Your Shadow Is the Centre

Around 1267, a Franciscan friar in Oxford wrote a very long book for the Pope. In the part on experimental science he took an astrolabe outside, measured the height of the sun and the height of a rainbow, and reported a number: the bow can never stand more than 42 degrees above the horizon.

Then Roger Bacon wrote down something stranger, and he wrote it as a finding rather than a fancy:

From these facts we learn by experiment that there are as many bows as there are observers.Roger Bacon, Opus Majus, Part VI, c. 1267 · trans. Robert Belle Burke, 1928, ii. 600

He did not leave it as an aphorism. He gave the test. Two people stand side by side, looking north at the bow. One walks west and the bow travels west with him. The other walks east and his bow travels east with him. Stand still and it stands still.

It is evident, therefore, that there are as many bows as there are observers; and for this reason it is impossible for two observers to see one and the same bow, although an ignorant person does not grasp this fact.Bacon, ii. 600–601

Note the last clause. In 1267 he had already met the reaction, and it is the same reaction now: people do not believe you.

And here is what should stop you. Bacon had no theory of refraction. He did not know what happens to light inside a raindrop — nobody did for another forty years, and nobody could calculate it for another three hundred and seventy. He got this by going outside and walking about.

Which means the oldest correct thing anybody wrote down about a rainbow is not about colour, and not about water. It is about where you are standing.

A rainbow is a circle. You usually see part of one, because the ground is in the way. And that circle has a centre, which is not the sun and not the rain.

Bacon put the axis of his cone through “the center of the eye and the center of the sun and the center of the bow to the nadir of the sun” — the point in the sky directly opposite the sun. Astronomy calls it the antisolar point now. There is a much simpler way to find it. Stand with the sun behind you and look at your own shadow. Follow it out to where the head is. That direction, extended, is the centre of your rainbow.

For the shadow of each observer divides the arc of the rainbow into two equal parts … and therefore each observer must see his own bow.Bacon, ii. 601

Seven hundred and fifty-nine years later, an open-access meteorology textbook says it without the thirteenth-century sentence structure and without changing anything:

Rainbows are circles or portions of circles that are centered on the antisolar point, which is the point corresponding to the shadow of your head or camera.Roland Stull, Practical Meteorology, ch. 22

The radius of that circle is fixed and it is not a round number. Measuring outward from the antisolar point, the red edge sits at about 42.7° and the violet edge at about 40°, using a refractive index for water of 1.3315. The band between them is the whole bow, and it is narrower than most people draw it.

So the geometry is indexed to your body. Not as a figure of speech. The coordinate system this thing is described in has its origin inside your head, and the number 42.7 is measured from the shadow you are casting.

Here is the part that should be famous and isn't. Bacon did not treat the un-shareability of the bow as a curiosity, a paradox, or a problem to be explained away. He used it as data.

His reasoning runs like this. If the bow were sitting out there in the cloud like a painted object, it would stay put. It would not care who was looking or from where, and it certainly would not chase people down a road. So the fact that it does follow each observer separately tells you something about the light.

From these arguments it follows, then, that the bow is seen only by reflected rays of the sun, because if it were seen by incident rays, the bow would be an object fixed in one place in the cloud, which would not vary according to the motion of the observers nor according to the number of the observers.Bacon, ii. 601

And he ran controls. This is 1267, and he ran controls. A cloud lit from behind takes a colour — white, black, or something between — and “the color of the cloud appears the same to different observers, nor does it follow the motion of the observer.” A stick or a fish seen through water is displaced by refraction, and “the image of an object seen by refraction does not follow the observer if he recedes, nor does it recede if he approaches.”

Three phenomena, one question asked of each, and only one of them answers differently. That difference is the finding.

Be exact about the scoring, because we are not going to launder it. Bacon's conclusion was wrong, or at least badly incomplete. He landed on reflection alone; a rainbow needs refraction going in, one reflection at the back of the drop, and refraction coming out again. He was reasoning from a mechanism he did not have.

The method was still right, and the observation was still right. He asked whether a thing follows the observer, he tested it against cases that don't, and he drew a conclusion about the light rather than about the people. Forty years later somebody supplied the mechanism, and Bacon's geometry survived intact into a textbook you can download today.

So: the very feature that makes a rainbow impossible to hand to anyone is the feature that told us what a rainbow is. The privacy was not the obstacle to the knowledge. It was the route in.

Between about 1304 and 1310, two people who never met and never read each other worked out what happens inside a single raindrop: light refracts on the way in, reflects once off the far inside wall, and refracts again on the way out.

One was Theodoric of Freiberg, a Dominican in Germany, in a treatise called De iride. The other was Kamāl al-Dīn al-Fārisī, working in Persia, in a commentary called the Tanqīḥ al-Manāẓir. They arrived at the same answer independently, and they had the same reason to: both were reading Ibn al-Haytham's Kitāb al-Manāẓir, the book on optics that had already reached Latin Europe and was reshaping how everybody thought about light.

Two people, one book, two continents, the same answer, and no contact. That is not a coincidence, and it is not genius twice. It is what a good enough shared source does.

The number came much later. Descartes published the calculation in Les Météores in 1637, three hundred and seventy years after Bacon measured the angle he could not explain. Work out how much a ray is bent as you move across the face of the drop and you find the bending has a limit: about 137.6° for red, 139.4° for violet. Subtract from 180 and you have 42.4 and 40.6 — Bacon's 42, arrived at from the other end.

And the reason a bow is bright is that limit. Near it, rays entering the drop at noticeably different places all come out heading in almost the same direction. They stop spreading out and start stacking up. Physics calls the resulting concentration a caustic — the same effect as the wobbling bright line at the bottom of a swimming pool.

What matters for this zine is where the pile-up is. Not at a place in the sky. At an angle from your eye — and an angle from your eye is not a location anyone else can stand in.

The colours are not this zine's argument. That a spectrum is a fan of every wavelength at once and never a ladder from a bad end to a good end is the whole business of Not a Line (No. 21), and it is done there. Here the dispersion is scaffolding. The load is carried by the geometry.

Ask how far away a rainbow is and the question fails. Not because the answer is large, and not because it is hard to measure. Because there isn’t one.

Les Cowley, who has run the standard reference on atmospheric optics for decades, puts it flatly: “Rainbows possess neither distance nor linear dimension.” The drops making your bow can be near or far, or both at once — “the water drops making a rainbow can be at any distance … but provided the drops are the same size the rainbow always looks the same. It looks and behaves as though it is at infinity.”

It cannot be touched, tickled, walked around or driven through. It moves as you move and is forever unapproachable.Les Cowley, Atmospheric Optics

Bacon had this too, in the vocabulary he had: the bow “seems to flee before him, because it always appears at the same distance,” and the fleeing happens “with a speed equal to that of a man running or riding as fast as possible.” There is no pot of gold, and the reason is not that somebody moved it. A rainbow is a direction, not a place.

Then there is what it is made of, which is nothing for long. Rain falls. Every drop currently sending you light is on its way to the ground, and it leaves your 42-degree cone within a second or so of entering it. The bow you are looking at is being built out of water that was not in it a moment ago and will not be in it a moment from now.

The arch holds still while every single thing constituting it is replaced. Not once — continuously, the entire time you are watching, for as long as the rain lasts.

Marked honestly: the sentences from Cowley and Bacon are quotations. The replacement claim is ours, and it is a deduction from two settled things — the cone is fixed relative to your eye and the sun, and the drops are in free fall through it. We have not measured a turnover rate and we are not asserting one.

Stull's textbook states the general case in one line: “At any one time for any one rain storm, different observers see different rainbows caused by different light rays interacting with different raindrops.”

Stand a metre from a friend. Your antisolar points are a metre apart, so your cones are a metre apart, so the drops handing you light are not the drops handing them light. Two bows. They look the same, and they are not the same.

Now shrink the distance. Your pupils are roughly 63 millimetres apart. Each one is the apex of its own cone, each cone has its own axis running to its own antisolar point, and each therefore collects light from its own set of drops. You are not looking at one rainbow with two eyes. You are looking at two rainbows, one per eye, and your brain is handing you a composite.

Same species. Same genome. Same organ, built twice from the same instructions, sixty-three millimetres apart. Still two different phenomena.

This is the spread that earns the zine, so here is what it adds. Our own Every Creature’s Own Sky (No. 15) argues that every creature is sealed in its own perceptual world — the bat’s echo, the mantis shrimp’s light — and that there was never a neutral way to sense. True, and there is a standard way to wriggle out of it: well, you and I are both human, so we must be seeing the same thing.

The rainbow closes that exit. Difference in what is perceived does not require any difference in the perceiver. Identical hardware, identical software, and position alone is sufficient. Two eyes in one skull cannot be talked into a shared vantage point, because there is no such thing.

Marked honestly: Stull's sentence is a textbook statement about different observers. The two-eyes case is us running his geometry inward — a deduction, not a separate measurement, and the difference between the two bows is far too small for you to notice. It is real and it is imperceptible, and both of those matter.

A rainbow needs four things, and only one of them is you. The sun behind you. Rain in front of you. An unobstructed path from the sun to that rain. And the sun below about 42° of altitude.

That last one is the interesting one, and it follows straight from the geometry. The bow is a ring 42° around the antisolar point, and the antisolar point is as far below the horizon as the sun is above it. Lift the sun past 42° and the entire ring sinks out of sight. Bacon again, from measurement: “when the sun is at an altitude of 42 degrees the rainbow does not appear in the sky … When the sun rises higher, the rainbow can nowhere appear.”

He then did the arithmetic for where he lived and where he had read about. At the latitude of Paris the equinox noon sun stands at 41°12′ — a hair under the limit, so a bow at midday is barely possible and about to stop being. And working with a latitude near 24°25′, he found the noon sun clears 42° even at the winter solstice, and drew the conclusion:

Therefore no rainbow can be formed at any noon of the entire year.Bacon, ii. 602 — for a latitude near the Tropic of Cancer

Sit with the shape of that. There are places and hours where the rain is falling, the sun is blazing, the air is clear, your eyes are working perfectly, and there is no rainbow, for you or for anybody. Nothing is wrong with the observer. The arrangement simply is not one a rainbow can happen in.

And the corollary, which is the useful half: the conditions are arrangeable. Wait three hours. Turn around. Walk to where the rain is. None of that changes you and all of it changes whether you see the bow — because access here was a fact about the geometry, and it always was.

We take the same care everywhere else in this collection. When somebody cannot do a thing in a room, the first question is what the room is doing, not what is wrong with them. A rainbow is a case where nobody argues, because the geometry is written down and checkable, and it would be plainly absurd to tell a person that the noon sun is their attitude problem.

On the arithmetic: Bacon uses an obliquity of 23°35′; the modern value is about 23°26′, so his figures run a few minutes off. The conclusion is unaffected, and we would rather show you the seam than sand it.

Everything so far has been optics. Here is the whole of it put together, and it is a fact about people.

There is no vantage point from which two people see one rainbow. If someone doubts your bow and walks over to check, the walking is what breaks it: arriving where you stand gives them their bow, with its own centre in their own shadow, built from their own drops. You cannot hand it over. You cannot point at it in a way that makes it theirs. The only route from your rainbow to another person is your account of it.

A rainbow is real, it is exactly predictable, it is described in textbooks to a fraction of a degree — and it is impossible to share. All four of those at once, with no tension between them.

Now notice what nobody does. In seven hundred and fifty-nine years since Bacon wrote it down, no one has ever concluded that rainbows are not real on the grounds that they cannot be shown to a second person. No one demands corroboration. No one suggests you might be exaggerating, or attention-seeking, or would see fewer of them with a better attitude. The unshareability is simply understood to be a fact about the geometry.

That courtesy is not extended evenly. If it were real, you could show me. Nobody else in the class has this problem. I’ve never seen you like that. Prove it. Every one of those treats privacy of access as evidence of unreality — and there is a settled, boring, four-hundred-page-textbook counterexample sitting in the sky perhaps twice a month.

Bacon met the resistance too, and filed it in one clause: “although an ignorant person does not grasp this fact.” He is not being unkind about eyesight. He means the person who has been told the geometry, and would still rather trust their assumption that a real thing must be a shared thing.

Down to is not because of. The loving is not warranted by the star stuff; it is measured by how far it reaches. And what reaches you here is that your account was always going to be the only way in, and that this is ordinary.

Testimony is not the weak substitute for evidence. For some real phenomena it is the only mechanism there has ever been — and we already know how to accept it without argument, because we do it every time it rains and the sun comes out behind us.

This argument has an obvious cheap version, and the cheap version is worse than saying nothing. So, plainly: