(nI is the speed in air relative to the speed in vacuum, etc. ) then our formula is easy. The index for any two materials i and j is n Using only Snells law, there is no basis for a prediction of this kind. But of course this prediction works. The relation(26.5)was known very early, and was a very strong argument for the principle of least time Another argument for the principle of least time, another prediction, is that if we measure the speed of light in water, it will be lower than in air. This is a prediction of a completely different type. It is a brilliant prediction, because all we have so far measured are angles; here we have a theoretical prediction which is quite different from the observations from which Fermat deduced the idea of least time. It turns out, in fact, that the speed in water is slower than the speed in air, by just the proportion that is needed to get the right index 26-5 A more precise statement of Fermats principle Actually, we must make the statement of the principle of least time a little more accurately. It was not stated correctly above. It is incorrectly called the principle of least time and we have gone along with the incorrect description for convenience but we must now see what the correct statement is. Suppose we had a mirror as in Fig. 26-3. What makes the light think it has to go to the mirror? The path of least time is clearly AB. So some people might say, " Sometimes it is a maximum time. It is not a maximum time, because certainly a curved path would take a still longer time! The correct statement is the following a ray going in a certain particular path has the property that if we make a small change(say a one percent shift) in the ray in any manner whatever, say in the location at which t comes to the mirror, or the shape of the curve, or anything, there will be no first- der change in the time; there will be only a second-order change in the time In other words, the principle is that light takes a path such that there are many other paths nearby which take almost exactly the same time The following is another difficulty with the principle of least time, and one which people who do not like this kind of a 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 the surface. The idea of causality, that it goes from one point to another, and another, and so on, is easy to understand. But the principle of least time is a completely different philosophical principle about the way nature works. Instead of saying it is a causal thing, that when we do one thing, something else happens, and so on, it says this: we set up the situation, and light decides which is the shortest time, or the extreme one, and chooses that path them against each other? The answer is, yes, it does, in a way. That is the feature But what does it do, how does it find out? Does it smell the nearby paths, and check 回 which is, of course, not known in geometrical optics, and which is involved in the idea of wavelength; the wavelength tells us approximately how far away the Hight must"smell "the path in order to check it. It is hard to demonstrate this fact on a large scale with light, because the wavelengths are so terribly short. But with radiowaves, say 3-cm waves, the distances over which the radiowaves are checking are larger. If we have a source of radiowaves, a detector, and a slit, as in Fig. 26-13, Fig. 26-13. The passage of radio. the rays of course go from S to D because it is a straight line, and if we close down waves through a narrow slit the slit it is all right-they still go. But now if we move the detector aside to D the waves will not go through the wide slit from S to ', because they check several paths nearby, and say, No, my friend, those all correspond to different times On the other hand, if we prevent the radiation from checking the paths by closing the slit down to a very narrow crack, then there is but one path available and the Although it can be deduced if the additional assumption is made that adding a layer of one substance to the surface of another does not change the eventual angle of refraction n the latter material