaps the course of events in the thirteenth century can be summarized by saying that the condemnation saved science from dogmatic Aristotelianism, while the lifting of the condemnation saved science from dogmatic Christianity. After the era of translation and the conflict over the reception of Aristotle, creative scientific work began at last in Europe in the fourteenth century. The leading figure was Jean Buridan, a Frenchman born in 1296 near Arras, who spent much of his life in Paris. Buridan was a cleric, but secular—that is, not a member of any religious order. In philosophy he was a nominalist, who believed in the reality of individual things, not of classes of things. Twice Buridan was honored by election as rector of the University of Paris, in 1328 and 1340. Buridan was an empiricist, who rejected the logical necessity of scientific principles: “These principles are not immediately evident; indeed, we may be in doubt concerning them for a long time. But they are called principles because they are indemonstrable, and cannot be deduced from other premises nor be proved by any formal procedure, but they are accepted because they have been observed to be true in many instances and to be false in none.”8 Understanding this was essential for the future of science, and not so easy. The old impossible Platonic goal of a purely deductive natural science stood in the way of progress that could be based only on careful analysis of careful observation. Even today one sometimes encounters confusion about this. For instance, the psychologist Jean Piaget9 thought he had detected signs that children have an innate understanding of relativity, which they lose later in life, as if relativity were somehow logically or philosophically necessary, rather than a conclusion ultimately based on observations of things that travel at or near the speed of light. Though an empiricist, Buridan was not an experimentalist. Like Aristotle’s, his reasoning was based on everyday observation, but he was more cautious than Aristotle in reaching broad conclusions. For instance, Buridan confronted an old problem of Aristotle: why a projectile thrown horizontally or upward does not immediately start what was supposed to be its natural motion, straight downward, when it leaves the hand. On several grounds, Buridan rejected Aristotle’s explanation that the projectile continues for a while to be carried by the air. First, the air must resist rather than assist motion, since it must be divided apart for a solid body to penetrate it. Further, why does the air move, when the hand that threw the projectile stops moving? Also, a lance that is pointed in back moves through the air as well as or better than one that has a broad rear on which the air can push. Rather than supposing that air keeps projectiles moving, Buridan supposed that this is an effect of something called “impetus,” which the hand gives the projectile. As we have seen, a somewhat similar idea had been proposed by John of Philoponus, and Buridan’s impetus was in turn a foreshadowing of what Newton was to call “quantity of motion,” or in modern terms momentum, though it is not precisely the same. Buridan shared with Aristotle the assumption that something has to keep moving things in motion, and he conceived of impetus as playing this role, rather than as being only a property of motion, like momentum. He never identified the impetus carried by a body as its mass times its velocity, which is how momentum is defined in Newtonian physics. Nevertheless, he was onto something. The amount of force that is required to stop a moving body in a given time is proportional to its momentum, and in this sense momentum plays the same role as Buridan’s impetus. Buridan extended the idea of impetus to circular motion, supposing that planets are kept moving by their impetus, an impetus given to them by God. In this way, Buridan was seeking a compromise between science and religion of a sort that became popular centuries later: God sets the machinery of the cosmos in motion, after which what happens is governed by the laws of nature. But although the conservation of momentum does keep the planets moving, by itself it could not keep them moving on curved orbits as Buridan thought was done by impetus; that requires an additional force, eventually recognized as the force of gravitation. Buridan also toyed with an idea due originally to Heraclides, that the Earth rotates once a day from west to east. He recognized that this would give the same appearance as if the heavens rotated around a stationary Earth once a day from east to west. He also acknowledged that this is a more natural theory, since the Earth is so much smaller that the firmament of Sun, Moon, planets, and stars. But he rejected the rotation of the Earth, reasoning that if the Earth rotated, then an arrow shot straight upward would fall to the west of the archer, since the Earth would have moved under the arrow while it was in flight. It is ironic that Buridan might have been saved from this error if he had realized that the Earth’s rotation would give the arrow an impetus that would carry it to the east along with the rotating Earth. Instead, he was misled by the notion of impetus; he considered only the vertical impetus given to the arrow by the bow, not the horizontal impetus it takes from the rotation of the Earth. Buridan’s notion of impetus remained influential for centuries. It was being taught at the University of Padua when Copernicus studied medicine there in the early 1500s. Later in that century Galileo learned about it as a student at the University of Pisa. Buridan sided with Aristotle on another issue, the impossibility of a vacuum. But he characteristically based his conclusion on observations: when air is sucked out of a drinking straw, a vacuum is prevented by liquid being pulled up into the straw; and when the handles of a bellows are pulled apart, a vacuum is prevented by air rushing into the bellows. It was natural to conclude that nature abhors a vacuum. As we will see in Chapter 12, the correct explanation for these phenomena in terms of air pressure was not understood until the 1600s. Buridan’s work was carried further by two of his students: Albert of Saxony and Nicole Oresme. Albert’s writings on philosophy became widely circulated, but it was Oresme who made the greater contribution to science. Oresme was born in 1325 in Normandy, and came to Paris to study with Buridan in the 1340s. He was a vigorous opponent of looking into the future by means of “astrology, geomancy, necromancy, or any such arts, if they can be called arts.” In 1377 Oresme was appointed bishop of the city of Lisieux, in Normandy, where he died in 1382. Oresme’s book On the Heavens and the Earth10 (written in the vernacular for the convenience of the king of France) has the form of an extended commentary on Aristotle, in which again and again he takes issue with The Philosopher. In this book Oresme reconsidered the idea that the heavens do not rotate about the Earth from east to west but, rather, the Earth rotates on its axis from west to east. Both Buridan and Oresme recognized that we observe only relative motion, so seeing the heavens move leaves open the possibility that it is instead the Earth that is moving. Oresme went through various objections to the idea, and picked them apart. Ptolemy in the Almagest had argued that if the Earth rotated, then clouds and thrown objects would be left behind; and as we have seen, Buridan had argued against the Earth’s rotation by reasoning that if the Earth rotated from west to east, then an arrow shot straight upward would be left behind by the Earth’s rotation, contrary to the observation that the arrow seems to fall straight down to the same spot on the Earth’s surface from which it was shot vertically upward. Oresme replied that the Earth’s rotation carries the arrow with it, along with the archer and the air and everything else on the Earth’s surface, thus applying Buridan’s theory of impetus in a way that its author had not understood. Oresme answered another objection to the rotation of the Earth—an objection of a very different sort, that there are passages in Holy Scripture (such as in the Book of Joshua) that refer to the Sun going daily around the Earth. Oresme replied that this was just a concession to the customs of popular speech, as where it is written that God became angry or repented—things that could not be taken literally. In this, Oresme was following the lead of Thomas Aquinas, who had wrestled with the passage in Genesis where God is supposed to proclaim, “Let there be a firmament above the waters, and let it divide the waters from the waters.” Aquinas had explained that Moses was adjusting his speech to the capacity of his audience, and should not be taken literally. Biblical literalism could have been a drag on the progress of science, if there had not been many inside the church like Aquinas and Oresme who took a more enlightened view. Despite all his arguments, Oresme finally surrendered to the common idea of a stationary Earth, as follows:

Afterward, it was demonstrated how it cannot be proved conclusively by argument that the heavens move. . . . However, everyone maintains, and I think myself, that the heavens do move and not the Earth: For God has established the world which shall not be moved, in spite of contrary reasons because they are clearly not conclusive persuasions. However, after considering all that has been said, one could then believe that the Earth moves and not the heavens, for the opposite is not self- evident. However, at first sight, this seems as much against natural reason than all or many of the articles of our faith. What I have said by way of diversion or intellectual exercise can in this manner serve as a valuable means of refuting and checking those who would like to impugn our faith by argument.11

We do not know if Oresme really was unwilling to take the final step toward acknowledging that the Earth rotates, or whether he was merely paying lip service to religious orthodoxy. Oresme also anticipated one aspect of Newton’s theory of gravitation. He argued that heavy things do not necessarily tend to fall toward the center of our Earth, if they are near some other world. The idea that there may be other worlds, more or less like the Earth, was theologically daring. Did God create humans on those other worlds? Did Christ come to those other worlds to redeem those humans? The questions are endless, and subversive. Unlike Buridan, Oresme was a mathematician. His major mathematical contribution led to an improvement on work done earlier at Oxford, so we must now shift our scene from France to England, and back a little in time, though we will return soon to Oresme. By the twelfth century Oxford had become a prosperous market town on the upper reaches of the Thames, and began to attract students and teachers. The informal cluster of schools at Oxford became recognized as a university in the early 1200s. Oxford conventionally lists its line of chancellors starting in 1224 with Robert Grosseteste, later bishop of Lincoln, who began the concern of medieval Oxford with natural philosophy. Grosseteste read Aristotle in Greek, and he wrote on optics and the calendar as well as on Aristotle. He was frequently cited by the scholars who succeeded him at Oxford. In Robert Grosseteste and the Origins of Experimental Science,12 A. C. Crombie went further, giving Grosseteste a pivotal role in the development of experimental methods leading to the advent of modern physics. This seems rather an exaggeration of Grosseteste’s importance. As is clear from Crombie’s account, “experiment” for Grosseteste was the passive observation of nature, not very different from the method of Aristotle. Neither Grosseteste nor any of his medieval successors sought to learn general principles by experiment in the modern sense, the aggressive manipulation of natural phenomena. Grosseteste’s theorizing has also been praised,13 but there is nothing in his work that bears comparison with the development of quantitatively successful theories of light by Hero, Ptolemy, and al-Haitam, or of planetary motion by Hipparchus, Ptolemy, and al-Biruni, among others. Grosseteste had a great influence on Roger Bacon, who in his intellectual energy and scientific innocence was a true representative of the spirit of his times. After studying at Oxford, Bacon lectured on Aristotle in Paris in the 1240s, went back and forth between Paris and Oxford, and became a Franciscan friar around 1257. Like Plato, he was enthusiastic about mathematics but made little use of it. He wrote extensively on optics and geography, but added nothing important to the earlier work of Greeks and Arabs. To an extent that was remarkable for the time, Bacon was also an optimist about technology:

Also cars can be made so that without animals they will move with unbelievable rapidity. . . . Also flying machines can be constructed so that a man sits in the midst of the machine revolving some engine by which artificial wings are made to beat the air like a flying bird.14

Appropriately, Bacon became known as “Doctor Mirabilis.” In 1264 the first residential college was founded at Oxford by Walter de Merton, at one time the chancellor of England and later bishop of Rochester. It was at Merton College that serious mathematical work at Oxford began in the fourteenth century. The key figures were four fellows of the college: Thomas Bradwardine (c. 1295–1349), William Heytesbury (fl. 1335), Richard Swineshead (fl. 1340– 1355), and John of Dumbleton (fl. 1338–1348). Their most notable achievement, known as the Merton College mean speed theorem, for the first time gives a mathematical description of nonuniform motion —that is, motion at a speed that does not remain constant. The earliest surviving statement of this theorem is by William of Heytesbury (chancellor of the University of Oxford in 1371), in Regulae solvendi sophismata. He defined the velocity at any instant in nonuniform motion as the ratio of the distance traveled to the time that would have elapsed if the motion had been uniform at that velocity. As it stands, this definition is circular, and hence useless. A more modern definition, possibly what Heytesbury meant to say, is that the velocity at any instant in nonuniform motion is the ratio of the distance traveled to the time elapsed if the velocity were the same as it is in a very short interval of time around that instant, so short that during this interval the change in velocity is negligible. Heytesbury then defined uniform acceleration as nonuniform motion in which the velocity increases by the same increment in each equal time. He then went on to state the theorem:15

When any mobile body is uniformly accelerated from rest to some given degree [of velocity], it will in that time traverse one-half the distance that it would traverse if, in that same time, it were moved uniformly at the degree of velocity terminating that increment of velocity. For that motion, as a whole, will correspond to the mean degree of that increment of velocity, which is precisely one-half that degree of velocity which is its terminal velocity.

That is, the distance traveled during an interval of time when a body is uniformly accelerated is the distance it would have traveled in uniform motion if its velocity in that interval equaled the average of the actual velocity. If something is uniformly accelerated from rest to some final velocity, then its average velocity during that interval is half the final velocity, so the distance traveled is half the final velocity times the time elapsed. Various proofs of this theorem were offered by Heytesbury, by John of Dumbleton, and then by Nicole Oresme. Oresme’s proof is the most interesting, because he introduced a technique of representing algebraic relations by graphs. In this way, he was able to reduce the problem of calculating the distance traveled when a body is uniformly accelerated from rest to some final velocity to the problem of calculating the area of a right triangle, whose sides that meet at the right angle have lengths equal respectively to the time elapsed and to the final velocity. (See Technical Note 17.) The mean speed theorem then follows immediately from an elementary fact of geometry, that the area of a right triangle is half the product of the two sides that meet at the right angle. Neither any don of Merton College nor Nicole Oresme seems to have applied the mean speed theorem to the most important case where it is relevant, the motion of freely falling bodies. For the dons and Oresme the theorem was an intellectual exercise, undertaken to show that they were capable of dealing mathematically with nonuniform motion. If the mean speed theorem is evidence of an increasing ability to use mathematics, it also shows how uneasy the fit between mathematics and natural science still was. It must be acknowledged that although it is obvious (as Strato had demonstrated) that falling bodies accelerate, it is not obvious that the speed of a falling body increases in proportion to the time, the characteristic of uniform acceleration, rather than to the distance fallen. If the rate of change of the distance fallen (that is, the speed) were proportional to the distance fallen, then the distance fallen once the body starts to fall would increase exponentially with time,* just as a bank account that receives interest proportional to the amount in the account increases exponentially with time (though if the interest rate is low it takes a long time to see this). The first person to guess that the increase in the speed of a falling body is proportional to the time elapsed seems to have been the sixteenth-century Dominican friar Domingo de Soto,16 about two centuries after Oresme. From the mid-fourteenth to the mid-fifteenth century Europe was harried by catastrophe. The Hundred Years’ War between England and France drained England and devastated France. The church underwent a schism, with a pope in Rome and another in Avignon. The Black Death destroyed a large fraction of the population everywhere. Perhaps as a result of the Hundred Years’ War, the center of scientific work shifted eastward in this period, from France and England to Germany and Italy. The two regions were spanned in the career of Nicholas of Cusa. Born around 1401 in the town of Kues on the Moselle in Germany, he died in 1464 in the Umbrian province of Italy. Nicholas was educated at both Heidelberg and Padua, becoming a canon lawyer, a diplomat, and after 1448 a cardinal. His writing shows the continuing medieval difficulty of separating natural science from theology and philosophy. Nicholas wrote in vague terms about a moving Earth and a world without limits, but with no use of mathematics. Though he was later cited by Kepler and Descartes, it is hard to see how they could have learned anything from him. The late Middle Ages also show a continuation of the Arab separation of professional mathematical astronomers, who used the Ptolemaic system, and physician-philosophers, followers of Aristotle. Among the fifteenth-century astronomers, mostly in Germany, were Georg von Peurbach and his pupil Johann Müller of Königsberg (Regiomontanus), who together continued and extended the Ptolemaic theory of epicycles.* Copernicus later made much use of the Epitome of the Almagest of Regiomontanus. The physicians included Alessandro Achillini (1463–1512) of Bologna and Girolamo Fracastoro of Verona (1478–1553), both educated at Padua, at the time a stronghold of Aristotelianism. Fracastoro gave an interestingly biased account of the conflict:17

You are well aware that those who make profession of astronomy have always found it extremely difficult to account for the appearances presented by the planets. For there are two ways of accounting for them: the one proceeds by means of those spheres called homocentric, the other by means of so-called eccentric spheres [epicycles]. Each of these methods has its dangers, each its stumbling blocks. Those who employ homocentric spheres never manage to arrive at an explanation of phenomena. Those who use eccentric spheres do, it is true, seem to explain the phenomena more adequately, but their conception of these divine bodies is erroneous, one might almost say impious, for they ascribe positions and shapes to them that are not fit for the heavens. We know that, among the ancients, Eudoxus and Callippus were misled many times by these difficulties. Hipparchus was among the first who chose rather to admit eccentric spheres than to be found wanting by the phenomena. Ptolemy followed him, and soon practically all astronomers were won over by Ptolemy. But against these astronomers, or at least, against the hypothesis of eccentrics, the whole of philosophy has raised continuing protest. What am I saying? Philosophy? Nature and the celestial spheres themselves protest unceasingly. Until now, no philosopher has ever been found who would allow that these monstrous spheres exist among the divine and perfect bodies.

To be fair, observations were not all on the side of Ptolemy against Aristotle. One of the failings of the Aristotelian system of homocentric spheres, which as we have seen had been noted around AD 200 by Sosigenes, is that it puts the planets always at the same distance from the Earth, in contradiction with the fact that the brightness of planets increases and decreases as they appear to go around the Earth. But Ptolemy’s theory seemed to go too far in the other direction. For instance, in Ptolemy’s theory the maximum distance of Venus from the Earth is 6.5 times its minimum distance, so if Venus shines by its own light, then (since apparent brightness goes as the inverse square of the distance) its maximum brightness should be 6.52 = 42 times greater than its minimum brightness, which is certainly not the case. Ptolemy’s theory had been criticized on this ground at the University of Vienna by Henry of Hesse (1325–1397). The resolution of the problem is of course that planets shine not by their own light, but by the reflected light of the Sun, so their apparent brightness depends not only on their distance from the Earth but also, like the Moon’s brightness, on their phase. When Venus is farthest from the Earth it is on the side of the Sun away from the Earth, so its face is fully illuminated; when it is closest to the Earth it is more or less between the Earth and the Sun and we mostly see its dark side. For Venus the effects of phase and distance therefore partly cancel, moderating its variations in brightness. None of this was understood until Galileo discovered the phases of Venus. Soon the controversy between Ptolemaic and Aristotelian astronomy was swept away in a deeper conflict, between those who followed either Ptolemy or Aristotle, all of them accepting that the heavens revolve around a stationary Earth; and a new revival of the idea of Aristarchus, that it is the Sun that is at rest.