In The Meteors, Descartes explained the rainbow using the law of refraction, which he published in The Dioptric. When light passes from air to water, it bends

$$\sin i = n \cdot \sin r,$$

where $i$ is the entry angle, $r$ the angle into the water and $n \approx \frac{4}{3}$ for the water. (The Dutchman Willebrord Snell had found the same law before, but did not publish it. That is why it is called Snell-Descartes law.)

The original figure of *The Meteors* (1637): sun rays enter the drops and reach the observer's eye, bottom left.
The original figure of The Meteors (1637): sun rays enter the drops and reach the observer’s eye, bottom left.René Descartes · public domain

Descartes imagined a ray of sunlight entering a spherical drop, reflecting once inside and exiting. Following the angles, the ray leaves, making an angle of

$$\theta = 4r - 2i.$$

He calculated this angle by hand for rays entering various positions of the droplet. Redoing the account with $n = \frac{4}{3}$:

Entry angle $i$ $r$ Exit angle $\theta = 4r - 2i$
10° 7,5° 9,9°
20° 14,9° 19,5°
30° 22,0° 28,1°
40° 28,8° 35,3°
50° 35,1° 40,3°
55° 37,9° 41,6°
60° 40,5° 42,0°
65° 42,8° 41,3°
70° 44,8° 39,2°
80° 47,6° 30,5°

Notice the column on the right: it rises, reaches a maximum of around 42° and falls again. Near the maximum, rays that enter at very different positions (from 50° to 70°) exit at almost the same angle. Light accumulates there, and so we see a bright arc at 42° from the point opposite the Sun.

Now do Descartes’ experiment. Move the entry point and follow the chart to the right, then turn on “Many rays” to see red and violet accumulating at slightly different angles:

A ray enters the droplet, reflects once inside and exits. The departure angle never exceeds about 42°.

As $n$ varies slightly with color (red light deviates less than violet), each color has its own maximum at a slightly different angle: red is on the outside and violet is on the inside. It was the first mathematical explanation of the rainbow.

All topics by René Descartes · ← The kissing circles theorem · Descartes and the idea of ​​function →

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