Maxwell: Mathematician review (Step 2)

Reviewer role: PhD mathematician. Checks derivations, notation, units, hidden assumptions. Scope: the four documents in the Maxwell folder (two Maxwell scans, Hunt 2012, Belrose 2014).

Source tags: [read] = from the document text layer; [read: page image p.N] = from a rendered page image (PDF page numbers, not printed page numbers); [web] = Internet source with URL; [memory] = recalled, with confidence.

No Source gap applies: all four PDFs have a text layer. The Step 1 note said equation pages were not rendered. I rendered the key ones (listed under "What I checked") and read them as images.

What I checked

• dynamicaltheoryo00maxw.pdf, PDF p.14 (printed p.466, summary of Part I), p.31-34 (printed pp.483-486: equations D, E, F, G, H, the list of twenty variables, the energy integral), p.47 (printed p.499: Part VI, V = v, the Weber-Kohlrausch, Fizeau and Foucault numbers), p.52 (printed p.504: absorption in a conductor, eqs 95-99). All [read: page image].
• dynamicaltheoryo00maxw.pdf text layer: paragraphs 73-75 (illustrative vs literal language, independence from the medium hypothesis), paragraphs 98-100 (normal vibrations cannot propagate), equation (A) (text layer only; garbled).
• MaxwellOnFaradaysLines.pdf text layer: "Summary of the Theory of the Electro-tonic State" and Laws I-VI. Page images not rendered; the six laws are in clean prose, so I did not need them.
• HeavisideOliver-Hunt.12.pdf text layer: the energy-flow passage, the sidebar derivation of the second circuital law, the Preece/Electrician passages. Belrose I did not reread beyond what Step 1 reported; it mostly paraphrases Hunt, so I treat it as a secondary source.
• Counts and numbers recomputed by hand (see below).

Corrections to Step 1

1. Claim 6 / facts.json (Hunt): Hunt says Heaviside recast "the 20 fundamental equations of Maxwell's Treatise" (1873) [read]. Step 1 and claims.md attribute the twenty-equation, potential-based form to the 1865 paper. Both are true, because the 1865 paper also has twenty equations in twenty unknowns [read: page image p.34], but the two should not be conflated. Hunt's sidebar refers to the Treatise form (A, Psi, E, H, D, B).
2. Claim 4 as stated is incomplete. Maxwell finds V = v only for air with mu = 1 ("by equation (46)" [read: page image p.47]); in general his velocity depends on mu and k. The match with light is for non-magnetic media. Also, the wave he derives "consists entirely of magnetic disturbances", with the magnetization in the plane of the wave [read: page image p.47]; the electric part enters through the displacement terms. Step 1's "transverse electric and magnetic disturbance" is a modern restatement, not what that paragraph says.
3. Claim 5 (ether). Step 1 presents it as Maxwell's framing. Maxwell is more careful. Paragraph 73: "I avoid any hypothesis of this kind" (the 1861-62 vortex model), and words such as "electric elasticity" are "illustrative, not explanatory". Paragraph 74: only the energy claim is meant literally. Paragraph 75: the conclusions are "independent of this hypothesis" [read]. So the ether is a "very probable hypothesis", not a premise of the equations. This matters for the verdict below.
4. Hunt's sidebar prints "curl E = grad(-dA/dt) - curl grad Psi". The first term should read curl(-dA/dt); "grad" is a typesetting slip. The result curl E = -mu dH/dt is correct given curl A = mu H. Step 1 reported the derivation as fine; it is, with this typo noted.
5. Step 1 key_claims for the 1855-56 paper say the analogy gives "electrostatic, magnetic and steady-current phenomena obey formally identical equations". I read the six-law summary: Laws I-IV cover magnetic and current relations, Laws V-VI induction and force. The "formally identical" statement is a fair paraphrase of the fluid-analogy sections that I did not reread; I leave it uncorrected but unverified.
6. Wikipedia-style transcriptions of the 1865 equations (F) as a positive Ohm's law do not match the printed paper. See Claim 3, sign issue.

Claim 1: Lines of force have an exact mathematical form (incompressible-fluid analogy)

a. Mathematics internally consistent? Yes, in the sense the paper claims. The analogy needs only that div of the "flow" vanishes in source-free regions (incompressibility) and that flow is driven by a potential through a linear resistance law. This is the same structure as steady current flow or heat conduction. The map is a dictionary between solutions of Laplace's equation and fluid streamlines. It asserts nothing about fluid. Hidden assumption: linear, isotropic (or at most linear) media. Maxwell explicitly allows this via his "equations of conduction".

b. Consistent with experiment? The analogy predicts field maps, e.g. flux tubes, the electrostatic force law between charges and the magnetic fields of steady currents. These agree with measurement because they restate Coulomb's law, Ampere's law and Faraday's lines. Specific experiments are classical (Coulomb, Oersted, Ampere, Faraday) [memory: high]. Nothing here is new empirically; the claim is about representation.

c. Tested? Not as a physical hypothesis, because Maxwell says it is not one [read: text layer, "no physical theory, only a kind of artificial notation"]. As a calculational device, it was used by Thomson and others (heat-flow analogy) [memory: medium; not needed for my verdict].

d. If true, what changes? Nothing: it is already absorbed. The fluid picture is dropped; the equations survive.

Verdict: Established (as a mathematical representation). The claim has no content beyond what the field equations already say; it is not a claim about nature. If forced to split: "analogy as a calculational tool" Established; "analogy as a physical fluid" Not testable as stated (and Maxwell disclaims it).

Claim 2: Induction is governed by a point-wise vector (electro-tonic state = vector potential A)

a. Mathematics consistent? Yes. From the six laws [read]: Law I says the circulation of the electro-tonic intensity around a small loop equals the magnetic flux through it. This is Stokes' theorem with curl A = B. Law VI says EMF equals the rate of change of the circulation, EMF = -dPhi/dt, with the sign left implicit in the text but fixed in 1865 [read: page image p.32, eq 35: P = -dF/dt - dPsi/dx]. Consistency check: Law I fixes A only up to grad of a scalar (gauge freedom); Maxwell notes in 1865 that Psi "is indeterminate as far as regards the solution of the above equations, because the terms depending on it will disappear on integrating round the circuit" [read: page image p.32]. So he sees the ambiguity for closed circuits but does not flag the gauge freedom of F, G, H itself. Hidden assumption: a specific gauge is chosen implicitly when he later sets Psi equal to the electric potential and drops divergence terms.

b. Experiment? Faraday's law of induction, tested to high precision. Aharonov-Bohm (1959; observed by Chambers 1960) shows A has physical effect in a region with B = 0 [memory: high]. The "electro-tonic" intuition that A is real in some sense survives only in this restricted way.

c. Tested? Faraday's law has been confirmed continuously since 1831. Whether A is an "entity at each point" rather than a calculating tool is, in classical theory, not testable (gauge freedom), and in quantum theory is answered with "the loop integral of A is physical" [memory: high].

d. If true: nothing changes. It is standard.

Verdict: Established (Faraday's law and the potential formulation). The strong form, "A is a directly measurable local quantity", is Not testable as stated in classical theory because gauge shifts change it without changing any measurement.

Claim 3: Field energy lives in the space around conductors; with displacement current the equations close (twenty equations, twenty unknowns)

a. Mathematics consistent? Mostly, with one real defect that I verified.

Counting (checked on the page image p.34): variables F,G,H; alpha,beta,gamma; P,Q,R; p,q,r; f,g,h; p',q',r'; e; Psi = 6 groups of 3 + 2 = 20. Equations: B, C, D, E, F, A each give 3 = 18, plus G and H = 20. The count is right. Caveats: (i) the twenty are not independent in content: (H) is a consequence of (A), (C), (G) when signs are consistent (take the divergence of C, which gives div p' = 0), so one equation is redundant; (ii) A and Psi are gauge-ambiguous, so the system is underdetermined until a gauge is chosen. "Twenty equations in twenty unknowns" is therefore a bookkeeping statement, not a well-posedness theorem. He gives no existence or uniqueness argument, which was normal for 1865. Boundary and initial conditions are "provided we know the conditions of the problem" [read: page image p.34].

Energy: Equation (37)-(38) [read: page image p.34]: E = (1/2) sum (F p' + G q' + H r') dV is rewritten, after integration by parts and "alpha, beta, gamma vanish at an infinite distance", as E = (1/8 pi) sum (alpha mu alpha + ...) dV. Checking the step: substituting 4 pi p' = dgamma/dy - dbeta/dz gives (1/8 pi) A . curl H; integration by parts moves the curl onto A, giving H . curl A = H . B; the boundary term is dropped because the fields fall off. This is correct provided the surface term vanishes (needs A x H to fall faster than 1/r^2). Maxwell states the decay assumption. The factor 1/(8 pi) is Gaussian-type units: the 4 pi sits in equation (C), so this is consistent, though it is not SI and a modern reader must convert. Units warning: there are two unit systems at once (electromagnetic units and the "electrostatic" ratio v), by design.

Where the energy lives: the integrand density is the field energy; but the integral only fixes the total. Adding any divergence-free term, or moving to the form A.J, gives the same total, so "energy is located in the field" is an interpretation of the density, not a theorem of the integral. This is the same ambiguity Heaviside and Poynting addressed with the flux S = E x H (Claim 6). Maxwell himself recognizes the alternative: "On the old theories it resides in the electrified bodies... On our theory it resides in the electromagnetic field" [read: text layer, para 74].

SIGN INCONSISTENCY (my finding; not found in Step 1). From the page images [read: page image p.31-34]: - (D): P = mu(gamma dy/dt - beta dz/dt) - dF/dt - dPsi/dx. The motional term is (v x B)_x; so P is the physical electric field E. - (E): P = k f, so f = P/k, parallel to P. - (F): P = -rho p, so p = -P/rho, antiparallel to P. - (A): p' = p + df/dt. - (G): e + div f = 0. (H): de/dt + div p = 0. (C): 4 pi p' = curl(H-vector). Then div p' must be zero by (C). But div p' = div p + d(div f)/dt = -de/dt (from H) + d(-e)/dt (from G) = -2 de/dt, not zero. So as printed, (A), (C), (G), (H) cannot all hold when e changes in time. The same sign shows up in Part VI, where the conduction term is put into the wave equation [read: page image p.52, eq 95]: d2G/dx2 = -4 pi mu (df/dt + q) = +4 pi mu ( (1/k) d2G/dt2 - (1/rho) dG/dt ). Correct physics (conduction current sigma E, with E = -dA/dt) gives d2A/dx2 = mu eps d2A/dt2 + mu sigma dA/dt. Maxwell's damping term has the opposite sign. Check on his own solution G = e^{-px} cos(qx + nt): substitute: p^2 - q^2 = -(4 pi mu/k) n^2 and -2 p q = -(4 pi mu/rho) n (Maxwell's sign), so p = +(2 pi mu/rho) n/q, which is his eq (97) [read: page image p.52]. For a wave of phase qx + nt the wave travels toward -x when n, q > 0, and e^{-px} with p > 0 then grows along the direction of travel. With the physical sign, p flips sign and the amplitude decays along the direction of travel. The magnitude |p| = 2 pi mu V/(rho i) is right, so his numerical result for the opacity of gold-leaf (eq 99) is unaffected; but the sign as printed amplifies rather than absorbs, contradicting his text "the undulation is thereby weakened". Scope: this is a sign-convention defect in (F) (a minus sign on resistance, with P taken as the electric field), not a conceptual error in the displacement-current idea. Correcting (F) to P = +rho p fixes both the continuity test (div p' = div p + d(div f)/dt gives de/dt... still needs (G) sign: with p = +P/rho, (H) as printed gives relaxation of e with the wrong sign, so (G) or (H) needs flipping as well). I could not fully pin down which single sign Maxwell intended; the minimal statement is that the sign set (A), (F), (G), (H) as printed is not mutually consistent. Wikipedia's transcription gives (F) with a positive sign [web: https://en.wikipedia.org/wiki/A_Dynamical_Theory_of_the_Electromagnetic_Field] and notes sign ambiguity in (G)/(H) across his papers, which supports that later transcriptions silently repaired this. Reservation: I checked this on a 110-130 dpi scan; the minus signs in (F) are clear in the images. Confidence: high that the printed signs are as I describe; medium that this is the only way to repair them.

b. Experiment? Field energy density B^2/(2 mu) (Gaussian: B^2/8 pi) agrees with measured inductive energy (e.g., stored energy of coils) [memory: high]. Displacement current is confirmed indirectly by existence of EM waves (Hertz, 1887-88) [memory: high; see Claim 4] and directly by magnetic fields measured between capacitor plates [memory: medium; not relied on].

c. Tested? Yes, extensively (radio, optics, circuits). Replicated indefinitely. Located energy: Poynting's flux is the standard consistent assignment; no experiment distinguishes "energy in the field" from other local assignments in a way that overturns it.

d. If true: nothing changes; it is the mainstream.

Verdict: split. - Field energy and displacement-current system: Established. - "Twenty equations, twenty unknowns" as a complete, well-posed system: Plausible, untested in the mathematical sense (no existence/uniqueness proof, redundancy and gauge freedom unaddressed). - The 1865 sign set for (A), (F), (G), (H): Internally inconsistent as printed (a sign defect; demonstrated above). Non-fatal and later repaired. - "Energy resides in the field" as a statement of location: Not testable as stated beyond total energy (the density is defined only up to a divergence-free addition); the Poynting form is the conventional choice.

Claim 4: Transverse waves at V = v match the speed of light, so light is electromagnetic

a. Mathematics consistent? Yes. Plane-wave reduction [read: page image p.47, text layer paras 92-95]: all quantities depend on w = lx + my + nz - Vt; combining (B), (C), (E) gives d2(alpha)/dw2 type equations with V^2 = k/(4 pi mu) (his eq 70-71), and "the wave consists entirely of magnetic disturbances, the direction of magnetization being in the plane of the wave"; a magnetic disturbance not in the plane cannot propagate as a plane wave. This is the standard derivation (it follows from div B = 0 and the wave equation). Paragraphs 98-100 correctly find the remaining unknowns (J and Psi) obey no wave equation (they depend on the time only through uniform increase), so no longitudinal wave: consistent with Gauss's law.

Units: v is "the number of electrostatic units in one electromagnetic unit", a ratio with dimensions of velocity. In air (mu = 1) V = v [read: page image p.47]. Numbers in the paper compared with the modern exact c = 299,792,458 m/s [web: https://physics.nist.gov/cgi-bin/cuu/Value?c]: - Weber-Kohlrausch v = 310,740,000: +3.65 % (310.74/299.792 - 1). - Fizeau V = 314,858,000: +5.03 %. - Foucault V = 298,000,000: -0.60 %. - Aberration 308,000,000: +2.74 %. Maxwell's wording: "agrees sufficiently well" and "seems to show". The spread among the light measurements (about 5 %, and Fizeau vs Foucault 5.7 %) is as large as the gap between light and v. A strict statistical test was not possible on those numbers; the identification is a strong coincidence argument of a "same number to a few percent from entirely different experiments" kind, plus the structural match (transverse waves, Faraday's earlier suggestion). He notes no use of electricity in the light measurements and none of light in the v measurement [read: page image p.47], so the two numbers are independent; this is the key logical point and it is right.

Also in Part VI: the relation between refractive index and electromagnetic constants, "the square of the index of refraction is equal to the product of the specific dielectric capacity and the specific magnetic capacity" [read: page image p.14, summary]. This is n^2 = epsilon_r mu_r. It is exact only at low frequency for non-dispersive media; Maxwell says "if the same character of elasticity is retained in dense transparent bodies" [read], a hidden assumption that fails for water (n^2 = 1.77 optically versus epsilon_r ~ 80 static) [memory: high; known textbook example]. Maxwell flags the assumption, so this is a limitation, not an error.

b. Experiment? Consistent: modern c matches 1/sqrt(mu0 eps0) exactly by definition of the SI system today and, before 2019, by measurement to high precision [memory: high]. Weber-Kohlrausch's number being 3.65 % high is within the era's accuracy.

c. Tested? Yes. Hertz generated and detected waves of finite speed in air 1887-88, which Hunt reports was hailed as the confirmation of Maxwell's field theory [read: Hunt, text layer, FitzGerald's address in 1888]. Later measurements of the speed of Hertzian waves along wires and in air gave c (Hertz's own early wire results were off; Lecher wires etc.) [memory: medium; not relied on for the verdict]. Replicated worldwide for 130+ years. Polarization is consistent with transverse nature.

d. If true: it is true; nothing to change.

Verdict: Established. Caveat on historical evidence: the 1865 numbers alone supported Plausible, with a ~5 % scatter; the decisive evidence is later.

Claim 5: These effects occur in an all-pervading ethereal medium

a. Mathematics consistent? The field equations do not contain the medium: they involve only fields, k, mu and rho. Maxwell says so: "The conclusions arrived at in the present paper are independent of this hypothesis" [read: text layer, para 75]. So the ether is not derived; at most it motivates the vocabulary (elasticity, momentum). One genuine internal tension: the equation set is not Galilean-invariant, and equation (D) contains a v x B term referring to the velocity of a conductor relative to something unstated. If that "something" is the ether, the equations are valid only in the ether frame; the paper does not say this and that is a hidden assumption about the reference frame. Heaviside's recast does not fix it either; Lorentz and Einstein do (Lorentz covariance) [memory: high].

b. Experiment? The ether as a medium with a definite rest frame is contradicted by Michelson-Morley (1887), and later Kennedy-Thorndike and others, all null for an ether wind [memory: high]. I did not web-verify these dates; verdict does not rest on exact dates.

c. Tested? Yes: many ether-drift experiments, null results [memory: high], and special relativity explains them. Replicated many times (also with modern resonator and optical clock tests) [memory: medium].

d. If true: it would require a preferred frame; relativity and the Lorentz symmetry of the Standard Model would need replacement or an undetectable-in-principle ether. That is why the claim is no longer held.

Verdict: Historically superseded but instructive. Split: the "medium as physical substance with a rest frame": Tested, not supported (Michelson-Morley and successors). The "equations without the medium": unaffected and Established. Per para 75, Maxwell did not make his equations depend on the ether, so this should be seen as a reasonable interpretive position of its day, not a flaw in the mathematics.

Claim 6: Heaviside recast the equations into four vector equations, derived S = E x H, and held that the action lies in the field

a. Mathematics consistent? Yes; I checked the sidebar [read: Hunt text layer]. Start: E = -dA/dt - grad Psi. Take the curl: curl grad Psi = 0 (identity for any twice-differentiable scalar), and the order of curl and d/dt can be swapped for smooth fields, giving curl E = -d(curl A)/dt = -mu dH/dt. (Hunt's printed "grad(-dA/dt)" is a typo for curl.) The four equations listed there: div(eps E) = rho, curl E = -mu dH/dt, div(mu H) = 0, curl H = k E + eps dE/dt; with k = conductivity: the conduction-plus-displacement term enters with matching signs, so the sign problem found in Claim 3 does not appear in Heaviside's set. That supports the view that Heaviside's set is the repaired one. Equivalence with Maxwell's twenty: eliminating A and Psi loses the gauge freedom and the redundancy (eight equations of the paper become four independent vector ones after using the identities). Eight vector equations to four is not a loss of content: the potentials are determined by the fields up to gauge. Hidden assumptions: the medium is linear (D = eps E, B = mu H), and the fields are smooth enough to differentiate.

S = E x H: derived from the field equations by taking H . curl E - E . curl H, which gives the identity div(E x H) = H . curl E - E . curl H [standard vector identity; checkable by expanding components]. Substituting the field equations gives -d/dt(eps E^2/2 + mu H^2/2) - k E^2, i.e. Poynting's theorem. The integral form is true; what is not forced is that S itself is the local energy flux, because curl of any vector can be added without changing the integral over a closed surface. This is the same ambiguity as in Claim 3. Hunt's statement that the energy "passes through the surrounding field and enters the wire through its sides" is therefore a consequence of the chosen S, which is the standard and well-motivated choice, but it is an interpretation of the local flux [read: Hunt text layer].

b. Experiment? The integrated form (radiated power equals the integral of E x H over a surface) is verified by every antenna and radio-power measurement [memory: high]. Heating of the wire by energy entering through its sides: standard textbook result (Griffiths example) [memory: high].

c. Tested? Yes, constantly. Replicated.

d. If true: nothing; this is mainstream. Priority: Hunt reports that Poynting published slightly before Heaviside [read]. Belrose's NEC-4 simulation (currents inside and outside a lattice tower) is a numerical electromagnetics computation, an application and not a test of Heaviside's interpretation [read: Step 1 summary, not independently rechecked].

Verdict: Established. Split: "four-equation vector form and S = E x H as an integral conservation law": Established. "The real action is in the field, not the wire", as a statement about the local location of energy: Not testable as stated (an interpretation consistent with the equations and with a measurable total).

Claim 7: Heaviside was blocked or dismissed by officials and journals, then vindicated

a. Mathematics? Mixed: the claim touches Heaviside's operational calculus. The objection reported by Hunt is that he "handled divergent series" cavalierly [read: Hunt]. That is a legitimate mathematical objection: formal manipulations of divergent series and of the operator p = d/dt can give wrong answers when unjustified. Heaviside's response, "Shall I refuse my dinner...", is a reasoning about results, not rigor. The method was later put on a rigorous footing (Bromwich, Carson, Wagner, then Laplace transform theory) [read: Belrose; [memory] high: Bromwich 1916]. The rejection by the Royal Society's referee was thus for a real rigor gap, though the method gave correct answers [read]. This is the one place in the Maxwell file where "outsider dismissed" is an incomplete story: part of the objection was mathematically sound.

b. Experiment? Inductive loading works; it was implemented by AT&T and others [read: Hunt]. Hunt notes Pupin patented it in 1899 [read]. Real-world success is documented in the article, not independently confirmed by me.

c. Tested? Yes, in practice [read: Hunt, "AT&T and other companies later used it with great success"]. Regarding the claim that Preece blocked publication and an Electrician editor was removed and the series cancelled: Hunt reports that Preece "declared the paper worthless and blocked it" and the editor, Biggs, was "abruptly removed" and his successor cancelled the series [read: Hunt text layer, partial lines; the exact wording is split across two columns in the extraction, so I quote the gist, not verbatim]. This is one historian's account; I did not cross-check a second source.

d. If true: nothing in physics changes; it is a sociological claim.

Verdict: Plausible, untested in my area (history is outside a derivation check). I cannot assign a scale verdict stronger than this from a mathematical standpoint. In words: documented in one reputable secondary source [read]; mathematically, the specific rigor objection to operational calculus was fair, and the loading claim is Established in engineering terms.

Summary table

Confidence and gaps

• The sign inconsistency in claim 3 is my own finding from page images; I recommend the arbiter look at PDF p.33-34 and p.52 of dynamicaltheoryo00maxw.pdf.
• Equation (A), (B), (C) I read from the text layer and from how they are used on p.52 (eq 95 quotes (A)); I did not render the pages with (B) and (C) themselves.
• Faraday 1856 paper: only the six-law summary was checked, not the fluid-analogy derivations or the electrostatic and magnetic sections.
• Web checks done (2 of ~6 budget): NIST value of c; Wikipedia transcription of the 1865 equations. Not web-checked: Michelson-Morley, Hertz, Aharonov-Bohm, Bromwich (all [memory], high or medium confidence; the verdicts do not depend on exact dates).