Tyndall-John: Mathematician review (Step 2)

Reviewer: PhD mathematician. Scope: derivations, arithmetic, units, calibration, hidden assumptions. Physics judgements belong to the physicist; I flag where the mathematics hands a question to physics. Tags: [read] = source text or the deep reads (which checked key numbers on page images); [calc] = my own calculation in the project venv (numpy); [web] = Internet source with URL; [memory] = recalled, confidence given.

Source note carried forward: everything in the folder is by Tyndall himself (an interested party in his own priority and in the success of his method). Plate I, journal p.1 of the 1866 paper, the unpublished galvanometer calibration for the 1866 paper, Tyndall's raw series and any error bars are not in the folder. No claim below rests on [memory] alone unless marked.

0. How I worked

I read facts.json's claims list (claims.md), all five deep reads in full (On Radiation; the 1866 Calorescence paper; Heat Ch. II, III, VI, IX, X, XI, XIV sections; New Fragments; Six Lectures), the Investigator's sites.md and all research notes. No file in research/files/ exists (none downloaded). I recomputed the arithmetic that the verdicts turn on (sections per claim). Web budget used: 2 searches plus 1 failed fetch (results at the end).

Units and conventions that matter for every claim (a short primer on how Tyndall's numbers are built)

1. The detector is nonlinear. The galvanometer needle deflection theta is proportional to heat only up to about 30 degrees; beyond that heat rises faster than theta (40 deg = 47 units, 50 deg = 80, 70 deg = 211, 75 deg = 276; unit = heat that moves the needle 0 to 1 degree) [read: Heat p.327-328, 334]. Every "per cent absorbed" must pass through that calibration. Where Tyndall quotes raw degrees the ratio is distorted (example in claim 1).
2. "Times air" is not a physical constant. Air's own absorption is below the detection limit, so Tyndall assumes it equals 1 degree for a whole atmosphere "probably less even" [read: Heat para 398, p.335], and in the 1-inch table assumes 1/30 of that for the four feeble gases [read: Heat para 421]. A ratio whose denominator is an upper bound is a lower bound for the true ratio; it grows if the air is purified further, as Tyndall himself says ("the purer the air became ... the greater ... the action of the aqueous vapour", Heat p.401). So 972, 5460, 70, 90 are measurements of "this apparatus, this source, this purity", not intrinsic ratios.
3. Two different units are in use. The gas tables compare each gas at 1 inch of mercury with air at the same 1 inch (On Radiation table; Heat p.349-350). The vapour tables and the olefiant/ether pressure tables compare with a whole atmosphere (30 in) of air [read: Heat p.335, 356-357]. Conversion: 1 inch CO2 = 972 equal-pressure units = 972/30 = 32.4 whole-atmosphere-air units, while 30 inches CO2 = 90 [read: Heat p.347-348] [calc]. Ratio 30-inch/1-inch absorption: CO2 2.8, CO 3.6, SO2 3.3, olefiant gas 4.8, H2S 5.6, N2O 5.7, NH3 6.6. Thirty times the gas gives only 2.8 to 6.6 times the absorption: saturation, not proportionality.
4. Sources differ. Boiling-water cube (about 100 C), Bunsen-heated copper plate (hotter, temperature not stated), platinum spiral, lime light, electric arc. A number from one source cannot be transferred to another or to the Earth's emission (about 288 K). Tyndall knows this ("the self-same gas exercises a different action upon different qualities of radiant heat", 1859 Note [web] https://archive.org/details/jstor-111604).

---

1. Claim 1: simple gases are transparent to radiant heat; compound gases and vapours absorb enormously; radiating power equals absorbing power

a. Is the mathematics internally consistent? Yes at the level of ordering and magnitude, with five bookkeeping defects, none fatal. - The On Radiation table (air 1; CO 750; CO2 972; HCl 1005; NO 1590; N2O 1860; H2S 2100; NH3 5460; olefiant 6030; SO2 6480) equals Heat's 1-inch table, and the independent Wikisource validated transcription confirms the page-image values (1590, 6030, 372) [read: deep read; web https://en.wikisource.org/wiki/On_Radiation_(Rede_Lecture), via Investigator]. One label differs (the Rede image reads "Hydrochloric", Heat's OCR "Hydrobromic" for 1005; unverified in Heat). - Prose "nitrous oxide 1800" versus table 1860 (Tyndall's rounding) [read: On Radiation p.23]. - Heat Table 7c: ether/olefiant ratios from the printed table are 2.38, 2.29, 2.22, 2.14 (1 to 4 inches), but the text says "two and two-third times"; olefiant at 4 in is 154 here and 157 in table 7a [read: Heat p.335, 339] [calc]. Misprints or different runs; the text does not say. - Heat 7b: "calculated" for 14 measures printed 29.8; 2.2 x 14 = 30.8 [calc]. 7d: 27.0 for 6 measures should be 27.6; 1/220 x 1/2.5 x 1/1000 = 1/550,000, printed 1/500,000 [calc]. - The Rede vapour table does not match Heat's. On Radiation gives (at "1/30 atmosphere" = 1 inch): CS2 47, iodide of methyl 115, benzol 186, amylene 321, sulphuric ether 440, formic ether 548, acetic ether 612 [read: On Radiation p.38 image]. Heat's table at 1.0 inch gives 62, 242, 267, 823, 870, 1075, 1195 [read: Heat p.356-357]. The ratio Heat/Rede ranges 1.3 to 2.6; Heat's CS2 value 47 corresponds to its 0.5-inch column. Different apparatus or source, not stated. A reader who mixes the two tables will be wrong by a factor 1.3 to 2.6. (New observation; not in Step 1.) - Radiation/absorption table ("reciprocity", Heat p.344): in raw needle degrees radiation/absorption = 0.67 (CO), 0.72 (CO2), 0.66 (N2O), 0.87 (olefiant). Converted to heat units with Tyndall's own calibration points (interpolated) the ratios become 0.67, 0.72, 0.48, 0.62 [calc, interpolation between his tabulated points, so approximate]. The rank order is identical in both columns, which supports "go hand in hand"; the claimed proportionality is loose (a factor of 1.5 spread) and the 61-degree entry lies in the strongly nonlinear range [read: Heat p.344; calc]. - Worked calibration example, para 395: olefiant gas 70 deg = 211 units against empty tube 75 deg = 276 units gives 76.4 per cent; Tyndall writes "seven-ninths ... about 80 per cent" (7/9 = 77.8) [read: Heat p.333-334] [calc]. Raw degrees would give 93 per cent. The calibration matters and is the dependency to audit (Melloni's shunt method, Heat Appendix to Ch. X; I checked its arithmetic: 3.333 x 15.5 = 51.7, 3.333 x 24.1 = 80.3, 3.333 x 36.7 = 122.3, all consistent [read: Heat p.359-361]). - Saturation law. Olefiant gas at 1..10 inches: 90, 123, 142, 157, 168, 177, 182, 186, 190, 193 [read: Heat p.335]. A single exponential A = Amax(1 - exp(-k p)) fits poorly (residual sum of squares 385); a two-band fit (63.6 with k = 2.12 per inch, plus 136.8 with k = 0.288 per inch) fits to within 2 [calc]. Meaning: the source spectrum contains a strongly absorbed part that saturates in about an inch and a weakly absorbed part that needs about 3.5 inches. This is the behaviour of a molecule with several bands of different strength. Tyndall's own gloss, "all absorbable rays being removed, the residual heat passes unimpeded" (para 399), is the same idea in words. CO2 and olefiant gas against a CO-flame source rise 48.0 to 74.3 per cent and 23.2 to 65.5 per cent from 1 to 10 inches: about +11.9 and +19.1 percentage points per unit of ln(pressure) over that range [read: Heat digest/OCR p.431, not image-checked; calc]. - Conservation-of-energy statement ("transference, not annihilation", para 410) is correct.

b. Consistent with established experiment? The qualitative result (N2, O2, H2 transparent; CO, CO2, H2O, NH3, hydrocarbons strongly absorbing) agrees with modern infrared spectroscopy: a vibration absorbs only if it changes the dipole moment, which symmetric diatomics lack [memory, high]. Independent corroboration inside the Investigator's record: Röntgen 1884 (different method: pressure rise of the irradiated gas) states moist air absorbs [web https://archive.org/details/sim_annalen-der-physik_1884_23_9, via sites.md]; Lecher and Pernter 1880-81 confirmed Tyndall for dry gases [web https://en.wikisource.org/wiki/Popular_Science_Monthly/Volume_20/March_1882/Sound_and_Radiant_Heat]; Very 1900 reviews the band identifications (Ångström, Rubens and Aschkinass, Paschen) [web https://archive.org/details/atmosphericradia00veryrich]; photoacoustic gas analysis (Bell 1880, Röntgen 1881) is the modern descendant. Priority: Franz 1855 and Eunice Foote 1856 reported gas diathermancy earlier [web https://royalsocietypublishing.org/doi/10.1098/rsnr.2018.0066]; Franz measured about 3.54 per cent for dry air (Very), so "air = zero" has a historic exception at the 3.5 per cent level for a different source. Against / limits: Magnus 1861-62 and Buff found no vapour action (claim 2); Macdonald (viXra:1504.0165, unrefereed, the author's claim) argues N2 and O2 also absorb because they are Raman active [web https://vixra.org/abs/1504.0165]. Mathematically that argument confuses a Raman-active vibration (polarisability change) with an infrared-active one (dipole change). Tyndall's own measurement bound is "less than 1 degree for a whole atmosphere" and he does not claim exact zero; the weak collision-induced absorption of N2 and O2 is real but tiny [memory, medium].

c. Tested? By whom? Replicated? Tested by Tyndall (tens of thousands of air runs by his own account, Heat para 389) and by Melloni-school and later workers; replicated for dry gases by Lecher and Pernter; qualitatively replicated by Röntgen's independent method; every later infrared spectrometer confirms the picture. No modern replication of the numbers in the table with the original apparatus is on record (the surviving RI tube is 698 mm, dated 1861, and the 1859 tube was 4 ft; [web https://www.rigb.org/explore-science/explore/collection/john-tyndalls-radiant-heat-apparatus]).

d. If true, what in mainstream physics changes? Nothing; it is part of it (molecular spectroscopy, Kirchhoff's law). In 1859 it overturned Melloni's and Magnus's belief that gases are "an open door".

Verdict: Established (ordering, the elementary/compound contrast, and absorption = emission in rank). The magnitudes ("972 times air") are apparatus-dependent lower bounds, not constants; say so.

---

2. Claim 2: atmospheric water vapour, far more than the air carrying it, controls how fast the Earth loses heat ("blanket"/"dam"); at least 10 per cent of terrestrial heat is intercepted within ten feet; CO2 is only a minor constituent

This claim has four parts; I rate them separately.

a. Is the mathematics internally consistent? - Ratios 15, 30-70, 72, 70, 90, "90 or 100": explained by Tyndall as improving purity of the dry reference; the ratio is therefore an instrument-limited lower bound, not a measurement (see primer item 2). 15 times (impure dry air) versus 90 times (purified) is a factor 6 from the denominator alone [read: Heat p.380, 401]. - Linearity table (humid air, 5 to 30 inches): observed 16, 32, 49, 64, 82, 98; calculated 16, 32, 48, 64, 80, 96. Observed/pressure = 3.2, 3.2, 3.27, 3.2, 3.28, 3.27 units per inch; maximum deviation 2 units (2 per cent) [read: Heat p.388 image; calc]. The statement "absorption proportional to the quantity of vapour" is supported at this level. Proportionality is expected here: 98 units out of about 780 is only 12.6 per cent absorption, the weak-absorption regime where absorbed fraction is linear in column amount. Note the contrast with olefiant gas, which saturates; the water-vapour data are all in the optically thin regime for his source. - Open-tube result: 45 deg = 62 units of 780, 62/780 = 7.95 per cent [calc, matches the printed 7.9]. The path over which the moist air was displaced is not stated (the C-to-D section of a 4 ft 3 in tube), so a per-foot absorption coefficient cannot be derived from the text [read: Heat p.386]. - "At least 10 per cent of terrestrial heat is intercepted within ten feet" (Heat para 480): derived from "over 10 per cent" in a 4 ft polished tube with a roughly 100 C laboratory source, then asserted for Earth. Two hidden assumptions: (i) transfer from a 373 K source to a 288 K surface (the Wien peak moves from 7.8 to 10.1 micrometres [calc]; water vapour's strength is band-dependent, so the fraction need not carry over); (ii) path length scaling from 4 ft to 10 ft: if the 7.9 per cent applied to a path of 21 inches, a naive exponential extrapolation to 120 inches gives 37.5 per cent; if to the full 51 inches, 17.6 per cent [calc]. Tyndall's "at least 10" is therefore a statement of the tube result relabelled, with the extrapolation unstated. The Rede lecture states a different thing ("a column of ordinary air ten feet long would intercept 10 to 15 per cent ... of the heat radiated from an obscure source", On Radiation p.41): a laboratory statement, no experiment cited. A modern estimate for 3 m of ordinary humid surface air against 288 K emission is of order a few per cent [memory, low confidence; band-model estimate from Hottel-type charts, not verified, no web check done]. I mark this as unresolved and testable by a line-by-line calculation (HITRAN), which I did not run. - Per-molecule claim "16,000": 80 x 200 = 16,000, but 200 atoms per vapour molecule is 0.5 per cent while his own composition figure is 0.45 per cent (1 in 222, giving 17,800 with the same 80); and 80 is a mid-range of 70-90. An order-of-magnitude figure with false precision [read: Heat p.401] [calc]. - Strachey series (March 1850, vapour tension 0.888..0.435 inch against night fall 6.0..16.5 F): least-squares slope -22.5 F per inch of vapour tension, correlation -0.98, nine points, one non-monotone step (12.6 then 12.1) [read: Heat p.391 image; calc]. Tyndall chose "a single representative series". Hidden assumptions: night cooling also depends on wind, cloud, ground conductivity, and on the dew-point floor: near-surface night temperature cannot fall much below the dew point because condensation releases latent heat, so a strong correlation between vapour tension and smaller night fall is expected even if vapour were radiatively transparent. The series therefore does not isolate the radiative mechanism. (Mathematical point about confounding; the meteorology belongs to the physicist.) - Source discrepancy not resolved: Heat gives Strachey's 4-25 March 1850 series (unnamed place); Tyndall's 1882 Bakerian describes Strachey's July 1866 paper as Madras data for 1841-1844 ([web] https://archive.org/details/philtrans03250633, via sites.md). Both can be in one paper; my web check for Strachey's 1866 paper found nothing usable [web search, no result]. Open. - "Dam" energy statement: "the dam finally overflows, and we give to space all that we receive from the sun" (Heat p.401-402) is a correct steady-state energy balance (outgoing = incoming, with a higher surface temperature when there is an absorbing layer). Correct as stated. - Water-layer arithmetic (On Radiation p.42-43) is inconsistent with Heat: the Rede says a half-tenth-inch layer stops "no single ray" of terrestrial heat (attributed to Melloni); Heat gives 11 per cent transmitted at 0.36 in (Argand lamp), complete opacity at 0.27 in to a hydrogen flame, and over 97 per cent cut off at about 0.04 in [read: deep read, On Radiation note]. The popular statement rounds up. - Rede "35 per cent for a 2-inch shell of ether-saturated air" has no stated experimental basis; Heat uses olefiant gas (2 in 33 per cent, 1 in 26 per cent, 0.1 in 2 per cent). Check against Heat's saturation behaviour: 2 in at 33 per cent and 1 in at 26 per cent shows sub-linear growth (ratio 1.27), consistent with saturation [read: deep read]. - Role of CO2: by Tyndall's composition (about 0.05 per cent CO2, 0.45 per cent vapour) and his own tables (CO2: 972 equal-pressure units; per molecule) CO2 is a strong absorber per molecule but about 9 times scarcer; he names it "minor" [read: Heat p.379, 394-406]. No quantitative split between vapour and CO2 appears in the folder. His CO2 table showing 48 per cent absorbed at 1 inch from a CO-flame source and a log-like growth with pressure (+12 percentage points per ln unit) is already a hint of the logarithmic dependence used in modern forcing formulae (5.35 ln(C/C0) W/m2, which gives 3.7 W/m2 for doubling [calc]; formula [memory, high; Myhre et al. 1998]). I treat that as a modern parallel, not as something Tyndall claimed.

b. Consistent with established experiment? The core (water vapour is the strongest greenhouse absorber; dry-air regions have large diurnal ranges) agrees with modern radiative physics [memory, high]. The disputed part in the 1860s-80s: Magnus 1861-62 (film/glass-plate/convection artefacts), Buff, and Lecher and Pernter (moist air "a little more transparent than a vacuum", Wien Sitzb. July 1880; Phil. Mag. Jan 1881 [web via sites.md/Wikisource]) found no vapour effect; Röntgen 1884 and Very 1900 sided with Tyndall (Very asks for "further elucidation" of the humid-air proportionality table) [web via sites.md]. The Investigator records Tyndall's own control (free-air jet, no walls; rock-salt plate removed; seven outdoor sites; rock-salt plates kept dry with fused CaCl2), which answers the film objection mathematically: a film cannot give an absorption proportional to the vapour pressure within 2 per cent over a 6-fold range and in an open jet.

c. Tested? Replicated? Tested by Tyndall (1861-1882), by Magnus (negative, with sources of error later argued by Tyndall and Very), Lecher and Pernter (negative), Röntgen (positive, independent method), and thereafter by band spectroscopy. The controversy lasted to 1884. Quantitative modern replication of Tyndall's 70-90 figure with his source is not on record.

d. If true, what changes? Nothing; it was a contribution to meteorology (Airy, Herschel, per letters in Epsilon, interested-party reports of approval).

Verdicts (split): - Water vapour is the dominant infrared absorber of the lower atmosphere and controls nocturnal cooling and desert/mountain ranges (strong form): Established. - "At least 10 per cent within ten feet" as a number for terrestrial radiation: Never properly tested as stated (a laboratory number relabelled for the Earth; the extrapolation is not shown; not dismissed, simply unexamined). - Ratios "70, 90, 16,000 times": Not testable as stated as constants (they depend on the purity of an assumed-1 reference); they are lower bounds. - CO2 as a "minor" constituent: Historically superseded but instructive: on the amount-only reasoning it is minor next to vapour; the band absorption of CO2 and its logarithmic growth are later central. Tyndall did not rule CO2 out; his tables show it strongly absorbing.

---

3. Claim 3: the invisible radiation of an intensely luminous source is several times the visible (arc 7.7 by spectrum area, 8 by filter; sun about 2), and the iodine-in-bisulphide filter removes light but passes most heat

a. Internal consistency. - The two methods give 7.7 and 8: 4 per cent apart, described as "almost exactly the same". The filter method infers "eight" from 89/11 = 8.1 (50 cells) [read: 1866 paper p.11; calc]. - Table I second column is the first times a constant factor 1.37 (the maximum is set to 100): the actual ratios second/first are 1.355 to 1.371 across thirteen rows [calc]. The deviations at the low-intensity rows (1.355 at 15.5) are about 1 per cent. Consistent. - Calibration dependency: the 2-inch bisulphide cell gives raw deflections 73/74 and 72/73 degrees but a transmission of 94.8 per cent after calibration (raw ratio 98.6 per cent): the calibration (not published in the paper) shifts the answer by 3.8 points. The five-pair iodine experiment gives a raw mean ratio of 41.75/44.28 = 94.3 per cent but Tyndall reports 86.8 per cent after calibration (a 7.5 point shift) [read: 1866 paper p.9-11 images; calc]. Because the heart of the result (11-13 per cent absorbed by iodine in the arc beam = visible content) is a difference of two large numbers after a nonlinear calibration, the 8:1 depends on a table the reader cannot see. The "transparent" readings drift from 47.0 to 42.0 across the sequence (lamp stability) [read: p.11]. - Logical step: the whole 11 per cent absorbed by iodine is assumed to be visible light ("it may be inferred"); his justification is Table III (zero absorption for dark sources, 3-4.6 per cent for flames and white-hot spiral) plus the 7.7 area ratio; he concedes absorption "a little way beyond the red" (p.11). Any extra-red absorption makes the true ratio larger than 8 only if subtracted from the visible estimate... more precisely, the filter method then overstates the visible content and understates 'invisible/visible'. The direction of the bias is toward a smaller true ratio from the filter and a larger from the area (the area method has its own bias: the visible limit is set by the eye). The two methods share only the eye's limit at the red end, so their agreement is meaningful. - Physical plausibility of 7.7 [calc]: a grey body's infrared (0.76-15 micrometres, rock-salt window) to visible (0.4-0.76) power ratio is 4.3 at 3500 K, 3.2 at 3800 K, 7.8 at 3000 K, 10.5 at 2800 K. The ratio 7.7 corresponds to about 3000 K. The carbon-arc crater is hotter (about 3800 K [memory, medium]) but Tyndall's "electric light" includes the incandescent carbon points at lower temperature, so 7.7 is physically reasonable, not a red flag. Prism-position area equals total energy provided the pile width is constant, so the planimetry is valid in principle. - Solar value "about twice": a 5772 K blackbody gives IR(0.76-15 micrometres)/visible = 1.07 [calc]; measured extraterrestrial solar spectrum is about 49 per cent IR and 43 per cent visible [memory, high] (ratio about 1.1), and sea-level values are lower after atmospheric absorption. Tyndall's "about 2" (from Müller's curves, not in the folder) is therefore too high by about a factor 2 by modern numbers. Cause unknown (Müller's curves are not available; instrument, prism and thermopile response). I flag it as an inconsistency for the physicist; it does not affect the arc ratio. - Six Lectures fig. 56: not legible; the 7.7 is the printed text only. - Safety remarks about eye experiments (platinum put at the retina's position glowing while the eye feels nothing) are descriptive; the 18 per cent / 10 per cent ox-eye figures (On Radiation p.32-34) are not reproducible from the text; the 3.0 x 10^10 arithmetic (1,500 x 20,000,000) is correct [read; calc].

b. Established experiments. Consistent with Planck's law and with later IR source measurements; combustion at a dark focus consistent with IR heating; the 63 per cent transmission of black glass (carbon in white glass) is plausible for a source whose energy is largely near-IR (no thickness given) [read: p.24].

c. Tested/replicated? Tested by Tyndall; publicly demonstrated at the RI from early 1862 (On Radiation p.30). Akin and Griffith (Oxford) independently got incandescence of platinum at a sunlit mirror focus (Phil. Mag. 29, 1865 [web via sites.md]). Demonstrations are repeatable in principle with modern IR-pass filters; I found no recorded modern repeat of the 7.7 figure.

d. Change in mainstream? None.

Verdict: Established for the arc (source-specific number; Tyndall's calibration not auditable). Solar "about 2": Never properly tested here (Müller's data not in folder; modern value about 1.1 [memory, high]).

---

4. Claim 4: the glow of platinum/carbon at the dark focus is a "transmutation" of heat rays into rays of higher refrangibility ("calorescence"), reversing Stokes's rule; blue glasses give a pink glow (possible gap in the spectrum)

a. Internal consistency. Tyndall's own paper contains two incompatible accounts that he does not reconcile: (i) every solid heated to incandescence emits the invisible rays "with augmented power" and the visible ones in addition (p.3, 7); (ii) the incident rays "are returned at a quicker rate" (p.17-18) [read: deep read §10 note]. In (i) the only requirement for visible emission is that the absorbing foil reaches the temperature where its Planck spectrum has visible content; energy bookkeeping is ordinary heating. No individual wave is converted; the foil absorbs, thermalises and re-emits a broad spectrum. His reply to Croll in January 1865 states exactly that: "you hit the mass with waves of slow recurrence, and ... obtain, in return from the mass, waves of quick recurrence" [web https://epsilon.ac.uk/view/tyndall/letters/Tyndall2216]. Stokes replied that the process "is not a little different from fluorescence" and that Tyndall's name was "not quite correct" ("if Calorescence = - Fluorescence we ought to have - Calorescence = Fluorescence") [web https://epsilon.ac.uk/view/tyndall/letters/Tyndall2215]. So the "reverse of Stokes's rule" is a misnomer: Stokes's rule governs a one-step photon process in a cold body; the foil is a thermal process. Tyndall's evidence for "exclusively extra-red" (no fluorescence at the focus, alum cell test, eye test) is sound for excluding visible light and (more weakly) ultra-violet; his reasoning that rate of return equals frequency of the emitted light treats the glow as monochromatic when he himself records a continuous spectrum. The mathematics of the thermal picture is a calculation: a 3 W/cm^2-class absorbed flux heating thin foil with radiative loss sigma T^4 reaches white heat at about 2000 K+, where about a few per cent to tens of per cent of Planck power is visible; no violation of Stokes or of energy. - Platinum not fused: Tyndall's explanation (platinum black dissipated, reflectivity returns, absorption collapses) is qualitatively self-consistent; no energy budget is given [read: p.19-20]. - Pink glow: the paper's own coloured-glass table says the blue glasses transmit "blue, a green band and a band in the extreme red" (mean blue: blue, yellowish-green, extreme red; light blue: bands in blue, green, red and dark red) [read: 1866 paper table, page image p.23]. A continuous glow viewed through a filter whose transmission is blue plus extreme red looks pink/purple. So the "pink glow" needs no gap in the emission; the table he printed explains it. He wrote: "if subsequent experiments should confirm this result, it would follow that there is a gap in the calorescence" and "further experiments ... are necessary to satisfy my own mind" [read: p.23-24]; no follow-up is on record in the folder or in the Investigator's searches (reprinted unchanged in later books; Wikisource, Fragments of Science per sites.md). The only control stated is "every care was taken". At the dim glows described (darkened room needed) human colour perception is poor, a further caution. (Optics argument; the filter-transmission explanation is mine, not Tyndall's.) - Table III iodine data and 7.7: see claim 3. - Colour absorption of papers (red paper barely charred; "the popular notion that dark colours are best absorbers" at variance): consistent with a source whose output is mostly at wavelengths where visible colour is irrelevant. Internally coherent.

b. Established experiments. The glow, the vacuum control (no combustion), and the prism spectrum of the glow are ordinary thermal radiation; no experiment contradicts them. Anti-Stokes (up-conversion) luminescence exists as a different, multi-photon or phonon-assisted effect in some materials [memory, high]; it is unrelated to a hot metal foil.

c. Tested? By whom? Reproduced by Akin and Griffith (published as a programme in BA Report 1863; Phil. Mag. 29, 1865), who claimed priority; Sabine, Armstrong and the RS establishment sided with Tyndall; Tait 1868 credits Akin with "Calcescence or Calorescence" [web via sites.md]. Akin later left physics (nevpont: died 1893 in Fiume; no physics papers after 1865 listed) [web https://www.kozakpeter.hu/palyakep/akin-karoly-e7791]; the competing explanations for his silence (the dispute, other interests, no laboratory post) are as listed by the Investigator, and I add no verdict on them. The priority question is historical and does not change the mathematics. Stokes's and Croll's letters are interested-party-free evidence on the interpretation (Croll shared the "negative fluorescence" idea).

d. Change in mainstream? Nothing, once read as heating. As stated ("transmutation", "negative fluorescence") it would require a one-step wavelength-raising process in a hot solid which no data support and which the thermal picture does not need.

Verdicts (split): the observation (dark rays heat foils to incandescence): Established. The interpretation as a reverse-Stokes "transmutation": Historically superseded but instructive (Tyndall himself moved to a thermal statement in 1865). The pink-glow gap: Never properly tested (not dismissed; Tyndall asked for confirmation; the filter table he printed offers a sufficient explanation).

---

5. Claim 5: absorption is a property of the individual molecule, selected by resonance of wave period with the periods of the atoms in the group, and unchanged when the same molecules are compressed into a shorter tube or condensed to liquid ("conservation of molecular action", "thermal continuity")

a. Internal consistency. - Scaling check: 38 in / 10.8 in = 3.52, matching "three and a half times"; 1 in long tube equals 3.5 in short tube; 2 in equals 7 in [calc]. Results: 18.4 vs 18.8; 25.7 vs 25.6 (lime light); 23.5 vs 23.4; 32.1 vs 32.6 (platinum spiral): differences 0.4, 0.1, 0.1, 0.5 percentage points [read: New Fragments p.103]. It is not stated which vapour (ether or hydride of amyl) each pair belongs to. - Liquid versus vapour: a 1 mm layer of ether (0.713 g/cm^3, M = 74.12) holds 9.62 x 10^-4 mol/cm^2; the vapour column 7.2 in Hg x 38 in at 293 K holds 9.66 x 10^-4 mol/cm^2 (ideal gas, n/A = pL/RT) [calc]; for hydride of amyl (pentane-like, 0.626 g/cm^3, M = 72.15) 8.68 x 10^-4 versus 8.86 x 10^-4 at 6.6 in Hg [calc]. The matching of molecule counts is correct to 0.5 and 2 per cent. Results: vapour 66.7 per cent against liquid 67.2 per cent (difference 0.5); lime light 33.3 against a value garbled in OCR (the text says "no difference whatever") [read: p.105]. - Hidden assumptions: (i) single tabulated examples chosen by Tyndall, no repeats or error bars; (ii) broadband sources and a thermopile integrating all wavelengths, so the invariance holds for the integrated absorption; (iii) in modern terms the integrated absorption of a line is independent of pressure only in the weak-absorption limit; at higher optical depth, pressure broadening changes the effective absorption at fixed column; Tyndall's range (a factor 3.5 in density) is small enough for the effect to be hidden within about 0.5 points [memory, medium]. And "liquid = vapour" is approximate: liquids shift and broaden bands, though for broadband heat the integrated value changes little for these ethers. - The resonance argument (singing into a piano) is qualitatively right (a wave drives modes whose natural frequency matches) but it is pictorial, not a derivation; it gives no formula for which gases absorb; the "loaded piano strings" explanation of why combination slows atomic periods into the infrared is an assertion. The statement that atoms (not whole molecules) are the seat of absorption rests on the observation that decomposition occurs ("the atoms [are] the seat of the absorption", New Fragments p.98): an inference, not a test. The "ether condensed among the atoms" idea (On Radiation p.51) has no content that can be calculated. The nitrite-of-amyl test (visible light decomposes the vapour, the more energetic ultra-red does not) shows that the energy per wave is not what decides the effect; the correct modern statement is that the photon energy is the relevant quantity for photolysis (blue photons carry more energy than red ones). Tyndall's remark that red waves "measured by the amount of energy which they carry, are immensely more powerful than the waves of blue" is wrong if energy per wave is meant: the energy per photon is proportional to frequency, so a blue photon carries about 1.7 times the energy of a red photon [memory, high]; the energy flux of a beam depends on intensity, not wavelength alone. This is a genuine error in the conceptual statement, though the experiment is consistent with the photon picture [read: New Fragments p.96] [calc: 700 nm / 420 nm = 1.67]. - Röntgen/Bell photophone sounds: Tyndall uses them as evidence that absorption is by gas molecules (not films) and as a test of "free molecules" ([web via sites.md], Bakerian 1881).

b. Established experiments. Beer-Lambert-type density independence of integrated absorption is a standard result; photoacoustic spectroscopy confirms molecular absorption in gases [memory, high]. The quantum basis (vibrational/rotational transitions with dipole change) replaces the resonance-of-periods picture [memory, high].

c. Tested? Replicated? Tested by Tyndall (New Fragments data; Phil. Trans. memoirs not in the folder); Forbes's 15 July 1859 letter asked whether absorption "follows the degree of condensation" and thus anticipates the test (Epsilon Tyndall1627 [web]). Independent replication: Röntgen's photothermal method (1881, 1884) and modern photoacoustic detection. No independent repeat of the 38 in / 10.8 in and ether liquid/vapour pairs is on record.

d. Change in mainstream? None: it is the molecular picture.

Verdicts (split): molecular (not film) origin and density/phase invariance within the tested range: Established (tested by Tyndall, supported by later work). Resonance of periods as the explanation and the "ether entanglement" of molecular shape: Historically superseded but instructive (the resonance idea survives as the quantum transition condition; the ether part does not). The statement that red waves are "immensely more powerful" than blue: Internally inconsistent with the energy-per-photon relation if read as energy per wave (I note it as a mistaken sentence, not a failure of the experiment).

---

6. Claim 6: radiant heat and light are the same kind of ether wave motion, differing in period (reflection, total reflection, invisible images, polarization, double refraction, magnetization of heat); heat is a mode of motion (Rumford, Davy, Mayer's method reconstructed to 771.4; Joule 772)

a. Internal consistency. I recomputed the Mayer-type arithmetic (Heat Ch. III). - Chain: 144 x 15 = 2,160 ft-lb per foot of lift; 1.29 oz x 0.24 = 0.3096 oz of water-equivalent; x 490 F = 151.7 oz = 9.48 lb; ratio 1.421 gives 6.67 lb at constant volume; difference 2.81 lb; 2,160/2.81 = 769. Tyndall prints 9.5, 6.7, 2.8 and 771.4 (= 2160/2.8) [read: Heat p.97-99; calc]. The result is a difference of two nearly equal numbers; rounding 2.81 to 2.8 moves the answer by 3 ft-lb. - Sensitivity to the specific-heat ratio gamma: J = 797 at gamma = 1.40; 783 at 1.41; 770 at 1.42; 769 at 1.421; 758 at 1.43 [calc]. The logarithmic sensitivity d ln J/d ln gamma = (1/gamma)/(gamma - 1) = 1.67: a 1 per cent error in gamma shifts J by 1.7 per cent. - Using exact physical constants (atmospheric pressure 14.696 psi = 2116.2 lb/ft^2, temperature interval 491.4 F, air density 0.08073 lb/ft^3) with gamma = 1.40, the same method gives 777.9 ft-lb [calc], which matches the modern value (about 778 ft-lb per Btu [memory, high]; Joule's 772 is the 1850 value). With the exact constants but Tyndall's gamma = 1.421 the result would be 750 [calc]. So Tyndall's 771.4 arises from two compensating errors: atmospheric pressure rounded up from 14.7 to 15 psi (raises J by 2.0 per cent) and gamma 1.421 instead of 1.40 (lowers J by about 2.5 per cent). The agreement with Joule ("relieves the mind of every shade of uncertainty", para 92) is accidental at the 1 per cent level. This refines a statement in claims.md: "about 797 with the modern ratio of specific heats 1.40" is correct arithmetic but is not "the modern value"; the modern value for the method is 778 once the pressure is not rounded. - Mayer: Tyndall states Mayer's first paper does not contain the calculation (para 83), so 771.4 is Tyndall's reconstruction with his own data. The deep reader's [memory] that Mayer's published value is about 365 kgf-m/kcal (about 665 ft-lb per lb per F) is unverified: my web check returned a source that states 365 appears in Mayer's 1845 booklet and that the link to 1842 is unconfirmed [web https://galileo.phys.virginia.edu/classes/152.mf1i.spring02/MayerJoule.htm (fetch failed on certificate), search result text; check failed]. So I carry it as unconfirmed. - Joule data: 1843 values 896..587 (mean 882.75); 1844 air values 823, 795, 820, 814, 760 (mean 802.4); 1845 paddle 890; "taking the mean 817" is not reproducible from the listed numbers (mean of the 1844 values and the 890 is 853.5), so the set averaged is unstated; 1849 values 772.692 (40), 774.083 (50), 774.987 (20), experiment-weighted mean 773.7; Joule fixed 772 "for reasons assigned in his paper" (not shown) [read: Heat p.100-101; calc]. - Faraday and the speeds: molecular speeds 1,514, 1,616, 6,050 ft/s scale as 1/sqrt(mass) within 0.2 per cent and agree with kinetic theory (about 1,512; 1,617; 6,030 ft/s) [read; calc/memory high]. Coefficient 0.00366 = 1/273 (0.003663); Fahrenheit 1/490 is 1/491.4 rounded; 273 C = 491.4 F, Tyndall uses 490 [read; calc]. - Rumford: 107 is a temperature in F (water after one hour), not a result; 23.5 versus 28 minutes is the goldbeater's-skin vessel (coating, not colour) [read: New Fragments deep read, which corrects an earlier mis-statement]. Energy check: 26.58 lb of water through 180 F = 4,784 lb-F-units (4,784 Btu); Rumford's own bookkeeping is not in the deep read beyond this, so I cannot check his wax equivalence. - Ice arithmetic in Ch. VI is consistent (3 ft per year for 1,860 years = 5,580 ft; latent heat about 1,000 units; iron 10:1; 10,000/2,000 = 5 lb) [read; calc]. - The identity-of-heat-and-light experiments are qualitative (needle deflections to 90 degrees, "150 of the lower degrees"); the nonlinear scale again makes "90 degrees" a large but not quantitative statement. The mathematics of polarization, double refraction and the Faraday effect is not presented (the Faraday rotation is described with no Verdet constant or angles).

b. Established experiments. The identity of radiant heat and light as electromagnetic waves of different frequency is established (Maxwell 1865, Hertz 1888 [memory, high]); Tyndall's tests (reflection, refraction, polarization, double refraction) are consistent. The "luminiferous ether" as a medium is not (Michelson-Morley, relativity [memory, high]).

c. Tested? The identity claims were tested by Tyndall with obscure heat filtered by iodine, extending work by Forbes (polarization), Knoblauch (double refraction), De la Provostaye and Desains (magnetization); the magnetization of obscure heat is claimed by Tyndall as a first; no independent confirmation in the folder (interested-party priority claim). Mechanical equivalent: confirmed by Joule and every later measurement (4.186 J/cal [memory, high]).

d. Change in mainstream? None beyond what Maxwell and Hertz gave; the ether language is superseded.

Verdicts (split): radiant heat and light are the same wave phenomenon: Established. Heat as motion and the mechanical equivalent: Established (but the 771.4 "coincidence" is not a confirmation, see above). The ether as a physical carrier: Historically superseded but instructive.

---

7. Claim 7: glacier ice moves without being viscous under tension because broken ice refreezes at contacts (regelation); Ice Age glaciers show abundant vapour and strong solar heat followed by effective condensation, not a weaker sun (Heat Ch. VI)

a. Internal consistency. No equations; the argument is verbal. Consistent points: the arithmetic (above); he concedes viscous-like behaviour under pressure and non-viscous under tension (para 235), so "not viscous" is a partial claim. Gaps: the "most delicate experiments" that failed to detect stretching are cited to no one (para 230) and no data, apparatus or date is given; Thomson's pressure-melting is not mentioned in pp.204-217. A mathematical aside: the pressure-melting shift of ice is -0.0075 K per atmosphere (Clausius-Clapeyron, T(v_w - v_i)/L with L = 334 kJ/kg) [calc], so ice under 10 atmospheres (about 100 m of ice) melts only 0.075 K lower: pressure melting alone is a very small effect. This is a hidden mathematical fact bearing on any "pressure melting" account, not on Tyndall's contact-refreezing account. "Heat as much as cold": the latent-heat bookkeeping implies that producing the ice requires an evaporation of water that stores solar heat; nothing in the text quantifies the required cooling ("an improved condenser" is named, not modelled), so the argument is not quantitative.

b. Established experiments. Regelation (Faraday 1850) is real [memory, high]. Modern glaciology treats glacier ice as a polycrystal that deforms by creep with a power-law (Glen's law, exponent about 3), i.e. non-Newtonian viscous flow, plus basal sliding [memory, high; not web-checked]. Forbes (viscosity) and Tyndall (regelation, fracture) held views Forbes called complementary [web https://www.sciencehistory.org/stories/magazine/under-pressure/, via sites.md; the article is secondary and reports Tyndall's attacks and Forbes's 1864 letter]. Tyndall's own words concede the compression case.

c. Tested? Tyndall's press-mould demonstrations (ice cup, lens, sphere) are reproducible and test regelation of fragments, not glacier flow. The "failed stretching" experiments are untraceable. Ice-age cause: no calculation or test in the folder.

d. Change in mainstream? None; the later view is that both mechanisms (and sliding) occur.

Verdicts (split): regelation exists and welds ice under pressure: Established. "Ice is not viscous; glacier flow is explained by fracture and regelation": Historically superseded but instructive (it has a true part, regelation; the strong form is replaced by creep). Ice-age cause (needs more vapour and a better condenser, not a weaker sun): Not testable as stated in the folder (no mechanism or number for the condenser).

---

8. Investigator findings that bear on mathematics (kept whatever the verdict)

• 1859 Note (Proc. RS 10): tube 4 ft x 3 in, first crude test negative, compensated two-pile method gave "very large effects"; no numbers [web https://archive.org/details/jstor-111604]. Bears on the sensitivity argument in Heat para 377: a crude null at 10 degrees deflection cannot detect 1/1000 absorption (needle would move 0.01 degree).
• Forbes to Tyndall, 15 July 1859, asking tube length and per cent stopped, and suggesting testing condensed gases [web https://epsilon.ac.uk/view/tyndall/letters/Tyndall1627]; the density question became the New Fragments tests (claim 5).
• Tyndall to Hooker (Epsilon 1941): "Magnus denies the action; but he has not a foot to stand upon"; Airy (Epsilon 2029) "we should be in a way for beginning Meteorology": Tyndall's reports of approval (interested party) and Airy's own letter.
• Stokes (13 Jan 1865) and Tyndall to Croll (14 Jan 1865): claim 4.
• Röntgen 1884, Very 1900, Lecher and Pernter 1880-81: claim 2; the record does not go silent; the question moves to new methods.
• Akin and Griffith, priority dispute 1863-65; the silence after 1865 is analysed under claim 4.
• Jackson 2019 on Eunice Foote (1856) and Franz (1855): priority for gas absorption (Foote reported absorption by CO2 and water vapour; Tyndall's 1859 announcement did not mention climate; the climate argument came 1861-65) [web https://royalsocietypublishing.org/doi/10.1098/rsnr.2018.0066]. Mathematics note: the rank-order of CO2 against air in Foote's cylinders is not in the folder; no judgement from me.
• Macdonald (viXra, unrefereed): addressed under claim 1 (Raman versus infrared activity).
• Rede statement that "the surviving RI tube is 698 mm" (rigb): shorter than 4 ft; the 1861 date is not the 1859 tube.
• Not reached (listed in sites.md): RI archive catalogue, Wellcome, Biodiversity Heritage Library, Gutenberg, St Andrews; Contributions to Molecular Physics (1873) would give the primary tables and error statements for claims 1, 2, 5, and the complete 1866 paper (jstor-108936) would supply journal p.1 and Plate I for claims 3-4. I did not obtain them (no downloads were made).

---

Corrections to Step 1 (claims.md, facts.json, deep reads, sites.md)

1. claims.md, Mayer line ("about 769 with unrounded intermediates and about 797 with the modern ratio of specific heats 1.40"): the arithmetic is right (I get 768.9 and 797.3), but it should add that with the exact pressure (14.696 psi) and gamma = 1.40 the same method gives 778, the modern value; the 771.4 agreement with Joule arises from two compensating errors (15 psi rounding up about 2 per cent; gamma 1.421 about 1.5 per cent high, with sensitivity 1.67) [calc]. The deep read's phrase "sits on a ratio that gives about 770 only because 1.42 was the accepted value" is incomplete for the same reason.
2. Heat deep read, Mayer's own value ("about 365 kgf-m/kcal ... verify on the web"): not confirmed; my web check could not confirm that 365 belongs to the 1842 paper (one source associates 365 with Mayer's 1845 booklet). Treat as unverified.
3. claims.md, Specific testable statements: the At-30-inches and At-1-inch tables use different reference units (air at the same pressure versus a whole atmosphere of air). The sentence "(at 1 inch pressure: CO2 972, ammonia 5,460, olefiant gas 6,030 times air ...)" in claim 1 is correct only as "air at the same 1 inch"; against a whole atmosphere of air the numbers are 32.4, 182 and 201. Add the conversion in the primer.
4. claims.md claim 1/facts.json describe On Radiation's vapour table as at "1/30 atmosphere" matching Heat; the values do not match Heat's vapour table (factor 1.3-2.6, e.g. acetic ether 612 against 1,195). The Rede table appears to derive from a different apparatus or source.
5. claims.md claim 2 merges two statements: "a ten-foot column of ordinary air intercepts 10-15 per cent of heat from an obscure source" (Rede, no experiment cited) and "at least 10 per cent of terrestrial heat is intercepted within ten feet" (Heat para 480, extrapolated from a 4 ft polished-tube result with a 100 C source). They are not the same measurement.
6. Heat deep read, humid-air table: confirmed (observed/pressure constant to 2 per cent). The deep read's Strachey note ("monotonic except one step") is confirmed; add that the slope is -22.5 F per inch of tension (r = -0.98, n = 9) and that dew-point control is an uncontrolled confounder.
7. sites.md (Corrections) leaves the Strachey March 1850 versus Madras 1841-44 point open; I could not close it (my search for the 1866 paper found nothing). Remains open; the Phil. Mag. 1866 text would settle it.
8. 1866 deep read, Table I: confirmed as small rounding (factor 1.355-1.371 against the stated 1.37). No change.
9. New Fragments deep read reports Tyndall's sentence that red waves are "immensely more powerful than the waves of blue" without comment; it is wrong if "energy per wave" is meant (photon energy scales with frequency). Add this flag.
10. Digests and OCR: I did not re-render pages; the deep reads' page-image corrections stand, supported independently by the Wikisource transcription for 1590, 6030 and 372.

Web checks done (budget about 6)

• Mayer 1842 value and method: search run; the result named 365 kgf-m/kcal in Mayer's 1845 booklet and left the 1842 link unconfirmed; the Virginia lecture page failed a certificate check. Retag [memory] "Mayer 365 in 1842" as unconfirmed. The result's "415" rough check was the search tool's own calculation and I do not rely on it.
• Strachey Phil. Mag. July 1866: not found.
• Everything else tagged [web] above comes from the Investigator's notes (sites.md, research/); I did not re-fetch them. [memory] items I did not check (modern solar IR fraction, Glen's law, Myhre formula, Hottel-type estimate, creep, crater temperature) are marked with confidence; none carries a verdict on its own except the solar "about 2" comparison (high confidence) and the ten-feet fraction (low confidence, therefore the verdict stays "Never properly tested").

Summary table