Mathematician's review: Frank W. Very, The Luminiferous Ether (1919)

Reviewer role: PhD mathematician (Step 2). I check derivations, units, notation and hidden assumptions. Physics-versus-experiment judgements are the physicist's and contrarian's job; where I use experimental facts I say so and tag them.

Source tags: [read] = the 1919 book's text layer; [read: page image p.N] = I rendered the page and read the image (PDF page numbers; printed page = PDF page minus 4); [web] = URL given; [memory] = recalled, confidence stated. Every [memory] fact below is high confidence (textbook constants and well-known dated experiments), so none needed a web check beyond the three archive.org papers I fetched.

Source note carried over from claims.md: the book is a Google Books scan, 63 of 64 pages OCR. I re-read PDF pp.37-38 as page images (equations), and read the OCR text of PDF pp.19-20, 24, 28, 33-34, 39-54 and 60 myself in addition to the deep read of pp.36-42. There is no source gap for the central text. The book contains almost no equations: the only displayed ones are the two on PDF p.38 and Einstein's photoelectric relation on PDF p.20. Most "mathematics" in this project is therefore numerical bookkeeping, and that is what I check.

Units and conventions used (for the reader)

CGS-Gaussian units: lengths in cm, mass in g, force in dyne (1 dyn = 1 g cm/s^2), energy in erg (1 erg = 1 dyn cm), charge in esu (electrostatic units; e = 4.774 x 10^-10 esu is the electron charge as Very quotes Millikan). Pressure and stress have units dyn/cm^2. A bulk modulus K (Very's "volume elasticity") is the pressure needed to compress a material by a given fraction: K = -P / (dV/V), so it also has units dyn/cm^2. A dimensional check asks whether both sides of an equation have the same units; an equation that fails is not a mere typo, it cannot be a statement about physical quantities. All arithmetic below was run in Python (CGS).

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Corrections to Step 1

1. The 10^73 "volume elasticity" is Fessenden's number, not Very's invention (corrects facts/claims/deep read and the Investigator's lead). I fetched the 1900 Science paper and read the page image of p.743 [web: https://archive.org/download/jstor-1628439/1628439.pdf, PDF p.5]. Fessenden writes: "we can calculate the volume elasticity, and we find it to be about 10^73. But I have previously shown that the density of the ether is about 0.66 and its rigidity about 6.10^20. Hence we can calculate the value of the compressional or gravitational wave, and find it to be approximately 5.10^36 cms. per second." The Investigator's note had these exponents as OCR-garbled; they are legible on the image. Very's own formula, (surface pressure)^(3/2) / (surface area), is therefore a second route to Fessenden's figure, and it is the weaker one (see Claim 3).
2. The deep read (section 4) and claims.md say the book gives no link between the quantum and the frequency. That is incomplete. On PDF p.20 (printed p.16) Very writes that the "energy of formation" of an ether-particle "is proportional to the frequency of the luminous vibration, but not to the radiant intensity", and he reads Einstein's equation h/e = ΔV/Δν as saying this [read: p.20]. On PDF p.40 he instead speaks of "the fixed value of the energy-constant" and of the quantum as "the energy required to overturn and change the sign of an electron" [read: deep read p.40]. So the book states both a frequency-proportional energy and a single fixed energy, and does not reconcile them. The link to the rotation rate 1.66 x 10^22 remains absent, as the deep read says.
3. The deep read's "none of these readings gives a ratio near 1,000" needs one qualification. With the corrected rate 2.512 x 10^22 rev/s, a ratio of about 1,000 holds for the K-alpha X-ray lines of the very heaviest elements (uranium K-alpha is about 98 keV, about 2.4 x 10^19 Hz) [memory, high]. It does not hold for "the lighter atoms" as Very says. This is a possible source of his "1,000", not a verified one.
4. The Investigator's lead that the radius/diameter slip "may be in Fessenden's original" is confirmed. Fessenden 1900 p.743 prints "about 3/4 x 10^-13 cms. in diameter" and "about 2.10^32 dynes" [read: page image, URL above]. Very quotes it faithfully; the slip, if it is one, is Fessenden's. I add why Fessenden used "dynes": his glass-prism experiment on the same pages stretched a prism 1 cm by 1 cm, so a force in dynes equals a stress in dyn/cm^2 numerically. His habit of writing "dynes" for stress is a unit shortcut from that case, not a different quantity.
5. Rotation rate: confirmed, with a candidate explanation. I confirm c/(2 pi r) with c = 2.999 x 10^10 cm/s and r = 1.9 x 10^-13 cm is 2.512 x 10^22, not the printed 1.660 x 10^22 [read: page image p.38]. The printed value corresponds to r = 2.875 x 10^-13 cm. One possible origin (my conjecture, not in the text): the "classical electron radius" e^2/(mc^2) is 2.78 to 2.82 x 10^-13 cm, which is 1.5 times Thomson's 2e^2/(3mc^2) = 1.86 x 10^-13 cm, and 2.512/1.5 = 1.67 x 10^22, within 1% of the printed figure. I cannot show this was Very's path.
6. Step 1's bookkeeping for claim 5 is right: 922 + 921 nuclear electrons plus 1 satellite = 1,844, net charge zero; uranium 439,000 is 238 x 1,844 = 438,872 (the figure uses mass number 238; the ratio of atomic weights, 238.03/1.008, would give about 435,500) [read: pp.28, 24].

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Claim 1. Light is a stream of discrete electron-sized "ether-particles" travelling at c, with no space-filling light-bearing wave medium

a. Internal consistency. The book gives no equations for the particle stream, so there is little to check, but the verbal model has three tensions. - Energy and the inverse-square law. On PDF p.52 the ether-particle "possesses radiant energy which diminishes with the inverse square of the distance ... simply by spreading its field of force over a wider area", yet "inherently, the energy of the ether-particle does not change until it is ready to be reabsorbed" [read: pp.52]. A single particle carrying one quantum (fixed hv) cannot also lose energy as 1/r^2. The two statements can be reconciled only if the "radiant energy" is a property of a field around the particle and not of the particle itself, but the book does not set out that structure. Number of particles through a sphere of radius r is constant; intensity falls as 1/r^2 because the area grows. That works for a stream of particles; the particle's own energy would not fall. So the 'spreading' sentence contradicts the 'unchanged' sentence unless it refers to a different quantity. - Reference frame for "speed c". The particle is "flung off" with the electron's surface speed c (PDF p.38). Speed c relative to what? If relative to the emitting electron, this is an emission theory (the particle's speed depends on the source motion). If relative to the aura, the aura is an absolute frame. The book never says which. Section 4 shows that the answer matters. - Polarisation and a fixed particle shape. A footnote (PDF p.33) argues the particle has a "polar axis" along its direction of travel, so a meridional oscillation explains polarisation [read: p.33 fn]. This is a verbal sketch, not a derivation. No treatment of interference or diffraction is offered; the book says only that the fields of force "interpenetrate and sometimes interfere" (PDF p.59). Superposition and fringe spacing are quantitative phenomena and need at least a rule of the form "path difference d gives phase 2 pi d / lambda"; there is none.

b. Experiments. (All [memory, high] unless noted.) Interference and diffraction (Young 1801; fringe spacing proportional to wavelength), the Foucault 1850 result that light is slower in water (which Very himself cites as showing the "included ether" varies), Fizeau 1851 (moving water drags light by a fraction 1 - 1/n^2, about 0.43 for water), Airy's 1871 water-filled telescope (aberration unchanged by water), de Sitter 1913 (double-star light would arrive scrambled if speed were c + source speed) and Sagnac 1913 (a light-speed difference around a rotating loop) all predate the book. The book mentions none of Fizeau, Airy, de Sitter or Sagnac (I searched the text for these names and for "relativity": no hits). Very's statement that the bound ether retards light "independently of its state of relative motion or rest" (PDF p.53) [read] means zero drag by moving matter, against Fizeau's nonzero measured coefficient.

c. Tested? The corpuscular-at-c-with-no-waves picture has been tested implicitly by every interference experiment; the quantum part of the idea (light comes in packets of energy hv) has been confirmed (photoelectric effect, Compton). The two parts must be kept apart: Very's quanta are not the issue; his lack of any wave-like rule is.

d. If true. Optics would need a particle-level rule reproducing fringes and diffraction patterns, and Maxwell's equations would be reduced to a statistical description of particle streams. Special relativity is not mentioned in the book at all.

Investigator's findings that bear on this claim (kept in the record). Fessenden 1913, "Transformation of Gravitational Waves into Ether Vortices": waves in a frictionless continuous ether shed identical vortices, so "every continuous medium theory must involve quanta" [web: https://archive.org/stream/jstor-1638835/1638835_djvu.txt, per sites.md; not re-read by me]. This is a vortex route to quantisation parallel to Very's. Very 1909 holds an early form: light is slowed in water by "minute, alternating deflections ... in the bound ether attached to the molecules" [web, per sites.md; I fetched the 1909 paper, https://archive.org/stream/jstor-1635576/1635576_djvu.txt, and checked the mass-equation passage only].

Verdict (split). - "Light is discrete quanta of energy": Established (as the photon concept; not Very's mechanism). - "Light is only a stream of particles at c with no wave rule, and no space-filling wave medium": Tested, not supported by experiments that predate the book and are not addressed in it (interference, Fizeau, Airy, de Sitter); the book's own energy statements are in tension (Internally inconsistent on the 1/r^2 versus fixed-quantum point). - "A universal medium (aura) fills space": Not testable as stated (no measurable property is given beyond the Fessenden numbers in Claim 3).

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Claim 2. The Planck quantum is the energy to gyroscopically overturn an electron and reverse its sign

a. Internal consistency. - Dimensional slip. Planck's constant h has units of action (erg s). The book calls it "Planck's energy-constant" and "least quantity of energy" (PDF pp.20, 39-41). The energy of a light-quantum is h times frequency (hv, in erg). A single "least quantity of energy" cannot produce light of different colours; only if the energy scales with a frequency does the idea work. The book's p.20 sentence ("proportional to the frequency") supports hv; the p.44 sentence ("fixed value of the energy-constant") does not (Correction 2). - Where the frequency comes from. If the energy were h times the electron's own rotation frequency, every quantum would have E = h x 1.66 x 10^22 s^-1 = 1.1 x 10^-4 erg = 69 MeV (corrected rate: 2.512 x 10^22 gives 104 MeV) [my arithmetic; h = 6.626 x 10^-27 erg s]. An optical photon is about 2 eV. The book therefore rightly says the radiant period "appears to depend on some function of the radius of the electronic orbit" (PDF pp.39-40), not the surface rotation. Then the 1.66 x 10^22 figure plays no role in the quantum, which is worth stating plainly: the two central numerical claims (rotation at c; quantum from reversal) are not linked by any equation. - A sound route the book does not take. Fessenden's 1914 note, which Very says his view "agrees in a general way" with, states: "It may easily be shown that the amount of work done in upsetting the gyroscope varies directly as the angular velocity of rotation of the body, i. e., ... is directly proportional to the frequency" [web: https://archive.org/stream/jstor-1640236/1640236_djvu.txt, read by me]. No derivation is given. My reading: the work to reorient a spinning body of fixed angular momentum L by turning its frame at rate Omega is of order L x Omega, which is dimensionally an energy and is linear in the frequency. If L were the electron's spin angular momentum, E = L Omega with L = h/(2 pi) is exactly the Planck-Einstein relation. So the form of the idea is dimensionally respectable. But Very's own electron has L of order (2/5) m a c = 2.1 x 10^-30 erg s (solid sphere, r = 1.9 x 10^-13 cm, surface at c) against h/(2 pi) = 1.05 x 10^-27 erg s: his electron's spin is about 500 times too small to be the "h" of the relation (about 250 times too small against the later-measured electron spin h/(4 pi)) [my arithmetic; electron spin value from memory, high; spin was not known in 1919]. So the idea works dimensionally with a different L and fails numerically with Very's.

b. Experiments. The relation E = hv is established (photoelectric effect, black-body spectrum, Compton scattering). No experiment tests "overturn with sign reversal" as the cause of a quantum. Electron spin is not a reversal of sign, and electric charge does not flip when light is emitted by an atom (charge conservation is exact in all measurements).

c. Tested? Not as stated: nothing in the book is measurable beyond "h".

d. If true. Emission would have to flip an electron's charge. Charge would not be conserved in each emission unless a partner flips oppositely. Very sees this and answers with pairs (PDF p.41, "positive and negative charges always appear simultaneously"), but nothing is derived.

Investigator's finding kept in the record. Fessenden 1914 (above) is the missing frequency link. Fessenden's note also says the positive nucleus "can be formed of a number of negative electrons", which bears on Claim 5.

Verdict: Not testable as stated. Within it: the idea that the energy scales with frequency is internally acceptable (and Established as Planck-Einstein); the claim that a charge-reversing overturn is the cause has no stated measurable consequence; if the quantum is tied to the surface rotation rate, it is Refuted by calculation (69-104 MeV per quantum against eV photons).

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Claim 3. The electron is a vortical sphere of aura rotating at c at its surface; its size is set by Fessenden's breaking strain; pressure about 10^33 "dynes"; volume elasticity 10^73

I re-did every number (CGS, Python).

a. Internal consistency.

• Dimensional check of the volume-elasticity formula. (dyn/cm^2)^(3/2) / cm^2 = dyn^(3/2) cm^-5, not dyn/cm^2. The step "volume to surface being that of the cube to the square" is the only justification and does not produce this expression. So Very's own formula is dimensionally inconsistent. Fessenden's route is dimensionally consistent and I can reconstruct it: if the stress P is balanced by an elastic strain with stored energy equal to the gravitational energy U of the corpuscle, then U = P^2 V / (2K), so K = P^2 V / (2U). With P = 2 x 10^32, V = (pi/6) d^3 for d = 7.5 x 10^-14 cm (2.2 x 10^-40 cm^3) and U = 10^-48 erg (Fessenden's gravitational energy), K = 4.4 x 10^72, which rounds to "about 10^73" [my reconstruction; Fessenden prints only the result]. Units: dyn^2 cm^-4 x cm^3 / (dyn cm) = dyn/cm^2, correct. Very's expression contains no gravitational energy, so it is not a derivation of the same quantity; it reproduces the exponent by arithmetic coincidence with the inputs he chose.
• Fessenden's constants cohere with each other. Rigidity mu = 6 x 10^20 and density rho = 0.66 give a transverse-wave speed sqrt(mu/rho) = 3.0 x 10^10 cm/s, which is c. Compressional speed 5 x 10^36 cm/s with rho = 0.66 gives K = rho v^2 = 1.65 x 10^73, matching "about 10^73" [my arithmetic from the figures on the page image]. Very imports only K.
• "Rotation at the speed of light at every point of the surface." On a sphere of radius r, latitude lambda, the circle of latitude has radius r cos(lambda). Speed c everywhere means angular velocity omega(lambda) = c / (r cos lambda), which grows without limit at the poles. Very notes "the angular velocity increases from the equator to the poles" but not the divergence. More basically, a continuous tangent velocity field of constant nonzero length on a sphere cannot exist (the "hairy ball theorem" of topology: every continuous tangent vector field on a sphere vanishes somewhere). At the poles the direction of "east" is undefined, so the stated flow has a singularity there. A rigid rotation instead has speed zero at the poles.
• Collapse or explosion. The electrostatic stress on a charged surface pulls outward (self-repulsion) [Maxwell stress; memory, high]. The book says the electron "is not to collapse entirely". The usual puzzle (Poincare 1906 [memory, high]) is the opposite: a charged sphere needs an extra inward stress to hold together. Very's phrase may mean that his "electric condensation" is a compression of the aura, which is Fessenden's picture; the sign question is not settled by the text.
• "1,000 times the frequency of lighter atoms": see Correction 3.

b. Experiments. (i) Electron size: no scattering experiment resolves a size; electron-electron scattering to well below 10^-16 cm shows no structure (modern, [memory, high]). 1.9 x 10^-13 cm is the classical radius from the mass formula, not a measured size. (ii) Spin: a body of radius a, mass m, surface speed c has L of order (2/5) m a c = 2.1 x 10^-30 erg s, about 250 times less than the measured h/(4 pi) = 5.3 x 10^-28 erg s [my arithmetic; spin not known until 1925]. Conversely, to give the measured spin with radius 1.9 x 10^-13 cm the surface would need to move at about 250 c [my arithmetic]. (iii) Electromagnetic mass: the 4/3 factor between momentum and energy mass [memory, high] is not discussed.

c. Tested? The pressure and elasticity numbers have no test; they are inferred, not measured. No one has measured a "breaking strain of the ether".

d. If true. A frictionless medium with K near 10^73 dyn/cm^2 and wave speed 5 x 10^36 cm/s (about 10^26 c) would have to exist; and the Lorentz-invariant structure of electrodynamics would have to be recovered from it. Modern bounds on gravitational-wave speed from the binary neutron star event GW170817 are within about 10^-15 of c [memory, high], which concerns the speed of metric waves, not Very's pulsation, but is the first place a reader will notice the 10^26 c figure.

A coherent fragment (steelman). If a galactic quarter-wave of radius R = 3.8 x 10^22 cm (about 40,000 light-years) is used with Fessenden's v = 5 x 10^36 cm/s, the pulsation frequency is v/(4R) = 3 x 10^13 Hz, the infrared, of the same order as the "periodicity comparable to that of light" the book claims (PDF pp.37-38). The book does not do this sum; I did. It shows Fessenden's constants and Very's galactic scale are at least order-of-magnitude compatible, which is the best the numbers do.

Investigator's findings kept. Fessenden 1900 (volume elasticity, density, rigidity, wave speed, glass-prism comparison with the Kerr effect: 30 x 10^6 against 25 x 10^6 dynes, which he calls "probably accidental"); Very 1909 (matter as "vacuities in an exceedingly dense ether kept from collapsing by the centrifugal pressure of a rapidly rotating shell", an early form of this claim); Fessenden 1900 "Inertia and Gravitation" (mass as electromagnetic inductance; test: gases should weigh slightly more than solids, not carried out as far as the Investigator found).

Verdict: Internally inconsistent in its own numbers (rotation rate off by 1.51; volume-elasticity formula fails the unit check and mixes two radii; the pressure goes the wrong way for the "revised" radius; speed c on the whole surface is topologically impossible without a singularity). The underlying Fessenden estimate is a different matter: it is dimensionally consistent and its constants cohere, though it rests on Fessenden's untested ether model. On angular momentum the rotating-sphere electron is Tested, not supported by later measured spin (Refuted by calculation, with the caveat that spin postdates the book).

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Claim 4. Michelson-Morley, Rayleigh and Brace double-refraction nulls, and stellar aberration, are explained by a bound-ether shell carried with each atom plus corpuscular light

a. Internal consistency. Two readings of "speed c" (Claim 1) lead to two outcomes. - Reading A: light travels at c relative to the aura, which is at rest in space. The Michelson-Morley (1887) arms were about 11 m (folded), wavelength about 5.9 x 10^-5 cm, Earth speed about 30 km/s. The expected shift on rotating the instrument is about 2 L v^2 / (lambda c^2) = 0.37 fringes; the observed shift was a few hundredths of a fringe at most [memory, high; my arithmetic for 0.37]. A bound shell around each atom is about the size of the atom's electrons; it cannot hold back the light on the vacuum or air path between mirrors, so Reading A predicts the fringe shift and the null is not explained. - Reading B: light leaves at c relative to the source (emission theory). A null follows, but then the speed of light depends on the source's velocity, and the double-star test of de Sitter (1913) [memory, high] excludes it. Airy's 1871 water-filled telescope also challenges a corpuscular aberration treatment (in water the particle speed c/n would change the aberration angle by a factor n). None of these tests is mentioned in the book. - The bound-ether mechanism for refraction (PDF p.53) is described verbally ("electromagnetic forces developed in the medium"); no refractive index formula is derived. Very notes only that refractive index can be predicted "with approximation" from atomic constitution. - Discriminating power. The null results of Rayleigh (1902) and Brace (1904) are of order v^2/c^2 = 10^-8 [memory, high]. They are equally consistent with any theory in which such effects cancel, including Lorentz's contraction and Einstein's relativity. Very reads them as showing "the ether concerned in refraction is bound to the earth". The data do not single this out.

b. Experiments. MM, Rayleigh and Brace nulls are established. The explanation is what is at stake. Stokes's 1845 aberration theory (flow of ether around the Earth) needs an irrotational flow and cannot also give Fresnel's coefficient; this is why Stokes drag lost. Very's route avoids ether flow by making light corpuscular, but then inherits Readings A or B.

c. Tested? The null experiments were tested many times. Very's bound-shell hypothesis has no test of its own; the Investigator found no independent test.

d. If true. Emission-type optics would have to replace relativity's treatment of light, and the double-star evidence would need another explanation.

Investigator's findings kept. Very 1909: the same bound-ether account of refraction (slower light in water is "no necessary change in the true ethereal velocity"). Fessenden 1900: glass stress and Kerr effect compared. Neither adds a test.

Verdict (split). The null results: Established. Very's explanation: Tested, not supported (Reading A fails the MM arithmetic by about 0.37 fringe; Reading B is excluded by double-star data and by Fizeau/Airy, none of which the book addresses). Whether the explanation is only underdetermined is a fair question; the point for a mathematician is that one of two unstated assumptions must be chosen and both fail.

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Claim 5. One kind of electron, no positive electron; all mass is electrical (hydrogen = 1,844 electrons; uranium nucleus about 439,000; 92 satellites)

a. Internal consistency. The bookkeeping closes: hydrogen is 922 right-handed + 921 left-handed + 1 satellite = 1,844, net charge 0 [read: p.28]. For uranium, 92 satellites in rings 6, 11, 15, 18, 20, 22 sum to 92 [read: p.24; 6+11+15+18+20+22 = 92]. The arithmetic of counting is correct. Two issues remain. - Mass additivity. The model needs M = N m_e. But in Thomson's formula m = 2e^2/(3 a c^2) the mass is electric field energy. Packing N charged spheres of radius a close together adds interaction energy of the same order as each electron's self-energy: a touching opposite pair (separation 2a = 3.8 x 10^-13 cm) has e^2/(2a) = 6.0 x 10^-7 erg = 0.37 MeV = 0.73 m_e c^2 [my arithmetic, e = 4.774 x 10^-10, a = 1.9 x 10^-13]. The binding is of the same order as the rest energy, so "mass is the sum of the electrons' masses" fails by order one, not the fraction of a percent that mass defects show. The book states no binding calculation. - Positive electron. "There is only one kind of electron" with positive charge as the opposite rotation means that an oppositely rotating electron has the same mass and opposite charge, which is the later-discovered positron (Anderson 1932) [memory, high]. Very says a freed electron "becomes indistinguishable" from other electrons (PDF p.27), i.e. a freed positive-sense electron turns negative. Free positrons are stable until they meet an electron, so that step fails experimentally, though not by anything in 1919 data.

b. Experiments (later than the book; [memory, high] unless noted). Proton charge radius is about 0.84 x 10^-13 cm; uranium nuclear radius is about 7.4 x 10^-13 cm (1.2 fm x 238^(1/3)). Packing 1,844 electrons of radius 1.9 x 10^-13 cm requires a cluster radius of at least 1.9 x 10^-13 x 1,844^(1/3) = 2.3 x 10^-12 cm (about 27 times the measured proton radius), and 439,000 electrons need at least 1.9 x 10^-13 x 439,000^(1/3) = 1.4 x 10^-11 cm (about 20 times the measured uranium nuclear radius) [my arithmetic]. Rutherford's 1911 scattering already put the gold nucleus below about 3 x 10^-12 cm [memory, medium; not needed for the verdict, which rests on the modern values]. Mass 1,844 vs measured 1,836.15 (proton/electron) or 1,837.15 (hydrogen atom/electron) [memory, high]: Very's figure is 0.4% high, within 1919 data, immaterial.

c. Tested? Not by Very. Fessenden (1914) holds the same view of a nucleus made of negative electrons; the Investigator found no test by either.

d. If true. Nuclei would be made of electrons, with nuclear forces purely electromagnetic. Mainstream physics instead has protons and neutrons (neutron 1932) with a separate short-range force.

Investigator's findings kept. Fessenden 1900: about 1,000 corpuscles in hydrogen, 200,000 in mercury (compare Very's 1,844); Fessenden 1914: nucleus "can be formed of a number of negative electrons".

Verdict: The counting is consistent. The model is Refuted by calculation on size and on the binding-energy budget (using nuclear sizes measured after the book), and Tested, not supported on "no positive electron" (positron). The 1919 reader could not have had these data; the Rutherford nuclear-size inference was available but the book does not address it.

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Claim 6. Gravity is the synchronous pulsation of electrons in the aura, confined to a "galactic cell"; galaxies repel at intergalactic distances

a. Internal consistency. The book gives no equation of the pulsation field. Its logic is: Bjerknes's pulsating-sphere analogy needs all pulsators in phase to avoid repulsion (PDF p.48); Leahy (1889) objected that an elastic medium would give repulsion beyond a quarter wavelength; Very answers that the quarter wavelength equals the radius of an aural cell, so beyond the cell there should be repulsion, which he claims to find in the "crowding" of galaxies (PDF pp.49-51) [read]. Turning an objection into a prediction is logically legitimate; it needs the prediction to be quantified. The crowding argument is the only evidence and I checked it with the book's own numbers: galaxies drawn as 1 mm dots are "a few centimeters apart", so separation s is about 30 diameters d; with random 1,000 km/s motions the mean free path is s^3/(pi d^2), about 8,600 d. For d = 80,000 light-years (the book's scale, PDF p.45) the path is 7 x 10^8 light-years, travelled at 1,000 km/s (0.0033 c) in about 2 x 10^11 years [my arithmetic]. "Frequent collisions are inevitable" (PDF p.46) is not true on the book's own numbers; the collision time exceeds the age of the universe (1.4 x 10^10 years [memory, high]) by about 15 times. So the argument for repulsion is quantitatively weak by its author's own scale.

b. Experiments. Galaxies do collide and merge (Antennae; Milky Way-Andromeda approach at about 110 km/s [memory, high]); galaxy clusters are gravitationally bound across about 1 Mpc, many "cells" wide (Zwicky 1933 [memory, high]). Large-scale repulsion exists in modern cosmology as cosmic acceleration, which is a property of expansion at gigaparsec scale, not a cutoff of Newtonian gravity at galaxy edges.

c. Tested? Not directly by Very. Hicks's remark that pulsating systems "would not influence each other, or would repel each other" is the origin of the idea (PDF p.49) [read]; the cell model is untested. Very's other work cited as pointing the same way: "On a Possible Limit to Gravitation" (Publ. Am. Astron. Soc. III, p.335) and the Nova Persei nebulosity paper (Investigator lead; neither read by me). His 1920 note on Nipher's electrical-gravity experiment (Science, paywalled) was not accessible; Nipher's claim is the physicist's/contrarian's to assess.

d. If true. Newtonian gravity would carry a cutoff at about galactic radius with a sign reversal beyond; gravitational lensing and cluster dynamics would contradict it, and the existence of a field speed of order 10^26 c (Claim 3) would need a role.

Investigator's findings kept. Fessenden 1913 (gravity waves shed vortices); Fessenden 1900 "Electrical Theory of Gravitation" (1901, not read); Very's Nipher note (1920); Very's 1910 exposure of the Norwood "meteorite" as a fraud after first reporting it (a record of how he handled evidence).

Verdict (split). Pulsation mechanism: Not testable as stated (no equations, no stated measurable consequence). Cell-limited gravity with galactic repulsion: Tested, not supported (mergers, bound clusters); the author's supporting collision argument fails on his own numbers.

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Claim 7. Matter is gradually converted to radiation and re-originates from the aura, so entropy increase is not one-sided

a. Internal consistency. No equations. The argument is verbal: if matter is destroyed and radiation is absorbed by the aura, "building-up" must also occur somewhere (PDF p.43). Two things are missing mathematically. (1) Energy conservation is respected by the verbal story; entropy is not addressed at all: the second law constrains a closed system's entropy, and a "building-up" phase needs an entropy sink (an external source) to avoid decreasing total entropy. Very names that source as "an infinite source of energy which is spiritual" (PDF p.44) [read], which is a metaphysical premise, not a quantity. (2) "Non-selective absorption by the aura" of starlight needs a rate. Without one it cannot be told apart from no absorption. Very's contention (footnote, PDF p.43) that this absorption exists but leaves no colour trace is unfalsifiable as stated.

b. Experiments. Matter-to-radiation conversion in stars and radioactivity is established; radiation-to-matter (pair creation above 1.022 MeV) was demonstrated in the 1930s, but with conserved charge and with entropy accounting; no process turns the aura's absorbed light into new atoms is known.

c. Tested? No.

d. If true. A cosmic entropy sink or mass source would break the second law for the universe as a closed system. (Steady-state cosmology, 1948, offered a rival version, but with continuous creation of matter and an expanding space.)

Investigator's findings kept. Very 1909 states claim 7 first ("if atoms are destroyed in radioactive transformations, there must be some other part of the universe where these atoms are reproduced"); Very 1913 "What becomes of the Light of the Stars?" (Popular Science Monthly 82:289, cited by the book) is the full paper and was not found/read.

Verdict: Not testable as stated.

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What I also checked from the Investigator's leads

• Very 1909 mass equation dm/m0 = v/(V - v) (Science 30:495) [web: https://archive.org/stream/jstor-1635576/1635576_djvu.txt; OCR partly garbled, so my reading is tentative]. It is dimensionless on both sides and diverges as v approaches V, which is the Kaufmann-experiment behaviour Very wanted. But it rises linearly with v at small speed (0.11 at v = 0.1 V), whereas the relativistic mass increase gamma - 1 is quadratic (0.005 at 0.1 V) [my arithmetic]. At v = 0.5 V Very's expression gives 1.0, relativity gives 0.155. Read as the general mass-speed law it does not match later (Bucherer 1908-09, Lorentz-Einstein) data; the passage concerns acceleration by light pressure, so I treat it as a limited-case equation and not a refutation.
• Fessenden's "work varies with angular velocity": see Claim 2. No derivation, plausible form.
• Investigator's still-open items (not reached in the time limit) do not alter my calculations: the second scan of the book (https://archive.org/details/luminiferousethe00veryrich) would settle the 1.660 x 10^22 print; the 1913 and 1918 New-Church Review papers may show whether the number came from an earlier slip.

Summary of the numerical checks

Verdict list

1. Light as particles: split. Quanta of energy: Established (not Very's mechanism). No-wave particle stream and "no light-bearing medium": Tested, not supported; internal inconsistency on energy versus 1/r^2. The aura: Not testable as stated.
2. Planck quantum as overturn energy: Not testable as stated; Refuted by calculation if tied to the rotation rate (69-104 MeV).
3. Vortical electron: Internally inconsistent (numbers, units, topology); Fessenden's underlying estimate is dimensionally consistent but untested.
4. MM and aberration by bound shell: nulls Established; the explanation Tested, not supported (fails under either reading of "speed c").
5. One electron, all mass electrical: counting consistent; model Refuted by calculation (size and binding energy); no positive electron Tested, not supported.
6. Gravity as pulsation in galactic cells: Not testable as stated (mechanism); cell-limited gravity with repulsion Tested, not supported; the supporting collision argument fails on its own numbers.
7. Matter-radiation cycle and entropy: Not testable as stated.