To Explain the World: The Discovery of Modern Science (30 page)

Using analytic geometry, we can find the coordinates of the point where two curves intersect, or the equation for the curve where two surfaces intersect, by solving the pair of equations that define the curves or the surfaces. Most physicists today solve geometric problems in this way, using analytic geometry, rather than the classic methods of Euclid.

In physics Descartes’ significant contributions were in the study of light. First, in his
Optics
, Descartes presented the 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 be
slowed 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 that light travels more easily in water than in air, so that for light
n
is greater than 1. For Descartes’ purposes his failure to explain the value of
n
didn’t really matter, because he could and did take the value of
n
from experiment (perhaps from the data in Ptolemy’s
Optics
), which of course gives
n
greater than 1.

A more convincing derivation of the law of refraction was given by the mathematician Pierre de Fermat (1601–1665), along the lines of the derivation by Hero of Alexandria of the equal-angles rule governing reflection, but now making the assumption that light rays take the path of least
time
, rather than of least distance. This assumption (as shown in Technical Note 28) leads to the correct formula, that
n
is the ratio of the speed of light in medium
A
to its speed in medium
B
, and is therefore greater than 1 when
A
is air and
B
is glass or water. Descartes could never have derived this formula for
n
, because for him light traveled instantaneously. (As we will see in
Chapter 14
, yet another derivation of the correct result was given by Christiaan Huygens, a derivation based on Huygens’ theory of light as a traveling disturbance, which did not rely on Fermat’s a priori assumption that the light ray travels the path of least time.)

Descartes made a brilliant application of the law of refraction: in his
Meteorology
he used his relation between angles of incidence and refraction to explain the rainbow. This was Descartes
at his best as a scientist. Aristotle had argued that the colors of the rainbow are produced when light is reflected by small particles of water suspended in the air.
⁶
Also, as we have seen in
Chapters 9
and
10
, in the Middle Ages both al-Farisi and Dietrich of Freiburg had recognized that rainbows are due to the refraction of rays of light when they enter and leave drops of rain suspended in the air. But no one before Descartes had presented a detailed quantitative description of how this works.

Descartes first performed an experiment, using a thin-walled spherical glass globe filled with water as a model of a raindrop. He observed that when rays of sunlight were allowed to enter the globe along various directions, the light that emerged at an angle of about 42° to the incident direction was “completely red, and incomparably more brilliant than the rest.” He concluded that a rainbow (or at least its red edge) traces the arc in the sky for which the angle between the line of sight to the rainbow and the direction from the rainbow to the sun is about 42°. Descartes assumed that the light rays are bent by refraction when entering a drop, are reflected from the back surface of the drop, and then are bent again by refraction when emerging from the drop back into the air. But what explains this property of raindrops, of preferentially sending light back at an angle of about 42° to the incident direction?

To answer this, Descartes considered rays of light that enter a spherical drop of water along 10 different parallel lines. He labeled these rays by what is today called their impact parameter
b
, the closest distance to the center of the drop that the ray would reach if it went straight through the drop without being refracted. The first ray was chosen so that if not refracted it would pass the center of the drop at a distance equal to 10 percent of the drop’s radius
R
(that is, with
b
= 0.1
R
), while the tenth ray was chosen to graze the drop’s surface (so that
b
=
R
), and the intermediate rays were taken to be equally spaced between these two. Descartes worked out the path of each ray as it was refracted entering the drop, reflected by the back surface of the drop, and then refracted again as it left the drop, using the equal-angles
law of reflection of Euclid and Hero, and his own law of refraction, and taking the index of refraction
n
of water to be
⁴
/
₃
. The following table gives values found by Descartes for the angle φ (phi) between the emerging ray and its incident direction for each ray, along with the results of my own calculation using the same index of refraction:

The inaccuracy of some of Descartes’ results can be set down to the limited mathematical aids available in his time. I don’t know if he had access to a table of sines, and he certainly had nothing like a modern pocket calculator. Still, Descartes would have shown better judgment if he had quoted results only to the nearest 10 minutes of arc, rather than to the nearest minute.

As Descartes noticed, there is a relatively wide range of values of the impact parameter
b
for which the angle φ is close to 40°. He then repeated the calculation for 18 more closely spaced rays with values of
b
between 80 percent and 100 percent of the drop’s radius, where φ is around 40°. He found that the angle φ for 14 of these 18 rays was between 40° and a maximum of 41° 30'. So these theoretical calculations explained his experimental observation mentioned earlier, of a preferred angle of roughly 42°.

Technical Note 29 gives a modern version of Descartes’
calculation. Instead of working out the numerical value of the angle φ between the incoming and outgoing rays for each ray in an ensemble of rays, as Descartes did, we derive a simple formula that gives φ for any ray, with any impact parameter
b
, and for any value of the ratio
n
of the speed of light in air to the speed of light in water. This formula is then used to find the value of φ where the emerging rays are concentrated.
*
For
n
equal to
⁴
/
₃
the favored value of φ, where the emerging light is somewhat concentrated, turns out to be 42.0°, just as found by Descartes. Descartes even calculated the corresponding angle for the secondary rainbow, produced by light that is reflected twice within a raindrop before it emerges.

Descartes saw a connection between the separation of colors that is characteristic of the rainbow and the colors exhibited by refraction of light in a prism, but he was unable to deal with either quantitatively, because he did not know that the white light of the sun is composed of light of all colors, or that the index of refraction of light depends slightly on its color. In fact, while Descartes had taken the index for water to be
⁴
/
₃
= 1.333 . . . , it is actually closer to 1.330 for typical wavelengths of red light and closer to 1.343 for blue light. One finds (using the general formula derived in Technical Note 29) that the maximum value for the angle φ between the incident and emerging rays is 42.8° for red light and 40.7° for blue light. This is why Descartes saw bright red light when he looked at his globe of water at an angle of 42° to the direction of the Sun’s rays. That value of the angle φ is above the maximum value 40.7° of the angle that can emerge
from the globe of water for blue light, so no light from the blue end of the spectrum could reach Descartes; but it is just below the maximum value 42.8° of φ for red light, so (as explained in the previous
footnote
) this would make the red light particularly bright.

The work of Descartes on optics was very much in the mode of modern physics. Descartes made a wild guess that light crossing the boundary between two media behaves like a tennis ball penetrating a thin screen, and used it to derive a relation between the angles of incidence and refraction that (with a suitable choice of the index of refraction
n
) agreed with observation. Next, using a globe filled with water as a model of a raindrop, he made observations that suggested a possible origin of rainbows, and he then showed mathematically that these observations followed from his theory of refraction. He didn’t understand the colors of the rainbow, so he sidestepped the issue, and published what he did understand. This is just about what a physicist would do today, but aside from its application of mathematics to physics, what does it have to do with Descartes’
Discourse on Method
? I can’t see any sign that he was following his own prescriptions for “Rightly Conducting One’s Reason and of Seeking Truth in the Sciences.”