DESCARTES, RENÉ DU PERRON (b. La Haye, Touraine, France, 31 March 1596; d. Stockholm, Sweden, 11 February 1650), natural philosophy, scientific method, mathematics, optics, mechanics, physiology.

    receive another determination. FIGURE 3 Moreover, although Descartes treated speed as a scalar quantity, determination was (operationally at least) always a vector, which could be resolved into components.18 When one body collided with another, only those components of their determinations that directly opposed one another were subject to alteration.

Imagine, then, says Descartes, a tennis ball leaving the racket at point A and traveling uniformly along line AB to meet the surface CE at B. Resolve its determination into two components, one (AC) perpendicular to the surface and one (AH) parallel to it. Since, when the ball strikes the surface, it imparts none of its motion to the surface (which is immobile), it will continue to move at the same speed and hence after a period of time equal to that required to traverse AB will be somewhere on a circle of radius AB about B. But, since the surface is impenetrable, the ball cannot pass through it (say to D) but must bounce off it, with a resultant change in determination. Only the vertical component of that determination is subject to change, however; the horizontal component remains unaffected. Moreover, since the body has lost none of its motion, the length HF of that component after collision will equal the length AH before. Hence, at the same time the ball reaches the circle it must also be at a distance HF = AH from the normal HB, i.e., somewhere on line FE. Clearly, then, it must be at F, and consideration of similar triangles shows that the angle of incidence ABH is equal to the angle of reflection HBF.

For the law of refraction, Descartes altered the nature of the surface met by the ball; he now imagined it to pass through the surface, but to lose some of its motion (i.e., speed) in doing so. Let the speed before collision be to that after as p:q. FIGURE 4 Since both speeds are uniform, the time required for the ball to reach the circle again will be to that required to traverse AB as p:q. To find the precise point at which it meets the circle, Descartes again considered its determination, or rather the horizontal component unaffected by the collision. Since the ball takes longer to reach the circle, the length of that component after collision will be greater than before, to wit, in the ratio of p:q. Hence, if FH:AH = p:q, then the ball must lie on both the circle and line FE. Let I be the common point.

The derivation so far rests on the assumption that the ball's motion is decreased in breaking through the surface. Here again Descartes had to alter his model to fit his theory of light, for that theory implies that light passes more easily through the denser medium. For the model of the tennis ball, this means that, if the medium below the surface is denser than that above, the ball receives added speed at impact, as if it were struck again by the racket. As a result, it will by the same argument as given above be deflected toward the normal as classical experiments with airwater interfaces said it should.

In either case, the ratio p:q of the speeds before and after impact depended, according to Descartes, on the relative density of the media and would therefore be constant for any two given media. Hence, since

FH:AH = BE:BC = p:q,

it follows that

(BE/BI):(BC/AB) = sin ? AHB: sin ? IBG = p:q = n (constant),

which is the law of refraction.

The vagueness surrounding Descartes's concept of “determination” and its relation to speed makes his derivations difficult to follow. In addition, the assumption in the second that all refraction takes place at the surface lends an ad hoc aura to the proof, which makes it difficult to believe that the derivation represented Descartes's path to the law of refraction (the law of reflection was well known). Shortly after Descartes's death, prominent scientists, including Christian Huygens, accused him of having plagiarized the law itself from Willebrord Snell and then having patched together his proof of it. There is, however, clear evidence that Descartes had the law by 1626, long before Golius uncovered Snell's unpublished memoir.19 In 1626 Descartes had Claude Mydorge grind a hyperbolic lens that represented an anaclastic derived by Descartes from the sine law of refraction. Where Descartes got the law, or how he got it, remains