Sunlight bends in air, and the bend is about one arcminute

Nobody proved that sunlight travels straight through air, because it does not: air bends light, and the bend has been measured since Tycho Brahe. The premise is conceded in full. The question’s force comes from a switch of regime. Refraction is a large, unruly, unpredictable effect along the ground, which is where the surveyor’s constant on p. 202 comes from, and a small, stable, routinely-corrected one for a sun high in the sky, which is where shadow angles are measured. For a sun 45° up it is about one arcminute — a sixtieth of a degree; an arcsecond is a sixtieth of that — and the argument needed degrees.

1 · What the book argues, and where

The question card at p. 55 is the short form. The claim is set up at p. 34, in a list of what globe defenders offer as proof: “Sun elevation angle measurements using shadows across latitude lines up with mathematical precision with the globe.” It is taken up at pp. 35–36, where Eratosthenes heads four proofs to be dismantled, and where two assumptions are named — that sunlight is effectively parallel on arrival, and that the Earth is a sphere — of which he writes, “He did not verify curvature before this experiment was conducted.” Page 36 then hands off explicitly: the inconsistent-sun-height objection rests on an unproven assumption “which is explained on page 202.”

Page 202 is the argument. Headed “What about the shadow-angle measurements with the Sun that ‘match’ globe predictions?”, it names the coordinated experiments — “10 or more participants at varying latitudes” — and their “one critical assumption”: “that a scientist has proven that the Sun’s light is completely straight over long distances.” The mechanism named is terrestrial refraction: surveyors and geodesists cannot assume their laser levels or sightlines are straight, and carry a constant to account for it. Beside the text is a table headed standard atmospheric refraction, k = 0.13, captioned: “This table proves we cannot assume sunlight travels in straight lines over long distances through the atmosphere.” The right-hand column below recomputes it from the surveyor’s formula, Δh = k D²/2R: the apparent lift of a target at distance D on an Earth of radius R.

The table is correct. Two observations about it, one trivial and one not.

The trivial one: the numbers are computed with k = 0.14 and the polar radius, not the k = 0.13 printed in the heading. The recomputed column above uses his figures rather than his caption, which is why it agrees to the last digit; at 0.13 the rows would read 1.0 cm, 25 cm, 1.02 m, 4.06 m, 25.6 m, 101.9 m. Nothing turns on it — 0.14 is a perfectly ordinary default, the one Topcon ships and WSDOT specifies — but the two do not match.

Read the bottom row for what it says, too. 110 m of displacement over 100 km sounds enormous, but expressed as the thing a shadow-angle measurement cares about, a direction, it is k D/2R = 3.8 arcminutes — for a ray which spent a hundred kilometres inside the surface layer. The whole atmosphere, crossed once by a sun 45° up, supplies about one. Run his own law far enough to matter and it says how far: to accumulate the 17° the flat model is short at the Arctic Circle in section 7, a ray would have to graze the ground for about 27,000 km — two-thirds of the way around the world. Apparent shift throughout, kD/2R, which is what a theodolite reads and the right analogue for an apparent sun elevation; the total bending of the ray is twice it.

The usual textbook presentation of Eratosthenes states the parallel-ray assumption in passing, puts no number on it, and does not mention refraction; asking for the number is fair. So the question can only be answered by measuring the bend, in the regime where the measurement is made, and showing what size of conclusion it can and cannot overturn.

2 · The Sun’s distance had already been measured

Neither of the two things p. 36 says Eratosthenes “assumed” — parallel sunlight and a spherical Earth — was an assumption. Both were prior results, reached by observations with nothing to do with shadow lengths.

The Sun’s distance had been measured, by a method that says nothing about the Earth’s shape. Aristarchus of Samos (c. 310–230 BC), a generation before Eratosthenes, saw that when the Moon is exactly half-lit the Sun–Moon–Earth angle at the Moon is a right angle, so measuring the remaining angle from Earth gives the ratio of the two distances. He measured 87° and concluded that the Sun’s distance is “greater than 18 times, but less than 20 times, the distance of the Moon,” with a diameter between 19/3 and 43/6 that of the Earth. Geometry plus one angle; no Earth model enters it. He was badly wrong — the true dichotomy angle is about 89.85°, which no naked eye can tell from 90°, so his answer came out roughly eighteen times too small — and it made no difference, because eighteen times too small was still an order of magnitude more than the measurement needed.

So the parallel-ray premise was a measured lower bound, an order of magnitude clear of where it would have started to matter, produced by someone doing something else entirely.

And the Earth’s shape was a prior conclusion too. Aristotle, around 350 BC, gave two observational arguments in On the Heavens: the Earth’s shadow on the Moon during a lunar eclipse is round every time, at every orientation, which a disc cannot manage except face-on; and southern constellations stand higher in the sky for southern observers than northern ones. Neither involves a shadow length or a sun elevation, so neither is circular with respect to what Eratosthenes went on to measure. They are inferences from repeated observation, not measurements with error bars, and section 7 concedes that a two-station shadow measurement cannot by itself decide between a far sun on a globe and a near sun over a flat plane. The narrow claim here is the one p. 36 disputes: sphericity was arrived at independently, from evidence of a different kind.

3 · Refraction is overwhelmingly a horizontal problem

Hirt and colleagues put two pairs of motorised total stations on a surveyed 800 m baseline, 1.8 m above grass, and took reciprocal vertical angles once a minute for 33 hours across five days. The refraction coefficient k ran from −4 to +16 on sunny days, against the textbook 0.13, fluctuating by 1 to 1.5 on ten- to thirty-minute timescales. For a grazing sightline the standard number can be wrong by two orders of magnitude, and in both directions.

Engineers who need light to travel straight over that kind of distance take the air out. SLAC’s two-mile linear accelerator was aligned through a 24-inch evacuated pipe at roughly 10–25 torr — a thirtieth of sea-level pressure or less — explicitly “to avoid distortion and translation of the image due to refraction of the light in the atmosphere.” Asked whether a 250 m system would need vacuum, Herrmannsfeldt answered: “you will need some kind of vacuum, yes… 250 m is too long for reliable work.”

A physicist who will not trust air over a 250 m horizontal sightline is saying the effect is not negligible in general. It is negligible for a particular geometry, and the reason the two regimes diverge so violently falls out of the same formalism that governs both. The bending a layer contributes is Δn · tan z, where n is the refractive index of air and z is the ray’s angle from the vertical, the zenith angle. Take a 2 m surface layer with 10 K of temperature drop across it: Δn = (n−1)·ΔT/T = 9.63 × 10⁻⁶. A ray at 60° elevation crosses that layer once and picks up 9.63 × 10⁻⁶ × tan 30° = 1.15 arcseconds. A horizontal sightline never leaves it: the ray curvature κ = 4.81 × 10⁻⁶ per metre acts over the whole 800 m, giving 790 arcseconds of direction change and about 1.5 m of apparent target rise.

Same air, same gradient, and a factor of about seven hundred in the bending — purely from how long the ray stays inside the disturbed layer.

That 10 K over 2 m corresponds to k = 30.7, beyond Hirt’s observed maximum, so it is an extreme rather than a typical case. Using 2 K gives k = 6.1, comfortably inside his measured range, and 0.23 arcseconds at 60° elevation. The conclusion does not depend on which is chosen. Our flat-surface test page handles the near-horizon case in full, including the surveyor’s k ≈ 0.13–0.17 and the seven-sixths-Earth rule; that page, not this one, is where the grazing-incidence argument properly lives.

Two things a surveyor would add, worked through in the method notes at the end of this page: the 0.13 was never a near-ground number in the first place — it belongs to the intermediate atmosphere from about 20–30 m up, where it is well behaved — and survey practice keeps sight lines short enough that even Hirt’s wildest excursions are worth millimetres. In the discipline that supplies the constant, refraction is the small term, a seventh of the curvature correction it is always paired with. Either way it is a value for short sight lines near the ground. A sunbeam at noon is neither short nor near the ground.

4 · Why “hundreds of miles of gas” is not the quantity

The phrase invites you to picture hundreds of miles of accumulated unknown. The physics deletes that unknown, exactly.

Model the atmosphere as horizontal layers of differing density and apply Snell’s law at each interface. The refracted angle leaving one layer is the incident angle entering the next, so the product n · sin z is identical at every point along the ray. Every intermediate layer cancels. Only the endpoints survive. A prepared reader will object that the derivation stacks flat layers, which is the shape under dispute. On concentric shells the invariant becomes Bouguer’s n r sin z = constant, and above roughly 10–15° of elevation the two agree to well inside the arcsecond figures used here. The plane-parallel form is the simpler one to show; nothing on this page depends on which is used.

The total bending does not depend on the density profile, the temperature profile, the scale height, the number of layers, or how many miles of air the ray crossed.

It depends on the refractive index of air where the observer is standing, and on the zenith angle. That is the whole of it, and it reduces to r = (n − 1)·tan z, which Young’s treatment states is good to better than one part in ten thousand while r stays under about half a degree — covering everything above roughly 10° of elevation, which is to say everything a noon shadow measurement ever uses.

The sign is known too. Where density falls monotonically upward — the ordinary case for a high sun on a clear day — the ray curves toward the denser air beneath it, the source appears higher than its true direction, and the shadow is shortened. Surface inversions and horizontal gradients, the conditions that produce mirages, belong to the ground-level regime of section 3. For a high sun, refraction is a correction with a known sign, not a scrambler.

And the index itself is not an astronomical quantity. The Ciddor dispersion relation gives n − 1 = 2.7715 × 10⁻⁴ at 589 nm for standard air at 15 °C and 101.325 kPa. NIST serves it through its Engineering Metrology Toolbox, because laser interferometric measurement of gauge blocks cannot be done without correcting for the optical path in air. It exists so that machine shops can agree on what a millimetre is.

This is why nobody mentions refraction above ten degrees. Not because it vanishes — shadow-angle work that wanted arcsecond accuracy would correct for it — but because it gets smaller, by a factor of about thirty-five from the horizon to 45°, and because the kind of uncertainty changes. Near the ground, what dominates is the shape of the vertical gradient in the first few metres of air, the profile, which nobody can model; it is what Hirt was measuring when k came out between −4 and +16. Above roughly ten degrees the invariant removes the profile from the problem entirely, and what is left depends only on the density of the air at the observer: a barometer reading and a thermometer reading, which every station has.

The near-horizon regime has an error nobody can compute. The high-sun regime has an error anyone can look up.

5 · The measured range of standard refraction

Here is the distribution, which is the thing p. 202 gestures at without giving. The middle column is ordinary conditions. The two beside it are the envelope of terrestrial weather: 840 hPa at +30 °C, and the record sea-level pressure of 1083.8 hPa recorded at Agata, Siberia in 1968, at −50 °C. Those extremes are bounds, not a forecast — no station experiences either, and certainly not both. The envelope captures bulk density, which under the invariant is the whole story for a high sun; near the horizon, where profile anomalies put the real bending outside any pressure-and-temperature envelope, section 3 applies instead.

Two independent routes agree on the middle column. The metrology-bench value fed into r = (n−1)·tan z gives 57.2″ at 45° elevation for 15 °C air and 60.3″ for 0 °C; Bennett’s 1982 navigation formula, fitted to sky observations for sextant correction rather than derived, gives 59.7″. The residual is mostly the few degrees of temperature separating the two conventions. Bennett is quoted here with its pressure–temperature factor set to unity, corresponding to about 1010 hPa and 10 °C; the envelope columns apply that factor at its extremes. The whole table is one formula and two numbers, so a reader can rebuild it.

6 · Many observers is the control, not the weakness

Page 202 treats the coordinated multi-latitude experiments as the thing that makes the proof vulnerable. It is the opposite: that design is the standard way of controlling for exactly the error being alleged, for four reasons that can each be checked.

• 1. The signal grows with baseline and the error does not. Noon sun elevation changes by one degree per degree of latitude — 3,600 arcseconds. Refraction does not care how far apart the stations are; it is bounded at each one by the local air. Put two stations ten degrees apart and the signal is 36,000 arcseconds against a worst-case differential of about 35. And note the book’s own word for what the atmosphere does: it creates “a constant distortion of the Sun’s light at long distances.” A constant is precisely the thing a difference between two stations deletes.
• 2. The errors are uncorrelated, so they average down. Different sites, different weather systems, different days, different instruments, different observers. Whatever refraction does at each station is independent of what it does at the others, which is the condition for the mean of N stations to converge as 1 / √N while the latitude signal stays put.
• 3. The dataset measures its own error bar. N stations give N−1 independent estimates of the same quantity. Their scatter is the empirical bound on the total error — refraction, instrument, timing, siting, all of it — obtained from inside the experiment. Nobody has to assume the air is well behaved; if it were not, the estimates would disagree, and by how much.
• 4. To fake the result, the errors would have to track latitude. Averaging kills random error, not systematic error, so the real question is whether refraction could be systematically correlated with latitude in the same shape as curvature. It would have to be, on every day the experiment is run, in both hemispheres, at every station.

That fourth one deserves a number rather than a rhetorical question, because there is a genuine latitude-correlated systematic.

A second systematic, which averaging also does not touch: if every observer uses the same flawed procedure, the flaw survives any number of observers. That is a real limitation of the design, and it argues for observers who are not coordinating on method — which is what the eighteenth-century arcs in section 7 supply, surveyed by rival teams to settle a different question.

7 · What it comes to, on the actual measurement

The arithmetic, with inputs so it can be redone. Alexandria sits at 31.20° N and Syene at 24.09° N, with the obliquity — the tilt of the Earth’s axis — about 23.72° in 240 BC. Noon solstice elevation is 82.52° at the first and 89.63° at the second. Bennett gives 7.79″ and 0.31″. Both stations are lifted; only the difference touches the answer, and the difference is 7.5 arcseconds against a signal of 7.11° — 25,596 arcseconds. That is a perturbation of 0.029 per cent, about 12 km on a 40,000 km circumference.

Set it against the measurement’s own error budget. Eratosthenes reported 252,000 stadia, and the stadion is known only to lie between about 155 and 160 m — a range of −2.4% to +0.8% against the true figure.

Refraction is about seventy times smaller than the uncertainty the ancient measurement already carried.

The parallel-ray assumption gets the same treatment. The astronomical unit is fixed at 149,597,870,700 m by IAU 2012 Resolution B2, because ranging measures it directly by time of flight. Across the 843 km between the two sites — the true separation, against the 787.5 km his own 5,000-stadia figure implies — the direction to the Sun differs by 1.16 arcseconds. So the two optical assumptions Q12 targets are worth 7.5″ and 1.16″ against 25,596″.

But two stations genuinely cannot decide it

This is the part of the objection with real force. Take his own datum — 5,000 stadia, which at 157.5 m to the stadion is 787.5 km — and 7.2° of angle. On a globe the circumference is 787.5 × 50 = 39,375 km. On a flat plane with a nearby sun, treating Syene as the subsolar point — the spot directly beneath the sun — the sun’s height is 6,234 km. Both fits are exact. Two observations, two models, one dataset, nothing to choose between them. Anyone presenting two sticks as proof of sphericity has claimed more than the data supports.

What resolves it is a third station and a fourth — section 6’s design. The discriminator is unusually clean, because the flat model’s one free parameter cancels out of it. On a plane with the sun at height H, ground distance to the subsolar point is x = H tan φ, so ground distance per degree of observed elevation is H sec²φ (π/180). H is an overall scale and divides out of every ratio.

And that quantity was measured, repeatedly, by people arguing about something else: the seventeenth- and eighteenth-century arcs were run to settle whether the Earth is oblate as Newton predicted or prolate as Cassini did, a dispute in which neither side had any stake in this one.

Converted at 1.949036 m to the toise du Pérou. Maupertuis’ Lapland figure is about 0.36% larger than the modern value, and was contested at the time. An error of 0.36% cannot rescue a prediction wrong by a factor of 6.19.

The last escape, and why refraction cannot take it

That derivation assumes rays travel straight. A reader can escape it by proposing that the air bends sunlight enough to fake the observations. This is the escape route the whole question is really about, so here is what it costs. The ground distance is taken from the three measured arcs, not from the modern ellipsoid, so that nothing in the calculation assumes the conclusion.

Calibrate the flat model’s sun height on the measured equatorial degree, where the two models agree best: H = 110,577 m ÷ (π/180) = 6,336 km. The ground distance from the equator to 66.3°, trapezoidal across the three surveys above, comes to 7,364 km. The flat model then predicts an equinox noon elevation there of arctan(6336 ÷ 7364) = 40.71°. Every observer there measures 23.70°.

The gap is 17.01 degrees. The atmosphere at that elevation supplies 2.25 arcminutes — 0.0375°. It is short by a factor of 454 on standard refraction; on the coldest polar air of section 6, 2.66′, the factor is 384.

Grant the flat model the largest refraction the atmosphere produces anywhere on Earth — the 34.5 arcminutes at the astronomical horizon, 0.575° — and it is still short by a factor of thirty.

And the escape has a consequence you can photograph

Whatever bends sunlight by 17° at the Arctic Circle and by nothing at the zenith must squash the mid-afternoon Sun into an oval by about a quarter. It does not, so nothing is bending sunlight by 17°. That is the test, and it does not care what the bending mechanism is.

The reply usually made at this point is that it is not atmospheric refraction at all — “electromagnetic acceleration”, light curving over a flat plane by some property of light itself, or a dome whose optics bend sunlight like a lens: a mechanism no barometer can price, so the magnitudes above are said not to apply. Grant it. Grant the air, or the dome, or the light itself, whatever bending the flat model needs, and refuse to argue magnitudes. The escape still fails, in a way anyone with a camera can check, because bending that large cannot be uniform. It has to be 17° at the Arctic Circle’s 23.7° noon sun and essentially nothing at the zenith, or the tropics would be as badly wrong as the Arctic. That is a mean gradient of about 0.26° of bending per degree of elevation.

Now put the Sun in it. The disc is 0.53° across, so its lower limb sits in air that must bend light 0.14° more than the air its upper limb sits in. The Sun would be squashed vertically by 26 per cent — visibly oval — every mid-latitude afternoon, not only at sunset.

The real differential across the disc at that elevation is 0.17 per cent. The Sun is round.

No mechanism escapes this. The oval follows from the bend varying with elevation, and the bend must vary with elevation or the tropics would be as badly wrong as the Arctic; the name given to the cause — air, field, dome — never enters the calculation. A filtered photograph of the Sun at 24° elevation, measured for height against width, is the whole test. Celestial Globes Exposed runs the same test against the bent starlight that chapter needs.

Computed from Bennett at the two limb elevations, 23.44° and 23.97°: 2.277′ against 2.221′, a difference of 0.056′. Everyone has seen the effect this argument requires — it is what flattens the Sun into an oval at the horizon, where refraction really does change fast with elevation. The flat-plus-bent-light model needs that flattening, at a hundred and fifty times its observed strength, in the middle of the sky. Solar diameter taken as 0.53°; the annual variation of a few per cent does not move the comparison.

8 · Where this page could be wrong, and what would change our mind

Q12 does not name an elevation. It says shadow angles, and this page answers the noon case, where the arcsecond figures hold. Serious shadow-angle work is done at local noon because that geometry is the well-behaved one, so restricting to it is not circular — but it is a choice of the favourable regime. p. 202 names refraction as the mechanism, which settles that branch, but it does not specify an elevation either. If the argument is specifically about long shadows near sunrise and sunset, the concessions get considerably larger and this page needs rewriting around Hirt’s k = −4 to +16 rather than around noon arcseconds.

His table is read off a page image, not extracted text. The p. 202 figure is a raster, so its six rows were transcribed by eye from the rendered page. They recompute exactly, but a misread digit is possible in a way it would not be with extracted text.

Horizontal homogeneity is assumed and unquantified. Every figure here assumes an atmosphere stratified horizontally. Real atmospheres have fronts. We could not find a measured bound on the residual from horizontal inhomogeneity at high sun elevations and do not assert it is zero. It is very unlikely to reach degrees; “unlikely” is not a number.

Several inputs are secondary-sourced. The 240 BC obliquity, the ancient station latitudes and Picard’s mean arc latitude are standard or approximate figures not traced to primary sources. Each is flagged at the point of use.

What would change our mind, specifically and in advance:

• — A consistent flat-model sun height across many stations. A published, replicable set of simultaneous solar elevations from four or more stations spanning at least 40° of latitude, in which the inferred H agrees between every station pair to better than 5%. Our claim is that it varies by about a factor of two across ordinary pairs and six or more by the Arctic Circle. One consistent dataset would end this page.
• — A meridian arc that grows the way the flat model needs. An independent survey showing ground distance per degree of measured noon elevation growing faster than about 1.05× between the equator and 60°. The flat model needs 4.00× there; the ellipsoid predicts 1.008×; the eighteenth-century arcs measured 1.0121× at 66.3°.
• — A larger refractive index of air. A bench measurement at one atmosphere in the visible giving n − 1 above about 1 × 10⁻³, roughly 3.6× the Ciddor value. It would force the whole error budget to be redone — and still leave the flat model a factor of about 100 short of the seventeen degrees it needs.
• — A failure of the invariant. A measured refraction at 45° elevation differing from (n−1)·tan z by more than a few arcseconds under stated pressure and temperature. Every quantitative claim here flows from n sin z = constant.
• — Circularity in the astronomical unit. Evidence that the AU as fixed by IAU 2012 B2 rests on an assumption about the Earth’s shape rather than on time-of-flight ranging. The 1.16-arcsecond bound would then be circular and we would drop it.

Sources & further reading

• P. E. Ciddor, "Refractive index of air: new equations for the visible and near infrared", Applied Optics 35, 1566–1573 (1996) — the dispersion relation giving n − 1 = 2.7715 × 10⁻⁴ for standard air at 15 °C, 101.325 kPa, 450 ppm CO₂, 589 nm. Served operationally by NIST's Engineering Metrology Toolbox (emtoolbox.nist.gov/wavelength/abstract.asp) for laser interferometric length metrology, which is why the number exists at all.
• G. G. Bennett, "The Calculation of Astronomical Refraction in Marine Navigation", Journal of Navigation 35(2), 255–259 (1982) — the refraction formula R = (0.28·P[hPa]/(T+273)) / tan(h + 7.31/(h + 4.4)), fitted to observations for sextant altitude correction. Establishes the refraction column used throughout this page.
• A. T. Young, "The Plane-parallel Model", aty.sdsu.edu/explain/atmos_refr/models/flat.html — states the n·sin z = constant invariant, the resulting approximation r = (n−1)·tan z, and its accuracy of better than one part in ten thousand while r stays under about half a degree.
• S. Y. van der Werf and W. H. Lehn, "Atmospheric refraction: a history", Applied Optics 44(27), 5624 (2005) — establishes that the first accurate measurements of atmospheric refraction were made by Tycho Brahe in the sixteenth century. We quote the abstract only; we did not obtain the full paper and quote none of Brahe's tabulated values.
• C. Hirt, S. Guillaume, A. Wisbar, B. Bürki and H. Sternberg, "Monitoring of the refraction coefficient in the lower atmosphere using a controlled setup of simultaneous reciprocal vertical angle measurements", J. Geophys. Res. Atmospheres, 2010JD014067 — 800 m baseline, 1.8 m above grass, 33 hours, ~2,000 epochs; measured refraction coefficient k from −4 to +16 against a textbook 0.13. Establishes the near-ground concession.
• W. B. Herrmannsfeldt, "SLAC Alignment System", proceedings.jacow.org/l66/papers/iv-05.pdf — the two-mile linac aligned through a 24-inch evacuated pipe at 10–25 torr to 0.025 mm, "to avoid distortion and translation of the image due to refraction of the light in the atmosphere". Establishes that engineers remove air from long horizontal sight lines. We found no statement in this paper about the linac's axis versus Earth's curvature and make no claim about it.
• Aristarchus of Samos, On the Sizes and Distances of the Sun and Moon (c. 310–230 BC) — the dichotomy method: hypothesis that at half-moon the Sun is “less than a quadrant by one-thirtieth of a quadrant” (87°), concluding the Sun’s distance is “greater than 18 times, but less than 20 times, the distance of the Moon,” with solar diameter between 19/3 and 43/6 Earth diameters. The true dichotomy angle is about 89.85°, so the result is roughly eighteen times too small. Quotations are the standard translated propositions as reported in secondary literature; we did not work from the Greek text. The implied solar distance of about 1,260–1,400 Earth radii follows from his lunar distance of roughly 70 Earth radii.
• Aristotle, On the Heavens II.14 (c. 350 BC) — the two observational arguments for a spherical Earth: the Earth’s shadow on the Moon is round at every lunar eclipse and every orientation, and southern constellations stand higher for southern observers. Cited for the arguments rather than quoted from the Loeb text, which we did not obtain.
• Surveying practice and the standard refraction coefficient: Noll & Rydlund, Procedures and Best Practices for Trigonometric Leveling in the U.S. Geological Survey (USGS TM 11-D3, 2020) — refraction “about 1/7 (0.143) the correction for Earth’s curvature”, table 4 tabulating it as negligible at 100 ft and 0.0009 ft at 500 ft, and the instruction to keep levelling lines under 500 ft. WSDOT Highway Surveying Manual ch. 3 — refraction offsets curvature by 14 per cent; differential levelling sights not to exceed 200 ft. Kharaghani, UNB Technical Report 132 (1987) — sight lines under 150 m with over a metre of ground clearance keep refraction inside Canadian first-order levelling specifications; 250 m by the reciprocal method. Instrument defaults differ by vendor: Leica TS16 uses k = 0.13, Topcon ships 0.14, Trimble Access offers 0.13, 0.142 or 0.2. The distance/height figures on this page are computed from Δh = kD²/2R, verified against the USGS table. We could not find a published measured distribution of k for ordinary survey conditions and do not assert one.
• Height dependence of the refraction coefficient: reported in the literature review of the Hirt et al. paper below — Brocks (1949) on Nanga Parbat giving k between +0.10 and +0.12, and Mavridis & Papadimitriou (1973) giving +0.12 to +0.20 between two Greek hills, with the conclusion of “fairly small variability of k in regions well above the surface”, and the statement that the Gaussian +0.13 “refers to the intermediate atmosphere” of roughly 20–30 m upward. Both campaigns reach us through Hirt’s citation of them; we did not open the primary papers. Hirt also states his own experiment was designed to replicate conditions “frequently found in geodetic practice”, so we do not describe it as an artificial worst case. Counter-case: Hennes, Zum Refraktionseinfluss auf terrestrische geodätische Messungen (2002) — at 100 m with a −0.2 K/m gradient, using 0.13 leaves 0.63 mgon of zenith error, exceeding instrument accuracy and common in summer over heated ground.
• Cleomedes, Caelestia i.7.49–52 — the surviving simplified account of Eratosthenes' measurement (his own On the measure of the Earth is lost): 5,000 stadia, 7.2°, 252,000 stadia. Modern assessment of the stadion length via the standard treatments; establishes the −2.4% to +0.8% error bar.
• Picard (Paris–Amiens, 1669), Bouguer and La Condamine (French Geodesic Mission to the Equator, 1735–44), and Maupertuis (Torne valley, Lapland, 1736–37; La Figure de la Terre, 1738) — degree-of-latitude measurements of 57,060, 56,734 and 57,422 toises, made to settle whether the Earth is oblate or prolate. Establishes the measured latitude dependence of the degree, from a dispute with no stake in the present one.
• Arc values: the Picard, Peru and Lapland degree lengths circulate with small variants between treatments (Peru as 56,734 / 56,748 / 56,753 toises, Lapland as 57,422 / 57,438). We use the commonest set; Todhunter, A History of the Mathematical Theories of Attraction and the Figure of the Earth (1873) is the standard compendium and is free. The choice is uncited to a specific edition. It does not matter: a 0.03% shift between variants sits against a prediction wrong by 6.19×.
• IAU 2012 Resolution B2, syrte.obspm.fr/IAU_resolutions/Res_IAU2012_B2.pdf — fixes the astronomical unit at 149,597,870,700 m exactly, on the stated grounds that modern range measurements make distance ratios unnecessary. Establishes solar horizontal parallax of 8.794″ without reference to the Earth's shape.
• Levi Miller, Globe Deconstruction? prerelease draft — p. 55, the text of Q12, quoted verbatim at the head of this page. The supporting argument is elsewhere in the book: p. 34 lists the shadow-angle claim among the proofs to be dismantled; pp. 35–36 are the Eratosthenes treatment, giving the Vsauce quotation and naming the parallel-ray and sphericity assumptions, with p. 36 cross-referring forward to p. 202; p. 202 carries the argument itself, headed “What about the shadow-angle measurements with the Sun that ‘match’ globe predictions?”, naming terrestrial refraction and the surveyor's k = 0.13. Separate arguments, not addressed here: p. 165 lists the local-Sun observational claims (crepuscular rays, cloud-top hotspots, sunset building shadows) and p. 131 extends the straight-line objection from sunlight to starlight. Page references are the book's own printed folios, checked against its page footers — p. 36's internal pointer to “page 202” lands on the shadow-angle page, which confirms the mapping. The k = 0.13 table on p. 202 is a raster image, transcribed from the page rendered at 200 dpi.