The Equator Flight Data Challenge, Answered

0 · The measurement he is asking for has already been made, at the gate, before every flight he has ever taken

Before any of the detail below, the thing that should be said first — and that the chapter, unlike the rest of the book, does not engage with.

The instrument this experiment would be validated against is the aircraft’s inertial reference system, a ring laser gyro package. It answers the question before the aeroplane moves. Not during the equatorial circuit, not from the flight recorder afterwards. On the ramp, during the ten-minute alignment that precedes every single departure.

Here is what it does. The accelerometers find the gravity vector: local down. The gyros find the Earth’s rotation vector, which is fixed in inertial space along the spin axis. Then it measures the angle between those two — two vectors and the angle between them, with nothing about the shape of the Earth entering the measurement. The label we put on the answer does use a model: calling it “co-latitude” is a WGS-84 reduction, and the latitude in question is geodetic. Reading it geocentrically instead would move Heathrow from 38.5° to 38.7°. The gap between the plumb line the accelerometers actually sense and the ellipsoid normal — the deflection of the vertical — is under a hundredth of a degree at lowland airports, twenty times smaller again.

Fly London to Cape Town and that angle changes by 85.4 degrees. Two stationary measurements, ten minutes each, at the two ends. No flight data required, and the route in between is irrelevant to it. And the axis is above the horizon at one end and below it at the other — not turning faster or slower, on the other side — a swing of 482.8° in daily carry between the two ends of one ordinary scheduled route. Every figure there is a total, not a rate, and every one of them is established while parked. A day here is the sidereal one, 23h 56m 04s for a full 360°, because the instrument measures against inertial space and not against the Sun.

The pattern even fails where the geometry says it must. North-finding depends on the horizontal component, which goes as cos φ and thins out toward the poles, so the Boeing 737 flight manual carries a hard limit: alignment must not be attempted above 78° 15′, where what is left has fallen from 15.04°/hr to 3.06. Above it you do not align at all — you align further south, where the tilt is still large enough to find, and carry that alignment north against GNSS on true-north grid procedures, because approaching the pole gravity and the rotation axis line up and there is no north left to find. Provenance of the 78° 15′ figure, the competing 78° 14.75′, and the separate ~82° maximum operating latitude that is set by magnetic variation rather than by the cosine: all in Sources. On a flat surface there is nothing for a cosine of latitude to be a function of. Nor is any of this a marginal signal: Cape Town’s vertical channel stands about 840 times clear of a navigation-grade gyro’s own drift, and the alignment is pass/fail rather than a quiet assumption — enter a position whose latitude disagrees with the one the unit has just derived from the rotation vector and the alignment is rejected and re-entered.

The book answers this, sixty pages later, and the answer does not reach the measurement

This is not new to him, and we should not pretend it is. At pp. 183–187, in the run headed “Evidence of Earth spin?”, he takes the ring laser gyroscope seriously and gives it a considered reply. He does not dispute that it reads something. He reattributes what it reads:

“So if the gyroscope measures anything while standing still to the observer, what you have detected is that the medium of light travel is drifting.”

That is an aether, and against this particular instrument it is the right objection to raise: a ring laser works by sending light both ways around a loop, so a drifting medium of light travel is a live rival explanation for what it reads. It fails three times over, each of them sufficient alone.

• 1. Instruments with no light in them read the same rotation. A spinning-mass marine gyrocompass finds true north from angular momentum. A gyrotheodolite does it with a suspended rotor, to arcseconds, underground. A hemispherical resonator gyro senses Coriolis forces on a vibrating quartz shell. A Foucault pendulum needs a wire and a weight, and precesses at 15.04°/hr × sin φ — the same sine of the same latitude. Not an optical path among them. The available reply is that the medium drags matter too. But a medium that drags spinning rotors, vibrating quartz and a pendulum bob by exactly the amount a rotating Earth would is no longer a rival to rotation — it is rotation under another name, and it still owes an explanation for why the amount goes as the sine of the latitude.
• 2. The substitution has been run on the aeroplane itself. Before ring lasers, airliners aligned on gimballed platforms spun by mechanical gyro wheels — the Delco Carousel IV of the early 747. Its own operations manual describes the same ramp procedure in the same terms: “During alignment, the gyros on the platform sense the earth’s rotation… the computer uses the resulting outputs to determine the platform’s relationship to true north,” performed only while parked, degrading above 80° of latitude. The same cosine, found by spinning metal, in an era that predates GPS entirely — so there was no satellite constellation for the answer to have been quietly seeded from. Those airframes were later refitted with ring-laser units dropped into the same bay, and went on navigating identically. Wheels and light, a generation apart, reporting the same rotation.
• 3. Grant the aether anyway and both observables survive it. His vortex turns about the vertical — his geometry, not our reading of it; the text names the Z-axis twice in plain words and the diagram stacks its speed rings vertically about a single source. Nor does it help to read that as one global axis through the pole rather than each observer’s own vertical: on a flat plane every local vertical is parallel to the global one, so the two readings make the identical prediction. And that prediction is that the sensed axis points straight down, at every airport on Earth, since the vertical is the only direction the vortex knows. The instrument finds it 38.5° from straight down in London, 88.6° in Singapore, and below the horizon altogether in Cape Town. A rotation about the vertical also contributes nothing at all to the horizontal channel, which is the whole of north-finding and the thing that runs out at 78°.

One thing his account gets right and deserves credit for: the Sagnac effect is what these gyros exploit, and asking whether the sensed rotation is the Earth’s or the medium’s is exactly the right question to ask of a ring laser. It is answered by walking away from ring lasers. His supporting data and the altitude test he proposes are taken up in §6, where they belong — the test is well posed, and the reason it cannot settle anything is not obvious.

1 · The challenge, stated fairly

The setup as printed: Quito to Pontianak along the equator, 8,250 miles at 560 mph, 14.73 hours, 119° of turn at 8.08°/hr, one steering input every 7.4 minutes, $715,000. Four things here are right, and they should be said first.

• — The experiment is sound. Turn rate is physical, it is recorded on every flight, and the two models genuinely disagree about it. This is a better argument than anything in the book resting on eyesight.
• — The globe’s prediction is exactly what he says. The equator is a great circle; flying it requires no turn at all. Not “a small turn” — zero, identically.
• — And he picked the right quantity. An aeroplane is not turned by the air. Lift acts perpendicular to the wings; to turn, you bank. So the flat model’s requirement is not a residual buried in a position log, it is a commanded action — a persistent one-directional bank, reversing on the return leg. Recorders log actions — though whether the certified recorder can resolve this one is §4.
• — And the analysis on p. 119 is correct. The attitude indicator’s pendulous vanes erect the gyro to local vertical, identically whether that vertical is normal to a plane or radial to a globe. “No shape debate points for either side” is right, and it is more careful than either camp usually manages. It comes back in §4, doing more work than he realised. Though the two jets he names carry no vane-erected instrument — a G800 or Global 8000 computes attitude in a laser inertial reference, which as §5 shows makes the modern “level” reference itself part of the Earth model.

2 · The number in the table

Where does 119° come from? Not from the airports. Mariscal Sucre is at 78.36°W and Supadio at 109.40°E — 187.8° apart eastbound, 172.2° westbound, about 13,000 or 11,900 miles. The route and the 8,250 miles are not describing the same flight.

It comes from the distance instead: 8,250 ÷ 24,901 × 360 = 119.3°, and 24,901 miles is the circumference of the equator on a globe. The figure presented as the flat model’s turning requirement was obtained by measuring the route on the globe.

Which raises the question the chapter has to answer before quoting any flat figure: how long is the equator on a flat Earth? p. 120 declines to say — the Gleason map is “just a distorted version of the globe,” its measured distances possibly “elongated to conform to a predefined Earth radius.” That is a more careful position than most flat-Earth writing, and it is also fatal to the table: the flat side of the comparison then has no scale, no distance and no angle. It also means pp. 118–120 distrust the distance scale that p. 121 turns round and trusts to four significant figures.

That is the Gauss–Bonnet theorem — turning supplied by the traveller, plus curvature enclosed by the circuit, always sums to 2π. A plane supplies none of it so the pilot supplies all; a globe supplies all so the pilot supplies none. Every disputed quantity — map scales, road lengths, who controls the tracking data — drops straight out. One exit closed: some flat maps are bipolar or rectangular, and on those the equator is not a loop, so eastbound circumnavigation is impossible — refuted by the Whitbread voyage the chapter itself cites on p. 121.

If a specific number is wanted for the partial leg, the only layout consistent with his own requirements — parallels as circles about the pole, spaced by true north–south distance — gives an equator of radius 6,225 miles and circumference 39,115 miles, longer than the globe’s by exactly π ÷ 2. On it, 8,250 miles is 75.9°, at 5.15°/hr, needing 76 one-degree inputs rather than 119. Anything that can tell 119° from 0° can tell 76° from 0°; only the number to look for changes.

3 · Why the equator, and what “turning” means

One definition first, because the whole dispute lives in it: turn here means yaw about the local vertical, which is what a rate gyro senses. The nose-down rotation of following the curve is §4’s business, not this.

“But you can just drive straight west without turning”

This is the objection to raise here, and it is the right one. Hold a heading of 270°: the compass does not move, nothing says turn. How can that be a turn?

You can indeed drive perfectly straight. You just stop going west when you do. Straight and due west are different instructions, and on a sphere they come apart. Stand ten metres from the North Pole and walk due west: you walk a circle twenty metres across, a lap every forty-five seconds, turning at eight degrees per second — compass reading 270° throughout. Nobody would say you had not turned. Walk south and the circle grows and the turn gentles — 30°/hr at 75°N, 8.1°/hr at 45°N, 2.2°/hr at 15°N — until at the equator it reaches exactly zero, the one latitude where hold the parallel and do not steer are the same instruction.

Your instinct is right about everything a pilot or driver would feel, though. At 45°N and 560 mph the bank needed to hold the parallel is 3.4 arcminutes, and nobody commands it — you command a heading, and the turn is what holding that heading means on a sphere. So both intuitions are correct about different things: “I am not turning” is right about the controls and the compass; “you are turning” is right about the path. The arbiter is a gyro, which holds against space rather than against a reference quietly rotating underneath it.

If the framing still feels like a definitional trick, close the circuit and it stops being one. Go right round a parallel and come back facing the way you started; something rotated through a full turn, and the only question is how much of it you supplied:

The globe column is 360° × sin φ and the gap is the curvature enclosed — the share of the turning the planet does for you. At the pole you circle a point, supply all of it, and the models agree exactly. At the equator the planet supplies all of it and they disagree by a full revolution. Which makes his route sharper than he argued: the lateral Coriolis deflection vanishes there too, where at 45°N a wings-level aircraft’s track curves at 21.3°/hr from Coriolis alone.

4 · The island of pilots, and what a flight would show

The chapter closes on pilots who have never seen a globe, protesting that they never point the nose down and never adjust for spin. His own p. 119 analysis answers them. The pendulous vanes erect the attitude indicator to local vertical continuously; on a globe local vertical rotates at exactly V÷R as you travel; an aircraft flown level against that indicator follows the curve automatically, with nobody pointing anything anywhere.

And the two directions are asymmetric in the flat model’s disfavour. Vertically, the aircraft is held on the curve by an equilibrium: constant pressure altitude, and an isobaric surface wraps the planet, so fly a Euclidean straight line and you climb off it and settle back. Horizontally there is no aerodynamic mechanism at all — whatever horizontal curvature exists has to come from bank. The flat model needs the kind of curve that cannot happen by itself, sustained for fourteen hours; the globe needs the kind that cannot be avoided.

Instrument by instrument at the equator, eastbound, here is what the flight itself could and could not show:

The heading row is the subtle one and the place a lazy rebuttal goes wrong. On the flat map “north” means toward the centre, so flying the equator circle keeps the pole exactly abeam and the heading indicator sits on 090° — the same reading a globe gives. Magnetic heading alone cannot separate the models. Each model on its own bookkeeping, which is the only fair way to read that row — an inertially-derived heading reads a clean 090° only where the constants in the mechanisation match the world being flown over, and what happens when they do not is exactly §5. True heading throughout: declination shifts from about 4°W at Quito to near zero at Pontianak, but that is a property of the magnetic field, identical under either model.

Which suggests a cleaner version of the experiment

Take the steering out of the loop. Roll-attitude hold at zero bank, no heading or lateral-navigation reference, wind logged throughout. The globe says you stay on the line. The flat map says 25 miles off in the first hour, 100 in two, 391 in four — southward, and southward on the return leg too. The chapter predicts “opposite adjustments on the return trip,” but flown straight, a flat equator curves away in the same direction whichever way you travel; it is the required bank that reverses, not the departure.

The one serious confound, ordinary crosswind, separates cleanly in the same log: wind drift accumulates linearly and shows up as drift angle against heading, while the flat departure grows as the square of distance and moves the pole-referenced heading itself. A hundred miles is a GPS readout, not an arcminute.

5 · Every flight is already the experiment

The chapter budgets $715,000 and hopes for a sponsor. It does not need one, and the reason is stronger than “airliners fly and seem fine.” A flight recorder logs a closed loop: what the aircraft was told to do; what it did, measured by gyros and accelerometers against inertial space rather than against any map; and where it ended up, cross-checked against GPS and ground stations.

So the route buys elegance, not validity: at the equator three terms go to zero at once, but any positioning leg supplies enough of the equation to solve for the shape. And the correction his island pilots say does not exist is the first row of that mechanisation — earth rate, 15.041°/hr × sin φ — applied several hundred times a second inside a box the size of a shoebox. Pilots never see it because of where it happens, not because it is absent. The whole loop is watched against GPS to about a nautical mile per hour.

The box measures the size of the Earth, with no windows

An inertial platform has to be Schuler tuned: its error loop is set so the instrument behaves like a pendulum whose length is the Earth’s radius. That gives a characteristic period,

T = 2π √(R ÷ g) = 84.4 minutes

and it is the period at which every inertial navigator’s error is observed to oscillate. Solve backwards for R and out comes 6,370 kilometres — from a sealed instrument bolted in a bay, with no view of the sky and no reference to anything outside the aircraft. Not read off a dial: it is the value that must be in the loop for the platform to hold vertical at all.

Mechanise it with the flat map’s 6,225 miles instead and the platform does not become unstable — and it does not tilt further and further either, because that loop is a feedback loop, and what a feedback loop does with a steady error is bounce it. The rate mismatch is V × (1/R_true − 1/R_assumed), which at 560 mph is 2.94° an hour of commanded-rate error. Left to accumulate that would be 2.9° of tilt after an hour and 5.9° after two. The loop does not let it: the tilt turns over at an amplitude of 0.66°, breathing on the 84.4-minute period. Which is worse for the flat mechanisation rather than better, because what it leaves is a signature and not a smear:

• — The position bookkeeping fails immediately. The navigator converts measured travel into angular position using the radius it was handed, so it reports 5.15° of movement per hour where the truth is 8.10° — a ratio of 0.6366, which is 2 ÷ π, the same constant §2 found sitting inside the chapter’s own table. Its position solution falls 203 miles behind the aeroplane in the first hour and keeps falling behind at that rate, against a GPS cross-check running the whole way and a certified drift envelope of about one nautical mile per hour.
• — And the residue is stamped with the right answer. Riding on that ramp is a position excursion of about 46 miles, oscillating with a period of 84.4 minutes — and that period is 2π √(R ÷ g) for the true R. The error a wrong radius produces arrives carrying the correct one.

Which also answers the obvious objection to all of this — that the box was programmed with the globe, so of course it reports one. The shape is a term in a closed loop, not a conclusion written into the output. Put the wrong term in and the loop stops closing, in a specific direction, at a specific rate, with a specific period. The globe is what makes it close.

6 · His altitude test, answered on its own terms

The book does more than object to the gyroscope. It proposes a mechanism and states a test, and the test is well constructed enough to deserve an answer rather than a wave.

The aether is reworked as a magnetic-toroidal field which “propagates as a vortex, with a rotation rate that varies based on vertical position along the Z-axis”, and which “increases in velocity as one moves down the Z-axis toward the ground” (pp. 186–187). The supporting data is a table of fibre-optic gyroscope readings attributed to Bob Knodel:

Printed immediately beneath that table, unprompted, is this: “I could not find the formal document with their exact measurements. We need multiple teams to perform this to see if the results are true.” That is the correct thing to say about data you cannot source, and he says it about his own central table rather than leaving a reader to discover it. We take him at his word and argue with the design, not the numbers.

The test itself: compare the measured drift at many locations along the same line of latitude and at different heights; if the values match closely the globe survives, and “if the values vary greatly, then the claim is falsified.”

His own table is the demonstration. The three stations span three quarters of a degree of latitude, two of them identical to the printed precision — not an oversight, but his stated protocol being followed. Across that span a rotating Earth allows the vertical component a total range of 0.15°/hr, from 9.43 to 9.59. The disagreement reported is “more than a full degree per hour” — his own units, stated in prose at p. 185 — some seven times wider than everything latitude has available at those three sites. Whatever produced it, a rotating Earth does not predict it and cannot be convicted by it.

Altitude is a real discriminator in principle — his vortex predicts a spread there and a rotating Earth predicts none — but it is the weakest one available, because a spread is exactly what a misbehaving instrument, a bad site or a sloppy method also produces. A positive result cannot tell those apart. Latitude is where the two accounts differ by something no instrument fault imitates: a large signal, varying as a known function, reversing sign at a known place. He picked the variable where winning proves least.

7 · The follow-up experiment, which is the better one

The chapter does not stop at the equator. It proposes a second flight, and this one goes almost unremarked: “start from the bottom of India, Africa, or Australia and fly a route that perfectly follows magnetic South.” That is the sharper test of the two, because southbound the models disagree not about a rate but about a topology: on a globe those tracks converge to a point and cross to the other side, while on any pole-centred flat map they spread apart without limit and never meet.

What discriminates is any leg with a substantial east–west component down there, where the map’s circumference has to stretch:

The last is the one to keep. Novolazarevskaya sits at 11.8°E and Progress at 76.4°E, both around 70°S — an east–west hop between two research stations, flown by Basler BT-67 in a working day on the DROMLAN network. On a pole-centred flat map those stations lie on a circle some 69,500 miles around, and the hop becomes 11,800 miles: a fortnight’s flying for an aircraft with under 2,000 miles of range. It is flown routinely, so the map is wrong by a factor of eight.

And the transpolar version has been flown commercially, with passengers and a published schedule. Pan Am Flight 50, 28–30 October 1977, a 747SP named Clipper New Horizons: San Francisco over the North Pole to London, London to Cape Town, then Cape Town to Auckland across the South Pole, and home — the whole circuit in 54 hours 7 minutes. The great circle between those two cities peaks at 74°S, but the flight did not take it; it went over the pole itself, which is why it is remembered, and that routing runs about 7,550 miles. The 747SP’s range was roughly 7,650, so the globe distance only just fitted. On the flat map the same pair of cities are 17,000 miles apart — beyond the range of any airliner ever built, and beyond that 747SP by a factor of more than two.

8 · The route that carries the most, which is not this one

The instinct behind the chapter’s choice is right: pick the route where the models disagree most. What it picks for is the turn rate — and a rate is the one thing a flat map can always answer, because the map has a free scale and rescaling it produces a different rate. §2 is that problem in miniature. The quantity a scale cannot touch is one that changes sign.

There are three models actually on the table, and only one of them is not a constant:

The disc is not his model — §0 covers what happens to a stationary Earth here — but it is the strongest rotating flat model anyone offers, and it deserves to be on the chart. Note what it predicts: the same reading everywhere. On a disc the spin axis is normal to the surface at every point, so the whole rate is vertical wherever you stand, and the tilt between axis and local vertical is zero at every point on the map. So the discriminator was never “does the gyro read something” — that is precisely the question his aether reply at pp. 183–187 is built to absorb. It is whether the reading varies, and whether it reverses.

Which makes the useful route a north–south one that crosses the equator. Here is what a gyro sees on real scheduled services, against the transatlantic leg for comparison:

And the measurement needs almost nothing: orientation and position at the start, orientation and position at the end, and the elapsed time. No steering log, no continuous recording, no charter. The steering record is worth having as a check that the route was flown rather than circled — but it is corroboration, not the measurement.

The honest cost of choosing this route. Look at the last column. Atlanta–Santiago is essentially a meridian, and its heading change is zero — on a meridian, a flat plane and a globe both hold the heading constant, so the curvature discriminator reads nothing on either. North–south buys the rotation law and throws away the geometry; east–west does the reverse. They are complementary legs rather than a better and a worse one, which is an argument for a closed circuit with one of each rather than for any single flight.

And what the reading has to survive. If the number comes from the inertial reference’s computed output, the box was handed sin φ before it reported sin φ, and the test is softer than it looks. Two ways out, and the second needs no instrument nobody will lend you: take the raw body rates rather than the computed heading; or fly the circuit in both directions, because the geometric term reverses with circuit direction and the rotation term does not. The two separate by symmetry alone, with no constant taken on trust from a manufacturer.

Margin is not the difficulty. A navigation-grade gyro’s bias stability sits at 0.003–0.010°/hr — and that, rather than output quantisation, is the real floor, because a bias does not average away. The London–Cape Town signal is 20.17°/hr: three orders of magnitude clear, consistent with the 840× figure §0 gives for the same city.

Which means the quantity this section is about changes sign inside a single downloadable file. Latitude runs −8.07° to +39.73°, so 15.041 sin φ runs −2.11 to +9.61°/hr, and integrated across the flight the platform has to account for 39.2° of rotation that a flat Earth does not have. Against a heading specification of 0.02°. We checked the file rather than trusting its header: 31,292 of 31,399 rows carry real values, the rest the −9999 fill; the sample interval is 1.00 s throughout; and True_Heading + Drift_Angle − Track_Angle has a median of 0.000° with a 95th percentile of 0.100°, which is an inertial solution behaving like one and not a magnetic heading wearing the label.

9 · What would settle it

• 1. A gyro on a table. A navigation-grade gyro, stationary, measures earth rate about the local vertical: 15.04°/hr × sin φ. Take the same instrument to two latitudes and the change is predicted in advance. No aircraft, no charter, no tracking data anyone has to trust.
• 2. The wings-level equatorial leg. Zero bank, no lateral reference, position and wind logged. Either the departure grows as the square of distance, or it does not happen.
• 3. And the full route, if it is ever funded. Worth flying for the public dataset alone. The number to record is accumulated inertial yaw against the local frame: 0.0° on the globe, 75.9° on the flat map’s own best layout.

Statute miles south of the parallel you started on, at 560 mph in still air. On the equator the aeroplane holds latitude 0.0000° and heading 090° for as long as you leave it alone. Begin the identical run at 45°N and the same untouched aeroplane is 577 miles south of its parallel after four hours, with its heading swung from 090° to 118° — and nobody moved a control. Computed in scripts/wings_level_drift.py; the simulator should reproduce these to within its own wind and trim noise.

What that is worth, stated carefully. The simulator is a globe by construction — X-Plane’s own developer documentation warns programmers that “true north is only the same as the negative Z axis for the point 0,0,0” and that the vertical “divergence increases as you go away from the reference point.” So flying it proves nothing whatever about the shape of the Earth, and we are not offering it as though it did. What it tests is us. §4 asserts that a globe behaves this way, and a reader is entitled to ask whether that is arithmetic or assertion. A flight simulator is an independent implementation of the same model, written by people with no stake in this argument, and it lets anyone check our claim about the model in an evening without taking a single number here on trust. The equator really is the one line you can fly hands-off and stay on — which is exactly why the chapter was right to pick it.

Read down the two columns rather than across them. Away from the equator the models merely disagree by an amount, and an amount can always be argued about — a different projection, a different scale. What cannot be argued about is the top row. A sphere has exactly one line that is both a parallel and a great circle, and an aeroplane left alone on it stays on it forever. A plane has no such line at any radius. So the thing to look for in the simulator is not that the numbers match a globe; it is that a single exact zero exists at all, in the one place a sphere requires it and no flat surface can put it. The flat column is the same azimuthal-equidistant layout §2 and §7 use, computed in the same script. Its 391 miles at the equator is the figure already quoted in §4, arrived at independently here, which is the internal check we would want a reader to make.

The honest bottom line

The chapter asks for raw data and numbers, and it is entitled to them. The numbers say the experiment was well chosen, that the equator is the right route for the reason he intuited and two more he did not state, and that his reading of the attitude indicator is correct. They also say the table’s 119° was obtained with the globe’s own circumference, and that the flat model as the chapter deliberately leaves it supplies no scale from which a rival figure could be computed.

None of which sinks the experiment, because the strongest form of it never needed a scale. One circuit of the equator is 360° of turning on any flat surface and 0° on a globe, and the difference is the curvature itself. The aeroplane is not the instrument that settles it — the gyro is, and it settled it decades ago, on every route rather than this one.

He wanted raw numbers from a black box. There is a black box on every aircraft in the sky; it has been logging exactly those numbers since the first jets crossed an ocean without a navigator, and one of the numbers it quietly reports — from a sealed case with no view of anything — is that the Earth is 6,370 km in radius.

Sources & further reading

• Levi Miller, Globe Deconstruction? — prerelease draft, 2026. Chapter “The Equator Flight Data Challenge,” pp. 117–123; the pendulous-vane discussion at p. 119; the route table at p. 121. The aether, vortex and Knodel material is at pp. 183–187, in the run headed “Evidence of Earth spin?”, with the degree-per-hour unit stated in prose at p. 185. The same challenge also appears as a question card at p. 46, printed as “Q4” — the draft's numbering runs Q1, Q2, Q4, Q4, Q5, with no Q3 and two consecutive Q4s, the first being this one and the second the reflections question at p. 47. It is indexed as Q3 on this site's contents page purely to keep twelve questions in order.
• Recorder specifications: 14 CFR Appendix M to Part 121, airplane flight recorder specifications (roll, pitch and heading at 0.5° resolution, ±2° accuracy, 1 Hz), and 14 CFR 121.344.
• Turn-rate and bank-angle figures computed for this page: globe (V÷R) tan φ, flat map V÷r with r = 69.17 (90−φ) miles — the equatorial degree used for the radial spacing too, which is what the chapter's own single-circumference framing implies, and which is what makes the π ÷ 2 in §2 exact rather than approximate. Coordinated-turn bank tan θ = Vω÷g. Equatorial radius 3,963.19 mi. Earth-rate figures: sidereal rate 360° ÷ 23h 56m 04s = 15.041°/hr; vertical component Ω sin φ, horizontal Ω cos φ; co-latitude angles from airport coordinates (Heathrow 51.47°N, Singapore 1.36°N, Johannesburg 26.14°S, Cape Town 33.97°S).
• Public-land survey practice: Standard Field Tables and Trigonometric Formulas (US General Land Office / BLM) — Table 12, azimuths of the tangent to the parallel; Table 13, offsets from the tangent to the parallel; Table 14, offsets from the secant. Also Manual of Surveying Instructions, 1973 on the running of true parallels and the 24-mile standard-parallel interval.
• Pan Am Flight 50, 28–30 October 1977, Boeing 747SP N533PA Clipper New Horizons — San Francisco–London–Cape Town–Auckland–San Francisco over both poles in 54h 07m; see Airways. Intra-Antarctic feeder legs are flown by the DROMLAN network between Novolazarevskaya, Progress and the other coastal stations.
• Inertial reference alignment: ARINC 704A, the industry characteristic defining the airborne IRS, cited by name for the mechanisation — it is sold rather than published, and we have not read it. D. Titterton & J. Weston, Strapdown Inertial Navigation Technology (2nd edn, IET, 2004), Ch. 10 §10.1 ff., “Inertial Navigation System Alignment”, is the standard treatment; its opening distinguishes “self-alignment, using gyrocompassing techniques” from the alignment of a slave system against a master reference. Chapter and section verified against a licensed excerpt of the chapter itself; the alignment equations sit further in, behind the publisher's paywall, and we have not read them line by line.
• The 737 alignment limit, in full, because §0 leans on it. Sources: operator summaries of the limitations section, a 737-800 limitations study guide (Smith & Allen, 1999–2004), and pilots quoting their own manuals on the 737 operational-latitude thread and the general IRS alignment thread. On the exact minutes we report rather than choose: the study guide and a line-pilot quotation of an operator's FCOM — the latter carrying 737-matching exception bands — both give 78° 15′, while a competing 78° 14.75′ is real but traces to the latitude of Longyearbyen, and appears alongside a 747-400 FCOM quotation giving a high-latitude alignment band of 70° 12.0′ to 78° 15.0′. Every one of those is somebody quoting a manual, not a manual; no Boeing document itself was opened. Whether the limitation lives in the AFM or the FCOM genuinely varies by operator, so “flight manual” on the page is deliberately loose. And it should not be confused with the separate 737 maximum operating latitude near 82° with its named longitude-band exceptions — that one is set by the magnetic variation tables, not by the cosine. The A330 FCOM states the corresponding handover directly: true heading instead of magnetic above 82°N, above 73°N between 90°W and 120°W, above 60°S.
• Mechanical predecessors: Delco Electronics, Carousel IV-A Inertial Navigation System Operations Manual — gyrocompass alignment by sensing Earth rate on a gimballed spinning-mass platform, parked only, degrading above 80° latitude, in an era ending well before GPS reached initial operational capability in 1993. Ring-laser retrofit of 747-300 fleets: Litton / Northrop Grumman LTN-92 announcements, describing the unit as a replacement for earlier “spinning wheel” inertial systems in the same ARINC 561 installation. The retrofit announcements do not name the outgoing system, so the page says spinning-mass rather than naming Carousel as the box replaced.
• Polar alignment: the geometric reason gyrocompassing fails near the poles — that gravity and the rotation vector become parallel there — is stated in Wang et al., “On inertial navigation and attitude initialization in polar areas”, Satellite Navigation (2020), which gives GNSS aiding as the practical substitute. On operating practice, high-latitude operations guidance states that navigation through the northern control area is by long-range inertial and GPS-based FMS “referenced to True North only.” The second is a pilot-authored operations reference compiling regulatory material, not a regulator's own page.
• §8 — the route arithmetic. Computed, not measured: scripts/equator_route_choice.py in this site's repository, spherical Earth at R = 6,371 km and a sidereal rate of 15.041°/hr, airport reference points as published.
• §8 — the flight data, and the paths that dead-end. The linked file is Hskping_DC8_20160817_R1.ict from NASA's ATom archive — ICARTT plain text, 6.4 MB, served without authentication; its own header names the LN-251 as the source of position and attitude and gives 0.02° on heading, pitch and roll. Every ATom flight follows the same URL pattern; Hskping_DC8_20160801_R1.ict is the high-latitude counterpart, 34.6°N to 79.3°N. We parsed the file rather than reading its header only, and the figures quoted in §8 are ours. Note for anyone retracing this: the archive's browse interface requires a login while download by filename does not, so the files are found by enumerating names.
• §8 — what is not obtainable, stated because it bounds the argument. The uncompensated body rates are defined by NSF/NCAR's RAF processing algorithms as BROLLR, BPITCHR and BYAWR, “the raw output of the IRU … rate gyro” at 50 Hz and 0.0039°/s, and a published production file header shows all three present in an archived C-130 file, twice over from two inertial units. They are nonetheless not clickable: the archive's data API returns “Missing API Key”, its OPeNDAP route redirects to a login, and the surviving non-JavaScript form is an email order queue. Two caveats on those rates even if obtained. “Raw” there means “as delivered by the unit, uncorrected by NCAR” — it is a statement about the file, not about the physics, and nobody outside the manufacturer sees inside the box. And whether ARINC 429 labels 326/327/330 carry Earth rate or have it removed by the internal mechanisation is not stated in any public document we could reach; ARINC 704A defines it and is sold rather than published. That is precisely why §8 offers the both-directions symmetry test, which does not depend on the answer. A third point argues against the raw rates regardless: Earth rate is 0.004178°/s against a transmitted resolution of 0.0039°/s, so the entire signal is about one least-significant bit of the word.
• §8 — the airliner corpus. NASA's DASHlink sample flight data, one regional-jet type, 35 tail numbers, 186 parameters at 1–16 Hz. It records TH (true heading) and MH (magnetic) as separate parameters, so the magnetic model need not enter an analysis; it also carries LATP, LONP, TRK, DA, PTCH and ROLL, and no angular rates of any kind — attitudes are recorded, their derivatives are not. Parameter names from the data dictionary in the alacer/flightdata conversion scripts, extracted from the MATLAB files themselves; a single source, and its own preamble records NASA's administrator saying no authoritative variable description exists. We have not downloaded a flight to confirm, and the smallest unit is an ~800 MB per-tail archive. No parameter carries a GPS name, which makes the position IRS- or FMS-derived on a reading of the naming convention — inference, not a documented statement.
• An earlier dead end recorded so it need not be repeated: the DC-8 Meteorological Measurement System, whose 1 Hz SONEX file states in its own header that the 5 Hz version carrying ROLL, HDG and PIT is “available upon request” — not posted, and sourced from a GPS-hybridised unit in any case.
• The simulator check. Drift figures computed in scripts/wings_level_drift.py — great-circle propagation on a spherical Earth, and straight-line propagation on the pole-centred flat map, both at 560 mph in still air. The script prints the two side by side, including the fact that the flat column has no zero at any latitude. On the recording side: X-Plane’s Data Output writes selected quantities to data.txt at a chosen rate, and its flight-data-recorder format defines named columns including time secon, lon degre, lat degre, ptch deg, roll deg and hdng TRUE — a format built so that real recorder data can be replayed, which makes it a ready container for the measurement §8 asks for. Microsoft Flight Simulator has no native equivalent and relies on third-party recorders through SimConnect. The caveat is the whole point and is stated in the section: these simulators implement a globe, so the exercise checks this page’s arithmetic rather than the Earth. The quoted warnings about true north and the local vertical are from X-Plane’s own coordinate-system documentation for plugin authors, where they appear as programming caveats rather than as claims about the world.
• Companion pages: Does the Earth Spin, or Does the Sky? for gyrocompasses, gyrotheodolites and the Coriolis correction; One Spin, Written Across Every Latitude for the 15.04 sin φ result measured independently.