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。

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。

It is easy to show that there are a numberof new things predicted by Fermat’s principle. First, suppose that there are threemedia, glass, water, and air, and we perform a refraction experiment andmeasure the index n for one medium against another. Let us call n12the index of air (1 ) against water (2 ); n13 the index of air (1 )against glass (3 ). If we measured water against glass, we should find anotherindex, which we shall call n23 . But there is no a priori reasonwhy there should be any connection between n12 , n13 , and n23. On the other hand, according to the idea of least time, there is adefinite relationship. The index n12 is the ratio of two things,the speed in air to the speed in water; n13 is the ratio of the speed inair to the speed in glass; n23 is the ratio of the speed in water to thespeed in glass. Therefore we cancel out the air, and get
很容易指出，通过费马原理，可以预告若干新的事物。首先，假设有三个介质，玻璃、水、和空气，我们做一个折射实验，测量一个介质对另一个介质的指数。让我们称n12是空气对水的指数；n13是空气对玻璃的指数。如果我们测量水对玻璃，我们就会发现另外一个指数，我们称之为n 23。但是，没有任何先天的理由，会让我们认为，n12 , n13 , 和n23之间，为什么会有联系。另一方面，依据最短时间这个想法，是有一个确定的关系。指数n12是两个事物的比率，空气中的速度和水中的速度；n 13是空气中的速度与玻璃中的速度的比率；n23是水中的速度与玻璃中的速度的比率。因此，消去空气，我们就得到：
(26.5)

In other words, we predict that theindex for a new pair of materials can be obtained from the indexes of theindividual materials, both against air or against vacuum. So if we measure thespeed of light in all materials, and from this get a single number for eachmaterial, namely its index relative to vacuum, called ni (n1is the speed in vacuum relative to the speed in air, etc.), then our formula iseasy. The index for any two materials i and j is
换句话说，我们预告了，一对新材料的指数，可以从独立的材料指数中得到，这两个指数，都是针对空气、或针对真空的。所以，如果我们测量了光在所有材料中的速度，由此，对于每一种材料，都能得到一个单独的数字，即它对真空的索引，称为ni（n1是真空中的速度，对于空气中的速度，等等。）然后，我们的公式，就容易了。对于任何两个材料i和j，其指数就是：
(26.6)
Using only Snell’s law, there is no basisfor a prediction of this kind.1But of course this prediction works. The relation (26.5)was known very early, and was a very strong argument for the principle of leasttime.
只使用Snell的规律，并不能为这种预告，提供任何基础（脚注1）。但当然，这个预告，确实行得通。关系（26.5），很久之前，就知道了，对于最短时间原理，它是一个强有力的论证。

Another argument for the principle of leasttime, another prediction, is that if we measure the speed of light inwater, it will be lower than in air. This is a prediction of a completelydifferent type. It is a brilliant prediction, because all we have so farmeasured are angles; here we have a theoretical prediction which isquite different from the observations from which Fermat deduced the idea ofleast time. It turns out, in fact, that the speed in water is slowerthan the speed in air, by just the proportion that is needed to get the rightindex!
另外一个关于最短时间原理的论证，或另外一个预测，就是，如果我们测量水中的光速，那么，它会比空气中的慢。这个预告，类型完全不同。它是一个辉煌预告，因为，迄今为止，我们所测，都是角度；这里，我们有了一个理论上的预告；费马推出最短时间想法，用的是观察，此预告与此观察，完全不同。结果就是，水中的光速比空气中的慢，要得到正确的指数，通过所需的比例就行{？}。

The following is another difficulty withthe principle of least time, and one which people who do not like this kind ofa theory could never stomach. With Snell’s theory we can “understand” light.Light goes along, it sees a surface, it bends because it does something at thesurface. The idea of causality, that it goes from one point to another, andanother, and so on, is easy to understand. 下面，是关于最短时间原理的另外一个困难，有些人，不喜欢这种理论，对他们来说，这个永远不能忍受。用Snell的理论，我们可以“理解”光。光往前走，看到一个表面，它弯曲，因为，在表面，它做了某事。因果关系的想法，即事情从一个点到另外一个点，再到另外一个点，如此等等，容易理解。 But the principle of least time is a completely differentphilosophical principle about the way nature works. Instead of saying it is acausal thing, that when we do one thing, something else happens, and so on, itsays this: we set up the situation, and light decides which is theshortest time, or the extreme one, and chooses that path. But what doesit do, how does it find out? Does it smell the nearby paths, andcheck them against each other? 但是，最短时间原理，是关于自然方式的，是一个完全不同的哲学原理。它所说的，并不是因果的事情：即当我们做了一件事情，另外一件事情，就会发生，如此等等；它所说的，是这个：我们设置一种情况，光将会决定，哪条路径，是最短时间的，或是极端的，并选择这个路径。但是，它做了什么呢？它是如何找到的呢？它能闻到附近路径的味道，然后，进行相互比较吗？The answer is, yes, it does, in a way. That is the feature which is,of course, not known in geometrical optics, and which is involved in the ideaof wavelength; the wavelength tells us approximately how far away thelight must “smell” the path in order to check it. It is hard to demonstratethis fact on a large scale with light, because the wavelengths are so terriblyshort. But with radiowaves, say 3 -cm waves, the distances over which theradiowaves are checking are larger. 答案是：是的，它是以某种方式，这样做的。当然，在几何光学中，我们并不知道这个特性，这个特性，牵扯到了波长这一概念；波长告诉我们，光对路径，大约可以“闻”多远，以便检查路径。在一个大尺度上，用光来演证这一事实，比较困难，因为，波长是令人可怕的短。但是，如果用无线电波，比如说3厘米的波，那么，无线电波所检查的路径，就要更大一些。Ifwe have a source of radiowaves, a detector, and a slit, as in Fig. 26–13,the rays of course go from S to D because it is a straightline, and if we close down the slit it is all right—they still go. But now ifwe move the detector aside to D′ , the waves will not go throughthe wide slit from S to D′ , because they check severalpaths nearby, and say, “No, my friend, those all correspond to differenttimes.” On the other hand, if we prevent the radiation from checking thepaths by closing the slit down to a very narrow crack, then there is but onepath available, and the radiation takes it! With a narrow slit, more radiationreaches D′ than reaches it with a wide slit!
如果我们有一个无线电波的源，一个探测器，和一条狭缝，如图26-13所示，源的光线，从S走到 D，因为，这是一条直线，如果我们关闭狭缝，没有问题，光仍会往前走。但现在，如果我们把探测器，移到 D′，光波将不会通过这个宽的狭缝{？}，从S到 D′ ，因为，它们检查到，附近有好几条路，于是说：“不，我的朋友，这些路径，都相应于不同的时间。” 另一方面，如果通过把狭缝，变得更狭，变成一个非常窄的缝隙，以防止辐射去检查路径，那么，将只会剩下一条道路，且辐射会选择它！一个窄的狭缝，比一个宽的狭缝，能让更多的辐射，到达D′！

Fig. 26–13.The passage of radiowavesthrough a narrow slit. 图 26-13 无线电波通过一个狭缝的路程。

Fig. 26–14.The summation of probabilityamplitudes for many neighboring paths. 图26-14 关于很多附近路径的振幅概率的总结。
Now let us show how this implies theprinciple of least time for a mirror. We consider all rays, all possible paths ADB, AEB , ACB , etc., in Fig. 26–3. Thepath ADB makes a certain small contribution, but the next path, AEB, takes a quite different time, so its angle θ is quite different.Let us say that point C corresponds to minimum time, where if we changethe paths the times do not change. So for awhile the times do change, and thenthey begin to change less and less as we get near point C(Fig. 26–14).So the arrows which we have to add are coming almost exactly at the same anglefor awhile near C , and then gradually the time begins to increaseagain, and the phases go around the other way, and so on. 现在，让我们指出，对于一个镜子，这是如何意味着最短时间原理的。我们考虑图26-3中的所有光线，即所有可能的路径ADB , AEB , ACB 等。路径ADB，只做了很小的贡献，但是，下一路径AEB，所取时间，完全不同，所以，它的角度θ，就完全不同。我们说，点C相当于最短时间，在这里，如果我们改变路径，时间不变。于是，有那么一会儿，时间确实在变，然后，当我们越来越接近点C时，时间的变化，就越来越小（图26-14）。于是，在点C附近，有那么一会儿，我们所要增加的箭头，到来时的角度，基本一样，然后，时间又开始增加，相位又开始往另外一个方向走，如此这般。Eventually, we have quite a tight knot. The total probability is thedistance from one end to the other, squared. Almost all of that accumulatedprobability occurs in the region where all the arrows are in the same direction(or in the same phase). All the contributions from the paths which have very differenttimes as we change the path, cancel themselves out by pointing in differentdirections. That is why, if we hide the extreme parts of the mirror, it stillreflects almost exactly the same, because all we did was to take out a piece ofthe diagram inside the spiral ends, and that makes only a very small change inthe light. So this is the relationship between the ultimate picture of photonswith a probability of arrival depending on an accumulation of arrows, and theprinciple of least time.
最终，我们会有一个相当紧的结。总的概率，就是从一端到另一端的距离，取平方。几乎所有积累的概率，都出现在一个区域，在此区域中，所有的箭头，几乎都在同一个方向（或者，是在同一个相位）。当我们改变路径时，有些路径，具有非常不同的时间，所有这些路径的贡献，都被其自己，通过指向不同的方向，而抵消了。这就是为什么，如果我们把镜子的大部分藏起来，它的反光，几乎仍是一样。因为，我们所做的，只把这个螺旋终端图像中的一部分，给去掉了，而这只能让光，发生很小的变化。所以，这就是，光子的带有到达概率的终极图像，与最短时间原理的关系；此到达概率，依赖于箭头的积累。