Lenses and Waves

130 CHAPTER 4
Therefore the distance between lines FE and HB must be twice a large as that
between AC and HB. As a result, the ball reaches point I on the circle. The
same is the case when instead of a cloth the ball hits the surface of a body of
water. For the water does not alter the motion of the ball any further after it
has passed the surface, according to Descartes. When the ball passes a
boundary where in some way or another its quantity of motion is augmented,
it reaches the circumference of the circle earlier and is deflected towards the
normal of the surface. Note that Descartes did not specify the change of the
perpendicular component, a point that is often overlooked. He did not know
that amount and he did not need to, for the two assumptions he used suffice
for the derivation of the sine law. 81
As Descartes took the motions of the ball to reflect the deflections of
light, he could now draw his main conclusion. Rays of light are deflected in
exact proportion to the ease with which a transparent medium receives them
compared to the medium from which they come. The only remaining
difference between the motion of a ball and the action of light is that a
denser medium like water allows rays of light to pass more easily. The
deflection caused by the passage from one medium into another ought to be
measured, not by the angles made with the refracting surface, but by the lines
CB and BE. Unlike the proportion between the angles of incidence and
refraction, the proportion between these sines remains the same for any
refraction caused by a pair of mediums, irrespective of the angle of
incidence. Et voilà, the law of sines.
Epistemic aspects of Descartes’ account in historical context
Both historically and intrinsically, Descartes’ account of refraction is a key
text in the transition from medieval perspectiva to seventeenth-century
optics. Yet, the line of inference is subtle and, at many points, implicitly
pursued. I will have to enlarge in some detail on its epistemic aspects in their
historical context.
At least three levels of inference can be distinguished in Descartes’
account. In the first place the level of mathematics. This holds the derivation
of the sine law from the two assumptions conveyed in the diagrams
accompanying his discourse. First, the passage to another medium alters in a
fixed ratio the quantity of motion. This ratio is represented by the radius of
the circle. Second, the parallel component of the direction of motion is
unaffected. This is represented by drawing horizontal lines in proportion to
the successive times to travel to and from the center of the circle. The
mathematical inference of Descartes’ account constituted a successful
culmination of perspectivist optics, in that Descartes was the first to derive a
law of refraction on the analytical groundwork laid by Kepler and his
forebears. He brought consistency to the analysis of reflection and refraction
by having the parallel component constant in all cases. More important, in
81 When both aspects of the motion are interpreted as speeds the assumptions can be written as: vr = nvi
and vi sini = vr sinr, which directly yield sini = n sinr. See Sabra, Theories of Light, 111.