Celestial Globes Exposed, Answered

1 · The claim, stated fairly

The argument runs from p. 143 to p. 145 of the printed draft. p. 143 sets up the objection accurately: measure Polaris's elevation from several latitudes, compute how high it must be above a flat plane, and the answers disagree. p. 144 offers the diagram — visual geometry against tangible geometry, with Angle A, what you see, always smaller than Angle B, what is physically there. p. 145 draws the conclusion: “The answer is linear perspective!” The accompanying figure shows straight sight-lines converging on Polaris, captioned as assuming elevation angles are tangible geometry — “This is undeniably wrong!”

Several things here are correct and should be said before anything else.

• — Linear perspective is real, and the hallway analogy is a fair description of it. Parallel lines do converge in angle with distance.
• — Refraction is real, and it genuinely does corrupt low-elevation measurements. Navigators correct for it on every sight.
• — The objection is often stated badly. “Do the trig, it doesn't work” is not an argument if the person hearing it thinks the angles are unreliable. The chapter is right that the reply usually skips a step.
• — And one calculation in the chapter is exactly right. p. 136 puts Polaris's angular size at 0.0000009° against an eye-resolution limit of 0.017°, and calls it 18,860× smaller. Checking it: Polaris A at roughly 46 solar radii and 433 light years subtends 3.2 milliarcseconds, or 8.95×10⁻⁷ degrees, and the ratio is 18,990. The small difference from his 18,860 is not an error of his — it is the distance he used, 430 light years against the 433 taken here, and Polaris's distance is genuinely disputed in the literature anyway. Nothing on this page turns on which figure is right. It is a correct piece of arithmetic, done properly.

So the disagreement is not about whether perspective exists. It is about how much of it there is.

2 · The multi-observer problem

Here is the structural difficulty, and it is the one most explanations of this kind run into. A single observer can never settle it. Whatever one person measures, some arrangement can be found that produces it, so one reading tests nothing at all. The constraint only bites when a single arrangement has to satisfy every observer at once.

That test needs no claim about perspective and no claim about refraction in order to run. It takes the angles exactly as measured and asks whether any one arrangement can produce all of them together. So it goes first, before anything about how eyes work — and it has a number attached that anybody can check without a telescope.

First, take the measurements at face value

Start where the chapter starts, with tangible geometry. Assume for the moment that the measured angles are exactly what they appear to be, and ask what shape of world they describe. No corrections, no perspective, nothing subtracted — just the numbers as they come off the instrument. If the flat model survives that, the argument about perspective never needs to happen.

The flat model gets one free parameter to play with: how high the lamp sits. So the fair way to run the test is to let it choose that height to suit itself, and then check the result somewhere else. Pick the height from one measurement, and see whether the others fall into line.

The ruler that never changes length

Polaris drops one degree for every 69 miles you travel south, everywhere, to within one percent.

It is 68.7 miles per degree at the equator and 69.4 at the pole — and that one percent is not slack in the measurement, it is the Earth's oblateness, found by the French Geodesic Missions of 1735–44 when Maupertuis measured an arc in Lapland and La Condamine measured one in Peru and the two came back different. Which is worth pausing on: that expedition was this experiment, run with quadrants and chains three centuries ago, and it was precise enough to detect a one-percent departure from a perfect sphere. The flat model needs a hundred-fold departure.

Set the oblateness aside as the small correction it is. The distance is measured independently of any star sight — by odometer, by survey, by flight time — and the relationship holds all the way from the Arctic to the equator. Now ask what a flat plane with a lamp above the pole would have to produce. Anchor the lamp at the height the model's own 45° measurement implies, 5,009 km, and walk outward.

The same thing as a rate, degree by degree:

Measured answer: 69 miles, at every one of those elevations. A lamp at any finite height has to make the rate climb as you walk away from it — twenty-five-fold across the usable range, a hundred-fold by 5°. Carried to its conclusion the model never gets to the equator at all: above a flat plane, a lamp at any finite height only sinks to the horizon at infinite distance. Stop half a degree short of that — at an elevation of 0.5°, a star still perfectly easy to see — and the flat model puts the “equator” 356,680 miles from the pole, half again as far as the Moon. The measured distance is 6,225.

The size that never changes either

There is a second multi-observer measurement, independent of the first, and it points the same way for a different reason. A lamp a few thousand miles up is not only at a different angle from different latitudes — it is at a different distance.

Travelling from the high Arctic to the tropics would put you almost exactly twice as far from the lamp. That has three separate consequences, and it is worth taking them in order of how easy they are to see.

First, it would get dimmer. Polaris is a point source to the eye — its disc is far too small to resolve — so a distance change shows up as brightness, not size. Double the distance and the light falls to a quarter:

A four-fold drop in brightness, a star and a half of magnitude, between Alaska and Ecuador. Polaris is a familiar second-magnitude star that generations of navigators have picked out by eye across the whole northern hemisphere; at 3.5 it would be an inconspicuous smudge in the tropics.

Second, the constellation would shrink — and this one needs no equipment at all, because it is a comparison between two stars rather than an absolute measurement. Polaris and Kochab, at the end of Ursa Minor, sit 16.6° apart:

Kochab circles Polaris once a night at that radius, so the honest way to state the prediction is as the size and shape of that circle:

Two effects compound here. The whole circle shrinks as 1/distance, and it is also foreshortened in the north–south direction by the viewing angle, since you are looking at a horizontal circle increasingly edge-on as you travel away from it. So the flat model does not merely require Ursa Minor to halve in size between Alaska and Ecuador — it requires the Kochab–Polaris circle to flatten into an ellipse on the way.

Neither happens. Constellations keep their shape and their angular dimensions everywhere they can be seen — to within the fraction of a percent that differential refraction costs near the horizon — and only their elevation and orientation change. The circumpolar stars trace an arc of the same angular radius from Alaska and from the tropics. From 10°N, Kochab is not actually circumpolar; the lower half of its circle is below the horizon, so the comparison is between the visible arcs, which is enough. And giving Kochab its own separate height to escape this buys one latitude and breaks all the others — the multi-observer problem again.

Third, and most precisely, the disc itself. Interferometry resolves Polaris at 3.2 milliarcseconds, and it measures 3.2 milliarcseconds from every observatory on Earth. This is the least accessible of the three and the most exact, and it agrees with the other two.

And linear perspective cannot be brought in to save this one, because the rescue is a vertical magnification — that is what §3 works out. A magnification changes angular separations by definition. It cannot bend Polaris's elevation down by 19° while leaving the 16° gap to Kochab untouched; whatever it does to the one, it does to the other. The two halves of the argument pull in opposite directions, and the sky sits stubbornly between them, unchanged.

3 · What the visual geometry would have to do

The chapter's answer to all of that is that the angles were never tangible in the first place. What a person measures is visual geometry — Angle A — and it comes out smaller than the physical Angle B. Feed A into the trigonometry and of course you get nonsense.

That is a real distinction and it deserves to be taken at full strength. But before it can be tested it has to be given physical content, because there is a step here that is easy to walk past.

The sharpest way to put it is a fact about light known since Fermat: no medium can know how far away the source is. A ray arriving at a point carries no label saying where it started. What it does next depends only on its direction there and the refractive index there — so a bending field is a function of position and direction, and of nothing else. That is what makes the Sun and Moon usable as test objects later: the field cannot be one thing for a star 433 light years away and another for a source eight light-minutes away, because it has no way of telling them apart.

And perspective is a property of a projection, not of a light ray. It is what happens to rays that have already arrived when they get flattened onto a picture — a retina, a sensor, a canvas. Two rails converge in a photograph because the projection squeezes them together, not because the rails moved. Perspective cannot change the angle at which a ray comes in, because it only operates after the ray is in.

And the angle at which the ray comes in is precisely what the measurement is. A theodolite is levelled against gravity — a spirit bubble, or in the precise instruments a pool of mercury — and it reads the angle between that level and the incoming ray. There is no image plane anywhere in the answer. Nothing in it depends on how eyes work.

Which brings the figure on p. 145 back into focus. It shows the problem without naming it. Those converging straight lines are dismissed as the wrong assumption — but look at what has to replace them. Not one curve. A set of curves, one per observer, each bent by a different amount, all of them produced by the same air. Air does not know who is looking through it.

So the question becomes answerable: how much bending is needed? Anchoring at 45°N again:

Note the direction as well as the size. Light bends towards denser air, and everywhere above the first few metres the denser air is underneath — so refraction makes objects appear higher, while every correction in that column runs the other way. The rescue needs starlight pushed down, by amounts two to three hundred times larger than the real effect that pushes it up.

Pick whichever anchor helps most — it still fails

Those three rows are samples. The measurement is not three points, though. It is 69 miles per degree continuously — at 71°N, at 43½°N, at 12°N, at every latitude anyone has ever stood at. So the required correction is not three numbers either. It is a function of where you are standing, and it has to come out right everywhere along the walk.

Written out in full, with the lamp still anchored at 45°N, it does something the three-row version hides:

The same air, the same star — lifting the ray for an observer in Norway, lowering it for one in Mexico, and passing through exactly zero in between.

Real refraction has a characteristic shape. It is zero at the zenith, grows as you look lower, and is largest at the horizon — and clear of the surface layer it bends only one way. So the friendliest possible requirement is one shaped like that: no sign change, nothing to correct overhead, growing smoothly toward the horizon. Ask what lamp height delivers it, and two facts drop out of the geometry that no choice can touch.

• — At the pole the correction is zero, for every height. Polaris is overhead; there is nothing to bend. That end is free.
• — At the equator the correction is a downward push, for every height. Polaris is observed on the horizon, and a lamp at any finite height above a flat plane stands above the horizon from anywhere. So the star must always be pushed down there. No height avoids it.

Between those two fixed ends, the height decides the shape. Below a certain height the curve rises first and falls later, and the sign flips on the way. At or above it, the correction is downward everywhere and grows smoothly from nothing at the pole to its maximum at the equator — monotonic, single-signed, refraction-shaped. That threshold is 3,963 miles, or 6,378 km.

So take it. Raise the lamp to one Earth radius, drop the 45° anchor entirely, and grant the chapter the most physically respectable field its own geometry can produce:

And the high-latitude agreement is genuinely good — that deserves saying plainly, because it is the honest reason the flat picture feels workable to anyone who has only ever looked at Polaris from the northern half of the northern hemisphere. At 80°N this lamp is out by six arcminutes. It is a real near-miss, and it is not luck: a lamp one Earth radius above a plane laid tangent at the pole reproduces a sphere of that radius, to first order, in the pole's neighbourhood. The flat model works precisely where a flat approximation to a sphere works, and comes apart as you leave it.

So the model has two settings and both are closed. Anchor it low and the field must change sign, which means the air has to know which measurement was used to set the lamp. Anchor it at or above one Earth radius and the field is beautifully behaved and points the wrong way at every latitude on Earth. The boundary between the two is the Earth's own radius, which is a strange place for a flat model to keep finding itself.

Does any of this assume the globe?

It is the right question to stop and ask, because if the answer were yes the whole section would be worthless. Refraction theory was built on a spherical Earth wrapped in a spherical atmosphere. The tables in the back of a nautical almanac were fitted to observations reduced with spherical astronomy. If “real refraction is 0.09°” only ever means “observed altitude minus globe-predicted altitude,” then it is a residual of the globe and quoting it against a flat model is circular. The argument would be assuming its own conclusion and dressing it up in arcminutes.

So set all of it aside — Oriani, the almanac, the standard tables, the lot — and rebuild the number from two things that have nothing whatever to do with the shape of the Earth.

• 1. The refractive index of air, measured on a bench. n − 1 = 0.000293 at 0 °C and one atmosphere. An interferometer, a gas cell, a laboratory. No sky, no stars, no assumed planet.
• 2. Snell's law across horizontal layers. On a flat plane the air stratifies horizontally — that is what a fluid does in a downward acceleration field, whichever account of the field you prefer, and it is the only arrangement that stays put rather than overturning.

Those two together give an exact conserved quantity along any ray:

n × sin z = the same number at every point on the path

where z is the ray's angle from vertical. This is not a small-angle approximation and not a series expansion. It is exact, at every angle short of the horizontal, and there is no Earth radius anywhere in it. Better still, the whole profile cancels. Whatever the air does in between — inversions, ducts, exotic lapse rates, layers stacked any way you like — only the two endpoints survive:

sin z_observed ÷ sin z_true = n at the source ÷ n at the observer

Three things follow, and not one of them needed a globe to get here.

The direction is fixed. Polaris sits above the air, where n = 1. The observer sits at the bottom of it, where n is larger. So sin z_observed comes out smaller than sin z_true, the apparent zenith distance shrinks, and the star appears higher. Always. Exactly. For every profile there is or could be. Getting it to appear lower needs n at the observer to be below one — air less optically dense than a vacuum, at visible wavelengths, in a neutral gas.

The magnitude comes out right. Run the formula at 45° elevation with bench-measured air and it returns 1.008 arcminutes. The almanac says 1.0. At 10° elevation the plane-parallel form returns 5.7 arcminutes where the almanac tabulates 5.3 and Bennett's standard formula gives 5.4 — the small excess being the curvature term a flat Earth does not have, which is the flat model being handed the larger of the two. Every “real refraction” column on this page uses the standard 5.4′ (0.090°) rather than the flat-friendly 5.7′. The tables are not an input to this argument. They are a check on it, and they pass.

And it has to be one number for everybody. This is the part that finishes it. The ratio in that formula is a property of the air and the two altitudes — the ground, and wherever Polaris is. It is one physical constant, the same for every observer looking at the same lamp. Here is what the flat model needs it to be, taking the steel-manned version with the lamp at one Earth radius:

One atmosphere. One lamp altitude. One ratio. The flat model needs it to hold five different values simultaneously, and the spread is not marginal — it is twenty per cent.

It is worth seeing what even a single one of those values would mean physically. A ratio of 1.198 puts the refractive index at Polaris's altitude at 1.198, against 1.000293 at the ground. Since n − 1 tracks density, that is air at roughly 870 kg/m³ — seven-tenths the density of water — sitting on top of ordinary sea-level air, permanently, without falling. And the observer at 80°N needs the same air to be at 45 kg/m³.

Two honest caveats. The identity assumes n varies with height only; an atmosphere shaped like a dome, with its optical layers tilted relative to the ground, would break it — at the cost of horizontal pressure gradients large enough to read on a barometer and to drive permanent hemisphere-scale winds. And it permits a turning point when a ray reaches horizontal, which is what ducting is; that is a real loophole for sightlines within a degree or so of the horizon, which is why the equator row is the soft one. It is not available at 45° or 30° of elevation, and those rows fail on their own.

And if it is a lens rather than a layer?

There is one more mechanism to put on the table, and it is the strongest one left, because it genuinely does escape the identity above. Everything so far assumed the air is stratified — optical layers lying flat, parallel to the ground. That assumption is what produced the exact invariant, and it is what made the profile cancel. So break it. Suppose the atmosphere works as a lens: a shaped body of air with curved optical surfaces, bending light the way a lens does rather than the way a stack of flat layers does.

The dichotomy here is complete, which is what makes it worth walking. Either the optical surfaces lie parallel to the ground everywhere, or they do not. If they do, n is a function of height alone and we are back in the case already closed. If they do not, the refractive-index gradient has a horizontal component — and light bends toward the gradient. There is no third arrangement.

So a lens escapes the invariant. What it buys with is the one thing a stack of layers can never do: it deflects light sideways.

And the model cannot choose where to put it. To move Polaris, the lens has to be centred on the pole — that is the axis of the whole system. Which means its deflection is radial about the pole: everything in the sky is pushed toward or away from the celestial pole, not toward or away from the horizon.

For Polaris those are the same thing, which is exactly why the idea is tempting. Polaris sits due north, so “toward the pole” and “down” coincide for it perfectly. But nothing else in the sky is due north.

• — Anything off the meridian gets shoved sideways. The angle between “up” and “toward the pole” is the parallactic angle, and at 45°N it reaches 45° for an object on the eastern or western horizon. A field strong enough to move Polaris several degrees moves that object comparably far horizontally.
• — Which is a clock error, and then a navigation error. A star transits when its azimuth is exactly north. Displace azimuth by a degree and, for the equinox Sun at 45°N, transit moves by 2.8 minutes. Three degrees is 8.5 minutes, and at sea four minutes of time is a degree of longitude — up to sixty nautical miles, less as you go north. Meridian instruments were the definition of time for two centuries and they resolve arcseconds. This proposal introduces degrees.
• — Constellations would be sheared, not merely squashed. Layered air compresses everything toward the horizon, so circles become ellipses with upright axes — the same orientation everywhere in the sky. A pole-centred lens compresses along the line to the pole, so the axes come out tilted, by a different angle in every direction you look. Two identical star pairs at the same altitude, one north and one east, would show different separations and different orientations.

Which is the multi-observer problem again, and now in its worst form. It no longer needs two people in different countries. It needs one person, one field, one evening, turning around.

The same conclusion from the spherical side — a ceiling known since 1787

None of what follows is load-bearing, since the previous subsection settled the question without it. But the classical result is worth having, both because it is the version in the literature and because it shows the two geometries agreeing. The obvious reply to a refraction argument is that refraction is variable, that strange things happen in strange air, and that the standard table is only an average. All true — near the horizon. It stops being true higher up, and not by accident.

Oriani's theorem, proved in 1787, shows that for altitudes above roughly 10 to 15° the refraction depends only on the refractive index at the observer — that is, on the local pressure and temperature. The vertical structure of the atmosphere cancels out of the result. Inversions, ducts, exotic lapse rates, layered temperature profiles: none of them can produce anomalous refraction up there, because the theorem says the layering does not enter the answer.

That is precisely why every dramatic refraction phenomenon on record — the Novaya Zemlya effect, Fata Morgana, looming, superior mirages — occurs at or below the horizon. They need a long, grazing path through a steep gradient. A ray arriving from 20° up crosses the atmosphere almost perpendicularly and has no such path available to it.

So the ceiling can be computed rather than guessed. Refraction scales as pressure over temperature, and those have record extremes:

One honest note on the second row before a meteorologist raises it. The −89°C record belongs to Vostok Station, which sits at 3,488 m where the air pressure is only about 620 hPa — so the refraction actually observed there was around 5.1′, below standard. Cold air is denser, thin air is not, and at Vostok the altitude wins. The row is kept as a bound on what that temperature could do at sea level, which is why it is labelled hypothetical; the third row is a combination that really occurs.

By 45° elevation standard refraction is down to 1 arcminute. It is a small-print correction in the back of a navigation almanac, and invisible to the naked eye.

Set that aside too. If the angles are compressed by those amounts, the compression has a shape: it must be gentle overhead and severe near the horizon, and we can read off exactly how severe. Expressed as a vertical magnification — by which is meant how much a small vertical gap in the true sky is stretched or squeezed by the time it reaches the eye, measured at each apparent elevation:

That is the field the rescue requires. It is not an add-on to the argument; it is the argument, written out with numbers in it.

Two things about that table look contradictory and are not. The field pushes every object down in absolute terms, and yet it stretches the gaps between objects near the horizon — because it pushes the higher of two objects down by less than the lower one. And on the 45°N fit the magnification falls back below 1 above about 62° of apparent elevation, which is the sign change this section opened with, seen from the other side. The rows quoted are the range the Sun and Moon actually occupy in §4.

Those magnifications come from the 45°N fit, which is the weaker of the two requirements — the Earth-radius version above needs 25.6° of correction at 10° elevation rather than 19.4°, and a correspondingly stronger field. Everything §4 does with this table is therefore an understatement of what the steel-manned version demands.

4 · Grant it anyway — and walk the Sun and the Moon through it

Set every one of those objections aside. Suppose the field exists, in exactly the shape §3 computed, however it manages it. That is not the end of the argument — it is the start of a different one, because a distortion field is not selective. It cannot be a thing that happens to Polaris. Everything crossing that patch of sky goes through it, and two of those things are looked at by everybody, every day.

A static field can always be tuned to a static object. Polaris never moves. The Sun and Moon cross that same sky twice a day, at every elevation from the horizon to overhead.

They are, in effect, the field's own test instruments — dragged through it continuously, in public, for free. And the field cannot let them through untouched, because the stretching is the whole point of it. The Sun and Moon are each about half a degree across. Their width is unaffected, since the distortion is vertical. Their height gets multiplied by the table above.

What is actually observed is the opposite, and far smaller. Refraction squashes the Sun vertically — wider than tall, not taller than wide — by 0.8% at 10° elevation and 0.2% at 20°. The only distortion anybody sees without instruments is the familiar oval Sun in the last degree before it sets — and that is a squash of about a fifth: standard refraction lifts the lower limb some 6 arcminutes more than the upper, off a disc 32 arcminutes tall. That is the entire observable catalogue of atmospheric distortion of the sky.

And there is a second reading of the same test, which settles the lens question from §3 at the same time. The Sun does not sit still at one elevation; it walks the whole sky, so it samples the field's orientation as well as its strength. Layered air and a pole-centred lens make different predictions about that orientation, and both are wrong in a way anybody with a camera can check.

And the Moon makes it worse still, because the Moon is not just a disc but a disc with markings. A vertical stretch of 3.3× would distort the maria into unrecognisable streaks at low elevation and let them relax back to familiar shapes overhead. Anybody who has photographed a rising Moon and compared it with one taken later the same night has already run this experiment.

5 · Three smaller things

The chapter's own arithmetic refutes its parallel-light claim

p. 140 argues that because Polaris is a tiny dot, its light must spread radially and cannot arrive as parallel rays — that textbook diagrams “visually blow Polaris up to the size of Earth” to make parallelism look plausible. But the correct calculation four pages earlier settles this the other way. Angular size is precisely the quantity that bounds how non-parallel a source's light can be. All light from Polaris arrives inside a cone 3.2 milliarcseconds wide — that is the spread, and it is the chapter's own figure. Rays reaching opposite sides of the Earth differ in direction by 6.4×10⁻⁷ arcseconds, roughly 1,500 times finer than the best optical interferometer can resolve.

The smaller Polaris is correctly shown to be, the more perfectly parallel its light must be. The textbook diagram is not a trick, it is simply undrawable: with Earth as a 1 cm dot, Polaris belongs 3.2 million kilometres off the edge of the page, about eight times the distance to the Moon.

Theodolites do not have eyes

§3 made this the hinge of the whole argument; here is the one-line version. The chapter's mechanism is explicitly perceptual — “how their eyes work,” “our brain translates these eye angles.” But nobody measures Polaris with an eye. A theodolite has no brain, no prior experience and no perspective processing, and it returns the same angle a person does. So the effect is not in the observer. And once it is out of the observer it is in the air — where it must also bend the line to a surveyor's target a kilometre away, whose height can be checked against a tape measure on the spot. Surveyors would have noticed.

Perspective cannot manufacture a second pole

And one person can photograph the whole problem in a single night. From Ecuador, a camera pointed north records the sky turning anticlockwise about Polaris just above the horizon while a camera pointed south records it turning clockwise about σ Octantis just above the opposite horizon — simultaneously, from the same spot, on the same evening. Two centres of rotation, turning opposite ways, at once. A plane has one centre, or none.

Worth stating carefully, because the obvious version of this point is weaker than it looks. “Polaris is invisible from the southern hemisphere” is not by itself decisive: on a flat map the far south is simply far out, and a star sinking to the horizon with distance is what the model already claims. Fair enough.

What perspective cannot do is produce a second centre of rotation. The southern sky turns about its own pole, in the opposite sense, with no northern counterpart. A vanishing point makes things disappear; it does not create a new pivot for everything else to turn around. A plane has one centre, and the sky has two.

6 · What would settle it

• — Photograph the Sun at 10° and at 45° on the same day, same lens. Measure the width and height in pixels. The rescue in §3 requires a 3.3× vertical stretch at the lower elevation; refraction predicts a 0.8% squash. These are not close, and the equipment is a phone and a tripod.
• — Drive 69 miles south and measure Polaris. Then do it again from a different latitude. If the drop per 69 miles changes with latitude, the flat model is in business. If it stays at one degree, it isn't.
• — Shoot a rising Moon and a high Moon. Compare the maria. Any vertical distortion of the required size is unmissable in the pattern, not just the outline.
• — Point a theodolite at a target of known height a kilometre away. Tape-measure the height independently. If visual angles are compressed, this is where it shows up at a scale anyone can check.

The invitation this page is answering

“I may be wrong about the stars, and I am open to changing my mind… Could someone more knowledgeable take my analysis to the next level?”
— Levi Miller, Globe Deconstruction, p. 131 (review draft)

That is a rarer sentence than it should be in this genre, and it deserves to be taken at face value. Everything above is offered as the answer to it — not as a demonstration that the chapter is foolish, which it is not, but as the next level: the arithmetic run out to where it stops working, the strongest available version of the rescue built rather than caricatured, and the places where the chapter is simply right marked as such.

The honest bottom line

Linear perspective is real, refraction is real, and the chapter is right that the usual reply skips a step. But both effects have known sizes, and the gap is not small: 19° is needed and 0.09° is available, pushing the wrong way. And the escape is not free. Bending starlight by that much means building a distortion field, and a field is not selective — the Sun and Moon cross it twice a day, at every elevation, in front of everybody with a camera. The model requires them to be eggs. They are circles. That is not a measurement anyone has to be trusted about, which is the best kind of answer to have.

Sources & further reading

• Refraction figures: G. G. Bennett, “The calculation of astronomical refraction in marine navigation,” Journal of Navigation 35 (1982), and A. T. Young's refraction pages for the flattened Sun and the sinking/looming glossary.
• The oblateness measurement: Maupertuis, La Figure de la Terre (1738), and La Condamine's Peru arc — the French Geodesic Missions of 1735–44, which are this experiment run three centuries early.
• Latitude by Polaris at sea: Nautical Almanac Pole Star tables and Bowditch, American Practical Navigator — three centuries of navigators using altitude = latitude with a fixed correction.
• Polaris's angular diameter: Nordgren et al., ApJ 118, 3032 (1999), NPOI interferometry, 3.28 mas. The distance remains disputed — Turner et al., ApJ 762, L8 (2013) against the Hipparcos value.
• The Moon illusion: Kaufman & Rock, Science 136 (1962); Ross & Plug, The Mystery of the Moon Illusion (2002).
• The argument's ancestry: Samuel Rowbotham, Zetetic Astronomy (1881) — the “law of perspective” chapters are where visual-against-tangible geometry originates.
• Levi Miller, Globe Deconstruction? (prerelease draft, 2026), section “Celestial Globes Exposed,” printed pp. 124–152. The Polaris height argument is at pp. 143–145; the angular-size calculation at p. 136; the parallel-light claim at p. 140.
• Atmospheric refraction values computed from Bennett's formula, the standard approximation used in navigation almanacs, scaled by pressure and temperature.
• Oriani's theorem and the stability of refraction above 10–15° altitude: A. T. Young, “Astronomical Refraction”, San Diego State University — the standard modern reference on the subject.
• Record atmospheric extremes used for the ceiling in §3: highest sea-level pressure 1083.8 hPa (Agata, Siberia, 31 December 1968); lowest recorded temperature −89.2 °C (Vostok Station, 21 July 1983).
• Polaris distance and radius: standard published values, 433 light years and approximately 46 solar radii, giving an angular diameter of 3.2 milliarcseconds.
• Companion pages: the angular-resolution chapter answered, and does the Earth spin, or does the sky.