The Refraction Of Light And The Circle

By Larry Hecht

The law for the reflection of a ray of light, was known since ancient times. Imagine a plane mirror, resting on a table-top. A beam of light, directed at the mirror, forms an angle with the mirror’s surface called the “angle of incidence.” About 2,000 years ago, scientists knew that the beam, after striking the mirror, would reflect off in the opposite direction, the reflected ray making the same angle with the mirror’s surface, as the incident ray.

A related phenomenon is the refraction of light: A ray of light, passing from one medium, such as air, to another, such as glass or water, is bent (refracted) as it crosses the interface between the two media. Imagine the smooth surface of water contained in a home aquarium tank. A ray of light strikes the surface, where we measure the angle of incidence. The light ray continues on, below the surface of the water, but its path has changed direction! It is bent, or refracted, such that the angle it makes with the surface of the water, measured downward from that surface — called {the angle of refraction} — is greater than the angle of incidence.

As we increase or decrease the angle of incidence, the angle of refraction also increases or decreases. But in what proportion? The most skilled investigators of the laws of optics from the Hellenic age, to the Islamic Renaissance, on to the early European Renaissance, could not discover the lawful relationship of angle of incidence, to angle of refraction. The answer was found by the Dutch republican scientist Willebrord Snell, a student of the famous Simon Stevin, in 1620. Perhaps the reason no one had found it earlier, is that the proportion is a transcendental one; that is, it expresses the relationship of a circular arc to a straight line, or chord of the circle. In geometry, this relationship is called the “sine.” It is the same proportionality discussed by Nicholas of Cusa, in the {De Docta Ignorantia}, where he demonstrates the incommensurability of straightness and curvature.

Precisely this incommensurable proportion, defines the lawful relationship between the angle of incidence and angle of refraction of a ray of light. Snell’s beautiful discovery, was to show, that no matter how the angle of incidence may vary, the ratio of the sine of this angle, to the sine of the refracted angle, remains the same. This is Snell’s Law of Refraction, also called the Law of Sines. The beauty and simplicity of it, and its relationship to Cusa’s crucial breakthrough, are unfortunately disguised by the poor teaching of geometry today, in which the trigonometric functions (sine, cosine, and tangent), are usually seen only as linear ratios; that is, as ratios of sides of a right triangle.

To see clearly, what a sine actually is, and also to better understand Snell’s Law, let us look at the description of the law given by Snell’s countryman, Christiaan Huyghens, in the closing chapter of his {Treatise on Light}, written in 1673. [The reader will have to draw this simple diagram. — ed.] In a circle whose center is O, draw a horizontal diameter CD. Let the circle represent the cross-sectional view of the air-water interface, such that the area above the diameter CD is air, and the area below, is water. Now designate a point, A, at about the two o’clock position on the circumference, from which a ray of light originates, and proceeds to the circle’s center, O. Here it encounters the surface of the water, where it is bent downward, so that its direction is toward a point B, at about the seven o’clock position on the circle. Angle AOD is the angle of incidence. Angle BOC is the (larger) angle of refraction. But also notice, that what we call angle AOD, is a measure of circular rotation: the arc AD. And, similarly, angle BOC is the arc BC.

The problem, to repeat, was to find the lawful relationship between these two angles, or arcs. From A, drop a perpendicular to the diameter CD. Do the same upwards from B. The lengths of these perpendiculars, are the sines of the angles AOD and BOC. Snell discovered, that whatever the incident angle AOD, the refracted angle, BOC, will adjust itself, such that the ratio of their sines, will remain constant. How does the ray of light, know how to do that?

An even greater “willfulness” on the part of the insensible light ray, was discovered during the remainder of the Seventeenth Century. First, Pierre de Fermat showed that the path which the light ray “chooses” from A to B, is the shortest possible in time — that is, takes the least time. This is true, anywhere along the extended line OB, not just where it intersects the circumference of the circle. Next, Jean Bernouilli investigated the refraction of a light ray, in a medium of continuously varying density, such as the atmospheric air, as it rises and thins above the earth’s surface. Being continuously refracted at each interface of the denser, with the less dense, air, the path of the light ray is a curve. Bernouilli discovered, with great excitement and delight, that the curve which the refracted light follows under such conditions, is the cycloid — the same curve which, he had just discovered, was the path of least time, for a falling body under the influence of gravitation.

We leave to future discussion, the investigation of this “higher willfulness” of inanimate objects.