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。

The most advanced and abstract theory ofgeometrical optics was worked out by Hamilton, and it turns out that this hasvery important applications in mechanics. It is actually even more important inmechanics than it is in optics, and so we leave Hamilton’s theory for thesubject of advanced analytical mechanics, which is studied in the senior yearor in graduate school. So, appreciating that geometrical optics contributesvery little, except for its own sake, we now go on to discuss the elementaryproperties of simple optical systems on the basis of the principles outlined inthe last chapter.
几何光学的最高级的和最抽象的理论，是由汉密尔顿得出的，结果就是，这个理论，在力学中，也有着重要的应用。实际上，它在力学中，比在光学中更重要，所以，我们将把汉密尔顿的原理，放在高级分析力学的主题中，在打四、或研究生院学习。于是，几何光学除了对光学本身外，{对其他学科}贡献很少，在搞明白这点之后，下面，我们将继续讨论简单光学系统的本质属性，此系统，基于上一章所列出的原理。

In order to go on, we must have onegeometrical formula, which is the following: if we have a triangle with a smallaltitude h and a long base d , then the diagonal s(we are going to need it to find the difference in time between two differentroutes) is longer than the base (Fig. 27–1). Howmuch longer? The difference Δ=s−d can be found in a numberof ways. One way is this. We see that s2−d2=h2, or (s−d)(s+d)=h2 . But s−d=Δ, and s+d≈2s . Thus
为了继续，我们应有一个几何公式，它是这样的示：如果我们有一个三角形，它有一个小的高度h，和一个长的底 d，那么，斜边s（我们要用它，来找出两个不同路径上时间的差别），就要比底长（图27-1）。能长多少呢？差Δ=s−d，可以用若干方法得到。一种方法就是这个。我们看到s2−d2=h2 , 或 (s−d)(s+d)=h2 。但s−d=Δ , 且s+d≈2s 。这样：
Δ≈h2/2s. (27.1)
This is all the geometry we need to discussthe formation of images by curved surfaces!
通过弯曲界面，可以生成图像，讨论这种生成所需要的全部几何学，就是这些。

Figure 27–1 图27-1

27–2The focal length of a sphericalsurface 27-2 球形表面的焦距
The first and simplest situation to discussis a single refracting surface, separating two media with different indices ofrefraction (Fig. 27–2). Weleave the case of arbitrary indices of refraction to the student, because ideasare always the most important thing, not the specific situation, and theproblem is easy enough to do in any case. So we shall suppose that, on theleft, the speed is 1 and on the right it is 1/n , where nis the index of refraction. The light travels more slowly in the glass by afactor n .
首先且最简单的情况，就是讨论一个单独的折射界面，两种介质，由它分开，每种介质，折射率都不同（图27-2）。我们把任意折射率这种情况，留给学生，因为，最重要的事情，总是想法，而不是具体的情况，此问题，在任何情况下去做，都足够容易。所以，我们将假设，左边，速度为1，右边，为1/n，这里n就是折射率。光在玻璃中，旅行地更慢，通过一个因子n{来计算能慢多少}。

Fig. 27–2.Focusing by a single refractingsurface. 图27-2 通过一个单独的折射界面的聚焦。

Now suppose that we have a point at O , ata distance s from the front surface of the glass, and anotherpoint O′ at a distance s′ inside the glass, and wedesire to arrange the curved surface in such a manner that every ray from Owhich hits the surface, at any point P , will be bent so as toproceed toward the point O′ . 现在，假设我们有一点O，距玻璃表面为s，另有一点O′，在玻璃里面，距离玻璃表面为s′，我们想把曲面，如此安排，即让从O发出的光，无论打到玻璃表面上的哪一点P，最后，都会被折弯，以前进到点O′。

For that to be true, we have to shape the surface insuch a way that the time it takes for the light to go from O to P, that is, the distance OP divided by the speed of light (the speedhere is unity), plus n⋅O′P , which is the time it takes to go fromP to O′ , is equal to a constant independent of thepoint P . This condition supplies us with an equation fordetermining the surface. 要做成此事，表面的形状，必须如此：让光从O到P所花时间，也就是说，距离OP除以光速（这里，速度是同一的），加上，从P到 O′所花的时间 n⋅O′P，是一个常数，独立于点P的。这个条件，给我们提供了规定界面的公式。

这里建议换个地方更…这个吧已经被民科占领了

Itis so much easier to fabricate a sphere than other surfaces that it isprofitable for us to find out what happens to rays striking a sphericalsurface, supposing that only the rays near the axis are going to be focusedperfectly. Those rays which are near the axis are sometimes called paraxialrays, and what we are analyzing are the conditions for the focusing ofparaxial rays. We shall discuss later the errors that are introduced by thefact that all rays are not always close to the axis.
制造一个球面，要比制造一个其他的界面，容易地多，对于我们来说，当光线打到球形界面时，要找出究竟发生了什么，比较有利；这里假设只有接近于轴的光线，才会完美聚焦。那些接近轴的光线，有时被称为近轴光线，近轴光线要聚焦，需要一些条件，我们正在分析的，就是这些条件。并不是所有的光线，都是接近于轴的，这个事实，会带来一些错误，稍后，我们将讨论这些错误。

Now it turns out, interestingly, that thesame lens, with the same curvature R , will focus for otherdistances, namely, for any pair of distances such that the sum of the tworeciprocals, one multiplied by n , is a constant. Thus a given lenswill (so long as we limit ourselves to paraxial rays) focus not only from Oto O′ , but between an infinite number of other pairs of points, solong as those pairs of points bear the relationship that 1/s+n/s′is a constant, characteristic of the lens.
现在，很有趣的结果就是，同样的透镜，具有同样的曲率R，将会为其他的距离聚焦，也就是说，对于任何一对距离，只要一个的倒数，加上另一个的倒数乘以n，是一个常数。这样，给定透镜（只要我们把我们限制在近轴光线），它将不仅会聚焦O到O′，而且，还可以聚焦无数其他的点对，只要这些点对，满足关系：1/s+n/s′是一个常数，即透镜的特性。

In particular, an interesting case is thatin which s→∞ . We can see from the formula that as one sincreases, the other decreases. In other words, if point O goesout, point O′ comes in, and vice versa. As point O goestoward infinity, point O′ keeps moving in until it reaches acertain distance, called the focal length f′ , inside thematerial. If parallel rays come in, they will meet the axis at a distance f′. Likewise, we could imagine it the other way. (Remember the reciprocity rule:if light will go from O to O′ , of course it will also gofrom O′ to O .) Therefore, if we had a light source insidethe glass, we might want to know where the focus is. In particular, if thelight in the glass were at infinity (same problem) where would it come to afocus outside? This distance is called f . Of course, we can alsoput it the other way. If we had a light source at f and the lightwent through the surface, then it would go out as a parallel beam. We caneasily find out what f and f′ are:
尤其当s→∞，这是一个有趣的情况。从公式我们可以看出，当一个s增加时，另一个就会减少。换句话说，如果点O往外走，那么，点O′就往里走，反之亦然。随着点O趋向于无穷大，点O′就继续往里走，直至到达某确定的距离，这被称为焦距f′，它在材料里面。如果平行光线进来，那么，它们将会在f′处，与轴相会。同样，我们可以想象相反的情况。（记住，反过来的规则：如果光会从O走到 O′，那么当然，它也会从从O′走到 O。）因此，如果在玻璃里面，我们有一个光源，我们也想知道，焦点在何处。尤其是，如果玻璃里的光，在无穷远（同样的问题），那么，它在外面的聚焦处，又会在什么地方呢？这个距离，被称为f。当然，我们也可以反过来想。如果我们在f，有一个光源，那么，光通过界面后，将会变成平行光束。我们可以很容易找到f和 f′:
n/f′ = (n−1)/R or f′ = Rn/(n−1), (27.4)
1/f = (n−1)/R or f= R/(n−1). (27.5)