How the Rainbow is an effect of the laws of refrangibility

The rainbow, or iris, is a necessary consequence of the properties of light.

There is nothing in the writings of the Greeks, Romans, or Arabs that suggests they understood the reasons for this phenomenon.

Lucretius says nothing of it; and with all the absurdities he spouts in the name of Epicurus on light and vision, it appears that his era, refined as it was in other respects, was steeped in deep ignorance of physics.

People knew that a thick cloud dissolving into rain must be exposed to the sun’s rays, and that our eyes must be between the sun and the cloud to see what was called the iris: “Mille trahit varias adverso sole colores” (a thousand hues drawn by the sun’s opposing light); but that was all they knew.

No one imagined why a cloud gives colors, how the nature and order of the colors are determined, why there are two rainbows layered on top of each other, or why we always see them in the shape of a semicircle.

Albert was surnamed the Great only because he lived when men were small in knowledge.

He imagined that the rainbow’s colors came from a dew between us and the cloud, and that these colors, reflected by the cloud, were then sent to us.

You may note that this same Albert the Great, along with the whole scholastic school, believed that light was an “accident.”

Finally, the famous Antonio de Dominis, Archbishop of Spalatro in Dalmatia, exiled from his bishopric by the Inquisition, wrote around the year 1590 his small treatise De Radiis Lucis et de Iride, which was only printed in Venice twenty years later. He was the first to show that the sun’s rays, reflected from the inside of raindrops, created this arc-like painting that had seemed an inexplicable miracle. He made the miracle natural—or rather, explained it using other marvels of nature.

His discovery was all the more remarkable given his many false ideas about how vision works. He stated in his book that the images of objects are in the pupil and that there is no refraction in the eye—a peculiar claim for a philosopher! He discovered the then-unknown refractions within raindrops that create the rainbow, yet denied the ones that occur in the fluids of the eye, which had just begun to be demonstrated. But let us leave his errors to focus on the truth he uncovered.

He observed, with uncommon insight, that each row or band of raindrops forming the rainbow reflects sunlight at different angles; he saw that the difference in these angles caused the difference in colors. He was able to measure the size of these angles: he took a very transparent crystal ball, filled it with water, and suspended it at a certain height, exposed to the sun’s rays.

Descartes, who followed Antonio de Dominis, corrected and surpassed him in some respects, and who perhaps ought to have cited him, also performed the same experiment. When the ball is suspended at such a height that the sunlight striking it forms an angle of 42 degrees and a few minutes between the sun, the ball, and the observer’s eye, the ball always displays a red color.

When the ball is suspended a bit lower and the angles are smaller, the other colors of the rainbow appear successively, with the largest angle producing red, and the smallest angle, about 40 degrees and 17 minutes, producing violet. This is the foundation for understanding the rainbow—but only the foundation.

Refrangibility alone explains this very ordinary, yet poorly understood phenomenon, of which very few beginners have a clear idea. Let’s try to make it understandable for everyone. Suspend a crystal ball full of water in sunlight; place yourself between the sun and the ball: why does the ball show you colors, and why specific ones? Beams of light, millions of rays, fall from the sun onto the ball. In each beam, there are primitive lines, homogeneous rays—many red, many yellow, many green, etc. All refract upon entering the ball; each one refracts differently depending on its type and the location of entry.

You already know that red rays are the least refrangible. The red rays in a given beam will converge at a particular point deep in the ball, while the blue and violet rays in that same beam will go elsewhere. The red rays also exit the ball at a different spot than the green, blue, or violet ones. It’s not enough to just consider where these rays enter and exit, but also the points where they strike and are refracted again before reaching your eye.

Let’s clarify this with more precision: imagine the ball as an assembly of an infinite number of flat surfaces (since a sphere is made up of infinitely small straight lines, and its surface is a multitude of planes).

Three red rays A, B, and C (see figure 34) fall in parallel from the sun on three such surfaces. Each is refracted according to its angle of incidence. Ray A strikes more obliquely than C, and all eventually converge at point R by different paths. Each ray reflects at R and refracts again as it exits, returning parallel and entering the eye, each at the correct angle for red light.

If there are enough of these red rays to stimulate the optic nerve, your sensation will be only red. These rays A, B, C are what we call “visible rays” or “effective rays” of this drop: each drop has its own visible rays.

Thousands of other red rays that strike different surfaces of the ball at different obliquities won’t reach point R and are lost to your eye; they may reach another observer in a different position. Similarly, orange, green, blue, and violet rays also enter the ball, but they refract more and land below point R. These cannot reach your eye unless you move the ball, reducing the angle to around 40 degrees and 17 minutes, at which point you will receive only the violet light.

Now imagine several layers or bands of raindrops; each drop acts like the crystal ball. Consider three bands of raindrops in the rainbow. It’s evident that angle POL (figure 35) is smaller than VOL, and that ROL is the largest. Thus, ROL corresponds to red, VOL to green, POL to violet. You will see red on the outer edge of the iris, green in the middle, violet in the inner band. Note that the last violet layer is always tinged with the whitish color of the cloud in which it fades.

You now understand that you only see the drops whose effective rays have reached your eyes after one reflection and two refractions, and at precise angles. Move your head, and you’ll see a new iris—every change of eye position reveals a new one.

Once you understand the first rainbow, it’s easy to grasp the second, outer rainbow, often seen embracing the first and called the “false rainbow” due to its fainter, reversed colors.

For both rainbows to appear, the cloud must be large and dense enough. This outer arc is also formed by sun rays entering raindrops, refracting, reflecting, and sending back primary rays—one drop gives red, another violet.

But everything in this large arc happens in the opposite direction to the smaller one. Why? Because your eye, which sees the effective rays from the top of the raindrops in the smaller arc, now receives rays from the bottom of the drops in the larger arc.

It’s not hard to conceive how rays reflect twice within the raindrops of the outer arc and, being refracted and reflected twice, result in a rainbow with reversed, less vivid colors. The rays enter the lower part of the drop, are partly refracted, partly lost. The remaining rays reflect inside, exit again—some lost at each step. This explains why the outer arc is dimmer—it loses more light through double reflection.

It also explains why the colors are reversed. In the smaller arc, your eye receives the least refrangible rays (red) at the top; in the larger arc, it receives the most refrangible (violet) at the top.

Finally, why is the rainbow always a circle (or part of one)? Because all these drops are equidistant from your eye and lie on the base of cones whose apex is your eye. Imagine the red rays forming a circle at 42° 2′, the violet rays forming one at 40° 17′. The drops form circles in the sky, but the earth cuts them off—you only see an arc.

Most of these insights were unknown to Antonio de Dominis and Descartes. They couldn’t guess that colors depended on refrangibility, that each ray contains a primitive color, or that differences in attraction cause different angles. Descartes, relying on invention, imagined spinning particles and vortices to explain the rainbow—brilliant but wrong. Just as he invented a false mechanism to explain the heart’s beating, he would have been the greatest philosopher of all time—if only he had invented less.