The Mathematics of Rainbows – Descartes’ Derivation

René Descartes explained the rainbow mathematically in 1637. Light enters a spherical raindrop, refracts, reflects internally once, then refracts out. The angle of deviation depends on the impact parameter. The rainbow occurs at the extremum of deviation (about 42° for primary rainbow). This concentration of rays creates a bright band. The secondary rainbow (two internal reflections) appears at 51°, with colours reversed. The mathematics uses Snell’s law (n₁ sin θ₁ = n₂ sin θ₂) and calculus to find the minimum. Rainbows are caustics – a concept in optics and catastrophe theory.