Lenses and Waves

THE 'PROJET' OF 1672 129
apply to Cartesian motion proper as well as to tendency to movement. When
a ball rebounds from the surface of an impenetrable body the following
happens. The quantity of its motion is unaffected because it remains moving
through the same medium - the air surrounding the body - and only the
direction changes. Regarding the parallel and perpendicular components of
the direction, Descartes noted that the body offers resistance only in the
direction perpendicular to its surface. Thus the parallel component is
unaltered.
To determine the path of the
ball after the impact, Descartes
switched to a derivation in which
he graphically mathematized the
assumptions just established
(Figure 41). In circle AFD radius
AB represents the path along
which the ball approaches the
surface where it rebounds from B
in some direction. As the quantity
of motion is constant, the ball
must traverse the same distance
Figure 41 Descartes’ analysis of reflection
after reflection. It thus reaches the circumference of the circle somewhere.
Since the parallel component of its direction is also constant, it follows that
the horizontal distance traversed after reflection must be equal too.
Therefore, BE is equal to BC. Under these conditions the ball can either
arrive at point D or point F on the circle. It cannot penetrate the body below
GE and so F is the only option left. “And thus you will easily see how
reflection occurs, namely according to an angle always equal to the one that
is called angle of incidence”, Descartes concluded without much further
ado. 80
Like reflection, refraction is understood as the combined effect on the
quantity and the direction of motion. The only difference is that in refraction
the ball penetrates the medium. In other words, it enters a medium of
different density. Therefore the quantity of motion changes. It does so at the
passing of the surface separating both mediums. This can be compared to
smashing a ball through a thin cloth. It loses part of its speed, say half. Again
only the perpendicular component of the direction of the motion is affected
and the parallel component remains unaltered. As in the case of reflection,
Descartes switched to a mathematical derivation in the form of a diagram to
determine the exact path of the ball after impact (Figure 40). As a result of
the loss of speed, it takes the ball twice as long to reach the circumference of
the circle after impact at B. However, as its determination to advance parallel
to the surface is unchanged, it moves twice as far to the right in this time.
80 “Et ainsy vous voyés facilement comment se fait la reflexion, a sçavoir selon un angle tousiours esgal a
celuy qu’on nomme l’angle d’incidence.” AT6, 96.