Reflection and refraction
The discussion above gives enough of the basic idea of                         geometrical optics—now we have to go a little further into the                         quantitative features. Thus far we have light going only in straight                         lines between two points; now let us study the behavior of light when                         it hits various materials. The simplest object is a mirror, and the                         law for a mirror is that when the light hits the mirror, it does not                         continue in a straight line, but bounces off the mirror into a new                         straight line, which changes when we change the inclination of the                         mirror. The question for the ancients was, what is the relation                         between the two angles involved? This is a very simple relation,                         discovered long ago. The light striking a mirror travels in such a way                         that the two angles, between each beam and the mirror, are equal. For                         some reason it is customary to measure the angles from the normal to                         the mirror surface. Thus the so-called law of reflection is                         \begin{equation}                         \label{Eq:I:26:1}                         \theta_i=\theta_r.                         \end{equation}                          That is a simple enough proposition, but a more difficult problem is                         encountered when light goes from one medium into another, for example                         from air into water; here also, we see that it does not go in a                         straight line. In the water the ray is at an inclination to its path                         in the air; if we change the angle $\theta_i$ so that it comes down                         more nearly vertically, then the angle of “breakage” is not as                         great. But if we tilt the 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? This also puzzled the ancients for a long                         time, and here they never found the answer! It is, however, one of the                         few places in all of Greek physics that one may find any experimental                         results listed. Claudius                         Ptolemy                         made a list of the angle in water for each of a number of                         different angles in air. Table 26–1 shows the angles in the                         air, in degrees, and the corresponding angle as measured in the water.                         (Ordinarily it is said that Greek scientists never did any experiments.                         But it would be impossible to obtain this table of values without                         knowing the right law, except by experiment. It should be noted,                         however, that these do not represent independent careful measurements                         for each angle but only some numbers interpolated from a few                         measurements, for they all fit perfectly on a parabola.) This, then, is one of the important steps in the development of                         physical law: first we observe an effect, then we measure it and list                         it in a table; then we try to find the rule by which one thing                         can be connected with another. The above numerical table was made in                         140 a.d., but it was not until 1621 that someone finally                         found the rule connecting the two angles! The rule, found by                         Willebrord Snell, a Dutch                         mathematician, is as follows: if $\theta_i$ is the angle in air and                         $\theta_r$ is the angle in the water, then it turns out that the sine                         of $\theta_i$ is equal to some constant multiple of the sine of $\theta_r$:                         \begin{equation}                         \label{Eq:I:26:2}                         \sin\theta_i=n\sin\theta_r.                         \end{equation}                         For water the number $n$ is approximately $1.33$.                         Equation (26.2) is called Snell’s                             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                         Snell’s law. Note the remarkable agreement with                         Ptolemy’s list.