Snell, a Dutch mathematician, is as follows: if i is the angle in air and 0, is the angle in the water, then it turns out that the sine of i is equal to some constan Table 26-2 multiple of the sine of 8, sin ]i= n sin 8, (26.2) A air Angle in water For water the number n is approximately 1.33. Equation (26.2) is called Snells 7-1/2° 20° law; it permits us to predict how the light is going to bend when it goes from air into water. Table 26-2 shows the angles in air and in water according to Snells 29 law. Note the remarkable agreement with Ptolemy's list 40-1/2° 26-3 Fermats principle of least time Now in the further development of science, we want more than just a formul first we have an observation, then we have numbers 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 thinking such that the law is evident The first way of thinking that made the law about the behavior of light evident ras discovered by Fermat in about 1650, and it 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 from one point to another, light takes the path which requires the shortest time Let us first show that this is true for the case of the mirror, that this simple principle contains both the law of straight-line propagation and the law for the mirror. So, we are growing in our understanding! Let us try to find the solution to the following problem. In Fig. 26-3 are shown two points, A and B, and a plane mirror, MM. What is the way to get from a to B in the shortest time? The answer is to go straight from A to B! But if we add the extra rule that the light has to strike the mirror and come back in the shortest time the answer is not so easy. One way would be to go as quickly as possible to the mirror and then B, on the path ADB. Of course, we then have a long path DB. If we move over a little to the right, to E, we slightly increase the first distance, but we greatly decrease the second one, and so the total path length, and therefore the travel time, is less How can we find the point C for which the time is the shortest? We can find it Fig.26-3.Illustration very nicely by a geometrical trick of least time We construct on the other side of MM' an artificial point B, which is the same distance below the plane MMas the point B is above the plane. Then we draw the line EB. Now because BFM is a right angle and BF= FB, EB is equal to EB. Therefore the sum of the two distances, AE EB, which is propor- tional to the time it will take if the light travels with constant velocity, is also the sum of the two lengths AE+ EB. Therefore the problem becomes, when is the sum of these two lengths the least? The answer is easy: when the line goes through point C as a straight line from A to B! In other words, we have to find the point yhere we go toward the artificial point, and that will be the correct one. Now if ACB is a straight line, then angle BCF is equal to angle B CF and thence to angle ACM. Thus the statement that the angle of incidence equals the angle of reflection is equivalent to the statement that the light goes to the mirror in such a way that it comes back to the point b'in the least possible time. Originally, the statement made by Hero of Alexandria that the light travels in such that it he mirror and to the other point in the shortest possible distance, so it is not a dern theory. It was this that inspired Fermat to suggest to himself that perhaps refraction operated on a similar basis. But for refraction, light obviously does not use the path of shortest distance, so fermat tried the idea that it takes the shortest Before we go on to analyze refraction, we should make one more remark about the mirror. If we have a source of light at the point b and it sends light to- vard the mirror, then we see that the light which goes to A from the point b comes to A in exactly the same manner as it would have come to A if there were an object at B, and no mirror. Now of course the eye detects only the light which enters it physically, so if we have an object at B and a mirror which makes the light com