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relation between the angles of incidence and refraction when light passes from medium A to medium B (for example, from air to water): if the angle between the incident ray and the perpendicular to the surface separating the media is i , and the angle between the refracted ray and this perpendicular is r , then the sine * of i divided by the sine of r is an angle-independent constant n :
    sine of i / sine of r = n
    In the common case where medium A is the air (or, strictly speaking, empty space), n is the constant known as the “index of refraction” of medium B . For instance, if A is air and B is water then n is the index of refraction of water, about 1.33. In any case like this, where n is larger than 1, the angle of refraction r is smaller than the angle of incidence i , and the ray of light entering the denser medium is bent toward the direction perpendicular to the surface.
    Unknown to Descartes, this relation had already been obtained empirically in 1621 by the Dane Willebrord Snell and even earlier by the Englishman Thomas Harriot; and a figure in a manuscript by the tenth-century Arab physicist Ibn Sahl suggests that it was also known to him. But Descartes was the first to publish it. Today the relation is usually known as Snell’s law, except in France, where it is more commonly attributed to Descartes.
    Descartes’ derivation of the law of refraction is difficult to follow, in part because neither in his account of the derivation nor in the statement of the result did Descartes make use of the trigonometric concept of the sine of an angle. Instead, he wrote in purely geometric terms, though as we have seen the sine had been introduced from India almost seven centuries earlier by al-Battani, whose work was well known in medieval Europe. Descartes’ derivation is based on an analogy with what Descartes imagined would happen when a tennis ball is hit through a thin fabric; the ball will lose some speed, but the fabric can have no effect on the component of the ball’s velocity along the fabric. This assumption leads (as shown in Technical Note 27) to the result cited above: the ratio of the sines of the angles that the tennis ball makes with the perpendicular to the screen before and after it hits the screen is an angle-independent constant n. Though it is hard to see this result in Descartes’ discussion, he must have understood this result, because with a suitable value for n he gets more or less the right numerical answers in his theory of the rainbow, discussed below.
    There are two things clearly wrong with Descartes’ derivation. Obviously, light is not a tennis ball, and the surface separating air and water or glass is not a thin fabric, so this is an analogy of dubious relevance, especially for Descartes, who thought that light, unlike tennis balls, always travels at infinite speed. 5 In addition, Descartes’ analogy also leads to a wrong value for n. For tennis balls (as shown in Technical Note 27) his assumption implies that n equals the ratio of the speed of the ball v B in medium B after it passes through the screen to its speed v A in medium A before it hits the screen. Of course, the ball would beslowed by passing through the screen, so v B would be less than v A and their ratio n would be less than 1. If this applied to light, it would mean that the angle between the refracted ray and the perpendicular to the surface would be greater than the angle between the incident ray and this perpendicular. Descartes knew this, and even supplied a diagram showing the path of the tennis ball being bent away from the perpendicular. Descartes also knew that this is wrong for light, for as had been observed at least since the time of Ptolemy, a ray of light entering water from the air is bent toward the perpendicular to the water’s surface, so that the sine of i is greater than the sine of r , and hence n is greater than 1. In a thoroughly muddled discussion that I cannot understand, Descartes somehow argues