Leonardo da Vinci wrote the rule down around 1500. Paint the far buildings bluer than the near ones, he told painters, and if one stands five times as far away, make it five times bluer. He was right about the blue. He could not have known why, because the reason was not worked out until 1871, by a physicist who was thinking about the sky and not about mountains at all.
Here is the surprise, and it is most of the answer in one sentence: the far mountain is not blue. The air in front of it is. Every mile between you and a ridge is full of sunlit air, and sunlit air is the same stuff the sky is made of. A mountain on the horizon is a mountain plus twenty miles of sky, laid over it like a veil.
On a clear day on Seattle’s Eastside you can watch it happen. The near hills are green and sharp. The next ridge is duller. The Cascades behind that are blue-gray, and the farthest peaks are barely a pale whisper against the sky, until it is hard to say where the mountain stops and the sky begins.
The rule came 370 years before the reason
Leonardo called it aerial perspective, and he was precise about it. Buildings seen above a wall all look the same size, he wrote, so a painter needs another way to say which one is far. “Make the nearest building above the wall of its real color, but the more distant ones make less defined and bluer. Those you wish should look farthest away you must make proportionately bluer; thus, if one is to be five times as distant, make it five times bluer.”
He had a theory to go with the rule, and it is worth reading, because it is nearly right. The blue of the sky, he wrote, “is not intrinsic color, but is caused by warm vapor evaporated in minute and insensible atoms on which the solar rays fall, rendering them luminous against the infinite darkness” beyond. Tiny things, lit by the sun, seen against dark. He climbed a peak in the Alps to check it, and he did an experiment at home: burn a little dry wood, and look at the smoke with a piece of black velvet behind it. The smoke turns “a beautiful blue.” Put a white cloth behind the same smoke and the blue is gone.
Hold on to that experiment. It is the whole of this post in a fireplace.
What he got wrong was the vapor. It is not water in the air that makes the blue. It is the air.
What the air does to light
Sunlight is every color at once. The molecules of nitrogen and oxygen it passes through are far smaller than a wave of light, and a passing wave shakes them. A shaken molecule gives off a little light of its own, in every direction. Physicists call that scattering, and it is the reason you can see a beam of light from the side at all.
The part that matters for painters is that short waves shake a molecule harder than long ones. Much harder. In 1871 John William Strutt, later Lord Rayleigh, worked out the rule: the scattering goes up with the fourth power of the shrinking wavelength. Halve the wavelength and you get sixteen times the scattering.
You can do the arithmetic for real colors. Red light has a wavelength of about 700 nanometers. Blue is about 450.
700 ÷ 450 = 1.56
1.56 × 1.56 = 2.4
2.4 × 2.4 = 5.9Blue light is scattered out of a sunbeam nearly six times as often as red. The air over your head is doing that to sunlight all day, and the blue it throws sideways reaches your eye from every direction at once. That is the sky.
Two years before Rayleigh, John Tyndall had seen it in a laboratory. Shining a strong lamp through a tube of air with a little fine dust in it, he found the beam glowed faintly blue from the side and came out the far end faintly orange. He had made a sky and a sunset in a glass tube, and Leonardo’s smoke was the same experiment, three and a half centuries earlier, without the lamp.
Airlight, and why the dark ridge goes blue first
Now put a mountain twenty miles away, and fill the twenty miles with that air, lit by the sun.
Some of the sunlight falling on all that air is scattered toward you. Physicists have a plain name for it: airlight. It arrives at your eye on top of the mountain’s own light, and it is blue for the reason the sky is blue. At the same time, some of the mountain’s own light is scattered out of its path on the way to you, and the blue part goes first. So the mountain dims, its edges soften, and a layer of sky is painted over it.
Which of the two effects wins depends on the mountain. A dark ridge, forested and in shadow, sends you very little light of its own, so the added sky dominates and the ridge goes blue. A brilliantly lit snowfield sends you a great deal, so the loss matters more than the addition, and its blue is stripped out on the way. Watch a snow peak on a clear day and look at what is beneath it. The forests at its foot go blue-gray long before the summit does, and the snow, if it shifts at all, goes a little warmer, not bluer.
That is Leonardo’s smoke, exactly. Against black velvet the smoke looks blue, because all you see is the light it scatters. Against a white cloth it looks like nothing, because the cloth’s own light swamps it.
Leonardo’s arithmetic, and where the air disagrees with it
“Five times as far, five times bluer” is a straight line. The air does not work in straight lines, and the way it actually works is something a painter can use.
Each mile of air does not add a fixed amount of sky. It takes a fixed fraction of whatever is left of the mountain’s own light and swaps it for sky. Say a mile of clear air swaps a tenth. After one mile, nine tenths of what you see is still mountain.
1 mile: 0.9 = 90% mountain, so 10% sky
2 miles: 0.9 × 0.9 = 0.81, so 19% sky
5 miles: 0.9 × 0.9 × 0.9 × 0.9 × 0.9 = 0.59, so 41% sky
20 miles: 0.12, so 88% skyLeonardo’s rule says the ridge five times as far should be five times bluer, fifty percent sky. The air says forty-one. The first few miles do most of the work, and the far ridges crowd up toward the color of the sky and never quite reach it. That is why the last ridge in a painting is more sky than land, and is still, faintly, there.
If that multiplying feels familiar, it is the same arithmetic as the coats of a wash in The light is behind the paint. Paint counts twice on every trip through it, and air counts once for every mile. Neither of them adds. Both of them multiply.
A tenth a mile is a made-up number, and that is the honest part. On a damp day it is more, on a dry cold one it is less, and nobody can tell you the figure for the view in front of you. What stays true at every setting is the shape of the curve.
The same air, lying down
At noon, sunlight reaches you through one thickness of atmosphere, straight down. At sunset it comes in sideways, along the ground, and crosses about thirty-eight thicknesses before it reaches your eye.
Thirty-eight veils, and every one of them is taking blue out of the beam and throwing it somewhere else. What is left when the light arrives is the part the air could not steal: orange, then red. A sunset is not the Sun changing color. It is the Sun seen through the far mountain’s air, thirty-eight times over.
The Moon does the same thing, and it is a better demonstration, because nothing about the Moon is bright enough to hurt. The Moon makes no light of its own. It sends back sunlight, and a Moon just clearing the horizon sends it to you along the same low, thirty-eight-veil path. So it comes up orange, climbs, and turns white by the time it is high, and the Moon itself has not changed at all. Only the amount of air in the way has. Next time it rises, watch for the twenty minutes in which it goes from copper to bone.
Where this stops being true
Not every haze is blue, and this is where painters get into trouble.
Rayleigh’s rule is for things far smaller than a wavelength of light, which is what a molecule is. Mist, smoke and dust are made of droplets and grains about the size of a wavelength or bigger, and those scatter every color about equally. That is why fog is white, why wildfire smoke turns the far ridges gray and brown rather than blue, and why a muggy August afternoon flattens the mountains into something the color of dishwater. Two different kinds of veil, two different colors, and a painter who mixes them up gets mud.
And there is a second place where the tidy story runs out, which is Rayleigh’s own paper. In 1871 he worked out the fourth-power rule for small particles floating in the air, dust or water, because that is what everyone assumed was scattering the light. It took him until 1899 to show that the molecules of the air are enough on their own, with nothing floating in it at all. Leonardo’s vapor survived in physics for almost thirty years after the rule that explains the sky was written down. Even a right answer arrives in pieces.
One more thing runs out, and it is the part a painter has to live with. No formula will tell you how blue this ridge should be today. It depends on the water in the air, the dust, where the Sun is, and what is behind the ridge. Leonardo noticed that the rule changed when the light came from the east, and left a note about it. Ayesa’s version, in the lesson this post is written for, is shorter: if it looks too pale, it is probably right.
Try it, three ways
- Leonardo’s smoke, without the fire. Fill a clear glass with water and stir in two or three drops of milk. In a dim room, shine a flashlight through it from one side. Looked at from the side, against something dark, the water goes faintly blue. Now look at the flashlight through the glass, end on. It has gone yellow-orange. You have the sky and the sunset in one glass, and the milk is doing what the air does.
- The card with a hole. On a clear day, punch a hole in an index card and hold it at arm’s length. Look at the farthest ridge you can find through the hole, then at the sky beside it, then at the nearest hill. Through the hole, with nothing around them to compare, the far ridge and the sky are nearly the same color. That is the eighty-eight percent, seen directly.
- The veils, multiplied. Mix one pale blue-gray wash. Paint a stripe of it and let it dry. Paint a second stripe with two coats, and a third with three. The step from one coat to two is bigger than the step from two to three. That is the curve above, and it is why the far ridges in a painting bunch together while the near ones stand apart.
Where this comes from
- The Notebooks of Leonardo da Vinci, translated by Jean Paul Richter, sections 295 to 302 — aerial perspective, the color of the atmosphere, and the smoke against black velvet (quoted here with American spelling)
- Rayleigh scattering — the fourth-power rule, Rayleigh’s 1871 paper, and his 1899 paper showing the molecules alone are enough
- Tyndall effect — the 1869 experiment with a beam through dusty air, blue from the side and orange at the end
- Aerial perspective — airlight, the loss of contrast with distance, and why distant objects shift toward the color of the sky
- Why is the sky blue? — NASA Space Place, the plain-language version, including why it is not violet
- Air mass — the path through the atmosphere at the horizon is about 38 times the path straight up
- Blue Ridge Mountains — a whole range named for the effect, with the trees adding a haze of their own
The two diagrams above are original drawings made for this post.