The flat earth (Zetetic astronomy): a primer

Project folder: Researchers/Flat-Earth. It holds 103 documents (facts.json). On the flat-earth side: Samuel Birley Rowbotham ("Parallax"), Zetetic Astronomy: Earth Not a Globe (1865) and its 3rd edition (1881); John Hampden's 1869 pamphlet; William Carpenter, One Hundred Proofs that the Earth is Not a Globe (1885); the Universal Zetetic Society's journal edited by Lady Elizabeth Blount (1893-97); Blount and Smith on the sun's motions (1904); Eric Dubay, 200 Proofs Earth is Not a Spinning Ball (2015); about twenty Flat Earth Society (TFES) wiki pages (2022-23); Rob Skiba's videos (2016-19, auto-captions); and lists of NASA and Army reports said to "admit" a flat earth. On the other side, and alongside: Alfred Russel Wallace's account of the 1870 Bedford Level wager (1905), H. Yule Oldham's 1901 repeat (Oldham was a Cambridge geographer; not R. D. Oldham the seismologist), Foucault (1851), Michelson and Gale (1925), Hirt et al. (2010) on refraction near the ground, the National Geospatial-Intelligence Agency (NGA) definition of WGS 84 (World Geodetic System 1984), the GPS interface specification IS-GPS-200N, Airy's 1856 pendulum measurements in Harton Colliery, the Koreshan Unity's 1897 "Rectilineator" survey (a concave-earth variant), fact-checks, and the NASA reports themselves.

Source note (repeated from claims.md and arbiter.md). There is no source gap for the claims, but the long primary works were read selectively. Rowbotham 1865 and 1881 were deep-read for the Bedford chapter and, after the quality check (QE), for his experiments against rotation and orbit and his sun-height method; the 1893-97 journal was deep-read twice (its Bedford material in the second read); the Koreshan book for the Rectilineator and the 1896-97 optical tests. Of the NASA, Army Research Laboratory (ARL) and Precise Time and Time Interval (PTTI) reports, 22 are in the folder: five were read in depth and 17 checked by phrase search and digest; ten more items on the Reddit and Steemit lists are not in the folder. Skiba's videos are machine captions, so numbers may be misheard. Not held: Clifton's 1904 negative, Oldham's full paper (only a 23-line abstract exists in the record), the Koreshan field notebooks, Skiba's balloon footage. Held as leads but not examined: Goudey's Earth Not a Globe (Boston, 1930) and Winship's Zetetic Cosmogony (Durban, 1899, with observations from Port Natal) [read: arbiter 9.6]. Where a statement below rests on recalled textbook knowledge rather than a document in the folder (stellar aberration and parallax, radar and laser ranging), it is tagged [memory].

Source tags: [read] = from a project document (usually via a deep read in deep/); [web] = Internet source with URL; [memory] = recalled, with confidence; [calc] = computed for this primer and reproduced by example.py.

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1. The idea in one paragraph

The modern flat-earth movement began in the English Fens. In 1838 Samuel Rowbotham waded into the Old Bedford River, a straight, still, six-mile drainage canal, held a telescope 8 inches above the water, and watched a boat with a 5-foot flag sail away from him. He reported that the flag stayed in view all the way, although the textbook rule ("8 inches times the square of the miles") said it should have dropped about 11 feet below his line of sight. From this and similar sightings he concluded that still water is not just level but flat, and built a whole cosmology on it: a disc-shaped earth centred on the North Pole and ringed by ice, a small sun and moon circling a few thousand miles overhead, sunset as an effect of perspective, and an earth that does not move. His followers added a 1,000-pound wager (1870), a concave-earth rival (the Koreshans, 1897), a photograph (1904), and, in the twenty-first century, "universal acceleration" in place of gravity, light that bends upward, and the claim that agencies hide the truth. The review found that the observations at the root of it are often real, that Rowbotham's arithmetic is mostly correct, and that the gap lies in one thing he set to zero: the bending of light in air near water. When the experiment was redesigned so that bending cancels out (Wallace, 1870; Oldham, 1901), the water showed the curve of a globe of the known size. Independent calculations from the flat model's own geometry (distances in the southern hemisphere, the height of the sun) fail without any appeal to refraction.

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2. What you need to know first

Small-angle geometry of a circle. For a circle of radius R, an arc of length d subtends an angle theta = d/R (radians) at the centre. For small theta, cos(theta) is about 1 - theta^2/2 and sec(theta) = 1/cos(theta) is about 1 + theta^2/2. Every curvature number in this project is one of these expansions. Read: any calculus text's chapter on Taylor series (e.g. Stewart, Calculus).

Level surfaces and plumb lines. Still water settles on a surface of constant gravitational potential energy per unit mass (an "equipotential"). A plumb line hangs perpendicular to it everywhere. On a sphere the plumb lines all point toward the centre, so two plumb lines a mile apart are not parallel: they differ in direction by 1/3,959 of a radian, about 52 arc-seconds. "Level" therefore means "perpendicular to the local plumb line", which is not the same as "straight". This single distinction runs through the whole project. Read: Halliday, Resnick and Walker, Fundamentals of Physics, the chapters on gravitation and on potential energy.

Refraction in a graded medium. Light travels more slowly in denser air. When the density changes smoothly with height, a ray bends toward the denser layer, just as it bends at a glass surface but continuously. Air near the ground is usually denser below, so nearly horizontal rays curve slightly downward and follow the earth part of the way. Surveyors describe this by a single number, the refraction coefficient k = R/r, where r is the radius of the ray's curve. Over warm ground (dense air above, light air below) k can be negative (rays bend up: the hot-road "puddle" mirage). Over cold water with warm air on top (an "inversion") k can be large and positive (rays bend down: "looming", objects seen beyond the horizon). Read: Hecht, Optics, the sections on Fermat's principle and on mirages; Hirt et al. (2010), cited below, for measured values.

Angles in arc-seconds. One degree = 3,600 arc-seconds (written "). One radian = 206,265". A surveyor's level of 1870 could resolve a few arc-seconds. The Bedford experiments turn on angles of 10" to 160".

Rotating reference frames. In a frame that rotates at angular velocity Omega, moving bodies appear to feel two extra ("fictitious") forces: the centrifugal force and the Coriolis force, 2m(v x Omega) for a body of mass m moving at velocity v. A Foucault pendulum's swing plane and a ring laser gyroscope both measure the component of Omega along the local vertical, Omega sin(phi) at latitude phi. Read: Taylor, Classical Mechanics, the chapter on non-inertial frames; Kleppner and Kolenkow, An Introduction to Mechanics, the chapter on non-inertial systems.

The equivalence principle. In a small, closed laboratory you cannot tell a uniform gravitational field from a uniform upward acceleration. This is true and is the starting point of general relativity. It is local: over larger regions real gravity has gradients (it weakens with height and points toward a centre), and an accelerating floor does not. Read: any modern-physics text's introduction to general relativity (e.g. Taylor and Wheeler, Spacetime Physics). No tensors are needed for anything below.

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3. The textbook view

Symbols used throughout: - R : radius of the earth, 3,958.8 statute miles (6,371 km). - d : distance from the observer to a target, measured along the surface (miles or feet). - h_e : height of the observer's eye above the water; h_t : height of the target above the water. - d_e : distance to the observer's own horizon (where the line of sight grazes the water). - k : refraction coefficient, R divided by the radius of the light ray. k = 0 for a straight ray; k = 1 for a ray that curves exactly as fast as the earth. - R_eff = R/(1 - k) : effective earth radius when refraction is included. - L : length of a chord between two equal-height marks; s : sagitta, the rise of the arc above the chord at its middle. - phi : latitude. Omega : the earth's rotation rate, 7.292 x 10^-5 rad/s (15.04 degrees per hour relative to the stars).

3.1 The curvature drop ("8 inches times miles squared"). A sphere falls away from the horizontal tangent at the observer by

drop(d) = R (sec(d/R) - 1) = d^2/(2R) + 5 d^4/(24 R^3) + ...

With d in miles and the drop in inches, d^2/(2R) = 8.00 d^2 [calc]. This is the rule every document quotes, and it is correct. At 1,000 miles it is 2.7 percent low [calc], and Rowbotham himself says it "requires to be modified after the first thousand miles" [read: Rowbotham 1881 p.10].

3.2 What is actually hidden. The tangent drop applies only to an eye at the waterline. A real eye at height h_e sees to its own horizon at d_e = sqrt(2 R h_e) (with h_e = 8 inches, d_e = 1.0 mile [calc]). Beyond that, a target at distance d is hidden up to the height

hidden(d) = (d - d_e)^2 / (2R).

A third quantity is the sagitta of a chord between two equal-height points, s = L^2/(8R), which equals the 8-inch rule applied to half the chord. The documents keep swapping these three (tangent drop, hidden height, sagitta), and most numerical disputes in the project come from that swap [read: mathematician.md section 0].

3.3 Refraction. If the ray has curvature k/R in the same sense as the earth, then relative to the ray the earth curves at (1 - k)/R. Every formula above holds with R replaced by R_eff = R/(1 - k): hidden heights and sagittas are multiplied by (1 - k), and the horizon moves out to sqrt(2 R h_e/(1 - k)). Gauss found an average k of about +0.13 from reciprocal sightings between survey stations [read: Hirt et al. 2010, p.2]. Near the ground it is not constant. Hirt et al. measured k from -4 to +16 along 1.8 m-high lines over grass, with +12 to +16 around sunset, and they cite +1 to +18 about 5 m above water; they write that the Gaussian +0.13 "is not suited for describing refraction effects in the lower atmosphere" [read: Hirt deep read]. Their eq. 2 gives the local k from the temperature gradient dT/dz (kelvin per metre, z upward), pressure p (hPa) and temperature T (K):

k = 503 (p / T^2) (0.0343 + dT/dz).

For a normal atmosphere (dT/dz about -0.0065 K/m) this gives k = 0.17 [calc].

That refraction near the ground changes with the time of day was noticed by surveyors long before Hirt. The flat-earth journal itself quotes General Roy's trigonometrical survey: thirty pickets 100 ft apart looked straight in the afternoon and concave upward in the morning [read: Earth Review p.34, second-hand via a correspondent citing Trig. Survey vol. 1 p.175, which is not in the folder]. Over 2,900 ft the earth's own bulge is only 0.05 ft, so the change is refraction [calc: arbiter 9.1e].

3.4 Rotation. A Foucault pendulum's swing plane turns at 15.04 sin(phi) degrees per hour (11.3 deg/h in Paris, zero at the equator, reversed in the south) [read: Foucault 1851 states the sine law; calc]. A light loop of area A, rotating at Omega, splits two counter-running beams by a fringe shift 4 A Omega sin(phi)/(lambda c), where lambda is the wavelength and c the speed of light (the Sagnac effect). For Michelson and Gale's 1925 loop (A = 207,836 m^2, phi = 41 deg 46', lambda = 570 nm) this predicts 0.236 fringe; they observed 0.230 +/- 0.005 over 269 readings [read: Michelson-Gale deep read; calc]. A modern 4 m x 4 m ring laser at 49.1 deg N beats at 348.6 Hz for the same reason [web: https://syrte.obspm.fr/jsr/journees2011/pdf/soffel.pdf, via the contrarian].

3.5 Gravity. Newton's inverse-square law makes gravity in open air weaken with height by 2g/R = 3.08 x 10^-6 s^-2 (about 0.31 milligal per metre; 1 milligal = 10^-5 m/s^2), the "free-air gradient" [calc from the textbook law]. Gravity varies with latitude from 9.7803 m/s^2 at the equator to 9.8322 m/s^2 at the poles [read: NGA WGS 84 deep read].

A measurement in the folder shows how gravity changes with depth. In 1856 George Airy swung two pendulums at the top and bottom of Harton Colliery, 1,256 ft (383 m) deep, interchanging them, and found gravity at the bottom larger by a ratio of 1.00005185 +/- 0.00000019 (Airy's probable error): a seconds pendulum gained 2.24 s a day below [read: Airy 1856 deep read, pp.35-37, 43-47]. This is a net figure, not the free-air gradient: going down brings you nearer the centre (free air alone would give 1.2 x 10^-4 of g over 383 m), and the layer of rock now above you pulls back up, leaving 5.2 x 10^-5 [calc: arbiter 9.4]. Explaining it needs both an inverse-square fall-off and attraction by mass. A floor accelerating at g would give only 4.2 x 10^-14 of g over the same height, about 10^9 times too little (example.py Part 6).

A second check at orbit height: the GPS specification IS-GPS-200N offsets the satellite clocks by Δf/f = -4.4647 x 10^-10 "for relativistic effects" [read: IS-GPS-200N deep read, PDF p.33]. Newtonian gravity's potential difference minus the speed term gives the same number to five digits; a constant g carried up to orbit would give 2.20 x 10^-9, five times too much (example.py Part 6). The specification comes from the US GPS operator, an interested party; the reproduction uses its own constants and a standard value of the earth's surface potential [memory, medium].

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4. Where the inventors depart

Fairness first. Each flat-earth argument starts from something true.

• "Water finds its level." True: still water lies on an equipotential and a spirit level reads it correctly. Rowbotham's step is from "level" to "flat", that is, from "perpendicular to the plumb line here" to "the same plane everywhere". Whether those differ over six miles is exactly what has to be measured.
• "I saw it when the globe says I should not." Often true as an observation. Distant objects are seen beyond the geometric horizon, and every party, including the mainstream optics expert and meteorologist quoted in Skiba's material, concedes it [read: Skiba Chicago deep read, section 5]. Rowbotham's arithmetic for a straight ray is right: his 11 ft 8 in for the 1838 flag reproduces exactly [calc].
• "Refraction is invoked, never measured." Also true for every historic sighting. Blount and Smith wrote in 1904 that the globe's defenders "blow both hot and cold", calling on refraction when an object is seen and on curvature when a ship disappears [read: Blount-Smith 1904 p.4]. As a logical point that is fair: k is a free parameter until it is measured.
• "We feel no motion." True: uniform motion cannot be felt (Galilean relativity), and the rotation is slow (Omega is about 7 x 10^-5 rad/s).
• "Gravity could be acceleration." True locally: the equivalence principle (TFES "Universal Acceleration").
• "The sun could be small and near." Not absurd in itself. A 32-mile sun at 3,000 miles subtends 0.61 degree, close to the observed 0.53; and the arbiter computed that a hot disc of solar temperature at that angular size would give about 1.8 kW/m^2, comparable to the measured 1.36, so sunlight's intensity does not rule it out [calc: arbiter section 1].
• "NASA documents say flat, nonrotating earth." True as a phrase: NASA TM X-2514 (1972) states it [read]. It is a stated modelling approximation, with its limits given.

The departures, side by side:

A symmetry worth knowing (mathematician.md 1.4): a flat plane where light bends upward at 1/R and a globe with straight light predict the same sightings along the surface. Sightings alone cannot separate them. Measurements that do not depend on light paths can: ground distances, flight times, gravity gradients, rotation sensors.

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5. Worked example: the Bedford Level, from Rowbotham's flag to Wallace's three marks

example.py reproduces every number here. Run it with .venv/bin/python review/Flat-Earth/example.py (from the altsci folder).

Part A. Rowbotham's 1838 sighting, done his way and with refraction

The set-up as he gives it: eye "about 8 inches above the surface", a flag whose top is 5 ft above the water, the boat sailing 6 statute miles from Welche's Dam to Welney Bridge, "about three o'clock in the afternoon of a summer's day". Result: "The flag and the boat were distinctly visible throughout the whole distance!" [read: Rowbotham 1881 pp.11-13]. (His 1865 edition tells it with a 3 ft flag and the observer seated in the water; the arbiter records both versions [read: arbiter section 4, item 12].)

Step 1. The eye's horizon. h_e = 8 in = 0.667 ft; R = 3,958.8 mi x 5,280 ft/mi = 2.090 x 10^7 ft.

d_e = sqrt(2 x 2.090 x 10^7 ft x 0.667 ft) = sqrt(2.787 x 10^7 ft^2) = 5,279 ft = 1.00 mile.

Step 2. Hidden height at the boat with a straight ray (k = 0). The boat is 6 - 1 = 5 miles beyond the horizon:

hidden = 8 in x 5^2 = 200 in = 16.67 ft.

The flag top (5 ft) lies 16.67 - 5 = 11.67 ft below the line of sight: Rowbotham's "11 feet 8 inches". His arithmetic is right.

Step 3. Now let the ray bend. Multiply by (1 - k) and move the horizon out to sqrt(2 R h_e/(1 - k)):

Step 4. Solve for the k at which the flag top just reaches the sight line: k = 0.612. For the boat's waterline to be visible too, k = 0.972 [calc]. The arbiter's table gives the same 0.612 [read: arbiter section 1].

So a globe accounts for the 1838 report only if the ray curved about 60 percent as fast as the earth for the flag, nearly 100 percent for the hull. That is far above the textbook 0.13 and inside Hirt's measured near-surface range.

Part B. What air would that take?

Invert Hirt's eq. 2 at p = 1013 hPa, T = 288 K:

dT/dz = k T^2 / (503 p) - 0.0343.

[calc] An inversion of a few tenths of a kelvin per metre is common over cold water on a calm day with warmer air above, which is the physical case for refraction. But nobody measured the gradient over the Old Bedford River in 1838, 1870 or 1904, so the case is plausible, not demonstrated. That is why the arbiter calls the single-ended sightings, as proof of flatness, Never properly tested rather than refuted.

Part C. Wallace's design: an experiment that refraction cannot fake

In March 1870 Wallace set three marks at the same height above the water: the parapet of Welney bridge, 13 ft 3 in up, where the telescope rested (The Field of 26 March 1870 gives 13 ft 4 in for all three signals and the telescope [read: arbiter 9.1a]); a black band on Old Bedford bridge about 6 miles away; and a red disc on a pole halfway between them, with a second disc 4 ft lower [read: Wallace 1905 p.4]. The predictions were agreed in advance by Hampden, his referee Carpenter and the umpire Walsh: over flat water the top disc would sit on the black band; over convex water it would stand "five feet eight inches" higher, "reduced a little by refraction to perhaps about five feet" [read: Wallace 1905 p.5-6].

Step 1. The bulge. With L = 6 mi = 31,680 ft:

s = L^2/(8R) = (31,680)^2 / (8 x 2.090 x 10^7) = 6.00 ft (k = 0); 6.00 x 0.87 = 5.22 ft (k = 0.13).

(Wallace's 5 ft 8 in corresponds to a chord slightly under 6 mi.)

Step 2. The angles in a levelled telescope. A mark at distance x, at the eye's height above curved water, lies x^2/(2 R_eff) below the eye's horizontal, so it appears at an angle of about x/(2 R_eff) below the cross-hair:

[calc] Uniform refraction shrinks both angles by (1 - k) but never changes their ratio, and a small tilt of the instrument moves both by the same amount. The sketches made through the Troughton level, drawn by Hampden's referee and signed by Wallace's, show exactly this pattern: marks 1, 2, 3 stepping down below the cross-hair [read: Wallace 1905 page image p.7].

Step 3. What a plane would show. On flat water all three marks are at one height. With a straight ray they sit on the cross-hair. With a downward-bending ray (k > 0) the telescope must aim slightly upward to catch a ray that arrives low, so the marks appear above the cross-hair, by k x/(2R): 10" and 20" at k = 0.13 [calc]. And the middle mark then sits below the line from A to C by k L^2/(8R) (0.78 ft at k = 0.13). On a globe it stands above that line by (1 - k) x 6.00 ft. The large-telescope sketch, drawn by Wallace's referee and signed by Carpenter, shows the pole's discs standing above the band [read: Wallace 1905 page image p.6]. The umpire's printed decision in The Field of 26 March 1870 gives curvature "to the extent of five feet, more or less" [web: https://people.wku.edu/charles.smith/wallace/zCarpenter1870Field1.pdf, via the Investigator]; Oldham's 1901 repeat over the Welney-Denver stretch saw the middle mark "stand up about six feet above the line of sight" [read: Oldham 1901 p.725-726].

The sketch also carries a number. The middle staff had two discs 4 ft apart, so they act as a scale bar at 3 miles. Scaled that way, the large-telescope sketch puts the upper disc about 1.5 disc-spacings above the far signal, about 6 ft (plus or minus 1.5 ft, because the woodcut is not exactly to scale) [calc: arbiter 9.1a, from The Field Fig. 6]. Wallace's own reading in The Field of 2 April 1870 was "about 5 1/2 ft. at the middle signal" [read: arbiter 9.1a; Wallace was the winner, so an interested party]. A globe predicts 6.0 ft (k = 0) or 5.2 ft (k = 0.13). This separation does not use the cross-hair, so the complaint that the cross-hair was not trustworthy does not touch it.

Step 4. Carpenter's reading. "Equal steps down" means a straight line only if the steps are equal in height. They are equal in angle. A straight sloping surface would put A, B and C on one line through the eye, and C would hide behind B. Equal angular steps mean drops proportional to x^2 (6 ft at 3 mi, 24 ft at 6 mi): a curve [calc; mathematician.md 1.2].

What the design does not exclude. A ray whose bending changed along the six miles, or an upward-bending ray (k < 0, the TFES "Electromagnetic Acceleration" picture), could in principle mimic a curve. On a plane with k = -1 the marks would also step down below the cross-hair and the middle one would stand 6 ft above the A-C line. Nothing in 1870 measured k. What would settle it is to repeat the trio with measured temperature profiles: the globe predicts an apparent bulge of (1 - k) x 6.00 ft, the plane -k x 6.00 ft, so the measured bulge plus k x 6.00 ft should equal 6.00 ft on a globe and 0 on a plane, whatever the weather.

Is k = -1 over a cool canal plausible? From Part B it needs dT/dz of about -0.20 K/m, i.e. air 2 K colder per 10 m of height, a strongly heated surface layer [calc from eq. 2]. That is the hot-afternoon state over land, not cool water in March. It would also have had to hold in Oldham's summer runs of 1900 and 1901. Read on a globe, the 6 ft scaled from the 1870 sketch means k below about 0.3 that afternoon at 13 ft; read on a plane, it needs k near -1 [calc: arbiter 9.1a; Part C above].

Part D. A check that needs no light ray at all

At an equinox the noon sun is overhead at the equator. On a plane, a place a surface distance s north of the equator sees the sun at elevation e with tan(e) = H/s, where H is the sun's height. The observed noon elevation is 90 deg minus the latitude. Using 69.09 miles per degree of latitude:

[calc] One sun cannot sit at three heights. The movement's own 1893 table gives nearly the same three numbers (3,585, 3,105, 2,390) [read: arbiter, claim 3; Earth Review p.173]. One of the journal's writers, Naylor, conceded that sun angles along a meridian "far more nearly correspond with the common theory than with Zeteticism" [read: Earth Review p.175, c.1895-96, via arbiter 9.3]. Rowbotham's own 1881 figure, a sun about 700 miles up from noon altitudes of 61 and 64 degrees at London and Brighton, fails on his own inputs: the two places differ by only 0.69 degree of latitude, so their noon altitudes on one day can differ by only about 0.7 degree, not 3; with consistent altitudes the flat model gives about 3,570 miles [calc: arbiter 9.3]. On a globe all three elevations follow from one distant sun (parallel rays) and plumb lines that tilt by the latitude. This is the kind of calculation the arbiter means by "Refuted by calculation": it uses the flat model's own geometry and does not depend on refraction.

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6. Evidence table

Verdicts are the arbiter's (arbiter.md section 8), in the Step 2 scale.

Kept in the record whatever the verdict (SOURCE HONESTY): Clifton's 1904 letter, written by the camera operator whom the zetetic account calls a "globularist" and who worked for Dallmeyer, the firm that sold the camera: he saw the sheet "though not sharply defined", in "very unfavourable" conditions with "an aqueous shimmering vapour", and wrote that "unless rays of light will travel in a curved path" the six miles present "a level surface" [read: Clifton letter in Blount 1905, research/files]. A press report says two flat-earth proponents measured 15 degrees per hour, the earth's rate, with a commercial laser gyroscope (Earth.com, Kottke; single source, no instrument record) [read: claims.md; arbiter claim 2]. A South Pole station page reports a pendulum turning "twice what is expected" (anecdote, no data) [read: Foucault TFES deep read]. Rowbotham's own 1872 preface calls some first-edition evidence "not entirely satisfactory" [read: 1881 p.v]. Rowbotham's own rebuttal of the 1870 test: the agreed design was three equal discs, one beside the telescope, but "at the end of six miles instead of a disc an oblong flag was placed ... and nothing whatever was fixed close to the telescope", so "the money was neither won nor lost"; he also names collimation error in spirit levels [read: Experimental Proofs 1870, pp.8-12, research/files]. The arbiter grants the procedural point in part (the cross-hair was a poor third point, and the near disc was not set up) but notes that with the telescope at the signals' own height, equal-height signals on a plane coincide however the instrument is tilted, so the separation in the sketch is not explained by the missing disc [read: arbiter 9.1b]. Naylor's August 1894 repeat on the same canal, by the flat side with a theodolite and a Dumpy level, reports Welney Bridge "a little below the cross hair" and the railway bridge 12 miles off visible above it; it gives heights but no instrument readings in figures, and on a globe it needs k of about 0.6 (from his boat, eye 2.5 ft) to 0.9 (from the bridge) [read: Earth Review pp.216-217, printed about 1896; calc: arbiter 9.1c]. The Koreshan Unity's own optical tests of 1896-97 (Summit canal, Lake Michigan, Naples Bay, the Gulf) report targets seen through telescopes a few inches above the water at 4.5 to 6.5 miles, which a globe can accommodate only with k of about 0.4 to 1.0; none records the air temperature [read: Koresh and Morrow 1898, pp.84-106; calc: arbiter 9.1d]. In the opposite direction, the flat-earth proponent Campanella's 2019 four-mile test, light shone through holes cut at the same height in two sheets, is reported as not seen at the far end, as curvature predicts; the hole height is not given [read: Earth.com, a press report; arbiter 9.1e]. The academic non-replies to Carpenter's 1885 challenge (Johns Hopkins, the Smithsonian, Proctor's postscript) were dismissal without engagement, not tests [read: Carpenter 1885 pp.33, 37-38]. The court case Hampden v. Walsh (1876) decided only wager law and made no finding that Wallace cheated; "cheat and swindler" are Hampden's words [web: https://people.wku.edu/charles.smith/wallace/zAnon1877ReportsCases.pdf; https://epsilon.ac.uk/view/wallace/letters/WCP6734, via the Investigator]. Hampden was also convicted over his letters to Wallace: a week's imprisonment for want of sureties at Stratford in July 1871, and a sentence at the Chelmsford Assizes in 1875, given as six months in Chelmsford Gaol by the critic Schadewald and the Chelmsford Chronicle [read: arbiter 9.6, summarising the Investigator's newspaper leads; web: https://epsilon.ac.uk/view/wallace/letters/WCP1947]. In April 1870 he wrote: "Either the Bible or science is a lie and a fraud" [read: Weston-super-Mare Gazette, 16 April 1870, research/files].

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7. What a learner should take away

What is solid. - The 8-inch rule, Rowbotham's construction for a straight ray, and most of his arithmetic. If you want to check a sighting, his method is the right first step. - The observations: objects are seen beyond the geometric horizon over calm water. That is a real optical effect, measured (Hirt: k up to +16 near the ground). - The logical point that refraction must be measured, not assumed. The 1904 "hot and cold" retort is good scientific criticism of how the globe side argued. - Wallace's experimental design. It is a model of how to build a test whose answer does not depend on an unmeasured nuisance parameter: compare a ratio, not an absolute.

What is wrong, and why the mistake is instructive. - Treating a prediction with a free parameter (k) as if the parameter were zero. The data from a single low sighting cannot decide between "flat" and "globe with k about 0.6"; Rowbotham read them as deciding. - "Level" read as "flat". Plumb lines a mile apart are not parallel. - Mixing the tangent drop, the hidden height and the sagitta. Dubay's list and Carpenter's figures disagree with each other by factors of 2 to 4 for this reason alone. - Reading equal angular steps as a straight line (Carpenter, 1870). - Choosing a light rule per argument. The 1881 cosmology needs straight light for the canal and some other rule for sunset; the 2023 wiki's upward-bending light (the same rule as the Koreshans' 1897 "curvilineating slightly upwards") explains sunset but would lower the Bedford flag. Each model of a flat earth needs one light rule that fits all its claims. - Ignoring the checks that do not use light at all: southern distances, flight times, sun elevations at several latitudes, the gravity gradient, rotation sensors.

What is genuinely open (and small). - How large k was on the days of Rowbotham's, Naylor's, the Koreshans', Clifton's and the Chicago sightings. Nobody measured it. The 1870 sketch points to a k near 0 at 13 ft over six miles, the low-eye sightings to 0.4 to 1.0; both can be true if the air near the water differs from the air at 13 ft. - What Clifton's 1904 negative showed (not held); what Oldham photographed, if anything. - Whether the Koreshan quadratic is a measurement or a repeated 0.12" instrument bias. The two magnitudes match too well to wave away; their short out-and-back closure test (if as described) argues against a down-bending bias, and their longer return does not [read: arbiter section 7]. - What the ten listed NASA/Army/CIA documents not in the folder say in context.

The cleanest experiment you could do. Wallace's trio over a long straight canal at three heights, with temperature sensors at two heights at each end and reciprocal vertical angles to measure k. Globe: measured bulge + 6.00 ft x k = 6.00 ft. Plane: it equals 0. Add one low-eye run (eye about 2 ft above the water) over the same stretch at the same time, so k near the water and at 13 ft are measured together [read: arbiter 9.1f].

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8. Check yourself

1. A lighthouse lamp is 100 ft above the sea. Your eye is 10 ft above the sea. With k = 0, how far away is the lamp when it just touches your horizon? With k = 0.13? Then use the same idea on a claim in the flat-earth journal (1894): a heliograph signal was seen 183 miles from Uncompahgre Peak (14,418 ft) to Mt Ellen (11,410 ft), where "nearly 6 miles" of curvature should have hidden it. What is the tangent drop at 183 miles, and could the two peaks see each other over a sea-level sphere with k = 0?
2. In Rowbotham's 1838 set-up, suppose a thermometer pair had shown dT/dz = -0.01 K/m (air cooling upward). Use Hirt's eq. 2 and Part A to say whether the flag top would be visible on a globe.
3. Why does Wallace's experiment not depend on the 13 ft 3 in height of the bridge, or on uniform refraction?
4. Carpenter said the three marks stepping down by equal amounts in the level telescope proved a straight line. Explain in two sentences why that inference fails, and say what a straight (sloping) water surface would actually show.
5. Michelson and Gale's loop sat at 41 deg 46' N. What fringe shift would the same loop give at the equator, and at 41 deg 46' S? What does a stationary earth predict at each?

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9. Further reading

Mainstream: - Hirt, Guillaume, Wisbar, Bürki and Sternberg, "Monitoring of the refraction coefficient in the lower atmosphere using a controlled set-up of simultaneous reciprocal vertical angle measurements", Journal of Geophysical Research 115, D21102 (2010). In the folder. - A. R. Wallace, My Life (1905), vol. 2, ch. 39, the Bedford Level wager in the winner's words. In the folder. - The Field, 26 March 1870, p.285: both referees' reports and the umpire's decision [web: https://people.wku.edu/charles.smith/wallace/zCarpenter1870Field1.pdf]. - H. Yule Oldham, "Experimental demonstration of the curvature of the earth's surface", Report of the British Association for the Advancement of Science (BAAS) 1901 (abstract). In the folder. - A. A. Michelson and H. G. Gale, "The effect of the earth's rotation on the velocity of light", Astrophysical Journal 61 (1925). In the folder. - L. Foucault, Comptes rendus 32 (1851), the pendulum note. In the folder. - National Geospatial-Intelligence Agency (NGA), NGA.STND.0036, World Geodetic System 1984 (2014); Defense Mapping Agency (DMA), TR 80-003, Geodesy for the Layman (1983). Both in the folder; the second is the gentlest introduction to how the earth's shape is measured. - Robert Schadewald, The Plane Truth (online chapters), a critic's history of the movement and the main secondary source for the Clifton and Voliva episodes [web, via the Investigator; sites.md]. - Christine Garwood, Flat Earth: The History of an Infamous Idea (2007) [memory, high], a historian's account. - Taylor, Classical Mechanics, chapter on non-inertial frames (Coriolis force, Foucault pendulum); Hecht, Optics, on refraction in graded media and mirages.

The inventors' own key works (all in the folder): - S. B. Rowbotham ("Parallax"), Zetetic Astronomy: Earth Not a Globe (1865; 3rd ed. 1881). Chapter II holds the Bedford experiments. - W. Carpenter, One Hundred Proofs that the Earth is Not a Globe (1885). - E. A. M. Blount (ed.), The Earth Not a Globe Review (1893-97); Blount and Smith, Zetetic Astronomy: The Sun's Motions North and South (1904); Lady Blount, Flat or Spherical? (1905) with Clifton's letter [web: http://library.tfes.org/library/, via the Investigator]. - Koresh (C. R. Teed) and U. G. Morrow, The Cellular Cosmogony (1898), the Rectilineator survey. - The Flat Earth Society wiki, pages "Bedford Level Experiment", "Universal Acceleration", "Electromagnetic Acceleration", "Michelson-Gale-Pearson Experiment" (2022-23). - E. Dubay, 200 Proofs Earth is Not a Spinning Ball (2015); R. Skiba, Testing the Globe videos (2016-19).

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Corrections to earlier documents

1. arbiter.md section 1, formula line: "Hidden height = (d - d_e)^2 / (2R(1-k))" should read (d - d_e)^2 (1 - k)/(2R), i.e. (d - d_e)^2/(2 R_eff) with R_eff = R/(1 - k). The arbiter's tabulated numbers (16.67, 9.1, k = 0.612, 12.15, 9.96) are computed with the correct form and are confirmed by example.py; only the printed formula is wrong.
2. arbiter.md claim 1, "What would settle it": "on a plane, zero whatever k is" and "A plane predicts no dependence of the offset on measured k at all" are not quite right. On a plane with a ray of curvature k/R, the middle mark appears offset from the A-C sight line by -k L^2/(8R) (below the line for k > 0, above it for k < 0), so the apparent offset does depend on k. The discriminating quantity is the k-corrected offset: apparent bulge + k L^2/(8R) equals L^2/(8R) (6.00 ft over 6 miles) on a globe and 0 on a plane, for any uniform k [calc: example.py Part 4]. The arbiter's recommended experiment (measure k alongside the trio) is unchanged; only the stated prediction needs this wording.
3. This primer, earlier version (QE revision, 2026-10-10): the 1901 repeat was attributed to "R. D. Oldham" twice. The author is H. Yule Oldham, M.A., a Cambridge geographer [read: Oldham title page; facts.json]; R. D. Oldham was a different man, a seismologist [memory, high]. Fixed in the header and Further reading.
4. This primer, earlier version: the agency-document count ("about 17 listed", "a dozen more", "the remaining dozen") was wrong. The Reddit list has 32 items and the Steemit list 15, largely overlapping; 22 reports are in the folder, 5 read in depth and 17 phrase-searched; ten listed items are not in the folder [read: qe.md section 4; arbiter 9.5]. The gravity gradient was tagged [memory]; section 3.5 now cites Airy's 1856 Harton measurement [read] and says it is a net figure, not the free-air gradient. The earlier version also left out Rowbotham's own 1870 rebuttal of the Wallace test, Naylor's 1894 repeat, the Koreshan optical tests, Campanella's light test and Hampden's convictions; they are now in "Kept in the record".

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Answers

1. Horizon distances with d_e = sqrt(2 R_eff h). k = 0: lamp 1.2246 sqrt(100) = 12.2 mi, eye 1.2246 sqrt(10) = 3.9 mi, total 16.1 mi. (1.2246 sqrt(h in ft) miles is sqrt(2R h) in convenient units.) k = 0.13: multiply by 1/sqrt(0.87) = 1.072: 17.3 mi. Refraction adds about 7 percent to visual ranges, which is roughly what nautical range tables include. Heliograph: the tangent drop at 183 mi is 183^2/(2 x 3,958.8) = 4.23 mi, not "nearly 6", and it is the drop below a tangent at sea level, which ignores the observers' heights. The horizon distances are 1.2246 sqrt(14,418) = 147.0 mi and 1.2246 sqrt(11,410) = 130.8 mi, 277.8 mi together, more than 183 mi, so the peaks see each other over a smooth sphere with a straight ray (terrain in between aside) [calc: example.py Part 6; arbiter 9.1e].
2. k = 503 x 1013/288^2 x (0.0343 - 0.01) = 6.14 x 0.0243 = 0.149. From Part A, k = 0.149 leaves the flag top about 8.7 ft below the sight line (between the 0.13 and 0.25 rows): invisible on a globe. A sighting under those measured conditions would be a real problem for the globe; that is why measuring k matters.
3. All three marks are at the same height h above the water, so h cancels: each mark's offset from the eye's horizontal is x^2/(2R) whatever h is (for h much smaller than the drops' scale this is exact enough). Uniform refraction replaces R by R/(1 - k) in both offsets and leaves the 1:2 ratio of angles, and the middle mark still stands above the A-C line by (1 - k) times 6 ft rather than falling below it as on a plane with k > 0.
4. Equal steps in angle, from a level at A, mean drops proportional to the square of the distance (offset = angle x distance, with the angle itself proportional to distance), which is a curve. If the water were a straight slope, the eye, B and C would be on one straight line and C would be hidden directly behind B: zero angular separation between them, not a second equal step.
5. The fringe shift is proportional to sin(phi): at the equator 0 (the loop's plane is parallel to the rotation axis); at 41 deg 46' S, 0.236 fringe again but with the opposite sign (the fast and slow beams swap). A stationary earth predicts 0 at all three sites. Ring lasers in both hemispheres show the sign flip, which is one of the arbiter's "what would settle it" items for claim 2.