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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.

DESCARTES: Mathematics and Physics.

Optics.

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