温伯格《解释世界》第十四章 复杂的牛顿

到了牛顿我们到了科学革命的巅峰。但这是多么奇异的历史角色！牛顿从来没有走出过连接伦敦，剑桥和他的出生地林肯郡间的狭窄区域，他甚至没见过大海，虽然他对潮汐很有兴趣。中年之前他从来没有亲近过女性，包括他的母亲。（注：牛顿五十几岁时雇用了同母异父妹妹漂亮的女儿凯瑟琳·巴顿为女管家，但是虽然他们是亲密的朋友，他们好像没有浪漫关系。牛顿去世时伏尔泰在英格兰，他讲述牛顿的医生以及“他所死于其手臂上的外科医生”向伏尔泰确认牛顿从来没有与女性有过亲昵行为（见伏尔泰《哲学信件》，鲍勃斯-美林教育出版社1961年， 第63页）。伏尔泰没有说明医生与外科医生是如何知道的。）他非常关注于与科学没有关系的事物，比如《但以理书》的年代。苏富比拍卖行1936年拍卖的一系列牛顿手稿中有六十五万字关于炼金术，一百三十万字关于宗教。对那些可能的竞争者牛顿有些狡猾，卑鄙。然而他将物理学，天文学和数学紧密结合在了一起，这些学科的关系从柏拉图以来一直困扰哲学家们。 描写牛顿的作家有时会强调他不是一名现代科学家。这其中最有名的是约翰•梅纳德•凯恩斯（1936年购得苏富比拍卖的部分牛顿文章）论述：“牛顿不是理性时代的第一人。他是最后一位魔法师，最后一位巴比伦人和苏美尔人，最后一位像几千年前为我们的智力遗产奠立基础的先辈那样看待可见世界和思想世界的伟大心灵。”(注：出自凯恩斯为1946年皇家学会会议写的发言稿--“牛顿其人”。凯恩斯在会议召开三个月之前去世，他的弟弟代为宣读。）但是牛顿不是魔术过去的天才继任者。他既不是魔术师，也不完全是现代科学家。他穿越了过去自然哲学与后来成为现代科学间的边界。牛顿的成就（即便不是他的观念和个人行为）为后世科学提供了遵循的范例，演变为现代科学。(?) 艾萨克·牛顿1642年圣诞日出生于林肯郡伍尔索普庄园一个家族农场。他的父亲是没有文化的自耕农，在牛顿出生之前不久去世。他的母亲属于上流阶层，母亲的一位兄弟毕业于剑桥大学，后来成为教士。牛顿三岁时母亲改嫁离开了林肯郡，把他托付给了外祖母。牛顿10岁时去了据伍尔索普八英里的格兰瑟姆国王中学，寄宿在一位药剂师家中。在格兰瑟姆他学习拉丁文，神学，算术，几何，一些希腊与以及希伯来语。 17岁时牛顿被招回家乡务农，但他很不适合农活。两年后他以一名减费生被送往剑桥三一学院，也就是说他需要通过侍候学院教员以及可以付费的学生来支付学费和住宿费。与伽利略在比萨一样，他从学习亚里士多德开始，但是不久他就转向自己所关注的事物。在第二年他开始在以前用来记录亚里士多德的笔记本上做笔记—即《一些哲学问题》，幸运的是一直保留至今。 1663年12月剑桥大学用议员亨利·卢卡斯的一笔捐款建立了一个卢卡斯数学教授席位，年薪100英镑。从1664年开始该职位由剑桥第一位数学教授艾萨克.巴罗担任，他比牛顿大12岁。在那阶段牛顿开始学习数学，部分从师于巴罗，部分自学，他获得文学学士学位。1665年剑桥遭受瘟疫侵袭，大学大面积关闭，牛顿回到伍尔索普家中，从1664年牛顿开始了他的科学研究，下面会介绍。 重新回到剑桥后，牛顿1667年被选为三一学院院士，每年有2英镑收入，还可以自由出入学院图书馆。他与巴罗密切合作，为巴罗的讲座准备文字版讲稿。1669年巴罗为了全身心投入神学辞去了卢卡斯席位。在巴罗的建议下牛顿获得了该席位。借助于他母亲的资助，牛顿开始有些出格，购买新衣服，家具，进行了一些赌博。 之前不久，刚刚在斯图亚特王朝于1660年复辟之后，几位伦敦人波义耳，胡克以及天文学家和建筑学家克里斯托弗·雷恩创建了一个学会，他们聚在一起讨论自然哲学和观察实验问题。最早只有一位外籍会员，即克里斯蒂安·惠更斯。1662年学会得到皇家认可，成为伦敦皇家学会，现为英国国家科学院。1672年牛顿被选为会员，后来成为会长。 1675年牛顿面临一个危机。任职院士八年以后，他到了剑桥学院院士在英国国教就任圣职时间。牛顿需要宣誓坚信三位一体，这对牛顿根本不可能，他不相信尼西亚会议做出的圣父和圣子为同一本体的学说。好在建立卢卡斯数学教授席位条文中有一条约定担任该席位的人不应该在教会在过于活跃，为此国王查理斯二世颁布法令担任卢卡斯数学教授席位者不需要就任圣职。这些牛顿可以继续留在剑桥。 下面让我们领略牛顿从1664年开始在剑桥开展的伟大研究工作。这些研究围绕着光学，数学，以及后来人们所说的动力学。他在这三个领域中取得的任何一项成就都足以使他成为历史上最伟大的科学家之一。 牛顿最重要的实验成就是在光学方面（注：牛顿对炼金术实验投入了很大精力。这也可以称为化学，因为那时这两种没有明确界限。就像在第九章对贾比尔·伊本·哈杨所作的评述，在十八世纪之前还没有完善的化学理论指导像把普通金属转化为金子这样的炼金术不可行。虽然牛顿从事炼金术方面并不代表他背弃科学，但这没有带来任何重要成就。）他在大学期间的笔记《一些哲学问题》表明那时已经开始关注光的特性。与笛卡尔相反，牛顿认为光不是施予眼睛的压力，否则的话我们跑动时天看起来应该更亮。1665年他在伍尔索普做出了他对光学最伟大的贡献—提出了他的光学理论。人们很早就知道光穿过弯曲玻璃后呈现色彩，但人们一般认为这些色彩是由玻璃以某种方式产生的。牛顿推测白光是由所有颜色光组成，光线在玻璃或水中的折射角略取决于颜色，红光弯曲小于蓝光，这样当光线穿过棱镜或雨滴后各种色彩光线会分开。（注：平镜不会将不同色彩光线分开，因为虽然不太色彩光线进入玻璃后弯曲角度稍有不同，但离开时又弯曲回初始方向。棱镜各边不平行，不同颜色光线进入玻璃后折射不同，它们离开棱镜角度与进入时的折射角不同，所以当这些光线离开棱镜弯曲回来时不同颜色的光线仍然以小角度分开。）这可以解释笛卡尔所不理解的彩虹色彩。为了证实他的想法，牛顿做了两个重要实验。首先当用一个棱镜把光线分离出蓝光和红光后，他把这些光线分别导向其他棱镜，发现不会再分散出不同色彩。接下来采用一种聪明的设置，他设法将白光折射后产生的不同色彩光线重新合并，发现这些光线组合在一起后产生白光。 反射角与颜色有关对伽利略，开普勒和惠更斯的天文望远镜镜片有不利影响，这些镜片对组成白光的各种颜色光线聚焦不同，这样会使远方目标图像变得模糊不清。为了解决色差问题，牛顿1669年发明了一个天文望远镜--先用曲镜而非玻璃镜片将光线聚焦。（光线后来又被一个平面镜导向目镜，所以并没有完全消除色差。）他用一个只有六英寸长的反射天文望远镜就可以得到40倍的放大效果。现在所有主要天文望远镜都采用源于牛顿发明的反射天文望远镜。我第一次访问位于卡尔顿的英国皇家学会现址，我有幸被带到地下室观看牛顿制作的第二个小天文望远镜。 1671年皇家学会秘书和引导者亨利·奥尔登伯格邀请牛顿发表他的天文望远镜设计。1672年早期牛顿向《皇家学会自然科学会报》提交了一封信件，描述了他的天文望远镜以及他对颜色的研究工作。这导致了对牛顿研究的原创性以及成就的争执，特别是与胡克间的争执，胡克从1662年起就一直任皇家学会实验负责人，从1664年起担任约翰·卡特勒爵士赋予的讲席职位。胡克不是一个简单的对手，他对天文学，显微镜，制表以及城市规划做出了卓越的贡献。他声称做了与牛顿一样的光的实验，不能证明什么—颜色只是由棱镜增加到白光之上。 1675年牛顿在伦敦做了他的光学理论演讲。他推测光与物质一样是由微粒构成，这与同时期胡克和惠更斯提出的光是一种波的学说不同。这里牛顿未能作出正确科学判断。即使在牛顿时期已经有许多观察表明光的波动特性。现代量子力学确实把光描述为是无质量粒子—光子的集成，但是一般光实验中光子数巨大，结果光确实表现出波的特性。 惠更斯在他1678年的著作《光论》中将光描述为一种在由大量极其靠近微粒构成的介质以太中的扰波。正如在海洋中水的波动不是水在海面的运动，而是水的扰动，所以同理惠更斯理论认为一束光的传播其实是在以太粒子中的扰动，而不是粒子自身传播。每个扰粒子又作为新的扰源，一起构成波的整体波幅。当然从十九世纪詹姆斯·克拉克·麦克斯韦的研究中我们已经知道（即使不算量子效应）惠更斯只对了一半—光是种波，但是是在电磁场中的扰波，不是物质微粒的扰波。 惠更斯应用这个光波动理论推导出光在均匀介质（或真空）表现的像以直线运行的结论，因为只有沿这些线所有扰动微粒产生的波才会完整的叠加。他重新推导了反射等角率以及斯涅尔折射定律，他的推导不需要费马的光线沿最少时间路径的预先假定。（见技术说明30）按照惠更斯的折射理论，光线以一倾角穿过两种介质界面发生弯曲，光速发生变化，类似于一列行进的士兵头兵进入沼泽地带后他们的行进速度变慢，行进方向随之发生改变。 下面离题说几句，光以有限速度运行对惠更斯波动理论非常重要，这点与笛卡尔的想法不同。惠更斯提出之所以难以观测到光以有限速度运行是因为光速太快。如果假定光穿行月亮和地球之间需要用一个小时，那么月食时月亮不会正好在太阳的反面，而是会滞后33度。事实上我们看不到这种滞后，惠更斯由此推断出光速比声速至少快10万倍。这是正确的，实际值是1百万倍。 惠更斯接着描述了丹麦天文学家奥利·罗默刚刚对木星卫星的观测。这些观测结果表明当地球和木星相向运行时伊奥运转周期较短，而当地球和木星远离时周期较长。（之所以关注于伊奥是因为其在木星所有的伽利略卫星中轨道周期最短—只有1.77天。）惠更斯用后来称之为“多普勒效应”对此做出了解释：当木星和地球运行靠近或远离时，伊奥每完成一次运转木星和地球间的距离也在缩短或拉长，所以如果光以有限速度运行，观测到的伊奥完整运转一周时间间隔就比如果木星和地球静止时的时间间隔短或长。伊奥视周期的微小变化应该等于木星和地球沿分离方向相对速度与光速的比值，相对速度以木星和地球是远离或相向运行而分别取正值或负值。（见技术说明31。）这样通过测量伊奥视周期的变化以及地球和木星的相对速度就可以计算光速。地球比木星运行快得多，所以地球速度起决定作用。那时对太阳系的大小还不了解，人们也不知道地球和木星相对速度数值，不过惠更斯仍然应用罗默测量数据可以计算出光穿行地球轨道半径用时11分钟，这个结果不需要知道轨道大小。换种说法，天文单位（AU）定义为地球轨道平均半径，惠更斯计算出的光速为1天文单位每11秒。现代值为1天文单位每8.32秒。 牛顿和惠更斯时代已经有了光波动实验证明：波伦亚耶稣会稣传教士弗朗西斯科·玛丽亚·格里马尔迪（里奇奥利的学生）发现的衍射，该发现于1665年他去世后发表。格里马尔迪发现一个细长不透明杆子阳光下的影子并不完全清晰，而是边部有条纹。之所以有条纹是由于光波波长与细长杆厚度相比不可忽略，但是牛顿认为这是杆表面的折射效应。十九世纪早期托马斯·杨发现干涉效应--即光波沿不同路径到达某点发生的加强或抵消的图样，这对多数物理学家来说解决了光的粒子说和波动说争论。前面讲过，二十世纪人们发现这两种观点并不冲突。爱因斯坦1905年意识到虽然多数情况下光像波，但光的能量以一份一份传播，后来称为光子，每个光子具有很小的能量和动量，正比于光的频率。 牛顿终于在十七世纪九十年代早期创作的《光学》（以英文写作）中呈现了他的光学研究成果。该书于1704年发表，那时他早已声名远扬。 牛顿不只是位伟大的物理学家，他也是极富创造力的数学家。他从1664年开始阅读数学著作，包括欧几里德的《几何原本》以及笛卡尔的《几何》。他很快就开始求解一系列问题，很多都涉及到关于无限的问题。比如他考虑到无限序列，例如x-x2/2+x3/3-x4/4+ …, 他证明其和等于（1+x）的对数。（注：这是（1+x）的自然对数，常数e=2.71828…对该值的指数等于1+x。这种独特定义的原因是由于自然对数具有一些特性比常用对数要简单得多，常用对手以10为底。比如用牛顿方程2的自然对数等于序列1-1/2+1/3-1/4+ …,如果用常用对数表达的话则复杂的多。） 1665年牛顿开始思考无穷小问题。他设想一个问题：假定我们知道时间t时的行进距离为D（t）,如何计算时间t时的速度？他分析非匀速运行时某时刻瞬时速度等于该时刻很小时间间隔内行进距离与行进时间的比值。通过引入很小时间间隔符合o，他把时间t时的速度定义为时间t到时间t+o之间行进距离与o的比值，即速度等于[D(t+o)-D(t)]/o。比如如果D(t)=t3，那么D（t+o）=t3+3t2o+3to2+o3。因为o很小，我们可以忽略o2和o3相，这样D（t+o）=t3+3t2o，那D(t+o)-D(t)= 3t2o，速度为3t2。牛顿称之为D(t)的“流数”，也就是后来被称为的“导数”—现代微积分的基本工具。 （注：计算中忽略o2和o3相看起来像是近似计算，这是误读。十九世纪数学家去掉了这个模糊的很小o的想法，他们采用了极限定义：速度为o足够小时[D(t+o)-D(t)]/o的值。我们后面还会看到，牛顿后来放弃了这个很小的想法，而用了现代的极限观点。） 牛顿接下来开始计算曲面面积。这是微积分的基础。需要计算其流数为用曲线方程描述的量。比如我们前面讲的，3x2是x3的流数，所以对抛物线y=3x2，在x=0与其他x间的面积等于x3。牛顿称之为“流数的反方法，”后来人们称之为“积分”。 牛顿很早就发明了微分和积分方法，但很长时间不为人所知。后来在1671年他决定与他的光学研究一起发表，但是除非有资助，否则伦敦书商不愿意发表此著作。 1669年巴罗将一份牛顿的手稿《运用无穷多项方程的分析学》给了数学家约翰·柯林斯。哲学家和数学家戈特弗里德·威廉·莱布尼兹（惠更斯的前学生，比牛顿小几岁，前一年已经独立发明了微积分的主体部分）1676年访问伦敦时看到了柯林斯对该手稿的拷贝。1676年牛顿在一些信件中向莱布尼茨展现了他的一些成果。1684年和1685年莱布尼茨发表了他的微积分成就，其中没有提及牛顿的贡献。在这些文章中莱布尼茨引入了|“微积分”一词，并且以现代符号做了表达，包括积分符号ʃ。 为了证实自己发现了微积分，牛顿在1704年《光学》版本两篇文章中描述了他自己的方法。1705年1月《光学》的一篇匿名书评暗示其方法来自莱布尼茨。正如牛顿猜测，这些书评是莱布尼茨所写。1709年《皇家学会哲学学报》发表了一篇约翰·基尔的文章支持牛顿的发现优先权，莱布尼茨1711年做出回复，对皇家学会大为愤怒。1712年皇家学会成立了一个匿名委员会调查此争议。两个世纪之后该委员会名单才公开，其中几乎所有成员都是牛顿支持者。1715年委员会做出报告称牛顿是微积分发现者。该报告是由牛顿起草。此结论获得一个匿名报告审查者的支持，这也是牛顿本人所写。 现代学者认为牛顿和莱布尼茨分别独立发现了微积分。牛顿比莱布尼茨早十年完成。但莱布尼茨发表了他的成就也功不可没。牛顿在1671年极力为他的微积分专著寻找出版社之后，他一直没有发表他的作品，直到由于与莱布尼茨的争端才被迫公开。公开发表是科学发现过程至关重要一步，它代表作者做出相信他的工作正确，适于其他科学家应用的判断。为此现今科学发现的荣誉通常赋予第一个发表的人。但是虽然莱布尼茨第一个发表了微积分，我们后面会看到是牛顿将微积分应用到科学，而不是莱布尼茨。莱布尼茨与笛卡尔一样是伟大的数学家，其哲学成就令人钦佩，但他对自然科学并没有做出重要贡献。 具有重大历史影响的是牛顿的运动和重力理论。导致物体落向地球的重力随着与地面距离增加而减小的观点早已存在。九世纪旅行家爱尔兰修道士邓斯·司各特提出过此设想，但并没有将这种力与行星运行联系起来。

With Newton we come to the climax of the scientific revolution. But what an odd bird to be cast in such a historic role! Newton never traveled outside a narrow strip of England, linking London, Cambridge, and his birthplace in Lincolnshire, not even to see the sea, whose tides so much interested him. Until middle age he was never close to any woman, not even to his mother.* He was deeply concerned with matters having little to do with science, such as the chronology of the Book of Daniel. A catalog of Newton manuscripts put on sale at Sotheby’s in 1936 shows 650,000 words on alchemy, and 1.3 million words on religion. With those who might be competitors Newton could be devious and nasty. Yet he tied up strands of physics, astronomy, and mathematics whose relations had perplexed philosophers since Plato. Writers about Newton sometimes stress that he was not a modern scientist. The best-known statement along these lines is that of John Maynard Keynes (who had bought some of the Newton papers in the 1936 auction at Sotheby’s): “Newton was not the first of the age of reason. He was the last of the magicians, the last of the Babylonians and Sumerians, the last great mind which looked out on the visible and intellectual world with the same eyes as those who began to build our intellectual inheritance rather less than 10,000 years ago.”* But Newton was not a talented holdover from a magical past. Neither a magician nor an entirely modern scientist, he crossed the frontier between the natural philosophy of the past and what became modern science. Newton’s achievements, if not his outlook or personal behavior, provided the paradigm that all subsequent science has followed, as it became modern. Isaac Newton was born on Christmas Day 1642 at a family farm, Woolsthorpe Manor, in Lincolnshire. His father, an illiterate yeoman farmer, had died shortly before Newton’s birth. His mother was higher in social rank, a member of the gentry, with a brother who had graduated from the University of Cambridge and become a clergyman. When Newton was three his mother remarried and left Woolsthorpe, leaving him behind with his grandmother. When he was 10 years old Newton went to the one-room King’s School at Grantham, eight miles from Woolsthorpe, and lived there in the house of an apothecary. At Grantham he learned Latin and theology, arithmetic and geometry, and a little Greek and Hebrew. At the age of 17 Newton was called home to take up his duties as a farmer, but for these he was found to be not well suited. Two years later he was sent up to Trinity College, Cambridge, as a sizar, meaning that he would pay for his tuition and room and board by waiting on fellows of the college and on those students who had been able to pay their fees. Like Galileo at Pisa, he began his education with Aristotle, but he soon turned away to his own concerns. In his second year he started a series of notes, Questiones quandam philosophicae, in a notebook that had previously been used for notes on Aristotle, and which fortunately is still extant. In December 1663 the University of Cambridge received a donation from a member of Parliament, Henry Lucas, establishing a professorship in mathematics, the Lucasian chair, with a stipend of £100 a year. Beginning in 1664 the chair was occupied by Isaac Barrow, the first professor of mathematics at Cambridge, 12 years older than Newton. Around then Newton began his study of mathematics, partly with Barrow and partly alone, and received his bachelor of arts degree. In 1665 the plague struck Cambridge, the university largely shut down, and Newton went home to Woolsthorpe. In those years, from 1664 on, Newton began his scientific research, to be described below. Back in Cambridge, in 1667 Newton was elected a fellow of Trinity College; the fellowship brought him £2 a year and free access to the college library. He worked closely with Barrow, helping to prepare written versions of Barrow’s lectures. Then in 1669 Barrow resigned the Lucasian chair in order to devote himself entirely to theology. At Barrow’s suggestion, the chair went to Newton. With financial help from his mother, Newton began to spread himself, buying new clothes and furnishings and doing a bit of gambling.1 A little earlier, immediately after the restoration of the Stuart monarchy in 1660, a society had been formed by a few Londoners including Boyle, Hooke, and the astronomer and architect Christopher Wren, who would meet to discuss natural philosophy and observe experiments. At the beginning it had just one foreign member, Christiaan Huygens. The society received a royal charter in 1662 as the Royal Society of London, and has remained Britain’s national academy of science. In 1672 Newton was elected to membership in the Royal Society, which he later served as president. In 1675 Newton faced a crisis. Eight years after beginning his fellowship, he had reached the point at which fellows of a Cambridge college were supposed to take holy orders in the Church of England. This would require swearing to belief in the doctrine of the Trinity, but that was impossible for Newton, who rejected the decision of the Council of Nicaea that the Father and the Son are of one substance. Fortunately, the deed that had established the Lucasian chair included a stipulation that its holder should not be active in the church, and on that basis King Charles II was induced to issue a decree that the holder of the Lucasian chair would thenceforth never be required to take holy orders. So Newton was able to continue at Cambridge. Let’s now take up the great work that Newton began at Cambridge in 1664. This research centered on optics, mathematics, and what later came to be called dynamics. His work in any one of these three areas would qualify him as one of the great scientists of history. Newton’s chief experimental achievements were concerned with optics.* His undergraduate notes, the Questiones quandam philosophicae, show him already concerned with the nature of light. Newton concluded, contrary to Descartes, that light is not a pressure on the eyes, for if it were then the sky would seem brighter to us when we are running. At Woolsthorpe in 1665 he developed his greatest contribution to optics, his theory of color. It had been known since antiquity that colors appear when light passes through a curved piece of glass, but it had generally been thought that these colors were somehow produced by the glass. Newton conjectured instead that white light consists of all the colors, and that the angle of refraction in glass or water depends slightly on the color, red light being bent somewhat less than blue light, so that the colors are separated when light passes through a prism or a raindrop.* This would explain what Descartes had not understood, the appearance of colors in the rainbow. To test this idea, Newton carried out two decisive experiments. First, after using a prism to create separate rays of blue and red light, he directed these rays separately into other prisms, and found no further dispersion into different colors. Next, with a clever arrangement of prisms, he managed to recombine all the different colors produced by refraction of white light, and found that when these colors are combined they produce white light. The dependence of the angle of refraction on color has the unfortunate consequence that the glass lenses in telescopes like those of Galileo, Kepler, and Huygens focus the different colors in white light differently, blurring the images of distant objects. To avoid this chromatic aberration Newton in 1669 invented a telescope in which light is initially focused by a curved mirror rather than by a glass lens. (The light rays are then deflected by a plane mirror out of the telescope to a glass eyepiece, so not all chromatic aberration was eliminated.) With a reflecting telescope only six inches long, he was able to achieve a magnification by 40 times. All major astronomical light-gathering telescopes are now reflecting telescopes, descendants of Newton’s invention. On my first visit to the present quarters of the Royal Society in Carlton House Terrace, as a treat I was taken down to the basement to look at Newton’s little telescope, the second one he made. In 1671 Henry Oldenburg, the secretary and guiding spirit of the Royal Society, invited Newton to publish a description of his telescope. Newton submitted a letter describing it and his work on color to Philosophical Transactions of the Royal Society early in 1672. This began a controversy over the originality and significance of Newton’s work, especially with Hooke, who had been curator of experiments at the Royal Society since 1662, and holder of a lectureship endowed by Sir John Cutler since 1664. No feeble opponent, Hooke had made significant contributions to astronomy, microscopy, watchmaking, mechanics, and city planning. He claimed that he had performed the same experiments on light as Newton, and that they proved nothing—colors were simply added to white light by the prism. Newton lectured on his theory of light in London in 1675. He conjectured that light, like matter, is composed of many small particles—contrary to the view, proposed at about the same time by Hooke and Huygens, that light is a wave. This was one place where Newton’s scientific judgment failed him. There are many observations, some even in Newton’s time, that show the wave nature of light. It is true that in modern quantum mechanics light is described as an ensemble of massless particles, called photons, but in the light encountered in ordinary experience the number of photons is enormous, and in consequence light does behave as a wave. In his 1678 Treatise on Light, Huygens described light as a wave of disturbance in a medium, the ether, which consists of a vast number of tiny material particles in close proximity. Just as in an ocean wave in deep water it is not the water that moves along the surface of the ocean but the disturbance of the water, so likewise in Huygens’ theory it is the wave of disturbance in the particles of the ether that moves in a ray of light, not the particles themselves. Each disturbed particle acts as a new source of disturbance, which contributes to the total amplitude of the wave. Of course, since the work of James Clerk Maxwell in the nineteenth century we have known that (even apart from quantum effects) Huygens was only half right—light is a wave, but a wave of disturbances in electric and magnetic fields, not a wave of disturbance of material particles. Using this wave theory of light, Huygens was able to derive the result that light in a homogeneous medium (or empty space) behaves as if it travels in straight lines, as it is only along these lines that the waves produced by all the disturbed particles add up constructively. He gave a new derivation of the equal-angles rule for reflection, and of Snell’s law for refraction, without Fermat’s a priori assumption that light rays take the path of least time. (See Technical Note 30.) In Huygens’ theory of refraction, a ray of light is bent in passing at an oblique angle through the boundary between two media with different light speeds in much the way the direction of march of a line of soldiers will change when the leading edge of the line enters a swampy terrain, in which their marching speed is reduced. To digress a bit, it was essential to Huygens’ wave theory that light travels at a finite speed, contrary to what had been thought by Descartes. Huygens argued that effects of this finite speed are hard to observe simply because light travels so fast. If for instance it took light an hour to travel the distance of the Moon from the Earth, then at the time of an eclipse of the Moon the Moon would be seen not directly opposite the Sun, but lagging behind by about 33°. From the fact that no lag is seen, Huygens concluded that the speed of light must be at least 100,000 times as fast as the speed of sound. This is correct; the actual ratio is about 1 million. Huygens went on to describe recent observations of the moons of Jupiter by the Danish astronomer Ole Rømer. These observations showed that the period of Io’s revolution appears shorter when Earth and Jupiter are approaching each other and longer when they are moving apart. (Attention focused on Io, because it has the shortest orbital period of any of Jupiter’s Galilean moons—only 1.77 days.) Huygens interpreted this as what later became known as a “Doppler effect”: when Jupiter and the Earth are moving closer together or farther apart, their separation at each successive completion of a whole period of revolution of Io is respectively decreasing or increasing, and so if light travels at a finite speed, the time interval between observations of complete periods of Io should be respectively less or greater than if Jupiter and the Earth were at rest. Specifically, the fractional shift in the apparent period of Io should be the ratio of the relative speed of Jupiter and the Earth along the direction separating them to the speed of light, with the relative speed taken as positive or negative if Jupiter and the Earth are moving farther apart or closer together, respectively. (See Technical Note 31.) Measuring the apparent changes in the period of Io and knowing the relative speed of Earth and Jupiter, one could calculate the speed of light. Because the Earth moves much faster than Jupiter, it is chiefly the Earth’s velocity that dominates the relative speed. The scale of the solar system was then not well known, so neither was the numerical value of the relative speed of separation of the Earth and Jupiter, but using Rømer’s data Huygens was able to calculate that it takes 11 minutes for light to travel a distance equal to the radius of the Earth’s orbit, a result that did not depend on knowing the size of the orbit. To put it another way, since the astronomical unit (AU) of distance is defined as the mean radius of the Earth’s orbit, the speed of light was found by Huygens to be 1 AU per 11 minutes. The modern value is 1 AU per 8.32 minutes. There already was experimental evidence of the wave nature of light that would have been available to Newton and Huygens: the discovery of diffraction by the Bolognese Jesuit Francesco Maria Grimaldi (a student of Riccioli), published posthumously in 1665. Grimaldi had found that the shadow of a narrow opaque rod in sunlight is not perfectly sharp, but is bordered by fringes. The fringes are due to the fact that the wavelength of light is not negligible compared with the thickness of the rod, but Newton argued that they were instead the result of some sort of refraction at the surface of the rod. The issue of light as corpuscle or wave was settled for most physicists when, in the early nineteenth century, Thomas Young discovered interference, the pattern of reinforcement or cancellation of light waves that arrive at given spots along different paths. As already mentioned, in the twentieth century it was discovered that the two views are not incompatible. Einstein in 1905 realized that although light for most purposes behaves as a wave, the energy in light comes in small packets, later called photons, each with a tiny energy and momentum proportional to the frequency of the light. Newton finally presented his work on light in his book Opticks, written (in English) in the early 1690s. It was published in 1704, after he had already become famous. Newton was not only a great physicist but also a creative mathematician. He began in 1664 to read works on mathematics, including Euclid’s Elements and Descartes’ Geometrie. He soon started to work out the solutions to a variety of problems, many involving infinities. For instance, he considered infinite series, such as x − x2/2 + x3/3 − x4/4 + . . . , and showed that this adds up to the logarithm* of 1 + x. In 1665 Newton began to think about infinitesimals. He took up a problem: suppose we know the distance D(t) traveled in any time t; how do we find the velocity at any time? He reasoned that in nonuniform motion, the velocity at any instant is the ratio of the distance traveled to the time elapsed in an infinitesimal interval of time at that instant. Introducing the symbol o for an infinitesimal interval of time, he defined the velocity at time t as the ratio to o of the distance traveled between time t and time t + o, that is, the velocity is [D(t + o) − D(t)]/o. For instance, if D(t) = t3, then D(t + o) = t3 + 3t2o + 3to2 + o3. For o infinitesimal, we can neglect the terms proportional to o2 and o3, and take D(t + o) = t3 + 3t2 o, so that D(t + o) − D(t) = 3t2 o and the velocity is just 3t2. Newton called this the “fluxion” of D(t), but it became known as the “derivative,” the fundamental tool of modern differential calculus.* Newton then took up the problem of finding the areas bounded by curves. His answer was the fundamental theorem of calculus; one must find the quantity whose fluxion is the function described by the curve. For instance, as we have seen, 3x2 is the fluxion of x3, so the area under the parabola y = 3x2 between x = 0 and any other x is x3. Newton called this the “inverse method of fluxions,” but it became known as the process of “integration.” Newton had invented the differential and integral calculus, but for a long while this work did not become widely known. Late in 1671 he decided to publish it along with an account of his work on optics, but apparently no London bookseller was willing to undertake the publication without a heavy subsidy.2 In 1669 Barrow gave a manuscript of Newton’s De analysi per aequationes numero terminorum infinitas to the mathematician John Collins. A copy made by Collins was seen on a visit to London in 1676 by the philosopher and mathematician Gottfried Wilhelm Leibniz, a former student of Huygens and a few years younger than Newton, who had independently discovered the essentials of the calculus in the previous year. In 1676 Newton revealed some of his own results in letters that were meant to be seen by Leibniz. Leibniz published his work on calculus in articles in 1684 and 1685, without acknowledging Newton’s work. In these publications Leibniz introduced the word “calculus,” and presented its modern notation, including the integration sign ∫. To establish his claim to calculus, Newton described his own methods in two papers included in the 1704 edition of Opticks. In January 1705 an anonymous review of Opticks hinted that these methods were taken from Leibniz. As Newton guessed, this review had been written by Leibniz. Then in 1709 Philosophical Transactions of the Royal Society published an article by John Keill defending Newton’s priority of discovery, and Leibniz replied in 1711 with an angry complaint to the Royal Society. In 1712 the Royal Society convened an anonymous committee to look into the controversy. Two centuries later the membership of this committee was made public, and it turned out to have consisted almost entirely of Newton’s supporters. In 1715 the committee reported that Newton deserved credit for the calculus. This report had been drafted for the committee by Newton. Its conclusions were supported by an anonymous review of the report, also written by Newton. The judgment of contemporary scholars3 is that Leibniz and Newton had discovered the calculus independently. Newton accomplished this a decade earlier than Leibniz, but Leibniz deserves great credit for publishing his work. In contrast, after his original effort in 1671 to find a publisher for his treatise on calculus, Newton allowed this work to remain hidden until he was forced into the open by the controversy with Leibniz. The decision to go public is generally a critical element in the process of scientific discovery.4 It represents a judgment by the author that the work is correct and ready to be used by other scientists. For this reason, the credit for a scientific discovery today usually goes to the first to publish. But though Leibniz was the first to publish on calculus, as we shall see it was Newton rather than Leibniz who applied calculus to problems in science. Though, like Descartes, Leibniz was a great mathematician whose philosophical work is much admired, he made no important contributions to natural science. It was Newton’s theories of motion and gravitation that had the greatest historical impact. It was an old idea that the force of gravity that causes objects to fall to the Earth decreases with distance from the Earth’s surface. This much was suggested in the ninth century by a well-traveled Irish monk, Duns Scotus (Johannes Scotus Erigena, or John the Scot), but with no suggestion of any connection of this force with the motion of the planets. The suggestion that the force that holds the planets in their orbits decreases with the inverse square of the distance from the Sun may have been first made in 1645 by a French priest, Ismael Bullialdus, who was later quoted by Newton and elected to the Royal Society. But it was Newton who made this convincing, and related the force to gravity. Writing about 50 years later, Newton described how he began to study gravitation. Even though his statement needs a good deal of explanation, I feel I have to quote it here, because it describes in Newton’s own words what seems to have been a turning point in the history of civilization. According to Newton, it was in 1666 that:

I began to think of gravity extending to the orb of the Moon & (having found out how to estimate the force with which [a] globe revolving within a sphere presses the surface of the sphere) from Kepler’s rule of the periodical times of the Planets being in sesquialterate proportion of their distances from the center of their Orbs, I deduced that the forces which keep the Planets in their Orbs must [be] reciprocally as the squares of their distances from the centers about which they revolve & thereby compared the Moon in her Orb with the force of gravity at the surface of the Earth & found them answer pretty nearly. All this [including his work on infinite series and calculus] was in the two plague years of 1665–1666. For in those days I was in the prime of my age for invention and minded Mathematicks and Philosophy more than at any time since.5

As I said, this takes some explaining. First, Newton’s parenthesis “having found out how to estimate the force with which [a] globe revolving within a sphere presses the surface of the sphere” refers to the calculation of centrifugal force, a calculation that had already been done (probably unknown to Newton) around 1659 by Huygens. For Huygens and Newton (as for us), acceleration had a broader definition than just a number giving the change of speed per time elapsed; it is a directed quantity, giving the change per time elapsed in the direction as well as in the magnitude of the velocity. There is an acceleration in circular motion even at constant speed—it is the “centripetal acceleration,” consisting of a continual turning toward the center of the circle. Huygens and Newton concluded that a body moving at a constant speed v around a circle of radius r is accelerating toward the center of the circle, with acceleration v2/r, so the force needed to keep it moving on the circle rather than flying off in a straight line into space is proportional to v2/r. (See Technical Note 32.) The resistance to this centripetal acceleration is experienced as what Huygens called centrifugal force, as when a weight at the end of a cord is swung around in a circle. For the weight, the centrifugal force is resisted by tension in the cord. But planets are not attached by cords to the Sun. What is it that resists the centrifugal force produced by a planet’s nearly circular motion around the Sun? As we will see, the answer to this question led to Newton’s discovery of the inverse square law of gravitation. Next, by “Kepler’s rule of the periodical times of the Planets being in sesquialterate proportion of their distances from the center of their Orbs” Newton meant what we now call Kepler’s third law: that the square of the periods of the planets in their orbits is proportional to the cubes of the mean radii of their orbits, or in other words, the periods are proportional to the 3/2 power (the “sesquialterate proportion”) of the mean radii.* The period of a body moving with speed v around a circle of radius r is the circumference 2πr divided by the speed v, so for circular orbits Kepler’s third law tells us that r2/v2 is proportional to r3, and therefore their inverses are proportional: v2/r2 is proportional to 1/r3. It follows that the force keeping the planets in orbit, which is proportional to v2/r, must be proportional to 1/r2. This is the inverse square law of gravity. This in itself might be regarded as just a way of restating Kepler’s third law. Nothing in Newton’s consideration of the planets makes any connection between the force holding the planets in their orbits and the commonly experienced phenomena associated with gravity on the Earth’s surface. This connection was provided by Newton’s consideration of the Moon. Newton’s statement that he “compared the Moon in her Orb with the force of gravity at the surface of the Earth & found them answer pretty nearly” indicates that he had calculated the centripetal acceleration of the Moon, and found that it was less than the acceleration of falling bodies on the surface of the Earth by just the ratio one would expect if these accelerations were inversely proportional to the square of the distance from the center of the Earth. To be specific, Newton took the radius of the Moon’s orbit (well known from observations of the Moon’s diurnal parallax) to be 60 Earth radii; it is actually about 60.2 Earth radii. He used a crude estimate of the Earth’s radius,* which gave a crude value for the radius of the Moon’s orbit, and knowing that the sidereal period of the Moon’s revolution around the Earth is 27.3 days, he could estimate the Moon’s velocity and from that its centripetal acceleration. This acceleration turned out to be less than the acceleration of falling bodies on the surface of the Earth by a factor roughly (very roughly) equal to 1/(60)2, as expected if the force holding the Moon in its orbit is the same that attracts bodies on the Earth’s surface, but reduced in accordance with the inverse square law. (See Technical Note 33.) This is what Newton meant when he said that he found that the forces “answer pretty nearly.” This was the climactic step in the unification of the celestial and terrestrial in science. Copernicus had placed the Earth among the planets, Tycho had shown that there is change in the heavens, and Galileo had seen that the Moon’s surface is rough, like the Earth’s, but none of this related the motion of planets to forces that could be observed on Earth. Descartes had tried to understand the motions of the solar system as the result of vortices in the ether, not unlike vortices in a pool of water on Earth, but his theory had no success. Now Newton had shown that the force that keeps the Moon in its orbit around the Earth and the planets in their orbits around the Sun is the same as the force of gravity that causes an apple to fall to the ground in Lincolnshire, all governed by the same quantitative laws. After this the distinction between the celestial and terrestrial, which had constrained physical speculation from Aristotle on, had to be forever abandoned. But this was still far short of a principle of universal gravitation, which would assert that every body in the universe, not just the Earth and Sun, attracts every other body with a force that decreases as the inverse square of the distance between them. There were still four large holes in Newton’s arguments:

1. In comparing the centripetal acceleration of the Moon with the acceleration of falling bodies on the surface of the Earth, Newton had assumed that the force producing these accelerations decreases with the inverse square of the distance, but the distance from what? This makes little difference for the motion of the Moon, which is so far from the Earth that the Earth can be taken as almost a point particle as far as the Moon’s motion is concerned. But for an apple falling to the ground in Lincolnshire, the Earth extends from the bottom of the tree, a few feet away, to a point at the antipodes, 8,000 miles away. Newton had assumed that the distance relevant to the fall of any object near the Earth’s surface is its distance to the center of the Earth, but this was not obvious.

2. Newton’s explanation of Kepler’s third law ignored the obvious differences between the planets. Somehow it does not matter that Jupiter is much bigger than Mercury; the difference in their centripetal accelerations is just a matter of their distances from the Sun. Even more dramatically, Newton’s comparison of the centripetal acceleration of the Moon and the acceleration of falling bodies on the surface of the Earth ignored the conspicuous difference between the Moon and a falling body like an apple. Why do these differences not matter?

3. In the work he dated to 1665–1666, Newton interpreted Kepler’s third law as the statement that the products of the centripetal accelerations of the various planets with the squares of their distances from the Sun are the same for all planets. But the common value of this product is not at all equal to the product of the centripetal acceleration of the Moon with the square of its distance from the Earth; it is much greater. What accounts for this difference?

4. Finally, in this work Newton had taken the orbits of the planets around the Sun and of the Moon around the Earth to be circular at constant speed, even though as Kepler had shown they are not precisely circular but instead elliptical, the Sun and Earth are not at the centers of the ellipses, and the Moon’s and planets’ speeds are only approximately constant.

Newton struggled with these problems in the years following 1666. Meanwhile, others were coming to the same conclusions that Newton had already reached. In 1679 Newton’s old adversary Hooke published his Cutlerian lectures, which contained some suggestive though nonmathematical ideas about motion and gravitation:

First, that all Coelestial Bodies whatsoever, have an attraction or gravitating power towards their own Centers, whereby they attract not only their own parts, and keep them from flying from them, as we may observe the Earth to do, but that they do also attract all the other Coelestial Bodies that are within the sphere of their activity—The second supposition is this, That all bodies whatsoever that are put into a direct and simple motion, will so continue to move forward in a straight line, till they are by some other effectual powers deflected and bent into a Motion, describing a Circle, Ellipsis, or some other more compounded Curve Line. The third supposition is, That these attractive powers are so much the more powerful in operating, by how much the nearer the body wrought upon is to their own Centers.6

Hooke wrote to Newton about his speculations, including the inverse square law. Newton brushed him off, replying that he had not heard of Hooke’s work, and that the “method of indivisibles”7 (that is, calculus) was needed to understand planetary motions. Then in August 1684 Newton received a fateful visit in Cambridge from the astronomer Edmund Halley. Like Newton and Hooke and also Wren, Halley had seen the connection between the inverse square law of gravitation and Kepler’s third law for circular orbits. Halley asked Newton what would be the actual shape of the orbit of a body moving under the influence of a force that decreases with the inverse square of the distance. Newton answered that the orbit would be an ellipse, and promised to send a proof. Later that year Newton submitted a 10-page document, On the Motion of Bodies in Orbit, which showed how to treat the general motion of bodies under the influence of a force directed toward a central body. Three years later the Royal Society published Newton’s Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), doubtless the greatest book in the history of physical science. A modern physicist leafing through the Principia may be surprised to see how little it resembles any of today’s treatises on physics. There are many geometrical diagrams, but few equations. It seems almost as if Newton had forgotten his own development of calculus. But not quite. In many of his diagrams one sees features that are supposed to become infinitesimal or infinitely numerous. For instance, in showing that Kepler’s equal-area rule follows for any force directed toward a fixed center, Newton imagines that the planet receives infinitely many impulses toward the center, each separated from the next by an infinitesimal interval of time. This is just the sort of calculation that is made not only respectable but quick and easy by the general formulas of calculus, but nowhere in the Principia do these general formulas make their appearance. Newton’s mathematics in the Principia is not very different from what Archimedes had used in calculating the areas of circles, or what Kepler had used in calculating the volumes of wine casks. The style of the Principia reminds the reader of Euclid’s Elements. It begins with definitions:8

Definition I Quantity of matter is a measure of matter that arises from its density and volume jointly.

What appears in English translation as “quantity of matter” was called massa in Newton’s Latin, and is today called “mass.” Newton here defines it as the product of density and volume. Even though Newton does not define density, his definition of mass is still useful because his readers could take it for granted that bodies made of the same substances, such as iron at a given temperature, will have the same density. As Archimedes had shown, measurements of specific gravity give values for density relative to that of water. Newton notes that we measure the mass of a body from its weight, but does not confuse mass and weight.

Definition II

Quantity of motion is a measure of motion that arises from the velocity and the quantity of matter jointly.

What Newton calls “quantity of motion” is today termed “momentum.” It is defined here by Newton as the product of the velocity and the mass.

Definition III

Inherent force of matter [vis insita] is the power of resisting by which every body, so far as [it] is able, perseveres in its state either of resting or of moving uniformly straight forward.

Newton goes on to explain that this force arises from the body’s mass, and that it “does not differ in any way from the inertia of the mass.” We sometimes now distinguish mass, in its role as the quantity that resists changes in motion, as “inertial mass.”

Definition IV

Impressed force is the action exerted on a body to change its state either of resting or of uniformly moving straight forward.

This defines the general concept of force, but does not yet give meaning to any numerical value we might assign to a given force. Definitions V through VIII go on to define centripetal acceleration and its properties. After the definitions comes a scholium, or annotation, in which Newton declines to define space and time, but does offer a description:

I. Absolute, true, and mathematical time, in and of itself, and of its own nature, without relation to anything external, flows uniformly. . . .

II. Absolute space, of its own nature without relation to anything external, always remains homogeneous and immovable.

Both Leibniz and Bishop George Berkeley criticized this view of time and space, arguing that only relative positions in space and time have any meaning. Newton had recognized in this scholium that we normally deal only with relative positions and velocities, but now he had a new handle on absolute space: in Newton’s mechanics, acceleration (unlike position or velocity) has an absolute significance. How could it be otherwise? It is a matter of common experience that acceleration has effects; there is no need to ask, “Acceleration relative to what?” From the forces pressing us back in our seats, we know that we are being accelerated when we are in a car that suddenly speeds up, whether or not we look out the car’s window. As we will see, in the twentieth century the views of space and time of Leibniz and Newton were reconciled in the general theory of relativity. Then at last come Newton’s famous three laws of motion:

Law I

Every body perseveres in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by forces impressed.

This was already known to Gassendi and Huygens. It is not clear why Newton bothered to include it as a separate law, since the first law is a trivial (though important) consequence of the second.

Law II

A change in motion is proportional to the motive force impressed and takes place along the straight line in which that force is impressed.

By “change of motion” here Newton means the change in the momentum, which he called the “quantity of motion” in Definition II. It is actually the rate of change of momentum that is proportional to the force. We conventionally define the units in which force is measured so that the rate of change of momentum is actually equal to the force. Since momentum is mass times velocity, its rate of change is mass times acceleration. Newton’s second law is thus the statement that mass times acceleration equals the force producing the acceleration. But the famous equation F = ma does not appear in the Principia; the second law was reexpressed in this way by Continental mathematicians in the eighteenth century.

Law III

To any action there is always an opposite and equal reaction; in other words, the actions of two bodies upon each other are always equal, and always opposite in direction.

In true geometric style, Newton then goes on to present a series of corollaries deduced from these laws. Notable among them was Corollary III, which gives the law of the conservation of momentum. (See Technical Note 34.) After completing his definitions, laws, and corollaries, Newton begins in Book I to deduce their consequences. He proves that central forces (forces directed toward a single central point) and only central forces give a body a motion that sweeps out equal areas in equal times; that central forces proportional to the inverse square of the distance and only such central forces produce motion on a conic section, that is, a circle, an ellipse, a parabola, or a hyperbola; and that for motion on an ellipse such a force gives periods proportional to the 3/2 power of the major axis of the ellipse (which, as mentioned in Chapter 11, is the distance of the planet from the Sun averaged over the length of its path). So a central force that goes as the inverse square of the distance can account for all of Kepler’s laws. Newton also fills in the gap in his comparison of lunar centripetal acceleration and the acceleration of falling bodies, proving in Section XII of Book I that a spherical body, composed of particles that each produce a force that goes as the inverse square of the distance to that particle, produces a total force that goes as the inverse square of the distance to the center of the sphere. There is a remarkable scholium at the end of Section I of Book I, in which Newton remarks that he is no longer relying on the notion of infinitesimals. He explains that “fluxions” such as velocities are not the ratios of infinitesimals, as he had earlier described them; instead, “Those ultimate ratios with which quantities vanish are not actually ratios of ultimate quantities, but limits which the ratios of quantities decreasing without limit are continually approaching, and which they can approach so closely that their difference is less than any given quantity.” This is essentially the modern idea of a limit, on which calculus is now based. What is not modern about the Principia is Newton’s idea that limits have to be studied using the methods of geometry. Book II presents a long treatment of the motion of bodies through fluids; the primary goal of this discussion was to derive the laws governing the forces of resistance on such bodies.9 In this book he demolishes Descartes’ theory of vortices. He then goes on to calculate the speed of sound waves. His result in Proposition 49 (that the speed is the square root of the ratio of the pressure and the density) is correct only in order of magnitude, because no one then knew how to take account of the changes in temperature during expansion and compression. But (together with his calculation of the speed of ocean waves) this was an amazing achievement: the first time that anyone had used the principles of physics to give a more or less realistic calculation of the speed of any sort of wave. At last Newton comes to the evidence from astronomy in Book III, The System of the World. At the time of the first edition of the Principia, there was general agreement with what is now called Kepler’s first law, that the planets move on elliptical orbits; but there was still considerable doubt about the second and third laws: that the line from the Sun to each planet sweeps out equal areas in equal times, and that the squares of the periods of the various planetary motions go as the cubes of the major axes of these orbits. Newton seems to have fastened on Kepler’s laws not because they were well established, but because they fitted so well with his theory. In Book III he notes that Jupiter’s and Saturn’s moons obey Kepler’s second and third laws, that the observed phases of the five planets other than Earth show that they revolve around the Sun, that all six planets obey Kepler’s laws, and that the Moon satisfies Kepler’s second law.* His own careful observations of the comet of 1680 showed that it too moved on a conic section: an ellipse or hyperbola, in either case very close to a parabola. From all this (and his earlier comparison of the centripetal acceleration of the Moon and the acceleration of falling bodies on the Earth’s surface), he comes to the conclusion that it is a central force obeying an inverse square law by which the moons of Jupiter and Saturn and the Earth are attracted to their planets, and all the planets and comets are attracted to the Sun. From the fact that the accelerations produced by gravitation are independent of the nature of the body being accelerated, whether it is a planet or a moon or an apple, depending only on the nature of the body producing the force and the distance between them, together with the fact that the acceleration produced by any force is inversely proportional to the mass of the body on which it acts, he concludes that the force of gravity on any body must be proportional to the mass of that body, so that all dependence on the body’s mass cancels when we calculate the acceleration. This makes a clear distinction between gravitation and magnetism, which acts very differently on bodies of different composition, even if they have the same mass. Newton then, in Proposition 7, uses his third law of motion to find out how the force of gravity depends on the nature of the body producing the force. Consider two bodies, 1 and 2, with masses m1 and m2. Newton had shown that the gravitational force exerted by body 1 on body 2 is proportional to m2, and that the force that body 2 exerts on body 1 is proportional to m1. But according to the third law, these forces are equal in magnitude, and so they must each be proportional to both m1 and m2. Newton was able to confirm the third law in collisions but not in gravitational interactions. As George Smith has emphasized, it was many years before it became possible to confirm the proportionality of gravitational force to the inertial mass of the attracting as well as the attracted body. Nevertheless, Newton concluded, “Gravity exists in all bodies universally and is proportional to the quantity of matter in each.” This is why the products of the centripetal accelerations of the various planets with the squares of their distances from the Sun are much greater than the product of the centripetal acceleration of the Moon with the square of its distances from the Earth: it is just that the Sun, which produces the gravitational force on the planets, is much more massive than the Earth. These results of Newton are commonly summarized in a formula for the gravitational force F between two bodies, of masses m1 and m2, separated by a distance r:

F = G × m1 × m2 / r2

where G is a universal constant, known today as Newton’s constant. Neither this formula nor the constant G appears in the Principia, and even if Newton had introduced this constant he could not have found a value for it, because he did not know the mass of the Sun or the Earth. In calculating the motion of the Moon or the planets, G appears only as a factor multiplying the mass of the Earth or the Sun, respectively. Even without knowing the value of G, Newton could use his theory of gravitation to calculate the ratios of the masses of various bodies in the solar system. (See Technical Note 35.) For instance, knowing the ratios of the distances of Jupiter and Saturn from their moons and from the Sun, and knowing the ratios of the orbital periods of Jupiter and Saturn and their moons, he could calculate the ratios of the centripetal accelerations of the moons of Jupiter and Saturn toward their planets and the centripetal acceleration of these planets toward the Sun, and from this he could calculate the ratios of the masses of Jupiter, Saturn, and the Sun. Since the Earth also has a Moon, the same technique could in principle be used to calculate the ratio of the masses of the Earth and the Sun. Unfortunately, although the distance of the Moon from the Earth was well known from the Moon’s diurnal parallax, the Sun’s diurnal parallax was too small to measure, and so the ratio of the distances from the Earth to the Sun and to the Moon was not known. (As we saw in Chapter 7, the data used by Aristarchus and the distances he inferred from those data were hopelessly inaccurate.) Newton went ahead anyway, and calculated the ratio of masses, using a value for the distance of the Earth from the Sun that was no better than a lower limit on this distance, and actually about half the true value. Here are Newton’s results for ratios of masses, given as a corollary to Theorem VIII in Book III of the Principia, together with modern values:10 Ratio Newton’s value Modern value m(Sun)/m(Jupiter) 1,067 1,048 m(Sun)/m(Saturn) 3,021 3,497 m(Sun)/m(Earth) 169,282 332,950

As can be seen from this table, Newton’s results were pretty good for Jupiter, not bad for Saturn, but way off for the Earth, because the distance of the Earth from the Sun was not known. Newton was quite aware of the problems posed by observational uncertainties, but like most scientists until the twentieth century, he was cavalier about giving the resulting range of uncertainty in his calculated results. Also, as we have seen with Aristarchus and al-Biruni, he quoted results of calculations to a much greater precision than was warranted by the accuracy of the data on which the calculations were based. Incidentally, the first serious estimate of the size of the solar system was carried out in 1672 by Jean Richer and Giovanni Domenico Cassini. They measured the distance to Mars by observing the difference in the direction to Mars as seen from Paris and Cayenne; since the ratios of the distances of the planets from the Sun were already known from the Copernican theory, this also gave the distance of the Earth from the Sun. In modern units, their result for this distance was 140 million kilometers, reasonably close to the modern value of 149.5985 million kilometers for the mean distance. A more accurate measurement was made later by comparing observations at different locations on Earth of the transits of Venus across the face of the Sun in 1761 and 1769, which gave an Earth–Sun distance of 153 million kilometers.11 In 1797–1798 Henry Cavendish was at last able to measure the gravitational force between laboratory masses, from which a value of G could be inferred. But Cavendish did not refer to his measurement this way. Instead, using the well-known acceleration of 32 feet/second per second due to the Earth’s gravitational field at its surface, and the known volume of the Earth, Cavendish calculated that the average density of the Earth was 5.48 times the density of water. This was in accord with a long-standing practice in physics: to report results as ratios or proportions, rather than as definite magnitudes. For instance, as we have seen, Galileo showed that the distance a body falls on the surface of the Earth is proportional to the square of the time, but he never said that the constant multiplying the square of the time that gives the distance fallen was half of 32 feet/second per second. This was due at least in part to the lack of any universally recognized unit of length. Galileo could have given the acceleration due to gravity as so many braccia/second per second, but what would this mean to Englishmen, or even to Italians outside Tuscany? The international standardization of units of length and mass12 began in 1742, when the Royal Society sent two rulers marked with standard English inches to the French Académie des Sciences; the French marked these with their own measures of length, and sent one back to London. But it was not until the gradual international adoption of the metric system, starting in 1799, that scientists had a universally understood system of units. Today we cite a value for G of 66.724 trillionths of a meter/second2 per kilogram: that is, a small body of mass 1 kilogram at a distance of 1 meter produces a gravitational acceleration of 66.724 trillionths of a meter/second per second. After laying out Newton’s theories of motion and gravitation, the Principia goes on to work out some of their consequences. These go far beyond Kepler’s three laws. For instance, in Proposition 14 Newton explains the precession of planetary orbits measured (for the Earth) by al-Zarqali, though Newton does not attempt a quantitative calculation. In Proposition 19 Newton notes that the planets must all be oblate, because their rotation produces centrifugal forces that are largest at the equator and vanish at the poles. For instance, the Earth’s rotation produces a centripetal acceleration at its equator equal to 0.11 feet/second per second, as compared with the acceleration 32 feet/second per second of falling bodies, so the centrifugal force produced by the Earth’s rotation is much less than its gravitational attraction, but not entirely negligible, and the Earth is therefore nearly spherical, but slightly oblate. Observations in the 1740s finally showed that the same pendulum will swing more slowly near the equator than at higher latitudes, just as would be expected if at the equator the pendulum is farther from the center of the Earth, because the Earth is oblate. In Proposition 39 Newton shows that the effect of gravity on the oblate Earth causes a precession of its axis of rotation, the “precession of the equinoxes” first noted by Hipparchus. (Newton had an extracurricular interest in this precession; he used its values along with ancient observations of the stars in an attempt to date supposed historical events, such as the expedition of Jason and the Argonauts.)13 In the first edition of the Principia Newton calculates in effect that the annual precession due to the Sun is 6.82° (degrees of arc), and that the effect of the Moon is larger by a factor 6⅓, giving a total of 50.0" (seconds of arc) per year, in perfect agreement with the precession of 50" per year then measured, and close to the modern value of 50.375" per year. Very impressive, but Newton later realized that his result for the precession due to the Sun and hence for the total precession was 1.6 times too small. In the second edition he corrected his result for the effect of the Sun, and also corrected the ratio of the effects of the Moon and Sun, so that the total was again close to 50" per year, still in good agreement with what was observed.14 Newton had the correct qualitative explanation of the precession of the equinoxes, and his calculation gave the right order of magnitude for the effect, but to get an answer in precise agreement with observation he had to make many artful adjustments. This is just one example of Newton fudging his calculations to get answers in close agreement with observation. Along with this example, R. S. Westfall15 has given others, including Newton’s calculation of the speed of sound, and his comparison of the centripetal acceleration of the Moon with the acceleration of falling bodies on the Earth’s surface mentioned earlier. Perhaps Newton felt that his real or imagined adversaries would never be convinced by anything but nearly perfect agreement with observation. In Proposition 24, Newton presents his theory of the tides. Gram for gram, the Moon attracts the ocean beneath it more strongly than it attracts the solid Earth, whose center is farther away, and it attracts the solid Earth more strongly than it attracts the ocean on the side of the Earth away from the Moon. Thus there is a tidal bulge in the ocean both below the Moon, where the Moon’s gravity pulls water away from the Earth, and on the opposite side of the Earth, where the Moon’s gravity pulls the Earth away from the water. This explained why in some locations high tides are separated by roughly 12 rather than 24 hours. But the effect is too complicated for this theory of tides to have been verified in Newton’s time. Newton knew that the Sun as well as the Moon plays a role in raising the tides. The highest and lowest tides, known as spring tides, occur when the Moon is new or full, so that the Sun, Moon, and Earth are on the same line, intensifying the effects of gravitation. But the worst complications come from the fact that any gravitational effects on the oceans are greatly influenced by the shape of the continents and the topography of the ocean bottom, which Newton could not possibly take into account. This is a common theme in the history of physics. Newton’s theory of gravitation made successful predictions for simple phenomena like planetary motion, but it could not give a quantitative account of more complicated phenomena, like the tides. We are in a similar position today with regard to the theory of the strong forces that hold quarks together inside the protons and neutrons inside the atomic nucleus, a theory known as quantum chromodynamics. This theory has been successful in accounting for certain processes at high energy, such as the production of various strongly interacting particles in the annihilation of energetic electrons and their antiparticles, and its successes convince us that the theory is correct. We cannot use the theory to calculate precise values for other things that we would like to explain, like the masses of the proton and neutron, because the calculation is too complicated. Here, as for Newton’s theory of the tides, the proper attitude is patience. Physical theories are validated when they give us the ability to calculate enough things that are sufficiently simple to allow reliable calculations, even if we can’t calculate everything that we might want to calculate. Book III of Principia presents calculations of things already measured, and new predictions of things not yet measured, but even in the final third edition of Principia Newton could point to no predictions that had been verified in the 40 years since the first edition. Still, taken all together, the evidence for Newton’s theories of motion and gravitation was overwhelming. Newton did not need to follow Aristotle and explain why gravity exists, and he did not try. In his “General Scholium” Newton concluded:

Thus far I have explained the phenomena of the heavens and of our sea by the force of gravity, but I have not yet assigned a cause to gravity. Indeed, this force arises from some cause that penetrates as far as the centers of the Sun and planets without any diminution of its power to act, and that acts not in proportion to the quantity of the surfaces of the particles on which it acts (as mechanical causes are wont to do), but in proportion to the quantity of solid matter, and whose action is extended everywhere to immense distances, always decreasing as the inverse squares of the distances. . . . I have not as yet been able to deduce from phenomena the reasons for these properties of gravity, and I do not “feign” hypotheses.

Newton’s book appeared with an appropriate ode by Halley. Here is its final stanza:

Then ye who now on heavenly nectar fare, Come celebrate with me in song the name Of Newton, to the Muses dear; for he Unlocked the hidden treasuries of Truth: So richly through his mind had Phoebus cast The radius of his own divinity, Nearer the gods no mortal may approach.

The Principia established the laws of motion and the principle of universal gravitation, but that understates its importance. Newton had given to the future a model of what a physical theory can be: a set of simple mathematical principles that precisely govern a vast range of different phenomena. Though Newton knew very well that gravitation was not the only physical force, as far as it went his theory was universal—every particle in the universe attracts every other particle with a force proportional to the product of their masses and inversely proportional to the square of their separation. The Principia not only deduced Kepler’s rules of planetary motion as an exact solution of a simplified problem, the motion of point masses in response to the gravitation of a single massive sphere; it went on to explain (even if only qualitatively in some cases) a wide variety of other phenomena: the precession of equinoxes, the precession of perihelia, the paths of comets, the motions of moons, the rise and fall of the tides, and the fall of apples.16 By comparison, all past successes of physical theory were parochial. After the publication of the Principia in 1686–1687, Newton became famous. He was elected a member of parliament for the University of Cambridge in 1689 and again in 1701. In 1694 he became warden of the Mint, where he presided over a reform of the English coinage while still retaining his Lucasian professorship. When Czar Peter the Great came to England in 1698, he made a point of visiting the Mint, and hoped to talk with Newton, but I can’t find any account of their actually meeting. In 1699 Newton was appointed master of the Mint, a much better-paid position. He gave up his professorship, and became rich. In 1703, after the death of his old enemy Hooke, Newton became president of the Royal Society. He was knighted in 1705. When in 1727 Newton died of a kidney stone, he was given a state funeral in Westminster Abbey, even though he had refused to take the sacraments of the Church of England. Voltaire reported that Newton was “buried like a king who had benefited his subjects.”17 Newton’s theory did not meet universal acceptance.18 Despite Newton’s own commitment to Unitarian Christianity, some in England, like the theologian John Hutchinson and Bishop Berkeley, were appalled by the impersonal naturalism of Newton’s theory. This was unfair to the devout Newton. He even argued that only divine intervention could explain why the mutual gravitational attraction of the planets does not destabilize the solar system,* and why some bodies like the Sun and stars shine by their own light, while others like the planets and their satellites are themselves dark. Today of course we understand the light of the Sun and stars in a naturalistic way—they shine because they are heated by nuclear reactions in their cores. Though unfair to Newton, Hutchinson and Berkeley were not entirely wrong about Newtonianism. Following the example of Newton’s work, if not of his personal opinions, by the late eighteenth century physical science had become thoroughly divorced from religion. Another obstacle to the acceptance of Newton’s work was the old false opposition between mathematics and physics that we have seen in a comment of Geminus of Rhodes quoted in Chapter 8. Newton did not speak the Aristotelian language of substances and qualities, and he did not try to explain the cause of gravitation. The priest Nicolas de Malebranche (1638–1715) in reviewing the Principia said that it was the work of a geometer, not of a physicist. Malebranche clearly was thinking of physics in the mode of Aristotle. What he did not realize is that Newton’s example had revised the definition of physics. The most formidable criticism of Newton’s theory of gravitation came from Christiaan Huygens.19 He greatly admired the Principia, and did not doubt that the motion of planets is governed by a force decreasing as the inverse square of the distance, but Huygens had reservations about whether it is true that every particle of matter attracts every other particle with such a force, proportional to the product of their masses. In this, Huygens seems to have been misled by inaccurate measurements of the rates of pendulums at various latitudes, which seemed to show that the slowing of pendulums near the equator could be entirely explained as an effect of the centrifugal force due to the Earth’s rotation. If true, this would imply that the Earth is not oblate, as it would be if the particles of the Earth attract each other in the way prescribed by Newton. Starting already in Newton’s lifetime, his theory of gravitation was opposed in France and Germany by followers of Descartes and by Newton’s old adversary Leibniz. They argued that an attraction operating over millions of miles of empty space would be an occult element in natural philosophy, and they further insisted that the action of gravity should be given a rational explanation, not merely assumed. In this, natural philosophers on the Continent were hanging on to an old ideal for science, going back to the Hellenic age, that scientific theories should ultimately be founded solely on reason. We have learned to give this up. Even though our very successful theory of electrons and light can be deduced from the modern standard model of elementary particles, which may (we hope) in turn eventually be deduced from a deeper theory, however far we go we will never come to a foundation based on pure reason. Like me, most physicists today are resigned to the fact that we will always have to wonder why our deepest theories are not something different. The opposition to Newtonianism found expression in a famous exchange of letters during 1715 and 1716 between Leibniz and Newton’s disciple, the Reverend Samuel Clarke, who had translated Newton’s Opticks into Latin. Much of their argument focused on the nature of God: Did He intervene in the running of the world, as Newton thought, or had He set it up to run by itself from the beginning?20 The controversy seems to me to have been supremely futile, for even if its subject were real, it is something about which neither Clarke nor Leibniz could have had any knowledge whatever. In the end the opposition to Newton’s theories didn’t matter, for Newtonian physics went from success to success. Halley fitted the observations of the comets observed in 1531, 1607, and 1682 to a single nearly parabolic elliptical orbit, showing that these were all recurring appearances of the same comet. Using Newton’s theory to take into account gravitational perturbations due to the masses of Jupiter and Saturn, the French mathematician Alexis-Claude Clairaut and his collaborators predicted in November 1758 that this comet would return to perihelion in mid-April 1759. The comet was observed on Christmas Day 1758, 15 years after Halley’s death, and reached perihelion on March 13, 1759. Newton’s theory was promoted in the mid-eighteenth century by the French translations of the Principia by Clairaut and by Émilie du Châtelet, and through the influence of du Châtelet’s lover Voltaire. It was another Frenchman, Jean d’Alembert, who in 1749 published the first correct and accurate calculation of the precession of the equinoxes, based on Newton’s ideas. Eventually Newtonianism triumphed everywhere. This was not because Newton’s theory satisfied a preexisting metaphysical criterion for a scientific theory. It didn’t. It did not answer the questions about purpose that were central in Aristotle’s physics. But it provided universal principles that allowed the successful calculation of a great deal that had previously seemed mysterious. In this way, it provided an irresistible model for what a physical theory should be, and could be. This is an example of a kind of Darwinian selection operating in the history of science. We get intense pleasure when something has been successfully explained, as when Newton explained Kepler’s laws of planetary motion along with much else. The scientific theories and methods that survive are those that provide such pleasure, whether or not they fit any preexisting model of how science ought to be done. The rejection of Newton’s theories by the followers of Descartes and Leibniz suggests a moral for the practice of science: it is never safe simply to reject a theory that has as many impressive successes in accounting for observation as Newton’s had. Successful theories may work for reasons not understood by their creators, and they always turn out to be approximations to more successful theories, but they are never simply mistakes. This moral was not always heeded in the twentieth century. The 1920s saw the advent of quantum mechanics, a radically new framework for physical theory. Instead of calculating the trajectories of a planet or a particle, one calculates the evolution of waves of probability, whose intensity at any position and time tells us the probability of finding the planet or particle then and there. The abandonment of determinism so appalled some of the founders of quantum mechanics, including Max Planck, Erwin Schrödinger, Louis de Broglie, and Albert Einstein, that they did no further work on quantum mechanical theories, except to point out the unacceptable consequences of these theories. Some of the criticisms of quantum mechanics by Schrödinger and Einstein were troubling, and continue to worry us today, but by the end of the 1920s quantum mechanics had already been so successful in accounting for the properties of atoms, molecules, and photons that it had to be taken seriously. The rejection of quantum mechanical theories by these physicists meant that they were unable to participate in the great progress in the physics of solids, atomic nuclei, and elementary particles of the 1930s and 1940s. Like quantum mechanics, Newton’s theory of the solar system had provided what later came to be called a Standard Model. I introduced this term in 197121 to describe the theory of the structure and evolution of the expanding universe as it had developed up to that time, explaining:

Of course, the standard model may be partly or wholly wrong. However, its importance lies not in its certain truth, but in the common meeting ground that it provides for an enormous variety of cosmological data. By discussing this data in the context of a standard cosmological mode, we can begin to appreciate their cosmological relevance, whatever model ultimately proves correct.

A little later, I and other physicists started using the term Standard Model also to refer to our emerging theory of elementary particles and their various interactions. Of course, Newton’s successors did not use this term to refer to the Newtonian theory of the solar system, but they well might have. The Newtonian theory certainly provided a common meeting ground for astronomers trying to explain observations that went beyond Kepler’s laws. The methods for applying Newton’s theory to problems involving more than two bodies were developed by many authors in the late eighteenth and early nineteenth centuries. There was one innovation of great future importance that was explored especially by Pierre-Simon Laplace in the early nineteenth century. Instead of adding up the gravitational forces exerted by all the bodies in an ensemble like the solar system, one calculates a “field,” a condition of space that at every point gives the magnitude and direction of the acceleration produced by all the masses in the ensemble. To calculate the field, one solves certain differential equations that it obeys. (These equations set conditions on the way that the field varies when the point at which it is measured is moved in any of three perpendicular directions.) This approach makes it nearly trivial to prove Newton’s theorem that the gravitational forces exerted outside a spherical mass go as the inverse square of the distance from the sphere’s center. More important, as we will see in Chapter 15, the field concept was to play a crucial role in the understanding of electricity, magnetism, and light. These mathematical tools were used most dramatically in 1846 to predict the existence and location of the planet Neptune from irregularities in the orbit of the planet Uranus, independently by John Couch Adams and Jean-Joseph Leverrier. Neptune was discovered soon afterward, in the expected place. Small discrepancies between theory and observation remained, in the motion of the Moon and of Halley’s and Encke’s comets, and in a precession of the perihelia of the orbit of Mercury that was observed to be 43" (seconds of arc) per century greater than could be accounted for by gravitational forces produced by the other planets. The discrepancies in the motion of the Moon and comets were eventually traced to nongravitational forces, but the excess precession of Mercury was not explained until the advent in 1915 of the general theory of relativity of Albert Einstein. In Newton’s theory the gravitational force at a given point and a given time depends on the positions of all masses at the same time, so a sudden change of any of these positions (such as a flare on the surface of the Sun) produces an instantaneous change in gravitational forces everywhere. This was in conflict with the principle of Einstein’s 1905 special theory of relativity, that no influence can travel faster than light. This pointed to a clear need to seek a modified theory of gravitation. In Einstein’s general theory a sudden change in the position of a mass will produce a change in the gravitational field in the immediate neighborhood of the mass, which then propagates at the speed of light to greater distances. General relativity rejects Newton’s notion of absolute space and time. Its underlying equations are the same in all reference frames, whatever their acceleration or rotation. Thus far, Leibniz would have been pleased, but in fact general relativity justifies Newtonian mechanics. Its mathematical formulation is based on a property that it shares with Newton’s theory: that all bodies at a given point undergo the same acceleration due to gravity. This means that one can eliminate the effects of gravitation at any point by using a frame of reference, known as an inertial frame, that shares this acceleration. For instance, one does not feel the effects of the Earth’s gravity in a freely falling elevator. It is in these inertial frames of reference that Newton’s laws apply, at least for bodies whose speeds do not approach that of light. The success of Newton’s treatment of the motion of planets and comets shows that the inertial frames in the neighborhood of the solar system are those in which the Sun rather than the Earth is at rest (or moving with constant velocity). According to general relativity, this is because that is the frame of reference in which the matter of distant galaxies is not revolving around the solar system. In this sense, Newton’s theory provided a solid basis for preferring the Copernican theory to that of Tycho. But in general relativity we can use any frame of reference we like, not just inertial frames. If we were to adopt a frame of reference like Tycho’s in which the Earth is at rest, then the distant galaxies would seem to be executing circular turns once a year, and in general relativity this enormous motion would create forces akin to gravitation, which would act on the Sun and planets and give them the motions of the Tychonic theory. Newton seems to have had a hint of this. In an unpublished “Proposition 43” that did not make it into the Principia, Newton acknowledged that Tycho’s theory could be true if some other force besides ordinary gravitation acted on the Sun and planets.22 When Einstein’s theory was confirmed in 1919 by the observation of a predicted bending of rays of light by the gravitational field of the Sun, the Times of London declared that Newton had been shown to be wrong. This was a mistake. Newton’s theory can be regarded as an approximation to Einstein’s, one that becomes increasingly valid for objects moving at velocities much less than that of light. Not only does Einstein’s theory not disprove Newton’s; relativity explains why Newton’s theory works, when it does work. General relativity itself is doubtless an approximation to a more satisfactory theory. In general relativity a gravitational field can be fully described by specifying at every point in space and time the inertial frames in which the effects of gravitation are absent. This is mathematically similar to the fact that we can make a map of a small region about any point on a curved surface in which the surface appears flat, like the map of a city on the surface of the Earth; the curvature of the whole surface can be described by compiling an atlas of overlapping local maps. Indeed, this mathematical similarity allows us to describe any gravitational field as a curvature of space and time. The conceptual basis of general relativity is thus different from that of Newton. The notion of gravitational force is largely replaced in general relativity with the concept of curved space-time. This was hard for some people to swallow. In 1730 Alexander Pope had written a memorable epitaph for Newton:

Nature and nature’s laws lay hid in night; God said, “Let Newton be!” And all was light.

In the twentieth century the British satirical poet J. C. Squire23 added two more lines:

It did not last: the Devil howling “Ho, Let Einstein be,” restored the status quo.

Do not believe it. The general theory of relativity is very much in the style of Newton’s theories of motion and gravitation: it is based on general principles that can be expressed as mathematical equations, from which consequences can be mathematically deduced for a broad range of phenomena, which when compared with observation allow the theory to be verified. The difference between Einstein’s and Newton’s theories is far less than the difference between Newton’s theories and anything that had gone before. A question remains: why did the scientific revolution of the sixteenth and seventeenth centuries happen when and where it did? There is no lack of possible explanations. Many changes occurred in fifteenth-century Europe that helped to lay the foundation for the scientific revolution. National governments were consolidated in France under Charles VII and Louis XI and in England under Henry VII. The fall of Constantinople in 1453 sent Greek scholars fleeing westward to Italy and beyond. The Renaissance intensified interest in the natural world and set higher standards for the accuracy of ancient texts and their translation. The invention of printing with movable type made scholarly communication far quicker and cheaper. The discovery and exploration of America reinforced the lesson that there is much that the ancients did not know. In addition, according to the “Merton thesis,” the Protestant Reformation of the early sixteenth century set the stage for the great scientific breakthroughs of seventeenth-century England. The sociologist Robert Merton supposed that Protestantism created social attitudes favorable to science and promoted a combination of rationalism and empiricism and a belief in an understandable order in nature—attitudes and beliefs that he found in the actual behavior of Protestant scientists.24 It is not easy to judge how important were these various external influences on the scientific revolution. But although I cannot tell why it was Isaac Newton in late-seventeenth-century England who discovered the classical laws of motion and gravitation, I think I know why these laws took the form they did. It is, very simply, because to a very good approximation the world really does obey Newton’s laws. Having surveyed the history of physical science from Thales to Newton, I would like now to offer some tentative thoughts on what drove us to the modern conception of science, represented by the achievements of Newton and his successors. Nothing like modern science was conceived as a goal in the ancient world or the medieval world. Indeed, even if our predecessors could have imagined science as it is today, they might not have liked it very much. Modern science is impersonal, without room for supernatural intervention or (outside the behavioral sciences) for human values; it has no sense of purpose; and it offers no hope for certainty. So how did we get here? Faced with a puzzling world, people in every culture have sought explanations. Even where they abandoned mythology, most attempts at explanation did not lead to anything satisfying. Thales tried to understand matter by guessing that it is all water, but what could he do with this idea? What new information did it give him? No one at Miletus or anywhere else could build anything on the notion that everything is water. But every once in a while someone finds a way of explaining some phenomenon that fits so well and clarifies so much that it gives the finder intense satisfaction, especially when the new understanding is quantitative, and observation bears it out in detail. Imagine how Ptolemy must have felt when he realized that, by adding an equant to the epicycles and eccentrics of Apollonius and Hipparchus, he had found a theory of planetary motions that allowed him to predict with fair accuracy where any planet would be found in the sky at any time. We can get a sense of his joy from the lines of his that I quoted earlier: “When I search out the massed wheeling circles of the stars, my feet no longer touch the Earth, but, side by side with Zeus himself, I take my fill of ambrosia, the food of the gods.” The joy was flawed—it always is. You didn’t have to be a follower of Aristotle to be repelled by the peculiar looping motion of planets moving on epicycles in Ptolemy’s theory. There was also the nasty fine-tuning: it had to take precisely one year for the centers of the epicycles of Mercury and Venus to move around the Earth, and for Mars, Jupiter, and Saturn to move around their epicycles. For over a thousand years philosophers argued about the proper role of astronomers like Ptolemy—really to understand the heavens, or merely to fit the data. What pleasure Copernicus must then have felt when he was able to explain that the fine-tuning and the looping orbits of Ptolemy’s scheme arose simply because we view the solar system from a moving Earth. Still flawed, the Copernican theory did not quite fit the data without ugly complications. How much then the mathematically gifted Kepler must have enjoyed replacing the Copernican mess with motion on ellipses, obeying his three laws. So the world acts on us like a teaching machine, reinforcing our good ideas with moments of satisfaction. After centuries we learn what kinds of understanding are possible, and how to find them. We learn not to worry about purpose, because such worries never lead to the sort of delight we seek. We learn to abandon the search for certainty, because the explanations that make us happy never are certain. We learn to do experiments, not worrying about the artificiality of our arrangements. We develop an aesthetic sense that gives us clues to what theories will work, and that adds to our pleasure when they do work. Our understandings accumulate. It is all unplanned and unpredictable, but it leads to reliable knowledge, and gives us joy along the way.