温伯格《解释世界》第十二章 实验开始

无人可以操控天体，所以第十一章介绍的天文成就是基于被动的观测。幸好太阳系行星的运行简单，人们应用不断完善的设备历经几个世纪的观测终于可以正确描述这些运动。但对其他问题需要超越观测，应用实验人工操控物理现象来测试或得出普遍性理论。

从某种意义上来说人们其实一直在做实验，比如尝试如何完成从大到冶炼小到烘烤蛋糕等等的一些事情。这里说实验的开始，我是指用于发现或测试有关自然普遍性理论的实验。

在这个意义上基本无法精确知道实验开始的时间。阿基米德可能用实验测试了他的静水力学，但是他的《论浮体》专著完全遵循纯粹数学推导风格，没有提及实验。希罗和托勒密做了测试他们的反射和折射的实验，但是几个世纪之后人们才重新继承他们的方法。

十七世纪开始出现人们热衷于公开应用实验结果来验证物理理论。在十七世纪早期静水力学中就以出现，比如伽利略1612年出版的《水中浮体对话集》中所展示。更为重要的是对落体运动的定量研究，这是牛顿力学的重要前提。对落体和大气压力特性的研究标志着现代实验物理的开端。

与其他很多别的东西一样，对运动的实验研究也是起源于伽利略。他对运动的结论出现在1635年完成的作品《关于两门新科学的对话集》，那时他还被监禁在阿切特里居所。教会的禁书审定院禁止出版他的作品，但是许多拷贝被偷运出意大利。1638年该书在基督新教大学城莱顿由路易斯·埃尔塞维尔出版社出版。《两门新科学》的场景仍然包括萨尔维亚蒂，辛普利西奥，和萨格莱多三人，担任同样的角色。

除了其他一些对话，《两门新科学》“第一天”包括重的物体与轻的物体下落速度一样的论点，这完全有悖于亚里士多德重的物体比轻的物体下落快的教条。当然由于空气阻力，轻的物体比重的物体下落会慢一点。在解决该问题方面，伽利略展示出他对科学家需要接受近似结果的了解，与希腊强调基于缜密数学做出精确阐述背道而驰。正如萨尔维亚蒂对辛普利西奥做出的解释：

亚里士多德说：“一个100磅重的铁球从100布拉恰下落比一个已经下落了1布拉恰的1磅重铁球着地快。”我说它们同时着地。你做个实验可以发现大的比小的早落2英寸；也就是说大的落地时小的还差2英寸。现在你想遮掩我的这2英寸，与亚里士多德的99布拉恰，只说我的微小误差，而不提他的巨大差错。

伽利略也证实了空气有重量；估算了其密度；讨论阻力介质中的运动；解释音乐和声；阐述无论摆动振幅多大，每次摆动时间一样。（注：事实上只有小角度情况下才如此，虽然伽利略并没有标注。他确实说了50度或60度（弧度）摇摆与小角度摇摆每次摆动时间一样，这说明伽利略实际上没有全部做了他所说的摇摆实验。）。几个世纪之后人们应用该原理发明了摆钟，精确测量了落体加速。

《两门新科学》“第二天”讨论不同形状物体强度。“第三天”伽利略又回到运动问题，做出了他最为令人关注的的贡献。第三天一开始他先回顾了匀速运动的一般属性，然后以与十四世纪墨顿学院的同样思路定义了匀加速运动：相同时间间隔内速度增加值相等。伽利略也给出了平均速度定理证明，与奥里斯姆的证明思路一致，但是他没有引用奥里斯姆和墨顿导师。与中世纪先驱不同之处在于伽利略超越数学定理，主张自由落体做匀加速运动，但是他没有去研究加速的原因。

第十章讲过人们曾经广泛认为持有另一个落体匀加速理论。这个理论是说自由落体的速度增加正比于下落距离，不是时间。（注：照字面意义解释的话这意味着从静止状态抛下的物体永远不会下落，因为以零起始速度在头一个无穷小时间结束时物体没有移动，这样若速度正比于距离则其速度仍为零。也许速度正比于距离学说只是为了适用于一段加速之后。）伽利略提出多种论据反对这一观点（注：伽利略其中一个论据不对。因为它适用于时间间隔内的平均速度，而不是间隔最后时间的速度）。但是最终还是需要用实验来验证哪种落体加速理论正确。

从静止状态下落距离等于增加速度的一半乘以耗时（根据平均速度定律），此增加速度本身也正比于耗时，那自由落体距离应该正比于时间的平方。（见技术说明25。）这就是伽利略着手要验证的。

自由落体速度太快，伽利略无法检验给定时间落体下落距离的结论，他想出一个减慢下落速度的主意，他用球沿斜面滚落来研究此问题。要想表明两者相关，他需要证实球从斜面滚落如何与自由落体有关。他指出球从斜面滚落的速度只取决于球滚落的垂向距离，与斜面倾斜角度无关。（注：如技术说明25所示。那里做了解释，虽然伽利略并不知道，球从斜面滚落的速度不等于自由落体下落同样垂向距离的速度。因为垂向下降释放的一些能量给了球的旋转。但是速度成正比，所以伽利略的定性结论落体速度正比于下落时间即使把到球的旋转考虑进去也不会改变。）自由落体可以看作是从垂直平面滚落，所以如果从斜面滚落的球的速度正比于耗时，那么自由落体也应该一样。斜面倾角小的话速度速度当然比自由落体速度小得多。（这正是使用斜面的原因），但两者速度成正比，所以沿斜面行进距离正比于同样时间自由落体下落距离。

在《两门新科学》中伽利略宣告滚落距离正比于时间平方。伽利略1603年在帕多瓦已经用与水平呈小于2度角的斜面做了实验，斜面每约1毫米划线做了标记。他用球行进中到达每个标记发出的等声音间隔来判断时间，从起点的距离为12=1，22=4，32=9，等等。在《两门新科学》介绍的实验他用的是水钟计相对时间间隔。现代重建的这个实验证实伽利略当时完全可以取得他所宣称的精度。

在第十一章介绍的《两大世界体系对话》中伽利略已经考虑了落体的加速度问题。在该对话的第二天，萨尔维亚蒂实际上宣称下落距离正比于下落时间平方，但是只给出草率解释。他也提到从100布拉恰高度落下的炮弹用5秒钟着地。这里很明显伽利略没有实测这个时间，只是作为一个示例。如果1布拉恰为21.5英寸，用现代重力加速度数值，重物下落100布拉恰用时3.3秒，不是5秒。但是伽利略显然没有认真地测量重力加速度。

《关于两门新科学的对话集》“第四天”讨论抛物运动轨迹。伽利略的观点主要基于他1608年做的实验。（技术说明26 有详细介绍）。一个球从不同初始高度斜面滚落，然后沿放置斜面的水平桌面滚动，最后从桌边滚下。通过测量球着地的距离，以及观察球在空中的路径，伽利略得出球轨迹是抛物线的结论。伽利略在《两门新科学》中并没有描述该实验，他只是给出抛物线的理论论据。关键之处（后来对牛顿力学至关重要）在于抛物运动分解的每种运动都分别受各个作用力的作用。当抛物体从桌边滚下或从加农炮射出，只有空气阻力改变其水平运行，所以水平距离非常近似正比于时间。另外一方面与任何自由落体一样，抛物同时也向下加速，垂向下落距离正比于下落时间的平方。垂直下落距离与水平距离的平方也成正比。什么样的曲线具有这种特性？伽利略应用阿波罗尼奥斯对抛物线的定义--圆锥体与平行于圆锥体表面的平面相切（见技术说明26）， 指出抛物体的路径是抛物线。

《两门新科学》描述的实验完全突破了以往。伽利略不再局限于自由落体这种亚里士多德所认为的自然运动，他转向人工运动—像让球从斜面滚落，或抛出物体。在此意义上可以说伽利略的斜面是今天我们人工生成自然界不存在粒子的粒子加速器的原型。

克里斯蒂安·惠更斯发展了伽利略的运动理论，他也许是介于伽利略与牛顿这些璀璨一代之间最为引人注目的人物。惠更斯1629年出生于一个高级文职人员家庭，其家族曾服务于荷兰共和国奥兰治王室。1645年到1647年惠更斯在莱顿大学学习法律和数学，但是后来他把全部时间都用来学习数学，最终走向了自然科学。与笛卡尔，帕斯卡，波义耳一样，惠更斯也是一位博学者，涉足数学，天文，统计，静水力学，动力学，以及光学等广泛领域。

惠更斯天文学方面最重要的工作是用天文望远镜研究土星。1655年他发现土星最大的卫星--泰坦，揭示出不只地球和木星有卫星。他也对伽利略注意到的土星特别的非圆形外貌做出了解释，这是由于围绕该行星的光环造成的。

1656-1657年惠更斯发明了摆钟。这是基于伽利略所观测到的摆每次摆动时间与摆动幅度无关。惠更斯意识到只有摆动很小时才会如此，他天才地发现即使对大振幅也可以保持振幅独立于时间的方法。以前简陋的机械钟每天会快慢5分钟，而惠更斯的摆钟每天快慢一般不会超过10秒，其中一个摆钟每天只慢0.5秒。

第二年惠更斯从确定长度摆钟周期推算出地表自由落体的加速度值。在后来于1673年发表的《关于钟摆的运动》中惠更斯指出“小幅震动时间与从摆动高度一半垂向下落时间的关系就像圆周长与半径关系一样。”也就是说摆从一边以小角度摆动到另一边用时等于𝞹乘以一个物体下落摆动一半长度的时间。（惠更斯不用微积分获得这个结果很不容易。）惠更斯应用这个原理，加上测量不同长度摆动的周期，他可以计算出重力加速度，伽利略用他当时的方法不可能做出准确测量。惠更斯表述为自由落体第一秒下落151/12“巴黎尺”。巴黎尺与现代英尺的比值估计在1.06到1.08之间。如果我们假设1巴黎尺等于1.07英尺，那惠更斯的结果是说自由落体头一秒下落16.1英尺，这样加速度为每秒32.2英尺/秒，与现代值32.17英尺/秒非常相符。（惠更斯是位很好的实验人员，他检查了落体加速度确实与他从观测钟摆推动出的加速度符合程度在实验误差之内。）我们后面会看到，这个测量（后来牛顿又重复了该测量）对将地球上的重力与维持月亮在其轨道运行之力联系起来至关重要。

里奇奥利从早期测量重物下落不同距离的时间本可以推导出重力加速度。为了准确测量时间，里奇奥利使用一个精心校准的钟摆来数一个太阳日或恒星日内敲击次数。令他吃惊的是他的测量证实伽利略下落距离与时间平方成正比的结论。从1651年发表的这些测量应该可以计算出（里奇奥利没有这样做）重力加速度为每秒30罗马尺/秒。好在里奇奥利记载了博洛尼亚安斯内力塔（不同重量重物由此塔抛落）的高度为312罗马尺。该塔现在仍然存在，其高度为323现代英尺，这样里奇奥利的罗马尺应该是323/312=1.035英尺。每秒30罗马尺/秒相当于每秒31英尺/秒，与现代值相当接近。确实如果里奇奥利知道惠更斯钟摆周期与物体下落一半长度的时间间的关系，那他就可以使用他校准的钟摆来计算重力加速度，而不需要在博洛尼亚塔上去抛什么重物。

1664年惠更斯被选为皇家科学院院士，附带相应的津贴，于是他搬到巴黎生活了20年。他在光学方面最伟大的著作《光论》1678年创作于巴黎，书中阐述了光的波动理论。该书1690年才出版，可能是因为惠更斯希望将书从法文译为拉丁文，但是到他1695年去世都没有时间去完成。第十四章我们还会讲到惠更斯波动理论。

惠更斯1669年在《学者杂志》发表的一篇文章中提出正确的刚体碰撞定律（笛卡尔理解有误）：守恒的是我们现今称为的动量和动能。惠更斯声明他用实验验证了这些结论，他可能是通过研究摆锤碰撞效应得出这些结论，因为摆锤碰撞初始和最终速度都可以精确计算。我们在第十四章还会看到，惠更斯在《关于钟摆的运动》中计算了曲线轨迹运动的加速度，这对后来牛顿的工作至关重要。

惠更斯的例子表明科学从对数学的仿效—从依赖于推导以及一些数学特征--已经有了多大进展。在《光论》前言中惠更斯解释说：

（在这部书中）看到的论证不像几何学中的论证那样确定，两者甚至差异甚大，因为几何学家用确定以及不容置辩的原理证明他们的命题，而这里的原理是有他们引出的结论来验证的；这些原理的本性不允许以其他方式论证。

这里对现代物理科学方法的描述不亚于任何人。

伽利略和惠更斯关于运动的研究用实验方法否认了亚里士多德的物理。同时期对气压的研究也一样。

真空不可能存在是十七世纪所质疑的亚里士多德教义之一。后来最终认识到像自吸这种大自然看起来像憎恶真空的现象其实是大气压力效应。意大利，法国和英格兰的三位人物在发现过程中起了关键作用。

佛罗伦萨挖井人早就知道提水泵排水扬程不会超过18布拉恰，即32英尺。（海平面实际值接近33.5英尺。）伽利略以及其他一些学者曾认为这体现出大自然憎恶真空的极限。佛罗伦萨人埃万杰利斯塔·托里拆利（他从事了几何，抛物运动，流体动力学，光学和早期微积分的研究）给出了不同的解释。托里拆利主张提水泵排水扬程极限是由于井内大气对水的压力只能支撑不超过18布拉恰高的水柱。这个压力通过空气传播，这样无论是否是平面都会受到正比于表面积的力。静止大气单位面积施加的力，即压力，等于直到大气顶部大气柱的重量除以气柱截面积。吃压力作用于境内水面，增加了水的压力，因而当泵降低浸于水中立管上部压力后，水在立管中上升，但上升高度受制于气压大小。

十七世纪四十年代托里拆利做了一些列实验来验证此观点。他推断因为汞柱重量是同体积水柱重量的13.6倍，那大气支撑（无论是通过大气作用于玻璃管所插的水银槽内的向下压力或作用于底部敞口处）的汞柱在顶部封闭的直立玻璃管内最大高度应该是18布拉恰除以13.6，或采用准确的现代值的话，33.5英尺/13.6=30英寸=760毫米。1643年他发现如果把一个高于此值顶部封闭的直立玻璃管充满汞，那部分汞将流出，直到汞的高度达到大约30英寸。这在顶部留下了真空，现在人们称之为“托里拆利真空。” 这种管子可以做为气压计来测量附近气压的变化。气压越高，其支撑的汞柱就越高。

法国博学之士布莱士·帕斯卡以基督教神学《思想录》以及支持詹森教派，反对耶稣会而闻名,不过他也对几何和概率论做出了贡献,探索了托里拆利研究的气动现象.帕斯卡推断如果底部敞口的玻璃管内汞柱是受大气压力所支撑，那么在高山上汞柱高度应该降低，因为那里空气稀薄,气压低。从1648年到1651年他做了一系列实验证实了此预测，他总结说：“所有那些归于（憎恶真空）的影响实际上是由于大气的重量和压力，这是唯一原由。”

为了对帕斯卡和托里拆利表达敬意，人们用他们的名字命名现代压力单位。一个帕斯卡等于作用于面积为1平方米之上1牛顿的力（牛顿单位的定义是使质量为1千克物体1秒钟获得1米/秒加速度的力）。1托等于支撑1毫米汞柱的压力。标准大气压是760托，比100000帕斯卡大一点。

罗伯特·波义耳在英国推进了托里拆利和帕斯卡的研究工作。波义耳是科克伯爵的儿子，自然是“上流社会” 缺席成员—属于他那时代控制爱尔兰的基督新教上层。他在伊顿公学求学，游历欧洲，在十七世纪四十年代肆虐英国的内战中为议会一方而战。他在他那阶层中很寻常，他沉迷于科学。他阅读了伽利略《两大世界体系对话》，他由此认识到1642年引发天文学革命的新观点。波义耳坚持对自然现象的自然解释，他宣称：“没有人（比我）更加认可，崇敬神的万能。（但是）我们争论之处不是上帝可以做什么，而是不超出自然范围自然能够做出什么。”但是想许多达尔文之前以及一些达尔文之后的学者一样，他主张动物和人类绝妙的能力表明他们必由仁慈的造物主所设计。

波义耳1660年在《关于空气弹性及其物理力学的新实验》中描述了气压。在他的实验中他应用了其助手罗伯特·胡克设计的新空气泵，关于胡克在第十四章会有更多介绍。通过将空气从容器中排出，波义耳证实声音的传播，火，以及生命都需要空气。他发现当将附近空气泵出后气压计中水银面下降，这有力支持了托里拆利的结论—即所谓憎恶真空其实是气压作用。通过使用水银柱来改变玻璃管中空气的压力和体积，保持没有空气进出以及温度不变，波义耳可以研究压力和体积的关系。在1662年《新实验》第二版中波义耳宣告压力与体积间的关系是压力与体积乘积保持不变，这就是现在所知的波义耳定律。

空气压力试验比伽利略的斜面实验更好地阐明新出现的实验物理锐意进取风格。自然哲学家不再只是依赖大自然向那些不经意的观察者露出真容。相反大自然被当作一个狡猾的对手，只有搭建精巧的人造环境才能揭示出其秘密。

No one can manipulate heavenly bodies, so the great achievements in astronomy described in Chapter 11 were necessarily based on passive observation. Fortunately the motions of planets in the solar system are simple enough so that after many centuries of observation with increasingly sophisticated instruments these motions could at last be correctly described. For the solution of other problems it was necessary to go beyond observation and measurement and perform experiments, in which general theories are tested or suggested by the artificial manipulation of physical phenomena. In a sense people have always experimented, using trial and error in order to discover ways to get things done, from smelting ores to baking cakes. In speaking here of the beginnings of experiment, I am concerned only with experiments carried out to discover or test general theories about nature. It is not possible to be precise about the beginning of experimentation in this sense.1 Archimedes may have tested his theory of hydrostatics experimentally, but his treatise On Floating Bodies followed the purely deductive style of mathematics, and gave no hint of the use of experiment. Hero and Ptolemy did experiments to test their theories of reflection and refraction, but their example was not followed until centuries later. One new thing about experimentation in the seventeenth century was eagerness to make public use of its results in judging the validity of physical theories. This appears early in the century in work on hydrostatics, as is shown in Galileo’s Discourse on Bodies in Water of 1612. More important was the quantitative study of the motion of falling bodies, an essential prerequisite to the work of Newton. It was work on this problem, and also on the nature of air pressure, that marked the real beginning of modern experimental physics. Like much else, the experimental study of motion begins with Galileo. His conclusions about motion appeared in Dialogues Concerning Two New Sciences, finished in 1635, when he was under house arrest at Arcetri. Publication was forbidden by the church’s Congregation of the Index, but copies were smuggled out of Italy. In 1638 the book was published in the Protestant university town of Leiden by the firm of Louis Elzevir. The cast of Two New Sciences again consists of Salviati, Simplicio, and Sagredo, playing the same roles as before. Among much else, the “First Day” of Two New Sciences contains an argument that heavy and light bodies fall at the same rate, contradicting Aristotle’s doctrine that heavy bodies fall faster than light ones. Of course, because of air resistance, light bodies do fall a little more slowly than heavy ones. In dealing with this, Galileo demonstrates his understanding of the need for scientists to live with approximations, running counter to the Greek emphasis on precise statements based on rigorous mathematics. As Salviati explains to Simplicio:2

Aristotle says, “A hundred pound iron ball falling from the height of a hundred braccia hits the ground before one of just one pound has descended a single braccio.” I say that they arrive at the same time. You find, on making the experiment, that the larger anticipates the smaller by two inches; that is, when the larger one strikes the ground, the other is two inches behind it. And now you want to hide, behind those two inches, the ninety-nine braccia of Aristotle, and speaking only of my tiny error, remain silent about his enormous one. Galileo also shows that air has positive weight; estimates its density; discusses motion through resisting media; explains musical harmony; and reports on the fact that a pendulum will take the same time for each swing, whatever the amplitude of the swings.* This is the principle that decades later was to lead to the invention of pendulum clocks and to the accurate measurement of the rate of acceleration of falling bodies. The “Second Day” of Two New Sciences deals with the strengths of bodies of various shapes. It is on the “Third Day” that Galileo returns to the problem of motion, and makes his most interesting contribution. He begins the Third Day by reviewing some trivial properties of uniform motion, and then goes on to define uniform acceleration along the same lines as the fourteenth-century Merton College definition: the speed increases by equal amounts in each equal interval of time. Galileo also gives a proof of the mean speed theorem, along the same lines as Oresme’s proof, but he makes no reference to Oresme or to the Merton dons. Unlike his medieval predecessors, Galileo goes beyond this mathematical theorem and argues that freely falling bodies undergo uniform acceleration, but he declines to investigate the cause of this acceleration. As already mentioned in Chapter 10, there was at the time a widely held alternative to the theory that bodies fall with uniform acceleration. According to this other view, the speed that freely falling bodies acquire in any interval of time is proportional to the distance fallen in that interval, not to the time.* Galileo gives various arguments against this view,* but the verdict regarding these different theories of the acceleration of falling bodies had to come from experiment. With the distance fallen from rest equal (according to the mean speed theorem) to half the velocity attained times the elapsed time, and with that velocity itself proportional to the time elapsed, the distance traveled in free fall should be proportional to the square of the time. (See Technical Note 25.) This is what Galileo sets out to verify. Freely falling bodies move too rapidly for Galileo to have been able to check this conclusion by following how far a falling body falls in any given time, so he had the idea of slowing the fall by studying balls rolling down an inclined plane. For this to be relevant, he had to show how the motion of a ball rolling down an inclined plane is related to a body in free fall. He did this by noting that the speed a ball reaches after rolling down an inclined plane depends only on the vertical distance through which the ball has rolled, not on the angle with which the plane is tilted.* A freely falling ball can be regarded as one that rolls down a vertical plane, and so if the speed of a ball rolling down an inclined plane is proportional to the time elapsed, then the same ought to be true for a freely falling ball. For a plane inclined at a small angle the speed is of course much less than the speed of a body falling freely (that is the point of using an inclined plane) but the two speeds are proportional, and so the distance traveled along the plane is proportional to the distance that a freely falling body would have traveled in the same time. In Two New Sciences Galileo reports that the distance rolled is proportional to the square of the time. Galileo had done these experiments at Padua in 1603 with a plane at a less than 2° angle to the horizontal, ruled with lines marking intervals of about 1 millimeter.3 He judged the time by the equality of the intervals between sounds made as the ball reached marks along its path, whose distances from the starting point are in the ratios 12 = 1 : 22 = 4 : 32 = 9, and so on. In the experiments reported in Two New Sciences he instead measured relative intervals of time with a water clock. A modern reconstruction of this experiment shows that Galileo could very well have achieved the accuracy he claimed.4 Galileo had already considered the acceleration of falling bodies in the work discussed in Chapter 11, the Dialogue Concerning the Two Chief World Systems. On the Second Day of this previous Dialogue, Salviati in effect claims that the distance fallen is proportional to the square of the time, but gives only a muddled explanation. He also mentions that a cannonball dropped from a height of 100 braccia will reach the ground in 5 seconds. It is pretty clear that Galileo did not actually measure this time,5 but is here presenting only an illustrative example. If one braccio is taken as 21.5 inches, then using the modern value of the acceleration due to gravity, the time for a heavy body to drop 100 braccia is 3.3 seconds, not 5 seconds. But Galileo apparently never attempted a serious measurement of the acceleration due to gravity. The “Fourth Day” of Dialogues Concerning Two New Sciences takes up the trajectory of projectiles. Galileo’s ideas were largely based on an experiment he did in 16086 (discussed in detail in Technical Note 26). A ball is allowed to roll down an inclined plane from various initial heights, then rolls along the horizontal tabletop on which the inclined plane sits, and finally shoots off into the air from the table edge. By measuring the distance traveled when the ball reaches the floor, and by observation of the ball’s path in the air, Galileo concluded that the trajectory is a parabola. Galileo does not describe this experiment in Two New Sciences, but instead gives the theoretical argument for a parabola. The crucial point, which turned out to be essential in Newton’s mechanics, is that each component of a projectile’s motion is separately subject to the corresponding component of the force acting on the projectile. Once a projectile rolls off a table edge or is shot out of a cannon, there is nothing but air resistance to change its horizontal motion, so the horizontal distance traveled is very nearly proportional to the time elapsed. On the other hand, during the same time, like any freely falling body, the projectile is accelerated downward, so that the vertical distance fallen is proportional to the square of the time elapsed. It follows that the vertical distance fallen is proportional to the square of the horizontal distance traveled. What sort of curve has this property? Galileo shows that the path of the projectile is a parabola, using Apollonius’ definition of a parabola as the intersection of a cone with a plane parallel to the cone’s surface. (See Technical Note 26.) The experiments described in Two New Sciences made a historic break with the past. Instead of limiting himself to the study of free fall, which Aristotle had regarded as natural motion, Galileo turned to artificial motions, of balls constrained to roll down an inclined plane or projectiles thrown forward. In this sense, Galileo’s inclined plane is a distant ancestor of today’s particle accelerators, with which we artificially create particles found nowhere in nature. Galileo’s work on motion was carried forward by Christiaan Huygens, perhaps the most impressive figure in the brilliant generation between Galileo and Newton. Huygens was born in 1629 into a family of high civil servants who had worked in the administration of the Dutch republic under the House of Orange. From 1645 to 1647 he studied both law and mathematics at the University of Leiden, but he then turned full-time to mathematics and eventually to natural science. Like Descartes, Pascal, and Boyle, Huygens was a polymath, working on a wide range of problems in mathematics, astronomy, statics, hydrostatics, dynamics, and optics. Huygens’ most important work in astronomy was his telescopic study of the planet Saturn. In 1655 he discovered its largest moon, Titan, revealing thereby that not only the Earth and Jupiter have satellites. He also explained that Saturn’s peculiar noncircular appearance, noticed by Galileo, is due to rings surrounding the planet. In 1656–1657 Huygens invented the pendulum clock. It was based on Galileo’s observation that the time a pendulum takes for each swing is independent of the swing’s amplitude. Huygens recognized that this is true only in the limit of very small swings, and found ingenious ways to preserve the amplitude- independence of the times even for swings of appreciable amplitudes. While previous crude mechanical clocks would gain or lose about 5 minutes a day, Huygens’ pendulum clocks generally gained or lost no more than 10 seconds a day, and one of them lost only about ½ second per day.7 From the period of a pendulum clock of a given length, Huygens the next year was able to infer the value of the acceleration of freely falling bodies near the Earth’s surface. In the Horologium oscillatorium—published later, in 1673—Huygens was able to show that “the time of one small oscillation is related to the time of perpendicular fall from half the height of the pendulum as the circumference of a circle is related to its diameter.”8 That is, the time for a pendulum to swing through a small angle from one side to the other equals π times the time for a body to fall a distance equal to half the length of the pendulum. (Not an easy result to obtain as Huygens did, without calculus.) Using this principle, and measuring the periods of pendulums of various lengths, Huygens was able to calculate the acceleration due to gravity, something that Galileo could not measure accurately with the means he had at hand. As Huygens expressed it, a freely falling body falls 151/12 “Paris feet” in the first second. The ratio of the Paris foot to the modern English foot is variously estimated as between 1.06 and 1.08; if we take 1 Paris foot to equal 1.07 English feet, then Huygens’ result was that a freely falling body falls 16.1 feet in the first second, which implies an acceleration of 32.2 feet/second per second, in excellent agreement with the standard modern value of 32.17 feet/second per second. (As a good experimentalist, Huygens checked that the acceleration of falling bodies actually does agree within experimental error with the acceleration he inferred from his observations of pendulums.) As we will see, this measurement, later repeated by Newton, was essential in relating the force of gravity on Earth to the force that keeps the Moon in its orbit. The acceleration due to gravity could have been inferred from earlier measurements by Riccioli of the time for weights to fall various distances.9 To measure time accurately, Riccioli used a pendulum that had been carefully calibrated by counting its strokes in a solar or sidereal day. To his surprise, his measurements confirmed Galileo’s conclusion that the distance fallen is proportional to the square of the time. From these measurements, published in 1651, it could have been calculated (though Riccioli did not do so) that the acceleration due to gravity is 30 Roman feet/second per second. It is fortunate that Riccioli recorded the height of the Asinelli tower in Bologna, from which many of the weights were dropped, as 312 Roman feet. The tower still stands, and its height is known to be 323 modern English feet, so Riccioli’s Roman foot must have been 323/312 = 1.035 English feet, and 30 Roman feet/second per second therefore corresponds to 31 English feet/second per second, in fair agreement with the modern value. Indeed, if Riccioli had known Huygens’ relation between the period of a pendulum and the time required for a body to fall half its length, he could have used his calibration of pendulums to calculate the acceleration due to gravity, without having to drop anything off towers in Bologna. In 1664 Huygens was elected to the new Académie Royale des Sciences, with an accompanying stipend, and he moved to Paris for the next two decades. His great work on optics, the Treatise on Light, was written in Paris in 1678 and set out the wave theory of light. It was not published until 1690, perhaps because Huygens had hoped to translate it from French to Latin but had never found the time before his death in 1695. We will come back to Huygens’ wave theory in Chapter 14. In a 1669 article in the Journal des Sçavans, Huygens gave the correct statement of the rules governing collisions of hard bodies (which Descartes had gotten wrong): it is the conservation of what are now called momentum and kinetic energy.10 Huygens claimed that he had confirmed these results experimentally, presumably by studying the impact of colliding pendulum bobs, for which initial and final velocities could be precisely calculated. And as we shall see in Chapter 14, Huygens in the Horologium oscillatorium calculated the acceleration associated with motion on a curved path, a result of great importance to Newton’s work. The example of Huygens shows how far science had come from the imitation of mathematics—from the reliance on deduction and the aim of certainty characteristic of mathematics. In the preface to the Treatise on Light Huygens explains:

There will be seen [in this book] demonstrations of those kinds which do not produce as great a certitude as those of Geometry, and which even differ much therefrom, since whereas the Geometers prove their Propositions by fixed and incontestable Principles, here the Principles are verified by the conclusions to be drawn from them; the nature of these things not allowing of this being done otherwise.11

It is about as good a description of the methods of modern physical science as one can find. In the work of Galileo and Huygens on motion, experiment was used to refute the physics of Aristotle. The same can be said of the contemporaneous study of air pressure. The impossibility of a vacuum was one of the doctrines of Aristotle that came into question in the seventeenth century. It was eventually understood that phenomena such as suction, which seemed to arise from nature’s abhorrence of a vacuum, actually represent effects of the pressure of the air. Three figures played a key role in this discovery, in Italy, France, and England. Well diggers in Florence had known for some time that suction pumps cannot lift water to a height more than about 18 braccia, or 32 feet. (The actual value at sea level is closer to 33.5 feet.) Galileo and others had thought that this showed a limitation on nature’s abhorrence of a vacuum. A different interpretation was offered by Evangelista Torricelli, a Florentine who worked on geometry, projectile motion, fluid mechanics, optics, and an early version of calculus. Torricelli argued that this limitation on suction pumps arises because the weight of the air pressing down on the water in the well could support only a column of water no more than 18 braccia high. This weight is diffused through the air, so any surface whether horizontal or not is subjected by the air to a force proportional to its area; the force per area, or pressure, exerted by air at rest is equal to the weight of a vertical column of air, going up to the top of the atmosphere, divided by the cross-sectional area of the column. This pressure acts on the surface of water in a well, and adds to the pressure of the water, so that when air pressure at the top of a vertical pipe immersed in the water is reduced by a pump, water rises in the pipe, but by only an amount limited by the finite pressure of the air. In the 1640s Torricelli set out on a series of experiments to prove this idea. He reasoned that since the weight of a volume of mercury is 13.6 times the weight of the same volume of water, the maximum height of a column of mercury in a vertical glass tube closed on top that can be supported by the air— whether by the air pressing down on the surface of a pool of mercury in which the tube is standing, or on the open bottom of the tube when exposed to the air—should be 18 braccia divided by 13.6, or using more accurate modern values, 33.5 feet/13.6 = 30 inches = 760 millimeters. In 1643 he observed that if a vertical glass tube longer than this and closed at the top end is filled with mercury, then some mercury will flow out until the height of the mercury in the tube is about 30 inches. This leaves empty space on top, now known as a “Torricellian vacuum.” Such a tube can then serve as a barometer, to measure changes in ambient air pressure; the higher the air pressure, the higher the column of mercury that it can support. The French polymath Blaise Pascal is best known for his work of Christian theology, the Pensées, and for his defense of the Jansenist sect against the Jesuit order, but he also contributed to geometry and to the theory of probability, and explored the pneumatic phenomena studied by Torricelli. Pascal reasoned that if the column of mercury in a glass tube open at the bottom is held up by the pressure of the air, then the height of the column should decrease when the tube is carried to high altitude on a mountain, where there is less air overhead and hence lower air pressure. After this prediction was verified in a series of expeditions from 1648 to 1651, Pascal concluded, “All the effects ascribed to [the abhorrence of a vacuum] are due to the weight and pressure of the air, which is the only real cause.”12 Pascal and Torricelli have been honored by having modern units of pressure named after them. One pascal is the pressure that produces a force of 1 newton (the force that gives a mass of 1 kilogram an acceleration of 1 meter per second in a second) when exerted on an area of 1 square meter. One torr is the pressure that will support a column of 1 millimeter of mercury. Standard atmospheric pressure is 760 torr, which equals a little more than 100,000 pascals. The work of Torricelli and Pascal was carried further in England by Robert Boyle. Boyle was a son of the earl of Cork, and hence an absentee member of the “ascendancy,” the Protestant upper class that dominated Ireland in his time. He was educated at Eton College, took a grand tour of the Continent, and fought on the side of Parliament in the civil wars that raged in England in the 1640s. Unusually for a member of his class, he became fascinated by science. He was introduced to the new ideas revolutionizing astronomy in 1642, when he read Galileo’s Dialogue Concerning the Two Chief World Systems. Boyle insisted on naturalistic explanations of natural phenomena, declaring, “None is more willing [than myself] to acknowledge and venerate Divine Omnipotence, [but] our controversy is not about what God can do, but about what can be done by natural agents, not elevated above the sphere of nature.”13 But, like many before Darwin and some even after, he argued that the wonderful capabilities of animals and men showed that they must have been designed by a benevolent creator. Boyle’s work on air pressure was described in 1660 in New Experiments Physico-Mechanical Touching the Spring of the Air. In his experiments, he used an improved air pump, invented by his assistant Robert Hooke, about whom more in Chapter 14. By pumping air out of vessels, Boyle was able to establish that air is needed for the propagation of sound, for fire, and for life. He found that the level of mercury in a barometer drops when air is pumped out of its surroundings, adding a powerful argument in favor of Torricelli’s conclusion that air pressure is responsible for phenomena previously attributed to nature’s abhorrence of a vacuum. By using a column of mercury to vary both the pressure and the volume of air in a glass tube, not letting air in or out and keeping the temperature constant, Boyle was able to study the relation between pressure and volume. In 1662, in a second edition of New Experiments, he reported that the pressure varies with the volume in such a way as to keep the pressure times the volume fixed, a rule now known as Boyle’s law. Not even Galileo’s experiments with inclined planes illustrate so well the new aggressive style of experimental physics as these experiments on air pressure. No longer were natural philosophers relying on nature to reveal its principles to casual observers. Instead Mother Nature was being treated as a devious adversary, whose secrets had to be wrested from her by the ingenious construction of artificial circumstances.