7–3Development of dynamics 7-3 力学的发展
While Kepler was discovering these laws,Galileo was studying the laws of motion. The problem was, what makes the planetsgo around? (In those days, one of the theories proposed was that the planetswent around because behind them were invisible angels, beating their wings anddriving the planets forward. You will see that this theory is now modified! Itturns out that in order to keep the planets going around, the invisible angelsmust fly in a different direction and they have no wings. Otherwise, it is asomewhat similar theory!) Galileo discovered a very remarkable fact aboutmotion, which was essential for understanding these laws. That is the principleof inertia—if something is moving, with nothing touching it and completelyundisturbed, it will go on forever, coasting at a uniform speed in a straightline. (Why does it keep on coasting? We do not know, but that is the wayit is.)
当开普勒在尝试发现这些规律的时候，伽利略则在研究运动的规律。这里的问题就是，究竟是什么，使得行星转着走？（在那时，有一种理论说，行星之所以转着走，乃是因为，在其后面，有看不见的天使，煽动着它们的翅膀，驱动着行星向前。你将看到这个理论，现在被修改了！结果就是，为了让行星转着走，看不见的天使，就应该在一个不同的方向飞，且它们没有翅膀，否则的话，它就是一种类似的理论了！）伽利略发现了一个关于运动的事实，非常引人注目，它对理解这些规律，具有本质性的意义。那就是惯性原理--如果某物在移动，且没有被碰到或完全没被干扰，那么，它就永远在运动，沿着一条直线，匀速滑行。（为什么它会一直滑行？我们不知道，但事情就是这样。）

Newton modified this idea, saying that theonly way to change the motion of a body is to use force. If the bodyspeeds up, a force has been applied in the direction of motion. On theother hand, if its motion is changed to a new direction, a force has beenapplied sideways. Newton thus added the idea that a force is needed tochange the speed or the direction of motion of a body. For example, if astone is attached to a string and is whirling around in a circle, it takes aforce to keep it in the circle. We have to pull on the string. In fact,the law is that the acceleration produced by the force is inverselyproportional to the mass, or the force is proportional to the mass times theacceleration. The more massive a thing is, the stronger the force required toproduce a given acceleration. (The mass can be measured by putting other stoneson the end of the same string and making them go around the same circle at thesame speed. In this way it is found that more or less force is required, themore massive object requiring more force.) The brilliant idea resulting fromthese considerations is that no tangential force is needed to keep aplanet in its orbit (the angels do not have to fly tangentially) because theplanet would coast in that direction anyway. If there were nothing at all todisturb it, the planet would go off in a straight line. But the actualmotion deviates from the line on which the body would have gone if there wereno force, the deviation being essentially at right angles to the motion,not in the direction of the motion. In other words, because of the principle ofinertia, the force needed to control the motion of a planet around thesun is not a force around the sun but toward the sun. (If there is aforce toward the sun, the sun might be the angel, of course!)
牛顿修改了这个想法，说唯一能改变物体运动的，是用力。如果物体加速了，那么，就是一个力被用在了它运动的方向之上。另一方面，如果其运动改变了方向，那么，就是侧面受了一个力。牛顿就这样增加了这个想法，即物体运动时，要改变速度和方向的话，都需要力。例如，一个石头上，捆一个绳子，绕着一个圆圈旋转，那么就需要一个力，保证石头在这个圆内。我们必须拉紧绳子。事实上，这个规律就是，由力所产生的加速度，反比于质量，或者说，力正比于质量乘以加速度。一个物体，，要产生被给予的加速度的话，质量越大，所需力就越大。（通过在其他的石头上绑同样的绳子，以同样的速度，绕同样的圆转，就可以测量质量了，以这种方式，就会发现，需要更多或更少的力，质量越大，需要的力就越大）。从这些考虑所导致的精彩结果，就是要保证行星在其轨道上运行，并不需要正切方向的力（天使不需要沿着切线飞），因为无论如何，行星都可以沿着那个方向滑行。如果没有任何东西打扰它的话，行星就会沿着一条直线走。但是，实际运动会偏离这条线—即没有任何力时这个物体本来应该走的直线，偏离本质上与运动程直角，而不是在运动的方向上。换句话说，由于惯性的原理，控制行星绕着太阳运动所需的力，不是绕着太阳的力，而是向着太阳的力。（如果有一个向着太阳的力，那么太阳当然就是天使了！）

7–4Newton’s law of gravitation 7-4 牛顿的万有引力规律
From his better understanding of the theoryof motion, Newton appreciated that the sun could be the seat ororganization of forces that govern the motion of the planets. Newton proved to himself(and perhaps we shall be able to prove it soon) that the very fact that equalareas are swept out in equal times is a precise sign post of the propositionthat all deviations are precisely radial—that the law of areas is adirect consequence of the idea that all of the forces are directed exactly towardthe sun.
对运动的理论，牛顿有了更好的理解，从此出发，他倾向于认为，对于所有左右着行星运动的力量来说，太阳可能是处于组织的地位。牛顿向他自己证明了（或许我们很快也能够证明）这一事实，即在同等时间里扫过同等的面积，是下述命题的一个精确标志，该命题就是，所有的偏离，都是精确地沿着径向的，即面积的规律，就是‘所有的力都精确地指向太阳’这一想法的直接后果。

Next, by analyzing Kepler’s third law it ispossible to show that the farther away the planet, the weaker the forces. Iftwo planets at different distances from the sun are compared, the analysisshows that the forces are inversely proportional to the squares of therespective distances. With the combination of the two laws, Newton concludedthat there must be a force, inversely as the square of the distance, directedin a line between the two objects.
其次，通过分析开普勒的第三条规律，有可能指出，距离越远，力就越弱。如果比较两个距太阳不同距离的行星，就可分析出，力反比于各自距离的平方。综合这两条规律，牛顿得出，必然有一个力，与距离的平方成反比，且在两个对象之间，沿着直线指向。

Being a man of considerable feeling forgeneralities, Newton supposed, of course, that this relationship applied more generallythan just to the sun holding the planets. It was already known, for example,that the planet Jupiter had moons going around it as the moon of the earth goesaround the earth, and Newton felt certain that each planet held its moons witha force. He already knew of the force holding us on the earth, so heproposed that this was a universal force—that everything pulls everythingelse.
牛顿是一个具有高度概括能力的人，他假设，当然只能假设，这个关系可以更普遍地应用，而不是仅用于太阳抓住行星。例如，已经知道，木星有月亮的绕着，正如地球有月亮绕着一样，所以，牛顿肯定，每个行星，都用一个力抓着其月亮。他已经知道了地球上的这个抓着我们的力，于是，他提议，有一个普遍的力--每个事物，都在拉着着其他的事物。

The next problem was whether the pull ofthe earth on its people was the “same” as its pull on the moon, i.e., inverselyas the square of the distance. If an object on the surface of the earth falls16 feet in the first second after it is released from rest, how fardoes the moon fall in the same time? We might say that the moon does not fall atall. But if there were no force on the moon, it would go off in a straightline, whereas it goes in a circle instead, so it really falls in from whereit would have been if there were no force at all. We can calculate from theradius of the moon’s orbit (which is about 240,000 miles) and how long it takes to go around the earth(approximately 29 days), how far the moon moves in its orbit in 1 second, and can then calculate how far it falls in one second.2This distance turns out to be roughly 1/20 of an inch in a second. That fits very well with the inverse square law,because the earth’s radius is 4000 miles, and if something which is 4000 miles from the center of the earth falls 16 feet in a second, something 240,000 miles, or 60 times as far away, should fall only 1/3600 of 16 feet, which also is roughly 1/20 of an inch. Wishing to put this theory of gravitation to a test bysimilar calculations, Newton made his calculations very carefully and found adiscrepancy so large that he regarded the theory as contradicted by facts, anddid not publish his results. Six years later a new measurement of the size ofthe earth showed that the astronomers had been using an incorrect distance tothe moon. When Newton heard of this, he made the calculation again, with thecorrected figures, and obtained beautiful agreement.
下一个问题就是，地球对人的吸引，是否与地球对月亮的吸引，是一样的，也就是说，与距离的平方成反比。如果地表上的一个对象，从静止状态被释放之后，第一秒钟下降了16英尺，那么，月亮在同样的时间里，能下降多少呢？我们可以说，月亮根本没有下降。但是，如果没有力作用于月亮，它就应该沿着一条直线走掉，而实际上，它是绕着一个圆在走，所以，它应该是从‘如果没有力量作用于它时应该在的那个地方’下降了。我们可以从月亮轨道的半径（约240，000英里）来计算：月亮绕地球一周需要多长时间（大约29天），月亮在其轨道中一秒能移动多远，然后，就可以计算它在一秒钟能下降多少（脚注2）。结果这个距离就是1/21英寸每秒。与距离的平方成反比规律，非常一致，因为地球的半径是4000英里，假设某物，距离地球中心4000英里，以每秒16英尺下降，假设另一物，在240,000英里外，或者说，其距离是前物的60倍，那么，该物应该下降16英尺的1/3600，约为1英寸的1/20。牛顿希望通过类似的计算，对这个重力的理论，进行测试，经过非常仔细的计算，他发现差异巨大，以至于认为，这个理论，与事实矛盾，所以，这个结果，并未发表。六年之后，一个新的关于地球尺寸的测量指出，天文学家所使用的对月距离，并不正确。得此消息，牛顿又用正确的数据，重做计算，所得结果，漂亮且一致。

Fig. 7–3.Apparatus for showing the independenceof vertical and horizontal motions. 图7-3 显示垂直和水平运动相独立的仪器。
This idea that the moon “falls” is somewhatconfusing, because, as you see, it does not come any closer. The idea issufficiently interesting to merit further explanation: the moon falls in the sensethat it falls away from the straight line that it would pursue if there wereno forces. Let us take an example on the surface of the earth. An objectreleased near the earth’s surface will fall 16 feet in the first second. An object shot out horizontallywill also fall 16 feet; even though it is moving horizontally, it still falls thesame 16 feet in the same time. Figure 7–3 showsan apparatus which demonstrates this. On the horizontal track is a ball which isgoing to be driven forward a little distance away. At the same height is a ballwhich is going to fall vertically, and there is an electrical switch arranged sothat at the moment the first ball leaves the track, the second ball isreleased. That they come to the same depth at the same time is witnessed by thefact that they collide in midair. An object like a bullet, shot horizontally,might go a long way in one second—perhaps 2000 feet—but it will still fall 16 feet if it is aimed horizontally. What happens if we shoot abullet faster and faster? Do not forget that the earth’s surface is curved. Ifwe shoot it fast enough, then when it falls 16 feet it may be at just the same height above the ground as it wasbefore. How can that be? It still falls, but the earth curves away, so it falls“around” the earth. The question is, how far does it have to go in one secondso that the earth is 16 feet below the horizon? In Fig. 7–4 we see theearth with its 4000 -mile radius, and the tangential, straight line path that the bulletwould take if there were no force. Now, if we use one of those wonderfultheorems in geometry, which says that our tangent is the mean proportional betweenthe two parts of the diameter cut by an equal chord, we see that the horizontaldistance travelled is the mean proportional between the 16 feet fallen and the 8000 -mile diameter of the earth. The square root of (16/5280)×8000 comes out very close to 5 miles. Thus we see that if the bullet moves at 5 miles a second, it then will continue to fall toward the earthat the same rate of 16 feet each second, but will never get any closer because theearth keeps curving away from it. Thus it was that Mr. Gagarin maintainedhimself in space while going 25,000 miles around the earth at approximately 5 miles per second. (He took a little longer because he was a littlehigher.)
这个月亮“下落了”的想法，在某种意义上，让人困惑，因为我们看到，月亮一点都未靠近。这个想法，对于评价进一步的解释，可能还是蛮有趣的，这种解释就是：如果没有其他的力，月亮就会沿着直线走，而下落的意义，是指从这条直线下落。让我们看一个地球上的例子。在靠近地表的地方，释放一个对象，在头一秒内，它会下落16英尺。水平地射出一个对象，也会降落16英尺；虽然它在做水平运动，但在同样的时间里，它仍下落了16英尺。图7-3中的仪器，演示的就是这个。在水平轨道上，有一个球，将被扔向前面。还有一个球，处于同样的高度，它将垂直降落，还有一个电子开关，保证当第一个球离开轨道时，第二个球也被释放。它们会来到同一高度这一事实，会通过它们在空中的碰撞，而得以证明。一个对象，比如一个水平射出的子弹，可能会在1秒内走很长的距离，比如2000英尺，但它仍然会降落16英尺，即便它是水平瞄准的。如果我们的射出速度越来越快，那么，会发生什么事情呢？不要忘记，地表是弯曲的。如果我们的射出速度足够快，那么，当它掉下16英尺时，它的高度，可能与以前正好一样。怎么会这样呢？它确实下落了，但是，由于地球是弯曲的，所以，它就是绕着地球在下落。现在的问题就是，它需要在一秒钟内走多快，才能保证地球总是在16英尺的水平之下？在图7-4中，我们看到，地球及其4000英里的半径，及一条切线、直线路径，如果没有任何力的话，子弹的将会沿着这条切线走。现在，我们使用几何中的一条精彩的定理，它说，直径被一条弦切成两部分，与此弦同等的切线，就是这两部分的比例中项，我们看到，所经过的水平距离，就是在16英尺和8000英里地球直径之间的比例中项。 (16/5280)×8000的平方根，非常接近5英里。这样，我们就看到，如果子弹移速是每秒5英里，那么，它就会持续地以16英尺每秒的下落速率落向地球，但是，它永远也不会靠近地球，因为，地球是弯曲的。加加林先生在太空中，就是这样，他绕地球走25000英里时，所用速度，就是大约每秒5英里。（他花的时间可能长点，因为他比较高一点。）

Fig. 7–4.Acceleration toward the center of acircular path. From plane geometry, x/S=(2R−S)/x≈2R/x, where R is the radius of the earth, 4000 miles; x is the distance “travelled horizontally” in one second; and S is the distance “fallen” in one second (16 feet). 图7-4 向一个环形路径中心的加速。从平面几何，x/S=(2R−S)/x≈2R/x，这里R是地球的半径，4000英里；x是一秒内水平走过的距离；S是一秒内“下落”的距离（16英尺）。

Any great discovery of a new law is usefulonly if we can take more out than we put in. Now, Newton used the secondand third of Kepler’s laws to deduce his law of gravitation. What did he predict?First, his analysis of the moon’s motion was a prediction because it connectedthe falling of objects on the earth’s surface with that of the moon. Second,the question is, is the orbit an ellipse? We shall see in a later chapterhow it is possible to calculate the motion exactly, and indeed one can provethat it should be an ellipse,3so no extra fact is needed to explain Kepler’s first law. Thus Newtonmade his first powerful prediction.
任何一条新规律的伟大发现，只有当我们投入的少，而得到的多时，它才有用。现在，牛顿使用开普勒的第二和第三规律，来推出他的万有引力规律。他预告了什么呢？首先，他对月亮运动的分析，是一个预告，因为，他把地球上对象的下落，与月亮的下落，联系了起来。其次，问题就是：轨道是一个椭圆吗？在后面的某章中，我们将会看到，精确地计算运动，是如何可能的，确实，我们可以证明，它应该是一个椭圆（脚注3），所以，就不需要额外的事实，来解释开普勒的第一规律了。就这样，牛顿做出了他的第一个有影响力的预告。

The law of gravitation explains manyphenomena not previously understood. For example, the pull of the moon on the earthcauses the tides, hitherto mysterious. The moon pulls the water up under it andmakes the tides—people had thought of that before, but they were not as cleveras Newton, and so they thought there ought to be only one tide during the day.The reasoning was that the moon pulls the water up under it, making a high tideand a low tide, and since the earth spins underneath, that makes the tide atone station go up and down every 24 hours. Actually the tide goes up and down in 12 hours. Another school of thought claimed that the high tide shouldbe on the other side of the earth because, so they argued, the moon pulls theearth away from the water! Both of these theories are wrong. It actually workslike this: the pull of the moon for the earth and for the water is “balanced”at the center. But the water which is closer to the moon is pulled morethan the average and the water which is farther away from it is pulled lessthan the average. Furthermore, the water can flow while the more rigid earth cannot.The true picture is a combination of these two things.
很多现象，以前不懂，万有引力的规律，可予解释。例如，月亮对地球的吸引，引起潮汐，迄今为止，这还是神秘的。月亮把它下面的水吸引起来，形成潮汐，以前人们也曾这么想过，但是，他们不如牛顿那么聪明，所以他们认为，一天只有一次潮汐。这个推理是这样的，月亮把它下面的水吸引起来，形成一个高的潮汐和一个低的潮汐，由于地球在旋转，所以，形成的潮汐，在一个地方的长起和落下，24小时一次。实际上，潮汐是在12小时内长起和落下。另外一个学派声称，高潮应该在地球的另一侧，他们是如此辩解的，因为月亮把地球从水中拽走。这两种理论全错。潮汐的工作原理，实际如此：月亮对地球的吸引和对水的吸引，在“中心”是平衡的。但是，离月近的水，受到的吸引，比平均值多，离月远的水，则少。另外，水可以流动，而相对更为固化的地球则不行。真正的图片，就是这两个事情的结合。

Fig. 7–5.The earth-moon system, with tides.图7-5 带有潮汐的地球-月亮系统。

What do we mean by “balanced”? What balances?If the moon pulls the whole earth toward it, why doesn’t the earth fall right“up” to the moon? Because the earth does the same trick as the moon, it goes ina circle around a point which is inside the earth but not at its center. The moondoes not just go around the earth, the earth and the moon both go around acentral position, each falling toward this common position, as shown inFig. 7–5.This motion around the common center is what balances the fall of each. So theearth is not going in a straight line either; it travels in a circle. The wateron the far side is “unbalanced” because the moon’s attraction there is weakerthan it is at the center of the earth, where it just balances the “centrifugalforce.” The result of this imbalance is that the water rises up, away from thecenter of the earth. On the near side, the attraction from the moon isstronger, and the imbalance is in the opposite direction in space, but again awayfrom the center of the earth. The net result is that we get two tidalbulges.
我们通过“被平衡了”，意味着什么呢？什么是平衡呢？如果月亮吸引着整个地球朝它走，为什么地球没有直接掉落到月亮上呢？因为地球对月亮，也做着同样的事，月亮绕着一个点做圆运动，此点在地球里面，但并不是地球的中心。月亮并不是直接绕着地球走，地球和月亮两个都是绕着一个中央位置在走，每个都向着这个共同的位置下落，如图7-5所示。这个总绕着共同中心的运动，就是那个平衡着相互掉落的原因。所以地球，也不是走一条直线，它也是绕着圆走。在远端的水，并“未被平衡的”，因为月亮对那里的吸引力，比对地球中心的要弱，在远端，它只平衡了“离心力”。这种未平衡的结果，就是水会涨起来，离开地球的中心。在近端，来自月亮的吸引力要更强些，所以，未平衡就是在空间中相反的方向，但同样是离开地球的中心。最终的结果就是，我们得到两个鼓起的潮汐。

7–5Universal gravitation 7-5 普遍的万有引力
What else can we understand when weunderstand gravity? Everyone knows the earth is round. Why is the earth round?That is easy; it is due to gravitation. The earth can be understood to be roundmerely because everything attracts everything else and so it has attracteditself together as far as it can! If we go even further, the earth is not exactlya sphere because it is rotating, and this brings in centrifugal effects whichtend to oppose gravity near the equator. It turns out that the earth should beelliptical, and we even get the right shape for the ellipse. We can thus deducethat the sun, the moon, and the earth should be (nearly) spheres, just from thelaw of gravitation.
在我们理解了万有引力之后，我们还能理解其他什么呢？每个人都知道，地球是圆的。为什么地球是圆的呢？这很容易；应归于万有引力。地球可被理解为圆的，仅仅是因为每个东西，都吸引着另外的东西，所以，地球就尽可能地把自己吸在了一起！如果我们走得更远些，地球并不是精确的球形，因为它在旋转，这就导致了离心的效果，在接近赤道的地方，该效果趋向于对抗万有引力。结果就是，地球应该是椭圆形的，对于这个椭圆，我们甚至可以得到其正确的形状。这样，我们就可以从万有引力规律推出：太阳、月亮、和地球，应该是（接近）球形的。

What else can you do with the law ofgravitation? If we look at the moons of Jupiter we can understand everythingabout the way they move around that planet. Incidentally, there was once acertain difficulty with the moons of Jupiter that is worth remarking on. Thesesatellites were studied very carefully by Rømer, who noticed that the moonssometimes seemed to be ahead of schedule, and sometimes behind. (One can findtheir schedules by waiting a very long time and finding out how long it takeson the average for the moons to go around.) Now they were ahead whenJupiter was particularly close to the earth and they were behindwhen Jupiter was farther from the earth. This would have been a verydifficult thing to explain according to the law of gravitation—it would havebeen, in fact, the death of this wonderful theory if there were no otherexplanation. If a law does not work even in one place where it ought to,it is just wrong. But the reason for this discrepancy was very simple andbeautiful: it takes a little while to see the moons of Jupiter becauseof the time it takes light to travel from Jupiter to the earth. When Jupiter iscloser to the earth the time is a little less, and when it is farther from the earth,the time is more. This is why moons appear to be, on the average, a littleahead or a little behind, depending on whether they are closer to or fartherfrom the earth. This phenomenon showed that light does not travel instantaneously,and furnished the first estimate of the speed of light. This was done in 1676.
你还能用万有引力规律，做其他什么事情呢？如果我们看木星的月亮，那么，所有月亮绕着行星运动的事情，就都可以理解了。附带地，对于木星的月亮，曾经有某些困难，值得说明。这些卫星被罗默仔细地研究过，他注意到，月亮有时似乎会先于时间表，有时则会落后于时间表（通过等待很长的时间，人们可以找到它们的时间表，且可发现，月亮绕一圈，平均需要多长时间。）现在，当木星离地球较近时，它们就先于时间表，离得较远时，就会落后于时间表。这件事情，依据万有引力的规律，非常难解，事实上，如果没有其它解释的话，这个奇妙理论就会死去。一个规律，哪怕是在一个它本应该起作用的地方，没有起作用，它也就是错的。但是，这个差异的原因，非常简单，且很漂亮，看木星的月亮，需要一些时间，因为，光从木星传到地球，需要时间。当木星离地球较近时，这个时间稍微少一点，离得较远时，稍多。这就是为什么月亮平均看起来，有时提前，有时落后，这依赖于，它们离地球较近还是较远。这个现象指出，光的传播，并非瞬时，且提供了对光速的第一次估计。这完成于1676年。

If all of the planets pull on each other,the force which controls, let us say, Jupiter in going around the sun is notjust the force from the sun; there is also a pull from, say, Saturn. This forceis not really strong, since the sun is much more massive than Saturn, but thereis some pull, so the orbit of Jupiter should not be a perfect ellipse,and it is not; it is slightly off, and “wobbles” around the correct ellipticalorbit. Such a motion is a little more complicated. Attempts were made to analyzethe motions of Jupiter, Saturn, and Uranus on the basis of the law ofgravitation. The effects of each of these planets on each other were calculatedto see whether or not the tiny deviations and irregularities in these motionscould be completely understood from this one law. Lo and behold, for Jupiterand Saturn, all was well, but Uranus was “weird.” It behaved in a very peculiarmanner. It was not travelling in an exact ellipse, but that was understandable,because of the attractions of Jupiter and Saturn. But even if allowance weremade for these attractions, Uranus still was not going right, so thelaws of gravitation were in danger of being overturned, a possibility thatcould not be ruled out. Two men, Adams and Le Verrier, in England andFrance, independently, arrived at another possibility: perhaps there is anotherplanet, dark and invisible, which men had not seen. This planet, N , could pull on Uranus. They calculated where such a planet would haveto be in order to cause the observed perturbations. They sent messages to therespective observatories, saying, “Gentlemen, point your telescope to such andsuch a place, and you will see a new planet.” It often depends on with whom youare working as to whether they pay any attention to you or not. They did payattention to Le Verrier; they looked, and there planet N was! The other observatory then also looked very quickly in the nextfew days and saw it too.
如果所有的行星，都相互吸引，那么，控制木星绕着太阳运行的力量，就不仅是来自太阳，还有来自土星的。这个力量，不是很强，由于太阳的质量，远大于木星，但还是有一些吸引，所以，木星的轨道，应该不是一个完美的椭圆，也确实如此；实际得轨道有一点偏，且绕着正确的椭圆轨道“晃动”。这样的一个运动，稍微有点复杂。基于万有引力规律，对木星土星和天王星的运动，做了一些分析。这些行星的相互间的影响，被计算了，以查看这些运动中的细微的偏差和不规则，是否可以从这一条规律出发，而被完全理解。瞧啊，对于木星和天王星，很好，但是，土星有点“奇怪”。它的表现方式，非常怪异。它并不是在一个精确的椭圆上运行，但这可理解，因为有木星和天王星的吸引。但是，即便这些吸引所带来的配额，可以允许，天王星还是不对，于是，万有引力规律就面临着被推翻的危险，这一可能性，无法排除。有两个人，英国的亚当斯，和法国的勒维利埃，他们各自独立地得到了另外一种可能性：或许还有另外一个行星，太黑看不见，人们从来没有观察到过。这个行星N，可能在吸引着天王星。他们计算了，这个行星应在何处，才能导致观察中的那个扰动。他们给不同的天文台发信息说：”先生们，把你们的望远镜，指向何处何处，就会看到一个新的行星。”通常，你是否会被注意到，取决于你与谁在一起工作。他们确实注意到了勒维利埃；他们查看了，行星N确实在那里！因此，另一个天文台，在接下来的几天内，也非常快地查看了，也看到了。

Fig. 7–6.A double-star system. 图7-6 一个双子星系统

Fig. 7–7.Orbit of Sirius B with respect toSirius A. 图7-7 就天狼星A而言的天狼星B的轨道。
This discovery shows that Newton’s laws areabsolutely right in the solar system; but do they extend beyond the relativelysmall distances of the nearest planets? The first test lies in the question, dostars attract each other as well as planets? We have definiteevidence that they do in the double stars. Figure 7–6 shows adouble star—two stars very close together (there is also a third star in thepicture so that we will know that the photograph was not turned). The stars arealso shown as they appeared several years later. We see that, relative to the“fixed” star, the axis of the pair has rotated, i.e., the two stars are goingaround each other. Do they rotate according to Newton’s laws? Careful measurementsof the relative positions of one such double star system are shown inFig. 7–7.There we see a beautiful ellipse, the measures starting in 1862 and going allthe way around to 1904 (by now it must have gone around once more). Everythingcoincides with Newton’s laws, except that the star Sirius A is not atthe focus. Why should that be? Because the plane of the ellipse is not inthe “plane of the sky.” We are not looking at right angles to the orbit plane,and when an ellipse is viewed at a tilt, it remains an ellipse but the focus isno longer at the same place. Thus we can analyze double stars, moving abouteach other, according to the requirements of the gravitational law.
这个发现指出，牛顿的规律，在太阳系，绝对正确；对于距离最近的行星，那里的距离，相对来说也较小，在那里，规律还正确吗？第一个测试，就是这个问题：恒星像行星一样，相互吸引吗？在双星座，我们有确定的证据，证明它们是这样。图7-6所示，就是一个双星，两个恒星靠的很近（在照片中，还有第三颗恒星，这样我们就知道这个图，没有被转过）。在这些恒星出现几年之后，它们又被显示我们。我们看到，相对于那个“固定的”恒星，这个双星的轴，旋转了，亦即，这两颗星，相互绕着旋转了。它们是否依据牛顿规律在旋转呢？对一个这种双星系统的相对位置的仔细测量，如图7-7。在那里，我们看到了漂亮的椭圆，测量始于1862年，然后，一直进行到1904年（到现在，它应该已经又走了一圈）。所有的事情，都是用牛顿定律来考虑，除了恒星Sirius A不是在焦点上，为什么会这样呢？因为椭圆的平面，并不在“天体的平面上。”我们并不是用直角来看轨道的平面，所以，当一个椭圆，是以一个倾斜的角度，被观察时，它就仍会保持一个椭圆，但焦点就不在同一个地方了。这样，我们就可以分析双星了，它们依据万有引力规律的要求，相互绕着运动。

Fig. 7–8.A globular star cluster. 图7-8 一个球形星簇。
That the law of gravitation is true at evenbigger distances is indicated in Fig. 7–8. If onecannot see gravitation acting here, he has no soul. This figure shows one ofthe most beautiful things in the sky—a globular star cluster. All of the dotsare stars. Although they look as if they are packed solid toward the center,that is due to the fallibility of our instruments. Actually, the distancesbetween even the centermost stars are very great and they very rarely collide.There are more stars in the interior than farther out, and as we move outwardthere are fewer and fewer. It is obvious that there is an attraction among thesestars. It is clear that gravitation exists at these enormous dimensions,perhaps 100,000 times the size of the solar system. Let us now go further, andlook at an entire galaxy, shown in Fig. 7–9. The shapeof this galaxy indicates an obvious tendency for its matter to agglomerate. Ofcourse we cannot prove that the law here is precisely inverse square, only thatthere is still an attraction, at this enormous dimension, that holds the wholething together. One may say, “Well, that is all very clever but why is it notjust a ball?” Because it is spinning and has angular momentum whichit cannot give up as it contracts; it must contract mostly in a plane.(Incidentally, if you are looking for a good problem, the exact details of howthe arms are formed and what determines the shapes of these galaxies has notbeen worked out.) It is, however, clear that the shape of the galaxy is due togravitation even though the complexities of its structure have not yet allowed usto analyze it completely. In a galaxy we have a scale of perhaps 50,000 to 100,000 light years. The earth’s distance from the sun is 8（1/3） light minutes, so you can see how large these dimensionsare.
即便是在更大的距离中，万有引力的规律，依然正确，如图7-8所示。万有引力在这里所起的作用，如果谁在看不到的话，那么他一定是没有灵魂。这个图形指出的，是天空中最美丽的事物之一，一个球形星簇。所有的点，都是恒星。虽然它们看上去，是趋向于中心的充实固体，那是因为，我们的仪器容易犯错。实际上，即便是在最中心，恒星间的距离，也非常巨大，它们之间，难得会碰撞。内部的恒星数，要比外部的多，随着我们向外移动，数目就会越来越少。很明显，在这些恒星中，有一个吸引力。这些巨型尺寸，大约是太阳系尺寸的100，000倍，很清楚，其中有万有引力的存在。现在，我们往更远的地方走，看整个星系，如图7-9。这个星系的形状，为这种凝聚的事情，指出了一个明显的趋势。当然，我们无法证明，这里的规律，依然精确地与平方成反比，我们只能证明，在这个巨大的尺寸中，有一个吸引力，把全部都抓住了。有人可能会说：“好，这很聪明，但是，为什么它不可能只是一个球呢？”因为它在自转，并且有角动量，在它收缩的时候，它不可能放弃这个角动量；它应该大部分收缩到一个平面中。（顺便说一下，如果你在寻找一个好的课题，那么注意，星系的肢体是如何形成的，及究竟是什么，规定着这些星系的形状等，这些细节问题，尚未得出。）然而，有一点很清楚，即星系的形状，要归于万有引力，即便其结构的复杂性，还不允许我们去完整地分析它。在一个星系中，我们所拥有的尺度，大约是50，000到100，000光年。地球到太阳的距离，约为8又1/3光分钟，所以你可以看到，这些尺寸，多么巨大。

Fig. 7–9.A galaxy (M81). 图7-9 一个星系。