Accord between different laws of Nature that seemed incompatible

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.

What is “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
• 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.

The speed of light:

• in the upper medium is V
• in the lower medium is W

A ray of light AR leaves point A and arrives at point B.

The refraction of light

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

…

or, equivalently,

…

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.

Accord between different laws of Nature that seemed incompatible - Wikisource, the free online library 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.