To Explain the World: The Discovery of Modern Science by Steven Weinberg Read Free Book Online Page A

that light travels more easily in water than in air, so that for light n is greater than 1. For Descartes’ purposes his failure to explain the value of n didn’t really matter, because he could and did take the value of n from experiment (perhaps from the data in Ptolemy’s Optics ), which of course gives n greater than 1.
    A more convincing derivation of the law of refraction was given by the mathematician Pierre de Fermat (1601–1665), along the lines of the derivation by Hero of Alexandria of the equal-angles rule governing reflection, but now making the assumption that light rays take the path of least time , rather than of least distance. This assumption (as shown in Technical Note 28) leads to the correct formula, that n is the ratio of the speed of light in medium A to its speed in medium B , and is therefore greater than 1 when A is air and B is glass or water. Descartes could never have derived this formula for n , because for him light traveled instantaneously. (As we will see in Chapter 14 , yet another derivation of the correct result was given by Christiaan Huygens, a derivation based on Huygens’ theory of light as a traveling disturbance, which did not rely on Fermat’s a priori assumption that the light ray travels the path of least time.)
    Descartes made a brilliant application of the law of refraction: in his Meteorology he used his relation between angles of incidence and refraction to explain the rainbow. This was Descartesat his best as a scientist. Aristotle had argued that the colors of the rainbow are produced when light is reflected by small particles of water suspended in the air. 6 Also, as we have seen in Chapters 9 and 10 , in the Middle Ages both al-Farisi and Dietrich of Freiburg had recognized that rainbows are due to the refraction of rays of light when they enter and leave drops of rain suspended in the air. But no one before Descartes had presented a detailed quantitative description of how this works.
    Descartes first performed an experiment, using a thin-walled spherical glass globe filled with water as a model of a raindrop. He observed that when rays of sunlight were allowed to enter the globe along various directions, the light that emerged at an angle of about 42° to the incident direction was “completely red, and incomparably more brilliant than the rest.” He concluded that a rainbow (or at least its red edge) traces the arc in the sky for which the angle between the line of sight to the rainbow and the direction from the rainbow to the sun is about 42°. Descartes assumed that the light rays are bent by refraction when entering a drop, are reflected from the back surface of the drop, and then are bent again by refraction when emerging from the drop back into the air. But what explains this property of raindrops, of preferentially sending light back at an angle of about 42° to the incident direction?
    To answer this, Descartes considered rays of light that enter a spherical drop of water along 10 different parallel lines. He labeled these rays by what is today called their impact parameter b , the closest distance to the center of the drop that the ray would reach if it went straight through the drop without being refracted. The first ray was chosen so that if not refracted it would pass the center of the drop at a distance equal to 10 percent of the drop’s radius R (that is, with b = 0.1 R ), while the tenth ray was chosen to graze the drop’s surface (so that b = R ), and the intermediate rays were taken to be equally spaced between these two. Descartes worked out the path of each ray as it was refracted entering the drop, reflected by the back surface of the drop, and then refracted again as it left the drop, using the equal-angleslaw of reflection of Euclid and Hero, and his own law of refraction, and taking the index of refraction n of water to be 4 / 3 . The following table gives values found by Descartes for the angle φ (phi) between the emerging ray and its