That is a simple enough proposition, but amore difficult problem is encountered when light goes from one medium intoanother, for example from air into water; here also, we see that it does not goin a straight line. 这个命题，足够简单，但是，当光从一个介质，进入到另一个介质，例如从空气到水中时，我们就会碰到一个更困难的问题；这里，我们又看到，光走的不是一条直线。 In the water the ray is at an inclination to its path in the air;if we change the angle θi so that it comes down morenearly vertically, then the angle of “breakage” is not as great. But if we tiltthe beam of light at quite an angle, then the deviation angle is very large.The question is, what is the relation of one angle to the other? 在水中，光线相对于它在空气中的路径，有所倾斜；如果我们改变角度θi，让光线更接近于垂直，那么，“损坏”的角度，就不是那么大。但是，如果我们让光线，倾斜一个相当大的角度，那么，偏差角，就会非常大。现在的问题是，两个角度之间的关系是什么？This also puzzled the ancients for a long time, and here they neverfound the answer! It is, however, one of the few places in all of Greek physicsthat one may find any experimental results listed. 这个问题，也让古人迷惑了很长时间，在这里，他们永远也没找到答案！在所有希腊物理学中，有实验记录的地方很少，而这却是其中之一。Claudius Ptolemy made a list of the angle in water for each of anumber of different angles in air. Table 26–1 showsthe angles in the air, in degrees, and the corresponding angle as measured inthe water. 光在空气中的不同角度，在水中，就有一个相应的角度，对此，克劳狄乌斯·托勒密，列出了一个表。表26-1显示了空气中的角度（单位是度），及在水中的相应的角度。(Ordinarily it is said that Greek scientists never did anyexperiments. But it would be impossible to obtain this table of values withoutknowing the right law, except by experiment. It should be noted, however, thatthese do not represent independent careful measurements for each angle but onlysome numbers interpolated from a few measurements, for they all fit perfectlyon a parabola.)
（通常，据说，希腊科学家永远也不做任何实验。但是，不知道正确的规律，而要得到这个表中的值，是不可能的，除非是通过实验。然而，应该注意，这些并不代表着：对每个角度的独立仔细的测量，而只代表着，从少数几个测量值，所得到的插值，因为，它们完美地符合一条抛物线。）

Fig. 26–2.A light ray is refracted when itpasses from one medium into another. 图26-2 当光束从一个介质，进入另一个时，会被折射。
表26-1 表26-2

This, then, is one of the important stepsin the development of physical law: first we observe an effect, then we measureit and list it in a table; then we try to find the rule by which onething can be connected with another. The above numerical table was made in140 a.d., but it was not until 1621 that someone finally found the ruleconnecting the two angles! The rule, found by Willebrord Snell, a Dutchmathematician, is as follows: if θi is the angle in air and θris the angle in the water, then it turns out that the sine of θiis equal to some constant multiple of the sine of θr :
在物理学规律的发展中，这是重要的一步：首先，我们观察到一个结果，然后，测量它，再把结果，列在一个表中；然后，我们尝试去找出规律，借此规律，一件事，可与另一件，联系起来。上面的数值表，完成于公元140年。但是，直到1621年，才有人最终找出了两个角度之间的关系。此规则，由斯涅耳(Willebrord snell)发现，他是荷兰的一位数学家，规则如下：如果θi是空气中的角度，θr是水中的角度，那么，结果就是，θi的正弦，等于某个常数，乘以θr的正弦。
sinθi=nsinθr. (26.2)
For water the number n isapproximately 1.33 . Equation (26.2)is called Snell’s law; it permits us to predict how the light isgoing to bend when it goes from air into water. Table 26–2 showsthe angles in air and in water according to Snell’s law. Note the remarkableagreement with Ptolemy’s list. 对于水，数n大约就是1.33。方程（26.2）被称为斯涅耳规律；它允许我们去预测，当光从空气中，进入水中时，会弯曲多少。表26-2指出，空气中的角度，和依据斯涅耳规律所得到的水中的角度。注意，它与托勒密的列表，高度一致。

26–3Fermat’s principle of least time 26-3 费马的最短时间原理
Now in the further development of science,we want more than just a formula. First we have an observation, then we havenumbers that we measure, then we have a law which summarizes all the numbers.But the real glory of science is that we can find a way of thinkingsuch that the law is evident.
现在，在科学的进一步发展中，我们想要的，不仅是一个公式，还有更多。首先，我们观察，然后，我们测量数据，总结这些数据，得到规律。但是，科学的真正荣耀，则在于，我们可以找到一种思考的方式，以让此规律，是显然的。

The first way of thinking that made the lawabout the behavior of light evident was discovered by Fermat in about 1650, andit is called the principle of least time, or Fermat’s principle.His idea is this: that out of all possible paths that it might take to get fromone point to another, light takes the path which requires the shortest time.
光的表现，有一定规律，那么，究竟是什么方式，才使得这个规律，成为显然的呢？人们进行了种种思考。第一种方式，是由费马发现的，大约在1650年，被称为最短时间原理，或费马原理。他的想法是这样：从一个点，到另一个点，可能的路径，有多种，在这些路径中，光所选择的路径，用时最短。

Before we go on to analyze refraction, weshould make one more remark about the mirror. If we have a source of light atthe point B and it sends light toward the mirror, then we see thatthe light which goes to A from the point B comesto A in exactly the same manner as it would have come to Aif there were an object at B′ , and no mirror. 在我们继续分析折射之前，关于镜子，我们还要再做一个说明。如果在点B，我们有一个光源，它发光到镜子，然后，我们看到，从点B来到A的光，与好像没有镜子，而在B′有一个对象，也发光来到A，是一样的。Now of course the eye detects only the light which enters itphysically, so if we have an object at B and a mirror which makesthe light come into the eye in exactly the same manner as it would have comeinto the eye if the object were at B′ , then the eye-brain systeminterprets that, assuming it does not know too much, as being an objectat B′ . 现在，眼睛当然只能探测到，实际进入到眼睛中的光，所以，如果我们在B处，有一个对象，而镜子，使得光来到眼睛，这是一种方式，另外，如果对象在 B′处，那么，光也会来到眼睛，这是另一种方式；两种方式，几乎一样；现在，对于眼-脑系统来说，假定它不知道实际情况，那么，它就会把这种情形，解释为，在B′处，有一个对象。So the illusion that there is an object behind the mirror is merelydue to the fact that the light which is entering the eye is entering in exactlythe same manner, physically, as it would have entered had there been anobject back there (except for the dirt on the mirror, and our knowledge of theexistence of the mirror, and so on, which is corrected in the brain).
所以，镜子后面，有一个对象，这只是一种幻觉，它只能归于，光进入眼睛的实际方式，与假设光在镜子背后时，光进入眼睛的方式，是一样的（除了镜子上有灰尘、及我们的关于镜子存在的知识等等，在脑子中被更正了）。

Fig. 26–4.Illustration of Fermat’sprinciple for refraction. 图26-4 关于折射的费马原理的示意图。

Fig. 26–5.The minimum time corresponds topoint C , but nearby points correspond to nearly the same time. 图26-5 与点C相应的最小时间，但是，附近的点，相应于几乎同样的时间。

In Fig. 26–4, ourproblem is again to go from A to B in the shortest time.To illustrate that the best thing to do is not just to go in a straight line,let us imagine that a beautiful girl has fallen out of a boat, and she isscreaming for help in the water at point B . The line marked xis the shoreline. We are at point A on land, and we see theaccident, and we can run and can also swim. But we can run faster than we canswim. What do we do? Do we go in a straight line? (Yes, no doubt!) However, byusing a little more intelligence we would realize that it would be advantageousto travel a little greater distance on land in order to decrease the distancein the water, because we go so much slower in the water. 在图26-4中，我们的问题，就再次又是，从A到 B的最短时间。要说明，最好的做法，并不是走一条直线，那么，让我们假想，一个漂亮姑娘，从船上落水，她在水中，点B处，尖叫呼救。标着x的线，是海岸线。我们在陆地上的点A处，看到这个事故，我们可以跑过去，也可以游泳过去，但我们跑的快，游的慢。我们该怎么做呢？我们要走一条直线吗？（当然，毫无疑问！）然而，稍微动一下脑子，我们就会意识到，如果我们在陆地上多走一点的话，会更好些，这样，可以减少在水中的距离，因为，我们在水中走，要慢的多。(Following this line of reasoning out, we would say the right thingto do is to compute very carefully what should be done!) At any rate,let us try to show that the final solution to the problem is the path ACB, and that this path takes the shortest time of all possible ones. （按照这个推理思路，我们可以说，要做的正确事情，就是要仔细地计算，应该怎么做！）无论如何，让我们尝试指出，这个问题的最终解，就是路径ACB，且这个路径，在所有可能路径中，用时最短。If it is the shortest path, that means that if we take any other, itwill be longer. So, if we were to plot the time it takes against the positionof point X , we would get a curve something like that shown inFig. 26–5,where point C corresponds to the shortest of all possible times.This means that if we move the point X to points near C, in the first approximation there is essentially no change in timebecause the slope is zero at the bottom of the curve.如果它是最短路径，那就意味着，如果我们选取任何其它路径，都会比它长。所以，如果路径是过点X，而我们要画走这条路径所花的时间，那么，我们将得到图26-5所示的曲线，在这里，点C相应于所有可能时间中最短的。这就意味着，如果我们把点X，移到C附近，那么，在一阶近似上，时间本质上将没有变化，因为，在曲线的底部，斜率是零。So our way of finding the law will be to consider that we move theplace by a very small amount, and to demand that there be essentially no changein time. (Of course there is an infinitesimal change of a second order;we ought to have a positive increase for displacements in either directionfrom C .) So we consider a nearby point X and wecalculate how long it would take to go from A to B by thetwo paths, and compare the new path with the old path. It is very easy to do.We want the difference, of course, to be nearly zero if the distance XCis short. 于是，我们要找到规律，就只能用这种方法，即考虑把这个位置，稍微移动一点，且要求，时间基本不变（当然，会有一个极小的二阶量的变化；从C往两边移，都应该有一个正增长。）于是，我们考虑附近的一个点X，这样，从A到B，就有两条路，我们计算，每条路，要花多长时间，然后比较它们。这很容易做。我们要的是差别，当然，如果距离XC很短，差别将接近于零。 First, look at the path on land. If we draw a perpendicular XE, we see that this path is shortened by the amount EC . Let us saywe gain by not having to go that extra distance. On the other hand, in thewater, by drawing a corresponding perpendicular, CF , we find that wehave to go the extra distance XF , and that is what we lose. Or, intime, we gain the time it would have taken to go the distance EC, but we lose the time it would have taken to go the distance XF .Those times must be equal since, in the first approximation, there is to be nochange in time. But supposing that in the water the speed is 1/n times as fast as in air, then we must have
首先，我们看陆地上的路径。如果我们画垂线XE，就会看到，这个路径，少了一个量EC。少走了这个额外的距离，这是我们所得。另一方面，在水中，画相应的垂线CF，就会发现，我们必须走额外的距离XF，这是我们所失。或者，从时间看，我们所得，就是EC上所花的时间，而我们所失，就是XF上所花的时间。这些时间，应该相等，由于在一级近似下，时间应该没有变化。但是，假设水中的速度，是空气中的1/n，那么，我们就应该有：
EC=n⋅XF. (26.3)
Therefore we see that when we have the right point, XCsinEXC= n⋅XC sinXCF or, cancellingthe common hypotenuse length XC and noting that
因此，我们看到，当我们有正确的点时，XCsinEXC = n⋅XC sinXCF，或者，消去共同的斜边XC，并注意到：
EXC=ECN=θi and XCF≈BCN′=θr (when X is near C),（当X 接近于 C）
we have
我们就有：
sinθi=nsinθr. (26.4)
So we see that to get from one point toanother in the least time when the ratio of speeds is n , the lightshould enter at such an angle that the ratio of the sines of the angles θiand θr is the ratio of the speeds in the two media.
所以，我们看到，要在最短的时间内，从一个点到另一个点，当速度的比率是n时，光就应该以这样的角度进入，以让sinθi与sinθr的比率，就是速度在两个媒介中的比率。

26–4Applications of Fermat’s principle26-4 费马原理的应用
Now let us consider some of the interestingconsequences of the principle of least time. First is the principle ofreciprocity. If to go from A to B we have found the path ofthe least time, then to go in the opposite direction (assuming that light goesat the same speed in any direction), the shortest time will be the same path,and therefore, if light can be sent one way, it can be sent the other way.
最短时间原理，有些后果，非常有趣，现在，让我们考虑之。首先，是互换的原理。如果从A到B，我们已经找到了最短时间的路径，那么，如果反过来走（假定在任何方向，光速都一样），最短时间，将会是同一个路径，因此，如果光可以在一个方向被发送，那么，反过来，它也可被发送。

Fig. 26–6.A beam of light is offset as itpasses through a transparent block. 图26-6 光束穿过透明快，会被偏移。

An example of interest is a glass blockwith plane parallel faces, set at an angle to a light beam. Light, in goingthrough the block from a point A to a point B(Fig. 26–6)does not go through in a straight line, but instead it decreases the time inthe block by making the angle in the block less inclined, although it loses alittle bit in the air. The beam is simply displaced parallel to itself becausethe angles in and out are the same.
一个有趣的例子，是一个玻璃块，它有平行的平面，以一定角度，对着一个光束。在光从点A到点B时（图26-6），并不会走一条直线，而是，进入玻璃时，光会降低一点，在稍微倾斜的玻璃中，会形成一个角度，虽然光在空气中，会失去一点{？}。光束只是相对于自己，简单地平移了一点，因为，入射角与出射角，是一样的。

Fig. 26–7.Near the horizon, the apparentsun is higher than the true sun by about 1/2 degree. 图26-7 在接近水平线的地方，显然的太阳，比真太阳，大约高1/2度。
Athird interesting phenomenon is the fact that when we see the sun setting, itis already below the horizon! It does not look as though it is below thehorizon, but it is (Fig. 26–7). Theearth’s atmosphere is thin at the top and dense at the bottom. Light travelsmore slowly in air than it does in a vacuum, and so the light of the sun canget to point S beyond the horizon more quickly if, instead of justgoing in a straight line, it avoids the dense regions where it goes slowly bygetting through them at a steeper tilt. When it appears to go below thehorizon, it is actually already well below the horizon. 第三个有趣的现象，就是当我们看到太阳落山时，实际上，它已经低于地平线了！它看上去，好像并不在地平线之下，但实际上是（图26-7）。地球的大气，顶部稀薄，底部浓厚。光在空气中旅行，比在真空中，要慢很多，于是，如果光走的不是直线，那么，太阳光要达到超过地平线的点S，它会以更快的方式走，因为，在浓厚的区域，它走的较慢，所以，它会避开这个区域，而是以一个较陡的倾斜，通过大气。当它看上去，似乎是向地平线下走时，而实际上，它已经低于地平线了。

As another important example of theprinciple of least time, suppose that we would like to arrange a situationwhere we have all the light that comes out of one point, P , collectedback together at another point, P′ (Fig. 26–9).That means, of course, that the light can go in a straight line from Pto P′ . That is all right. But how can we arrange that not onlydoes it go straight, but also so that the light starting out from P toward Qalso ends up at P′ ? We want to bring all the light back to what wecall a focus. How? If the light always takes the path of least time,then certainly it should not want to go over all these other paths. The onlyway that the light can be perfectly satisfied to take several adjacent paths isto make those times exactly equal! Otherwise, it would select the one ofleast time. Therefore the problem of making a focusing system is merely toarrange a device so that it takes the same time for the light to go on allthe different paths!
作为另外一个最短时间的重要例子，假设我们要安排这样一种情况，在那里，我们让所有的光，都发自同一个点P，然后，在另一个点P′，把它们又汇集起来（图26-9）。当然，这就意味着，光从P到P′，可以走一条直线。这是完全正确的。但是，让光不仅走直线，而且，要让从P到Q的光，最终也汇集于P′，我们该如何安排呢？我们想把所有的光，都收集到一起，这我们为聚焦。怎么做呢？如果光总是想走最短时间的路径，那么，这些其他的路径，它当然就不想走。要让光，满意地去选择几个临近的路径，唯一的方式，就是让所有这些时间，完全一样。否则，光就会选，最短时间的路径。因此，问题如何做一个聚焦系统，就变成了，安排这么一个设备，它可以让光，走不同的路，但是，所花时间一样。

Fig. 26–9.An optical “black box.” 图26-9 一个光学“黑盒子”。

This is easy to do. Suppose that we had apiece of glass in which light goes slower than it does in the air (Fig. 26–10).Now consider a ray which goes in air in the path PQP′ . That is alonger path than from P directly to P′ and no doubttakes a longer time. 这很容易做。假设我们有一片玻璃，光在其中走的，比在空气中慢（图26-10）。现在，考虑空气中的一束光线，它走的路径为PQP′。这个路径，比直接从P到P′要长，毫无疑问，所花时间也更长。But if we were to insert a piece of glass of just the rightthickness (we shall later figure out how thick) it might exactly compensate theexcess time that it would take the light to go at an angle! In thosecircumstances we can arrange that the time the light takes to go straightthrough is the same as the time it takes to go in the path PQP′ . 但是，如果我们塞的这个玻璃片，厚度刚好（我们后面将会弄清楚有多厚），那么，这个玻璃，就可以刚好补偿，光按一定角度走，所花的额外时间！在这些情形下，我们就可以安排，让光走直线所花的时间，与走路径PQP′所花的时间，一样。Likewise, if we take a ray PRR′P′ which is partlyinclined, it is not quite as long as PQP′ , and we do not have tocompensate as much as for the straight one, but we do have to compensatesomewhat. We end up with a piece of glass that looks like Fig. 26–10. 同样，如果我们选择光线PRR′P′，它是部分倾斜的，不像 PQP′那么长，所以，对它的补偿，也不会像对直线的那么多，但是，我们确实要做某些补偿。最终，我们的玻璃片，看上去，就像的图26-10那样。With this shape, allthe light which comes from P will go to P′ . This, ofcourse, is well known to us, and we call such a device a converging lens.In the next chapter we shall actually calculate what shape the lens has to haveto make a perfect focus. 用这个形状，所有从P发出的光，都会来到P′。当然，这我们都知道，我们称这种设备，为凸透镜。我们将实际计算，要完美聚焦，透镜的形状，应为什么样子。

Fig. 26–10.A focusing optical system. 图26-10 一个聚焦的光学系统。

The same principle works for gathering thelight of a star. The great 200 -inch Palomar telescope is built on thefollowing principle. Imagine a star billions of miles away; we would like tocause all the light that comes in to come to a focus. Of course we cannot drawthe rays that go all the way up to the star, but we still want to check whetherthe times are equal. Of course we know that when the various rays have arrivedat some plane KK′ , perpendicular to the rays, all the times inthis plane are equal (Fig. 26–12). 对于收集一个恒星的光，同样的原理，也可适用。伟大的200英尺的帕洛玛（Palomar）望远镜，就是基于如下原理建造的。想象有一个恒星，在数十亿英里之外；我们希望，让所有到来的光线，都来到焦点。当然，各条光线，如何从恒星过来，我们无法画出；但是，我们还是想检查，所花时间，是否相等。当然我们知道，当不同的光线，到达某个平面KK′ 时，KK′垂直于光线，这个平面上的所有时间，都是相等的。The rays must then come down to the mirror and proceed toward P′in equal times. That is, we must find a curve which has the property that thesum of the distances XX′+X′P′ is a constant, nomatter where X is chosen. An easy way to find it is to extend the lengthof the line XX′ down to a plane LL′ . Now if we arrangeour curve so that A′A′′=A′P′ , B′B′′=B′P′, C′C′′=C′P′ , and so on, we will have our curve,because then of course, AA′+A′P′=AA′+A′A′′will be constant. Thus our curve is the locus of all points equidistant from aline and a point. Such a curve is called a parabola; the mirror is madein the shape of a parabola. 因此，光线就应该来到镜子，并在相等的时间里，前进到 P′。也就是说，我们必须找到这样一条曲线，它有这样的属性：无论X怎么选，距离之和XX′+X′P′，应是一个常数。要找到它，有一个简单的方式，就是延长XX′的长度，到平面LL′ 。现在，如果我们安排我们的曲线，让A′A′′=A′P′ , B′B′′=B′P′, C′C′′=C′P′，等等，那么，我们将会拥有我们想要的曲线，因为这时，AA′+A′P′=AA′+A′A′′，将是一个常数。这样，我们的曲线就是这样，所有曲线上的点，到某条线和某个点的距离，是等距的。这种曲线，被称为抛物线；镜子就是按抛物线的形状，制造的。

Fig. 26–12.A paraboloidal mirror.图26-12 一个抛物面镜。

However, the importance of a powerfulprinciple is that it predicts new things.
然而，一个强有力的原理的重要性，就是它可以预告，新的事情。