Parallax: Physicist review (Step 2)

Project: Zetetic Astronomy. Earth Not a Globe! (1865 edition) by "Parallax" (Samuel Birley Rowbotham), and Gleason's 1892 "New Standard Map of the World".

Reviewer role: PhD physicist. I state the mainstream view, test each claim against it, and keep the Investigator's findings next to my judgement.

Source tags. [read] = read in the project documents (extracted text, page numbers are PDF pages; book page = PDF page - 5). [calc] = my own calculation (numpy, scratch script). [web] = a source with URL. [memory] = recalled, with confidence.

Source status. No source gap: the book's text layer is OCR with number errors. The pendulum section (PDF pp.138-159) was deep-read from page images by Step 1. Everything else in the book rests on OCR text. I re-read the pages that carry my calculations myself: PDF pp.16-24 (Bedford Level, horizon, balloons), 37-39 (vanishing point), 46 (Polaris), 56-58 (southern circumference), 75-79 (tubes, Sun distance), 80-84, 160-180 (sea-level sightings), and the Neptune passage near p.131. I did not re-read the eclipse, tide and scripture sections; for those I rely on the Step 1 summary and digests, and say so.

Some units to fix first. 1 statute mile = 5,280 ft. R = 3,959 mi is the Earth's radius. "8 inches per mile squared" is the book's rule: a globe's surface falls below a tangent line by 8 in x (distance in miles)^2 [calc: R^2/(2R) gives 8.00 in]. "k = 0.13" is the standard refraction coefficient: air bends light slightly toward the ground, so the globe's drop looks about 13 percent smaller than the pure geometry says [memory: high].

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Corrections to Step 1

1. The Southsea to Ryde sight was 4.5 miles, not 43 miles. facts.json and claims.md say "43-mile Southsea to Ryde". The OCR reads "43 statute miles" on PDF p.173 and "4i MILES" in the diagram on p.172. The book's own arithmetic settles it: "the centre of the arc would be 40 inches higher" and the steamer would climb "for 2 1/4 miles". (4.5/2)^2 x 8 in = 40.5 in [calc], which only fits 4.5 miles. At 43 miles the figure would be about 1,230 ft. [read: PDF pp.172-173; calc]. Claim 2 as written in claims.md overstates the sight tenfold.
2. The Investigator's note that the book's 16 ft 8 in "is the bare 8 in/mile-squared rule, with no allowance for eye height" is wrong. The book does allow for eye height. It says the highest point of the water is one mile from an 8-inch eye (the tangent point), so the boat is 5 miles beyond it, and 5^2 x 8 in = 200 in = 16 ft 8 in. It then subtracts the 3 ft flag: "13 feet below the line-of-sight" [read: PDF p.17]. So the figure is the drop at 6 miles from the eye, not at "5 miles" as facts.json says. My independent geometry agrees: hidden at 6 miles 16.7 ft for the water line, 13.7 ft for the flag top with no refraction, and 14.1 / 11.1 ft with k = 0.13 [calc]. The Investigator's 13.7 ft and 11.1 ft figures are right; only the comment on the book's method is wrong. The book's real weakness is different (see Claim 2).
3. The book's Southampton "officer" data are internally impossible. Altitudes 45 and 38 degrees, "800 statute miles" apart, same instant, same meridian [read: PDF p.79]. On any shape, two stations on one meridian differ in the Sun's altitude by their difference in latitude. 7 degrees of latitude is about 483 miles, not 800 [calc]. The book gives no name, date or instrument. Step 1 recorded the figures without comment.
4. Ordnance Survey London/Ackworth data fit the globe. Not an error in Step 1, but an omission: Step 1 reported the "under 4,000 miles" result without checking it. London (latitude 51.5 degrees [memory: high]) altitude 55 deg 18 min gives solar declination 16.8 degrees; Ackworth (53.65 degrees [memory: medium]) altitude 53 deg 2 min gives 16.7 degrees [calc]. Both stations see the same declination, as a distant Sun and a round Earth require. Details under Claim 4.
5. Step 1's note that the "Foucault figures agree with the sine rule" is confirmed [calc]: Paris 11.29 deg/h (31.9 h), London 30.7 h, Cairo 48.0 h, Calcutta 62.5 h, Edinburgh 29.0 h. The 11 m = "30 feet" conversion is off (11 m = 36 ft). The deep read is accurate.
6. Wikipedia's temperature-gradient number for the canal (0.11 degrees C per metre) is the Investigator's [web]. I checked it independently: for light to follow the Earth's curvature exactly (k = 1) at P = 1013 hPa, T = 290 K, the air must warm upward at about 0.13 K per metre [calc, standard refractivity formula, memory: high]. Same order. Retagged [web: https://en.wikipedia.org/wiki/Bedford_Level_experiment] for the 0.11 figure; my 0.13 is [calc].

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Mainstream background in four paragraphs (for the reader)

• Shape. The Earth is an oblate spheroid, flattened about 1 part in 300 by rotation, with mean radius about 6,371 km [memory: high]. A flat map can show it only with distortion. The north-pole-centred "azimuthal equidistant" map keeps distances from the pole true and stretches everything else; the UN emblem uses it [memory: medium, from facts.json].
• Light bends in air. A near-horizontal ray follows a slightly curved path because air density changes with height. Over water in sunshine the air just above the surface can be colder than the air above it (an "inversion"). Then rays bend strongly downward, objects "loom" up, and hidden things come into view. Mirage is the extreme case. This is the main mainstream explanation of every "I saw further than the globe allows" report [memory: high].
• Foucault. A long pendulum swings in a plane that stays fixed against the distant stars. The ground turns under it. At latitude phi the plane turns at Omega x sin(phi), with Omega = 15 degrees per hour. The effect is the Coriolis term (the sideways apparent force on a moving body in a rotating frame) [memory: high]. It is zero at the equator and reverses sense in the southern hemisphere [web: https://en.wikipedia.org/wiki/Foucault_pendulum].
• Distances in space. The Sun is about 1.5 x 10^8 km away (measured by radar and spacecraft), the Moon about 384,000 km (laser ranging), the nearest stars several light-years (annual parallax, first measured by Bessel in 1838) [memory: high].

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Claim 1: The Earth is a stationary plane, north pole at the centre, south a ring of ice

a. Is the mathematics internally consistent?

No. Two checks, both from the book's own numbers.

Southern distances. The book argues that the circuit of the Earth at about 45 degrees south must be about 22,000 to 26,000 miles (from sailors' distances Cape of Good Hope to Port Jackson to Cape Horn, 22,000 nautical miles, and Ross's Cape Horn to Port Phillip, 9,000 miles over 143 degrees of longitude) against about 14,300 for a globe [read: PDF pp.56-57]. It then says that on a plane a degree of longitude at 45 deg S is 69.44 statute miles, "a difference of nearly 20 miles" from the globe's 49.52 [read: PDF p.57]. - On the book's own picture (north pole at the centre, the equator a circle halfway out, distances along meridians true) a point at 45 deg S lies 135 degrees of arc from the pole, 135 x 69.05 = 9,320 miles out. The circle through it is 2 pi x 9,320 = 58,600 miles round, and one degree of longitude there spans 162.7 miles [calc]. The book's 69.44 is just the equator's value; it ignores that on a pole-centred disc the circles widen as you go south. - So the book's own measured circuit (22,000 to 26,000 miles) is less than half of what its own disc requires (58,600). The data the book cites to refute the globe refute the disc more strongly. - The figure "should only be 14,282 miles" for a globe at 45 degrees is also off: 21,600 nautical miles x cos 45 deg = 15,274 nautical miles, or 17,600 statute miles [calc]. [read: PDF p.56; OCR digit error possible, I did not render the page image.]

Sun height. See Claim 4: no single Sun height fits the noon altitudes on a plane.

b. Consistent with established experiments?

No. Examples the reader can check: - Voyage and flight times in the south. On the disc (equidistant map), Sydney (about 34 deg S, 151 deg E) and Santiago (about 33 deg S, 71 deg W) [memory: high] lie about 8,550 miles from the pole each, 138 degrees of longitude apart. The straight-line distance is about 15,900 miles [calc]. The real distance is about 7,000 miles (about 14 hours by air). Flight times in the southern hemisphere fit a sphere and do not fit a pole-centred disc [memory: high on typical flight durations, medium on exact numbers]. - Southern sky. From Chile, South Africa and Australia the stars circle one point, the south celestial pole, found near Sigma Octantis. A disc has no such single point seen from three continents. Polaris is never visible from the southern hemisphere at all [memory: high]. The book cites a single Times report (13 May 1862, "Captain Wilkins", Polaris and the Southern Cross at latitude 23 deg 58 min, presumably south) [read: PDF p.46]. On a globe Polaris at that latitude would be 24 degrees below the horizon; refraction lifts objects by well under 1 degree, so it cannot be refraction. It is a one-newspaper report with no instrument, no star measurement and no corroboration found. The mainstream reading is a misidentified star or a wrong latitude; that is my judgement, not a demonstrated fact. On a plane model Polaris should be easy to see from all southern lands, and it is not. - Circumnavigation and the Antarctic. The book says circumnavigation is possible on a disc [read: PDF pp.39-43]. True as geometry (sail round the pole). But a disc makes the east-west trip at 55 deg S longer than at 35 deg S by a factor of about 1.5 and the equator trip a fixed 25,000 miles; real voyages show circuits shrinking toward the south, as on a globe [memory: high].

c. Tested? By whom? Replicated?

Not by Rowbotham as a disc. His own measurement proposal (a degree of longitude in Australia, PDF p.58) was, per the book, "yet to be settled". Geodetic measurements of the length of a degree of longitude at many latitudes have been made since the 18th century (the French Geodesic Mission to Peru and Lapland, 1735-44) and give a sphere (flattened a little) [memory: high]. I did not find, in the project files or the Investigator's notes, any report that Rowbotham or his followers carried out the Australian measurement [read: PDF p.58; Investigator's notes: nothing found].

d. If true, what would have to change?

Almost everything with a distance in it: satellite orbits and GPS, the whole southern-hemisphere geodetic survey, navigation tables, time zones, the observed southern sky and the measured gravity map. A disc with true meridian distances cannot be reconciled with measured southern distances at all.

The Gleason map and patent (Investigator lead, kept in the record)

The map is drawn with the north pole at the centre. Mainstream cartography treats that as a valid projection of a globe; it says nothing by itself about shape [memory: medium]. Gleason's US patent 497,917 (filed 15 Aug 1892, granted 23 May 1893, assigned to Buffalo Electrotype and Engraving Co.; the patent holder is an interested party) describes a time-chart whose latitudes and longitudes "correspond to those of all other first class geographical globe maps", and describes the departure from a globe as a distortion ("extorsion") from "straightening out of the meridian lines" [web: https://patents.google.com/patent/US497917A/en, as read by the Investigator]. The printed slogan "As It Is" and Gleason's book Is the Bible from Heaven? Is the Earth a Globe? (1890; 2nd ed. 1893) are his other statements. In the patent he describes a globe-derived layout; on the map he labels it "Scientifically and Practically Correct, As It Is". Both are in the record; they sit uneasily together. I did not read Gleason's book. [read: map captions per facts.json; web per Investigator.]

Other lead: The Anti-Newtonian (anonymous, 1819), a north-centred disc with an ice wall, which Schadewald calls the probable unacknowledged source of Rowbotham's layout [web: https://www.cantab.net/users/michael.behrend/ebooks/PlaneTruth/pages/Chapter_01.html, via Investigator; I did not read it]. This bears on priority, not on whether the model holds.

Verdict, Claim 1

• Plane-with-pole-centre geography: Refuted by calculation (Sun altitudes, Claim 4; Sydney-Santiago distance above) and Internally inconsistent (book's own southern circuit versus its 69.44 miles per degree).
• The Gleason/azimuthal disc as a map: Not testable as stated. It is a legitimate projection; "as it is" is the claim that fails.

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Claim 2: Level water proves it (Bedford Level, long sightings, rising horizon)

a. Is the mathematics internally consistent?

The geometry is right for what it computes. My check of the Bedford numbers: 8 in eye, 3 ft flag, 6 miles, no refraction: water line hidden by 16.7 ft, flag top by 13.7 ft [calc]. The book's 6 ft for the middle of a 6-mile arc is correct (3^2 x 8 in = 6.0 ft) [calc].

Problems that are internal to the argument: - The result is claimed for a grazing line of sight. An eye 8 inches above the water, sighting over six miles, travels through the lowest, most strongly layered air, where refraction is largest. The book allows for this nowhere. It says only that surveyors ignore curvature [read: PDF pp.54-56]. (Surveyors use "back-and-fore sight" so that curvature cancels, which is not evidence that it is zero. Rowbotham's own text states this: "one 'sight' would compensate for the other" [read: PDF pp.54-55].) - A refraction coefficient of 1 flattens the canal exactly. That needs about +0.13 K per metre of air warming with height [calc]. A cold canal under warm summer sunshine can do this. The book reports a hot-weather bathing experiment [read: PDF p.16: "it being the summer season"], so nothing in the setting rules it out. Nobody measured the temperature profile. - The book's own wave explanation (Section 18) cuts both ways. It says wave crests block a line of sight at the vanishing point of the sea [read: PDF pp.164-165]. Then a bather at 8 inches could not see the "boat down to the water's edge" at six miles over a ruffled surface. The account describes "nearly stationary" water but gives no wave height, time of day, weather, or whether the observation was repeated [read: PDF p.16]. - The sea-level sightings are inconsistent among themselves. Eddystone: the light has a nominal 13 nautical-mile range, and from an eye at 5 ft and 14 statute miles the globe predicts the lantern is only marginally visible (an object needs about 85 ft of height, or about 71 ft with k = 0.13) [calc]. The committee did not see even the vane [read: PDF pp.167-168]. That is a failure to see something the globe says is borderline visible, and the flat Earth says should be easy. Flat-Earth "level water" would predict they should have seen it. The book accepts this and calls on waves. The same waves are standard in mainstream optics. The observation therefore does not discriminate. - Misprint caution. The text layer gives the Eddystone vane height as "1010 feet"; I assume 101 ft (OCR) but did not render the page.

b. Consistent with established experimental results?

Mixed, in the following sense. Some of the book's observations are real and well known. They do not mean what the book says. - The horizon "rises to eye level". What is real: the visible horizon is slightly below true horizontal by the "dip" angle, about 1.76 arc-minutes x sqrt(height in metres) [memory: high]. For 110 ft this is 11 arc-minutes [calc]; in the book's Camera-house mirror test a mirror viewed from 10 inches would shift the horizon image by only 0.032 inch [calc]. The committee could not have seen a difference, and both quoted reports say the horizon "rose and fell with the eye" [read: PDF pp.170-171]. That is exactly what a dip too small to see by eye predicts. From a balloon at 1.5 miles the dip is 1.4 degrees [calc], too small to look like a drop in the ground; human perception makes the whole scene look like a bowl (the "watch-glasses" description from Wise and Mayhew [read: PDF pp.22-23]). Sextants and navigators correct for dip routinely [memory: high]. - Surveying. Geodetic levelling, tunnels and long bridges (e.g. the Channel Tunnel alignments, long-span suspension bridge towers whose tops are further apart than their bases by about the geometry's amount) all include the curvature term [memory: high on principle, medium on any specific example, not checked here]. - Pass/fail on the Bedford Level. The book's 8-inch observation (summer 1838 or earlier, per the Investigator) was made once, by the person with the thesis. The same canal was used by others under better design: - Wallace, 5 March 1870: three markers 13 ft 4 in above water, 3 miles apart on six miles of the Old Bedford. The middle one appeared more than 5 ft above the line joining the end ones. The geometry predicts 6.0 ft, or 5.2 ft with k = 0.13 [calc]. The umpire, J. H. Walsh, editor of The Field, awarded the wager to Wallace. [web: Nature 1, 581 (7 Apr 1870), https://www.nature.com/articles/001581a0; Cambrian 15 Apr 1870, https://newspapers.library.wales/view/3330809/3330814/29/; Wikipedia, https://en.wikipedia.org/wiki/Bedford_Level_experiment; all via Investigator.] Hampden and Carpenter, the other party and his referee, claimed the result for themselves and said the opposite. They were interested parties. The Field also was one (its editor was the umpire). - H. Yule Oldham, 1901: three poles at equal height; the middle one stood about 6 ft higher, via theodolite. [web: Wikipedia, citing BA Report 1901 pp.725-6.] Consistent with 6.0 ft [calc]. - Clement Stretton, 20 Jan 1905, Ashby Canal: about 8 in of dip at 1 mile [web: Wikipedia, citing English Mechanic 80:546]. That is exactly the 8 in per mile-squared [calc]. - Lady Blount and Edgar Clifton, 11 May 1904: telephoto photograph of a white sheet 6 miles away with the camera 2 ft above the water; the photographer "was surprised" to get the image. This supports the book. [web: Wikipedia, citing The Earth vol.5 nos.49-50; primary not read.] It is a result with no measurement of air temperature, no repeat, and by an interested party (the Universal Zetetic Society's head, Lady Blount). A 6-mile sight at 2 ft eye height needs the same strong refraction as the 1838 case (a grazing ray), so it cannot separate "flat" from "strong downward refraction". - U. G. Morrow, 1896, Illinois Drainage Canal (Koreshan Unity): same method, concluded a concave Earth (the surface curving up) [web: Wikipedia, via Investigator]. This is a related lead: the same sort of line-of-sight survey has been used to "prove" a plane (Rowbotham), a convex Earth (Wallace) and a concave one (Morrow). Which is right depends on the refraction nobody measured. A concave result implies k greater than 1 across the survey. I did not read Morrow's report. - W. H. Le Fevre, president of the Balloon Society, 1882, said he had often seen the Earth from a balloon look concave, more so the higher he rose [web: Denbighshire Free Press 4 Feb 1882, https://newspapers.library.wales/view/3609549/3609555/55/]. A first-hand aeronaut's statement, consistent with Wise (1850s) and Mayhew in the book itself [read: PDF pp.22-23]. It records an appearance. It is a real, widely reported perceptual effect (the horizon stays level with the eye while the ground below drops away). It does not by itself fix the shape; the measured dip angle does.

c. Tested, by whom, replicated?

The strong form ("water does not curve") has been tested by Wallace (1870), Oldham (1901) and Stretton (1905) with curvature found, and contested by Hampden/Carpenter and by Blount/Clifton (1904) with a flat result at grazing heights. The one design that separates the two readings (several heights on the same day, with air temperature measured at two or three heights) was not carried out in the records I have. None of the sources the Investigator found, as far as I know, includes temperature profiles.

The Southsea to Ryde sight (4.5 miles, May 1864 [read: PDF pp.172-173]): the telescope was reported to cut the same part of a steamer's funnel all the way across. Globe prediction: the line should cross the funnel about 3 ft lower at mid-channel (40 in, or 35 in with k = 0.13) [calc]. The book gives no funnel height, telescope height, or whether anyone watched for a 3-ft shift. It is one account by the thesis's author, not measured, not repeated by others as far as I know.

The Grimsby tower "seen 60 miles out at sea" comes from a Times paragraph about the Queen's visit of 1854 [read: PDF p.179], not a measurement. The book's calculation subtracts only a 10-ft observer; for a 300-ft tower and a 10-ft deck the geometric limit is about 24 nautical miles (1.17 sqrt(300) + 1.17 sqrt(10) nautical miles [memory: high formula]) [calc]. The paragraph reads like press rhetoric. I give it no weight as an observation; it is the book's, weighted as printed.

d. If true, what would have to change?

Surveying, geodesy and navigation, which all use curvature and refraction models that work; and the refraction mechanism itself, which would need a separate reason for the flat result. Note an asymmetry: if light bends strongly enough to cancel the globe's curvature over a canal, then that bending is itself a large effect that mainstream physics already includes. The claim would not require new physics, only that the canal's air was in a strong inversion on the day.

Verdict, Claim 2 (split)

• The 1838 Bedford observation as evidence of no curvature: Never properly tested in the form claimed, a single unrepeated observation with no air profile; and it is consistent with known refraction (needs +0.13 K/m or more). It was dismissed with reasoning, not by a controlled replication under the same grazing geometry.
• The strong form (water surface is flat over miles): Tested, not supported (Wallace 1870, Oldham 1901, Stretton 1905, with raised sight lines; geodetic work generally).
• Horizon rising to eye level, balloon "concavity": Established as a real appearance; not evidence of flatness (the dip is measurable and matches a sphere, Established for the sphere).
• Southsea-Ryde, Eddystone, Nab light-ship, Grimsby: Not testable as stated, since distances, heights and conditions are not given well enough to compute against; the Nab and Eddystone reports have usable numbers and fit a variable-refraction mixture better than either clean model.

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Claim 3: Perspective, not curvature, makes ships sink hull first

a. Is the mathematics internally consistent?

No. The book's "vanishing point" is the eye's resolution limit: an object is just visible when it subtends about 1 arc-minute, i.e. at 3,000 times its own diameter, "a shilling... 250 feet" [read: PDF p.37]. Then it applies the rule to an object's height above the ground: a hull 10 ft high "vanishes at 3,000 times 10 ft, or nearly six statute miles" [read: PDF p.38]. Four problems [calc]: 1. A resolution limit depends on the size of the thing seen. A ship's hull is long and large: a 300-ft hull side-on at 6 miles subtends about 11 arc-minutes, well resolved. The thin mast is the smallest feature. A resolution limit would hide the mast first. The observation is the reverse. 2. The rule has no dependence on observer height. Real hull-down distances change a lot with it. From a mast-head at 100 ft, a ship's hull stays in view tens of miles farther than from the beach at 6 ft; the geometric horizon grows as sqrt(height) (3.0 miles at 6 ft, 12.2 miles at 100 ft, with k = 0 [calc]). The book concedes that above 6 ft the telescope "may not have power enough" [read: PDF p.39], which makes the rule unfalsifiable for any elevated observer. 3. The rule says a 3-ft flag vanishes at 3,000 x 3 ft = 1.7 miles, yet Claim 2 has the same flag seen at 6 miles with a telescope [calc, read: PDFs p.16, p.37]. A telescope of power 30 pushes the limit back by 30 times. So the book's own explanation of why the hull is lost (naked-eye limit) is what the telescope should repair at all distances; yet it also says the telescope fails over the sea because of waves (PDF pp.164-165). The explanation is used both ways. 4. A resolution limit makes things fade into haze at a distance; hull-down ships show a sharp horizon line crossing the ship at one height. I did not find any sharp-edge observation in the book.

b. Consistent with established experiments?

The geometric prediction is well supported: hidden height versus distance follows the sphere's geometry; the dip angle used with sextants and the range tables for lighthouses (Findlay's Lighthouses of the World 1862 is quoted in the book itself, PDF p.166: "to this elevation 10 feet is added for the height of the deck") assume it [read: PDF p.166]. Lighthouse range tables are built on curvature and refraction.

The book's own report that on calm days the Nab light-ship's hull was "distinctly visible" at 8 miles from an eye 32 in above water, but on less calm days "no portion of it could be seen from the same elevation" [read: PDF p.163]. The globe, with k = 0, hides 24 ft (matching the book's 24 ft) and with k = 0.13 about 20 ft [calc]. The bulwarks are said to be about 10 ft high, so "greater part of the hull visible" needs far stronger refraction than standard. This is an anomaly for the standard-refraction calculation. Under mainstream physics it is explained by day-to-day variation in near-surface temperature layers (looming); the book's explanation is waves. The report is the thesis author's account, from May 1864, with no meteorological data. I rate it as a real unresolved observation in the record, not as a replicated test.

c. Tested?

Telescope restoring hulls: asserted for ships and the Plymouth experiments in 1864; the Plymouth committee (names given in the book: "Parallax", "Theta" (Mr. Henry), Messrs Osborne, Richards, Rickard, Mogg, Evers, Pearce [read: PDF p.167]) and the Plymouth Athenaeum debates are the book's own; R. A. Proctor's account of the Plymouth tests, as given by Schadewald, is the critic's side [web: Schadewald Ch.1, via Investigator; not read by me]. No neutral replication is on record in my materials.

d. If true, what would have to change?

Optics of the eye (the rule would need to apply to heights, not sizes), the lighthouse and navigational tables, and refraction theory. The perspective explanation does not need any new physics, only a corrected use of an old rule, and it fails on its own numbers.

Verdict, Claim 3

• Perspective-limit mechanism: Refuted by calculation (resolution applies to angular size; a hull is larger than a mast; rule independent of eye height).
• "A telescope restores the hull": Never properly tested as a controlled, recorded experiment; the sea results in the book are mixed and are explained by refraction in mainstream physics.
• Circumnavigation and loss of a day on a plane: logically possible on a disc; Not testable as stated as an argument for a plane. The Polaris point is addressed under Claim 1 (Refuted by calculation; single unsupported newspaper report).

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Claim 4: Sun, Moon and stars are small and near and move over a fixed Earth

a. Is the mathematics internally consistent?

The plane-trigonometry formula is fine: with two stations a distance d apart on a meridian, Sun altitudes a1 (nearer) and a2, height H = d / (cot a2 - cot a1) [calc]. The book's data: - London 55 deg 18 min and Ackworth 53 deg 2 min, 151 miles: H = 2,508 miles [calc]. The book says "under 4,000". - Southampton officer 45 deg / 38 deg, 800 miles: H = 2,858 miles [calc]. Different from 2,508 by 14 percent, and the data are impossible (Correction 3).

A single-height Sun predicts the same H from every pair of stations. Take a later check [calc]. At the equinox the Sun is over the equator. Noon altitude is 90 degrees minus latitude (an observed fact anyone can measure). On the book's disc a station at latitude L is L x 69.05 miles from the sub-solar point, so H = (L x 69.05) tan(90 - L):

H changes by a factor of 1.6 between 15 and 60 degrees. So there is no single Sun height that fits the observed altitudes. On a distant Sun and a sphere the same altitudes follow from latitude alone. The altitude pair in the book (London 55.3, Ackworth 53.0) gives the same declination (16.8, 16.7 degrees), which is just what the sphere requires.

b. Consistent with experiment?

No. - Moon distance by parallax (La Caille at the Cape, Lalande at Berlin, 1751) gave about 60 Earth radii, consistent with everything since, down to laser ranging of the Apollo reflectors [memory: high]. The book's method, applied to the Moon, would give a different answer for each pair of stations. - Star parallax (Bessel, 61 Cygni, 1838) and aberration (Bradley, 1729) are annual effects in the stars' apparent positions that follow from an Earth in orbit [memory: high]. The book's parallel-tube test [read: PDF pp.75-76] uses two parallel tubes one yard apart and says "a distinct period of time will elapse" between the signals. This is stated as an expectation, not reported as an experiment. A delay is expected on any model: two tubes 0.9 m apart east-west are carried past a star at the ground's rotation speed, 290 m/s at 51.5 degrees north, so the star crosses them about 3 ms apart [calc]. A test of orbital parallax needs a baseline of 2 AU and angle precision around an arc-second, not a yard. - Eclipses. Book (via Step 1 summary, not re-read by me): the Moon is seen red in totality and has been seen eclipsed while the Sun is above the horizon (Paris 19 July 1750) [read: Step 1 digest]. The mainstream account: the red light is sunlight refracted through the Earth's atmosphere into the shadow, and horizon refraction (about 34 arc-minutes [memory: high]) lets observers see both bodies near the horizon when they are in fact about 1 degree below the line [memory: high]. The Earth's shadow on the Moon always has a circular edge, whichever way the Earth is turned, which only a sphere gives [memory: high, since Aristotle]. - Moonlight does not warm a thermometer. The book quotes De La Hire and Forbes with focusing mirrors or lenses (300 to 6,000 times) and no thermometer rise [read: Step 1 summary]. Full moonlight is about 2 x 10^-6 of sunlight [calc, 0.25 lux against 1.3 x 10^5 lux]. Even 3,000 times that is about 8 W per square metre, little more than a thermometer can show in a breeze with a lossy lens [calc]. A more sensitive thermocouple did detect it: Piazzi Smyth, from Guajara on Tenerife in 1856, found his instruments "sensibly affected" by the Moon's rays [web: https://en.wikipedia.org/wiki/Charles_Piazzi_Smyth; search summary]. Lord Rosse later measured the Moon's heat with a thermopile [memory: medium; not checked]. The "cold moonlight" argument was a real limitation of 1850s thermometers and no longer holds. Moon phases and eclipses already show the Moon is lit by the Sun [memory: high].

c. Tested? Replicated?

The Sun-height triangulation is fully testable and has been: the data in the book itself fit the globe (same declination from two stations). The tube test is described but not reported as run. The book's own sources for the Southampton figures are an unnamed officer. I did not find any independent record of it.

d. If true, what would change?

Stellar distances, the whole astrophysics chain (spectra, luminosity, stellar structure), spacecraft navigation, radar ranging of the Sun's distance (accurate to metres through Venus and planet radar), and eclipse prediction (which has been accurate to seconds for centuries with the globe model) [memory: high].

Verdict, Claim 4

• Near Sun and stars (under 4,000 and 6,000 miles): Refuted by calculation (no single height fits equinox noon altitudes; the book's own London/Ackworth data fit a distant Sun; Southampton data impossible).
• Lunar eclipse not from Earth's shadow, Moon self-luminous: Tested, not supported (circular shadow edge, phases; moonlight heat detected by thermocouple in 1856).
• Parallel-tube test: Never properly tested (described, not reported); does not discriminate as stated (3 ms delay expected on every model).

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Claim 5: The Foucault pendulum does not prove the Earth rotates

I rely on Step 1's deep read (PDF pp.138-159) and re-did its numbers.

a. Is the mathematics internally consistent?

• The book's Foucault figures are consistent with the sine rule (Correction 5). This also means the author's quoted sources state a regular law (rate = 15 deg/h x sin latitude), which sits badly with his first conclusion that the effect is "variable... not uniform... proves nothing" [read: PDF p.158]. The book offers no pendulum run of his own [read: deep read]. His "inconstant" conclusion rests on the Jones case, one of the unnamed reports, and the general assertion without source [read: PDF p.154].
• The Jones case [read: PDF pp.154-156]. Predicted 1 deg in 5 min (0.20 deg/min; the rule gives 12.05 deg/h at Manchester = 0.20 deg/min [calc]). Observed "nearly the third degree" in "quite seven minutes", about 0.43 deg/min, 2.1 times predicted. Jones had himself said in advance that atmosphere and "some slight lateral bias at starting" would spoil precision [read: PDF p.155]. A sideways error at release makes the swing slightly elliptical, and the ellipse itself turns (Airy precession, G. B. Airy 1851). The rate is about (3/8) x (a b / L^2) x omega0 with a, b the semi-axes of the path, L the length and omega0 = sqrt(g/L) [memory: medium; the web check of the formula failed, https://en.wikipedia.org/wiki/Foucault_pendulum does not give it]. For Jones's pendulum (L = 7.62 m, 360-inch circle so a = 1.46 m [read: PDF p.154; calc]) a minor semi-axis b = 6 mm gives 0.22 deg/min, exactly the excess of 0.23 deg/min [calc]. So a 6-mm sideways miss out of a 1.46-m amplitude (0.4 percent) accounts for the whole discrepancy. Whether it was added to or subtracted from the Earth effect depends on which way round the bob circulated; the report gives none. The excess is therefore explainable by a plausible, tiny release error; I do not claim it was that, since there is only one reported interval.
• The ball-and-socket argument. The book says Foucault's own suspension was a ball-and-socket joint that twists. The only description of such a joint in the pages is for Jones's apparatus [read: deep read, p.155]. Kamerlingh Onnes's 1879 thesis did study how imperfect suspension and wire elasticity turn linear motion elliptical within about an hour [web: https://en.wikipedia.org/wiki/Foucault_pendulum]. "Imprecise construction can cause additional veering which masks the terrestrial effect" [same page]. So suspension errors are a real source of scatter, and careful set-ups (burning a thread to release, symmetrical suspension) exist to remove them. That is a reason to be careful, not a reason the effect is not there. A random twisting cannot produce a steady turn with the sine-of-latitude dependence and a reversed sense in the southern hemisphere [web: same page].

b. Consistent with established experiments?

• Rate vs latitude: the sine law has been checked at many sites [memory: high; the book itself quotes Paris, Edinburgh and Cairo as predictions, not measurements, deep read §8].
• Opposite sense in the southern hemisphere, zero at the equator [web: https://en.wikipedia.org/wiki/Foucault_pendulum]. A magnetic or twisting effect has no reason to reverse at the equator.
• The "magnetic meridian" claim [read: PDF p.153; Dundee writer, Liverpool Journal 17 May 1851]: the pendulum "tends to the magnetic meridian". It is a second-hand claim from one writer in one newspaper; no numbers; Step 1 correctly noted that. I treat it as an observation that was reported and not followed up in the sources I have. Mainstream physics says it is wrong for brass bobs (non-magnetic [read: PDF pp.144, 154]), and it would predict a pendulum that settles at an azimuth; a Foucault pendulum goes round through every azimuth. The Foucault plane keeps turning past the meridian line, as the book's own table of 2-, 4-, 8-hour positions shows [read: PDF p.146].
• Independent confirmations of rotation: the gyrocompass, the Coriolis deflection of cyclones, and the Michelson-Gale (1925) ring-interferometer measurement of the Earth's rotation [memory: high]. The book does not address any of them (written 1865).

c. Tested? By whom? Replicated?

Foucault, Paris 1851 (with a 2-m, then 11-m, then Pantheon pendulum), repeated under Arago [read: deep read §3]; Jones, Manchester 1851 (discussed above); many others since, as standard demonstrations [memory: high]. The book reports no test by the author. The claim that "some runs showed no change or wrong direction" has no source in the pages [read: deep read §5]. The Dehaut postscript (Poinsinet de Sivry, 1782, a pendulum compass) treats the invariance of the plane as established [read: PDF p.159].

d. If true, what would have to change?

If the effect were an artefact of suspension, the sine-of-latitude rule, its sign reversal south of the equator and agreement with the Coriolis term in cyclones would all be coincidences. One could not use the effect to find latitude.

A fair point from the book. The author's second objection (the pendulum cannot show "that it is the Earth which moves underneath the pendulum, and not the pendulum which moves over the Earth" [read: PDF p.158]) has a legitimate core. Motion is relative, and in general relativity a rotating universe around a fixed Earth is equivalent in its dynamical effects (Thirring 1918) [memory: medium]. But that is an argument for "the heavens turn round a fixed globe", not for a plane. It does not rescue the plane: the sine-of-latitude dependence is the geometry of a sphere (or any curved surface) with a polar axis.

Verdict, Claim 5 (split)

• "The plane's turn is too irregular to be evidence": Tested, not supported. The regular law and its latitude dependence are measured; the Jones excess is explained by a 0.4 percent release error.
• "Earth under pendulum vs pendulum over Earth": Not testable as stated in pure kinematics; decided by dynamics (Coriolis effects elsewhere); it does not favour a plane.
• Magnetic-meridian claim: Never properly tested by neutral replication in the sources I have (one second-hand report); contradicted by the continuing turn of the plane.
• Twisting ball-and-socket as the cause: Plausible, untested as a source of scatter (supported by Onnes); Tested, not supported as the cause of the main effect.

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Claim 6: Orthodox astronomy rests on assumed premises and has failed its tests (Neptune, air-gun, mast-drop, tides)

a. Is the mathematics internally consistent?

• Neptune. The book says Le Verrier's period (217 years), distance (more than 3,000 million miles) and mass (38 Earths, via digest) differ from the planet found [read: PDF pp.130-131; digest for the mass]. Check [calc]: a Le Verrier mass of 1/9,300 of the Sun is 35.8 Earth masses (the actual is about 17); period 217 yr vs actual 165 yr. These are real differences [memory: medium on the exact Le Verrier numbers]. But the prediction that mattered was the position: "Le Verrier located Neptune within one degree of its discovery position" and Galle found it in under an hour on 23-24 Sep 1846; the orbit size was wrong because both Adams and Le Verrier used Bode's law to fix the distance [web: https://en.wikipedia.org/wiki/Discovery_of_Neptune]. The disagreement is in the unconstrained orbital elements, not in Newton's inverse-square law (position was right).
• Air-gun and mast-drop. The book says a ball fired straight up would fall back "within a few feet" though the ground would move about 5,600 ft in 8 s [read: Step 1 summary of PDF pp.72-74; I did not re-read these pages]. On a rotating Earth the ball keeps the ground's eastward speed (inertia: Galileo's ship argument). The surface moves 290 m/s at 51.5 degrees north, so over 8 s the ground goes about 7,600 ft, not 5,600 [calc; the book's figure is probably for another latitude]. The ball goes with it. What is left is the Coriolis shift: about 3.8 cm east for an 8-second up-and-down flight [calc, using (4/3) Omega cos(lat) g (T/2)^3]. No 1860s air-gun test could resolve 4 cm against wind and drag. Drop experiments from tall shafts were done: Reich's mine shaft (Freiberg, about 158 m, about 28 mm east predicted [calc]; Reich measured about that, as I recall [memory: medium; web check failed on this page]).
• Tides. The book (via Step 1) says a floating Earth rocks about once every 12 hours under atmospheric pressure [read: Step 1 summary of PDF pp.111-115; not re-read]. Real tides follow the Moon's passage: the interval between high waters is about 12 h 25 min, a lunar-day period, which makes high tide about 50 minutes later each day [calc: 12.46 h, memory: high]. A pressure rocking linked to the Sun's day would cycle every 12.00 h. So this is a numbers check the reader can do from a tide table.

b. Consistent with experiments?

Neptune was found where Newtonian celestial mechanics said, which counts for Newton, not against. The earlier discovery of Uranus's anomalies, and the 1846 find, are the standard successes. The air-gun/drop results follow Galilean relativity and the small Coriolis correction.

c. Tested?

Neptune: tested by observation (Galle); the "differs from the planet found" point is real but about orbital elements. Reich-type drop tests: by Reich (Freiberg), earlier Guglielmini (Bologna, 1791) [memory: medium]. I did not find these in the project's materials.

d. If true, what would change?

Celestial mechanics: planetary ephemerides accurate over centuries, spacecraft trajectories, tide tables.

Verdict, Claim 6 (split)

• Neptune as a failure of Newtonian theory: Tested, not supported. Position predicted to under 1 degree; wrong orbit elements came from an assumption (Bode's law), which the book is right to flag as an assumption. Historically instructive.
• Air-gun and mast-drop as proof of no motion: Refuted by calculation (inertia; 3.8 cm expected shift).
• Tides as a floating Earth rocked by air pressure: Refuted by calculation (period 12 h 25 min tracks the Moon; not 12 h).
• "Orthodox astronomy rests on assumed premises": Not testable as stated as a general charge. The author's own premises (the plane, a vanishing-point rule) are assumed in the same way.

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Claim 7: Scripture is literally true on the form of the Earth, so a proved plane answers atheism and the "plurality of worlds"

This rests on a theological premise (PDF pp.178-229, Step 1 summary; I did not re-read these pages). I judge physics only.

a. Mathematics

Not applicable.

b. Experiments

The physics content depends entirely on Claims 1-6, which fail.

c. Tested?

Not applicable.

d. If true

The "plurality of worlds" (other planets with inhabitants) is a question for astronomy; a plane Earth with a near Sun would remove other worlds, but Claim 4 is refuted.

Verdict: Not testable as stated (religious interpretation). Note for the reader: the author is an interested party with a religious motive, and he says so (Section 14).

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Other Investigator findings kept in the record (weighed, not dropped)

1. The inventor's own work. - The 1849 pamphlet (LoC QB638 .R87, 16 pp.) gives the founding experiment with a theodolite, a boat sent 6 miles on the "New Bedford", and "a fall of 24 feet" [web: Schadewald Ch.1, quoting the pamphlet]. The 1865 book has an eye 8 in above the Old Bedford and a flag 3 ft high [read: PDF p.16]. Schadewald calls the accounts contradictory [web, via Investigator; not read by me]. If the accounts differ, the 1838/39 observation is less reliable as a single clean experiment. I cannot verify the pamphlet. - A "Parallax" treatise in The Field's possession with Carpenter's name on the title page described an earlier canal experiment "with a result exactly the reverse" of Wallace's [web: Cambrian 15 Apr 1870]. This is a statement by The Field (an interested party, since its editor was the umpire). It is probably the 1849 or 1851 pamphlet. Unidentified. - 1882 lecture to the Balloon Society as "Dr. S. Birley" ("Parallax"), describing "many experiments which he had made" [web: Denbighshire Free Press 4 Feb 1882]. No details of the experiments found. - The 3rd edition (1881, 430 pp.) is online at sacred-texts.com [web]; I did not read it. A deep read of its standing-water chapter would help. The Jones/Foucault treatment in the 1865 edition cannot be assumed to be unchanged.

2. Supporting information. Le Fevre (1882), Blount/Clifton (1904), Hampden and Carpenter (1870) are discussed under Claim 2 with their stakes: Le Fevre as an aeronaut (a first-hand report of an appearance, which is real); Blount as the movement's president; Hampden and Carpenter as the wagering party and referee. Carpenter's Theoretical Astronomy Examined and Exposed (1864) and One Hundred Proofs (1885) are followers' works I did not read.

3. Related work. Morrow/Koreshan 1896 (concave Earth, the same method, opposite conclusion): kept. The Anti-Newtonian 1819: kept. Gleason's book and patent: Claim 1. Plymouth 1864 (Proctor's account): critic's account, not read. Charles Fort's account of Oldham 1901 in New Lands: not read.

4. Historical record. - RAS, 8 Dec 1848. The 1849 pamphlet says its argument was "a paper read before the Royal Astronomical Society". De Morgan wrote that no account appears in the Notices and that he read some of it aloud at the close of the meeting "for your amusement" [web: Schadewald Ch.1, quoting A Budget of Paradoxes]. De Morgan was the RAS Secretary, an institutional party. This is documented in one printed source; the RAS minute book would decide it. It says something about how the work was received. It does not bear on whether the physics is right. - Hampden libel conviction (Times, 8 Mar 1875; 12 months), and the court's later finding that the wager was void [web: Wikipedia, via Investigator]. Legal fact about the dispute with Wallace; it does not bear on the physics. - Inquests over Rowbotham's phosphorus remedies (per Wikipedia, Garwood, Schadewald; primary reports not found) [web]. Bears on the inventor's reliability as a medical vendor and is not evidence about the flat-Earth physics. Kept for the record. - Not an absence of record. Rowbotham was lecturing in 1882, was president of the Zetetic Society in 1883 and died in 1884; the movement continued through the Universal Zetetic Society (1893) [web, via Investigator]. I do not see an anomaly to analyse. - Records not reached by the Investigator (British Newspaper Archive, HathiTrust, NASA ADS MNRAS 9, Espacenet for Gleason's foreign filings, Cambridgeshire Archives for Manea Fen 1838-41, Scribd for Blount 1904): would help on the date and conditions of the 1838/39 observation and on whether Blount's 1904 result carried temperature data. library.tfes.org was blocked (TLS error, not bypassed) and holds Blount's 1904 scans.

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Web verification

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Summary of verdicts

What would settle the open points. For Claim 2: repeat the grazing-height sight on the Old Bedford at three eye heights (for example 8 in, 3 ft, 13 ft) on the same day, with air temperature measured at two heights along the path; curvature plus the measured refraction should reproduce all three. For Claim 5: one Foucault run of 30 minutes or more with a thread-burn release, recording the plane over time. For the 1838 observation: the primary account and the date.