A rainbow is a curved band of colors in the sky, and it needs sunshine and water drops in the air, such as rain. To see one, remember that the sun must be behind you and the water drops in front of you. The ancient Greek thinker Aristotle explained the rainbow as sunlight reflecting off a cloud, which is not right. The strip of sky between the two bows of a double rainbow is named for Alexander of Aphrodisias, a Greek philosopher who studied it around 200 AD in connection with Aristotle's incorrect rainbow theory. The dark strip is called Alexander's band, and it is dark because almost no sunlight is scattered toward your eyes from that part of the sky.

Around 1300, scholars in Persia and Germany worked out that a rainbow comes from light refracted and reflected inside single raindrops. Theodoric of Freiberg, in Germany, studied rainbows with a globe of water, treating it as a magnified raindrop. In Persia, Kamal al-Din al-Farisi studied the rainbow by filling a large glass vessel with water and putting it in a camera obscura, to stand in for one giant raindrop. In 1637 René Descartes explained the rainbow by tracing light through a raindrop, and with Newton he set up the basic ray theory of the rainbow. Sunlight enters a raindrop, bends, bounces off the back of the drop, and bends again as it leaves, heading back toward the sun's side of the sky. Light bends when it goes from air into water. This bending is called refraction.

Theodoric also explained the double rainbow: the extra reflection reverses the order of the colors and weakens the light. A double rainbow happens when light bounces twice inside the raindrops instead of once, so the second bow is outside the first and its colors are in reverse order. Sometimes two rainbows show at once, a bright one and a fainter one above it with its colors flipped. The secondary rainbow comes from light that reflects twice inside the drop and leaves at about 50 to 51 degrees, wider than the primary bow. In the main rainbow the colors always come in the same order, with red on the outside and blue and violet on the inside, and a third-order, or tertiary, rainbow is a different case. It is centered on the sun itself, so you have to face the sun to look for it, and it is very faint.

The ray theory puts the bow at a definite place, and the main bow forms where you look about 42 degrees away from the point exactly opposite the sun. That spot, called the antisolar point, is the center of the rainbow's circle. The light that makes the main rainbow leaves a drop turned about 138 degrees from the direction it came from, which is the same as 42 degrees from the point opposite the sun. Red light leaves turned about 42 degrees from the sunlight's path and blue-violet light about 40 degrees, so the colors line up in bands. A rainbow is round because every direction that is 42 degrees from the antisolar point makes a circle around it. A rainbow's light reaches you from the direction of the raindrops, not from the sun.

Before Newton, many people thought a prism changed sunlight and made the colors. In the mid-1660s Isaac Newton used glass prisms to show that white light is made of a spectrum of colors. He let a beam of sunlight pass through a prism onto a wall, and then showed that a second prism brings the colors back together into white light. In his crucial experiment, a board with a hole let only one color, such as red, pass into a second prism. That light bent but stayed the same color, so a prism does not make colors, it only bends each color by a different amount. A prism is a piece of glass that splits a beam of sunlight into rainbow colors.

Newton also used a lens to bring a spread-out spectrum back into one beam, and the merged beam was white. Passed through another prism it split into a spectrum again, and he explained the rainbow as sunlight refracted and reflected by raindrops that act as tiny prisms. Newton's picture treated the drops like prisms, adding the spreading of colors to Descartes's raindrop picture. Newton told the Royal Society about his prism findings in 1672, but he had made the discoveries years earlier, in the 1660s. A rainbow's colors and the colors that come out of a prism are the same, in the same order. People had seen prisms make colors before Newton, and his new idea was that white light is a mix of all the colors. The prism separates them because each color bends by a different angle.

The index of refraction of water changes with wavelength, which is why the drop spreads sunlight into colors. Different colors of light bend by different amounts when they enter water, so the colors spread apart, and red has the longest wavelength of visible light and violet the shortest. Dispersion means light is spread out by wavelength, and reflecting off the back of a drop does not disperse light, because reflection does not depend on wavelength. The same difference in bending by color shows up in lenses. Lenses bend each color by a different amount, so a lens telescope shows colored fringes around a star, and this blur is called chromatic aberration. Newton built a telescope with a curved mirror instead of lenses, because lenses split starlight into colors and mirrors reflect all colors equally.

More exactly, raindrops are not prisms: they are nearly round, and the light separates because it is bent twice at curved surfaces of the drop. Light leaves a raindrop in many directions, but the most light is concentrated near 42 degrees, so that is where the bright rainbow appears. One special ray, the rainbow ray, has the smallest angle of deviation of all rays entering the drop, and much of the sunlight is focused along it. Because the rays bunch up near that smallest angle, the light there is concentrated, and the concentration is what makes the bright arc of the rainbow. Big raindrops make bright, vivid rainbows, and small drops make paler ones, and the sky inside a rainbow is brighter than the sky outside it.

Words to know

• dispersion: spreading by wavelength
• aberration: optical blur
• tertiary: third-order
• spectrum: band of colors