To broaden our knowledge of the world, we use different methods that rarely lead us to the same truths. Nevertheless, it would be remarkable if certain fundamental truths of Philosophy could be illustrated by geometric arguments or algebraic proofs.
One such remarkable example can be found in one of the most important subjects of physics.
The most beautiful discoveries since the Renaissance, indeed since the beginnings of all science, are the laws governing light, whether moving through a uniform medium, or being reflected from an opaque surface, or changing direction upon entering another transparent medium. These laws are fundamental to the science of optics and colors.
I will not presume to treat the whole of such a vast subject, but will limit myself to a few well-known truths. As I have said, these laws are the basis of an admirable science, one that allows an old man with weakened eyes to see as well as he did in his youth, or even better; a science that extends our vision into the furthest reaches of space and into the smallest parts of matter, allowing us to discover many things that had been concealed.
The ancient Greeks knew the laws that govern the propagation of light in a uniform medium and upon its reflection. However, the law governing the refraction of light as it passes from one transparent medium to another was unknown until the last century. Snell discovered it, Descartes tried to explain it and Fermat criticized his explanation. Since then, many great geometers have researched the problem, although no one has yet found a way of harmonizing the law of refraction with more fundamental laws that Nature must obey.
Here are the laws that govern light.
The first law states that light moves in a straight line in a uniform medium.
The second law states that when light encounters a medium it cannot penetrate, it is reflected and the angle of reflection equals the angle of incidence. In other words, upon reflection, light makes an angle with the surface that is equal to the angle at which the light encountered the surface.
The third law states that when light passes from one medium to another, it is bent and the sine of the angle of refraction always has the same ratio to the sine of the angle of incidence. For example, if a ray of light passes from air to water, it is bent so that the sine of the angle of refraction is three-quarters of the sine of the angle of incidence, regardless of what the angle of incidence is.
The first law is the same for both light and material bodies; they both move in a straight line, as long as they are not deflected by an outside force.
The second law is also the same as that governing the reflection of an elastic ball from an impenetrable surface. Mechanics shows that such a ball is reflected from such a surface so that its angle of reflection equals its angle of incidence, as observed for light.
But the third law still requires a plausible explanation. The passage of light from one medium to another exhibits behavior that is totally different from a ball moving through different media. Every explanation of refraction has some problems that have not yet been overcome.
I will not cite all the great men who have worked on this problem. Their names would form a long list, a useless ornament in this article, and a review of their various systems would be an immense undertaking. However, I will group their explanations of the reflection and refraction of light into three classes.
The first class consists of explanations that seek to derive the behavior of light from purely mechanical laws, i.e., the basic laws governing the motion of material objects.
The second class consists of explanations that augment the mechanical laws with an attraction between light and matter, or something that produces an equivalent effect.
Finally, the third class consists of explanations derived from purely metaphysical principles, i.e., from laws to which Nature herself seems subjugated by a superior Intelligence that always produces an effect in the simplest possible manner.
Descartes and those who followed him belong to the first class. They modeled the reflection of light as the motion of a ball that, encountering an impenetrable surface, rebounds to the same side from which it came. Similarly, they modeled the refraction of light as a ball that, encountering a penetrable surface, continues to progress, albeit with a changed direction. Although the arguments used by this great Philosopher to explain refraction are imperfect, Descartes still deserves credit for trying to derive the laws of light from the simplest mechanical laws.
Several mathematicians have noted gaps in Descartes' logic and have sought an improved explanation.
Newton despaired of deriving the law of refraction from those governing the motion of a ball when it passes between two media of different resistance. Therefore, he proposed an attraction between light and matter that increases proportionally to the amount of matter present, which was able to account for refraction exactly and rigorously. Mr. Clairaut has written an excellent article on this topic. He lays out the failings of the Cartesian theory, and subscribes to the attraction theory between light and matter, although he proposes that an "atmosphere" surrounding the matter causes the apparent force of attraction. He derives the law of refraction with the clarity that is typical of all the subjects he has researched.
Fermat was the first to recognize the failings of Descartes' explanation, and also seemed to despair of explaining the refraction law from purely mechanical laws governing balls encountering obstacles or passing through a resistant medium. However, he also did not resort to "atmospheres" surrounding material bodies or other forms of attraction between light and matter, although he was certainly aware of the attraction theory and found it tolerable. Rather, Fermat sought to explain refraction using a totally different and purely metaphysical principle.
Everyone knows that, when light or another body travels from one point to another in a straight line, it follows the path of shortest distance and time.
One also knows (or at least it is easy to show) that, when light is reflected, it likewise travels along the path of shortest distance and briefest time. The equality of the angles of incidence and reflection result from requiring a body to travel from one point to another along the path of shortest distance and briefest time that involves a reflection from a given plane. For if the angles are equal, the sum of the two paths by which the ball travel and return is shorter in distance and briefer in time than any other sum of paths making unequal angles.
Hence, both the direct and reflected motion of light seem to depend on a metaphysical law stating that Nature always acts in the simplest possible manner to produce its effects. Whether a material body should travel from one point to another without encountering an obstacle, or encountering an impenetrable obstacle, Nature always leads it along the path of shortest distance and briefest time.
To apply this principle to refraction, let us consider two transparent media separated by the plane of their common surface. Suppose that the light's point of departure is within one medium, whereas its point of arrival is in the other medium, but that the line joing the two points is not perpendicular to the surface separating the media. Let us also assume that the light travels with different speeds in the different media. Hence, the straight line joining the two points is still the path of shortest distance, but is no longer the path of briefest time. Since the travel time depends on the speeds of light in the two media, the path of briefest time should be longer in length in the medium where the light moves more quickly, and shorter in the medium where light moves more slowly.
This seems to occur when light passes from air into water. The ray is bent such that a longer path is taken in air and a shorter path in water. If we assume, as seems reasonable, that light moves more quickly in air than it does in water, then the bent path taken by the light causes it to move from its departure point to its arrival point in the briefest possible time.
This path-of-least-time principle of Fermat seems reasonable, and accounts for the refraction of light as well as its direct propagation and reflection. Fermat himself had no difficulty in believing that light traveled more easily and more quickly in less dense media than in media of higher density. At first glance, who would assume that light moves more easily and more quickly in glass and water than it does in air or vacuum?
Several celebrated mathematicians also agreed with Fermat's principle, particularly Leibniz, who gave the problem an elegant mathematical analysis. He was charmed by the metaphysical principle as an example of the final causes to which he was so attached, and considered it beyond doubt that light moves more quickly in air than in water or glass.
Nevertheless, Descartes believed exactly the opposite, that light moves more quickly in denser media and, although his reasoning was perhaps inadequate, that failing does not stem from his assumption about the speed of light. Every theory of refraction makes some assumption about the relative speed of light in different media, some agreeing with Descartes and some agreeing with Fermat and Leibniz.
If it is assumed that light moves more quickly in denser media, the entire theory of Fermat and Leibniz is destroyed. For, in that case, the bent path of light upon refraction would correspond neither to the path of shortest distance nor to the path of briefest time. A path that traveled longer in a medium of slower speed would definitely not arrive in the shortest possible time. The article of Mr. de Mairan on reflection and refraction describes the history of the dispute between Fermat and Descartes, as well as the confusion and inability to harmonize the law of refraction with the metaphysical principle.
After meditating deeply on this topic, it occurred to me that light, upon passing from one medium to another, has to make a choice, whether to follow the path of shortest distance (the straight line) or the path of least time. But why should it prefer time over space? Light cannot travel both paths at once, yet how does it decide to take one path over another? Rather than taking either of these paths per se, light takes the path that offers a real advantage: light takes the path that minimizes its action.
Now I have to define what I mean by "action". When a material body is transported from one point to another, it involves an action that depends on the speed of the body and on the distance it travels. However, the action is neither the speed nor the distance taken separately; rather, it is proportional to the sum of the distances travelled multiplied each by the speed at which they were travelled. Hence, the action increases linearly with the speed of the body and with the distance travelled.
This action is the true expense of Nature, which she manages to make as small as possible in the motion of light.
Let there be two media, separated by a common surface (represented by the line CD), such that the speed of light in the upper medium is V and in the lower medium is W. Let there be a ray of light AR that leaves from a given point A and arrives at a given point B.
To find the point R at which the light is bent, I seek a point that minimizes the action, i.e., should be minimized or, equivalently,
Since , and are constants, minimization yields the equation
In other words, the ratio of the sine of the angle of incidence to the sine of the angle of refraction equals the inverse ratio of the speeds at which light moves in each medium.
Thus, the refraction of light agrees with the grand principle that Nature always uses the simplest means to accomplish its effects. From this principle, can be derived whenever light passes from one medium to another, the ratio of the sine of the angle of refraction to the sine of the angle of refraction equals the inverse ratio of the speeds at which light moves in each medium.
But this "budget", this expense of action that Nature minimizes in the refraction of light, is it also minimized in the direct propagation and reflection of light? Yes, it always has the smallest possible value.
In both cases (direct propagation and reflection), the speed of light remains constant. Hence, the path of least action is the same as the path of shortest distance and the path of briefest time. However, those latter two paths are merely a consequence of the path of least action, a consequence that Fermat and Leibniz took as the fundamental principle.
Having discovered the true principle, I then derived all the laws that govern the motion of light, those concerning its direct propagation, its reflection and its refraction. I reserve for particular members of our Assembly the geometrical demonstration of my theory.
I know the distaste that many mathematicians have for final causes applied to physics, a distaste that I share up to some point. I admit, it is risky to introduce such elements; their use is dangerous, as shown by the errors made by Fermat and Leibniz in following them. Nevertheless, it is perhaps not the principle that is dangerous, but rather the hastiness in taking as a basic principle that which is merely a consequence of a basic principle.
One cannot doubt that everything is governed by a supreme Being who has imposed forces on material objects, forces that show his power, just as he has fated those objects to execute actions that demonstrate his wisdom. The harmony between these two attributes is so perfect, that undoubtedly all the effects of Nature could be derived from each one taken separately. A blind and deterministic mechanics follows the plans of a perfectly clear and free Intellect. If our spirits were sufficiently vast, we would also see the causes of all physical effects, either by studying the properties of material bodies or by studying what would most suitable for them to do.
The first type of studies is more within our power, but does not take us far. The second type may lead us stray, since we do not know enough of the goals of Nature and we can be mistaken about the quantity that is truly the expense of Nature in producing its effects.
To unify the certainty of our research with its breadth, it is necessary to use both types of study. Let us calculate the motion of bodies, but also consult the plans of the Intelligence that makes them move.
It seems that the ancient philosophers made the first attempts at this sort of science, in looking for metaphysical relationships between numbers and material bodies. When they said that God occupies himself with geometry, they surely meant that He unites in that science the works of His power with the perspectives of His wisdom.
From the all too few ancient geometers who undertook such studies, we have little that is intelligible or well-founded. The perfection which geometry has acquired since their time puts us in a better position to succeed, and may more than compensate for the advantages that those great minds had over us.