Morton-Wall: Mathematician review (Step 2)

Reviewer: PhD mathematician. Date 2026-10-08. Project scope: E. J. Houston, "On a New Connection for the Induction Coil" (J. Franklin Inst. June 1871, pp. 417-419) and W. H. Wahl, "A Large Induction Coil" (same volume, pp. 212-216, on Henry Morton's Ritchie coil). Tags: [read] = I read it myself in the extracted text or a page image; [web] = web source with URL; [memory] = recalled, with confidence; [computed] = my own arithmetic (assumptions stated).

Method: I re-read both articles in the extracted text (PDF pp.594-596 and 319-323) and rendered Wahl's pp.320-321 as page images (150 dpi) to check numbers. I read Houston's Jan 1876 paper (opening, lines 4183-4330 of the saved OCR) and the Investigator's research notes. I did not re-read the Houston-Thomson April 1876 paper or Morton 1867 beyond the Investigator's summaries; those are relied on as [read by Investigator].

The mathematical content of this project is small. Houston gives no equations, no tables and no measured quantities except lengths and distances. Wahl gives a specification sheet. So the useful mathematics is: (1) checking the arithmetic and units in the sources, (2) building the simple circuit model that Houston's words imply, and (3) seeing what that model does and does not predict.

Corrections to Step 1 (and to earlier documents)

1. claims.md claim 5 and facts.json (Wahl entry, key claim 1) say "443 miles of secondary wire, 250 lb". That is an OCR error and is wrong. The page image of Wahl p.320 (j.213) reads "containing but 44⅓ miles of wire, 40 inches in length, and weighing about 250 pounds", and p.321 (j.214) reads "The secondary wire is 234,100 feet, or about 44⅓ miles, and ·007 of an inch in diameter, and weighs 44½ pounds" [read: page images pp.320-321]. Check: 234,100 ft / 5,280 ft per mile = 44.34 miles [computed]. "443" is "44⅓" with the fraction misread as a digit; the "443 pounds" in the text layer is "44½ pounds". The 250 lb is the whole coil, not the wire. The text layer error is in the PDF's own OCR, so every digest and fact copy that quotes it is affected. Evidence the correct figure is 44⅓: the page image, the feet figure, and the weight check below. Stale copies to search for: claims.md claim 5, facts.json Wahl summary and key_claims, the digest chunks that cite "443 miles", and any later file derived from them.
2. Same page: the condenser is 325 square feet of tinfoil, not "825". [read: page image p.321]. "100 square feet in permanent connection with the primary circuit, and three buttons throwing on 100, 75 and 50 feet" adds to 325 [computed], which confirms 325. If the text layer's "825" was copied anywhere, fix it.
3. Weight check on the corrected figures (new finding, minor). A 200 ft primary of 0.1655 in copper wire weighs 16.7 lb by volume x density [computed, copper 8,960 kg/m3, high confidence], matching the stated 17 lb. For the secondary, 234,100 ft of 0.007 in bare copper weighs 35.0 lb [computed], against the stated 44½ lb, 27% more. The primary check shows the method is sound, so either the "·007" is the bare diameter and the stated weight includes silk covering and paraffin paper, or the true bare diameter is nearer 0.0079 in. Wahl does not say. This does not threaten any claim; it is a loose end in a 1871 specification. The secondary's dc resistance from the stated figures is about 48 kilo-ohms [computed, bare 0.007 in copper].
4. Houston 1876 battery wording (Investigator's lead 1, "Corrections" bullet 1). The Investigator reads "One pole of the battery was placed in metallic connection with a gas pipe and the other ... with a large insulated conductor" as meaning the coil's secondary poles. I read the same OCR passage [read: JFI 1876 p.60]. It literally says battery, and the same paragraph says the white sparks "passed between the platinum points of the coil", which in a Ruhmkorff coil usually names the interrupter contacts, though the secondary electrodes may also be platinum-tipped [memory, medium]. If taken literally it would be a primary-side connection, a different circuit from 1871. I think the Investigator's reading (secondary) is the more likely, because Houston says the 1875 test repeated the 1871 one, but it is a reading, not a fact. The unit "coupled for an intensity of ten" means ten cells in series [memory, medium].
5. Deep read (Houston) holds up. I verified every quotation I used against the text layer, and the three framing corrections it lists (no battery in 1871; 30 ft applies to the engine only; the one-square-foot result is the two-insulated-conductors arrangement with neither pole earthed) are correct [read: PDF pp.594-596]. One refinement: the deep read says the 1871 text never says why the break handle shock occurred. That is true, and the electrical reason is simple (section "Side observations" below).
6. Investigator's other corrections accepted: Morton's 1867 paper is found; Wahl substituted for Houston in 1872-73 (Edmonds 1902, [read by Investigator]). The Investigator's "Not yet tried" list and the unread Thomson/Snyder records stay open.

The model behind Houston's words (a worked example)

Everything below is a modelling reading of Houston's text. He did not write any of it. Read it as "what would the physics say if the coil is treated as a simple source with capacitance".

Setup. A coil's secondary has some self-capacitance C_s between its two terminals. At each interruption of the primary, a store of energy E (set by the primary: E ≈ ½ L_p I², with inductance L_p and current I at break) is transferred into the secondary and appears as a voltage rise across C_s. The spark jumps when the voltage reaches the breakdown voltage V_b of the gap. Breakdown voltage of a long air gap grows roughly in proportion to gap length d: V_b ≈ k d. For long gaps k is about 4.5-5 kV/cm for a positive rod gap (streamer criterion) and lower for very long gaps where leaders form [web: https://archive.nptel.ac.in/content/storage2/courses/108104048/lecture8/slide2.htm, as relayed by a web search summary; range 3.4-11.2 kV/cm in a 2022 numerical study cited there]. I use "voltage ∝ length" only as a rough rule.

What a large conductor does. Connect one pole to earth (a gas pipe) and the other to a big conductor, or, in Houston's tin-ware arrangement, put an insulated conductor of capacitance C_x on a pole. Seen from the spark gap, C_x sits in parallel with C_s. Total capacitance C = C_s + C_x. Two regimes follow.

• Regime 1, gap set shorter than the coil's maximum. The terminal voltage must climb to V_b before the spark. The charge delivered is Q = C V_b and the spark energy is ½ C V_b². Both grow in proportion to C (energy) and to C (charge) as long as the coil can still reach V_b. This is the "more quantity" effect. It is the same thing a Leyden jar across the gap does.
• Regime 2, energy-limited. Once ½ C V_b² reaches the available E, the coil can no longer reach V_b. The maximum spark voltage falls as V_max = sqrt(2E / C), so the maximum spark length falls as 1/sqrt(C), while the charge per spark rises as Q = sqrt(2 E C). Length x charge is then constant at 2E/k. That is the "thicker, shorter, white" result Houston saw, and Wahl's trade-off between a long thin spark and a short loud one.

Numbers [computed, assumptions stated]. An isolated conductor of area A has capacitance about that of a sphere with the same area: C = 4 pi eps0 sqrt(A / 4 pi). With eps0 = 8.85e-12 F/m: - 1 sq ft (the small conductor of Houston's 5 in result): C = 9.6 pF. - 6 sq ft: 23 pF. 20 sq ft: 43 pF. - 300 sq ft ("several hundred square feet", Houston's proposed maximum): about 166 pF. These are isolated-in-space values. A real conductor on glass jars near a floor, walls and pipes has more, perhaps two to five times as much [memory, medium]. So in 1871 a conductor of a few hundred square feet adds order 100-1000 pF, compared with a coil secondary whose own capacitance, for a 44-mile winding, is plausibly of order 100 pF [memory, low; not checked, and not needed for any verdict]. That is why such conductors have a visible effect on a big coil and tiny ones do not.

Comparison, a Leyden jar of 14 sq ft on Wahl's coil: a 2 mm glass wall with relative permittivity 5 gives C = 29 nF [computed; thickness and permittivity assumed] which is about 300 times the large conductor. That is why Wahl's jar cuts a 21 in spark to 2 in while Houston's loads only trim 6 in to about 5 in. Morton's secondary condenser of 20 coated panes in cascade (series) gives a far smaller capacitance, one pane / 20: with panes of 6 x 8 in foil on 2 mm glass, 820 pF each [computed, assumed glass], so about 41 pF for 20 in series. A small capacitance gives a spark nearly as long as the unloaded one (Wahl: 14 in of 21 in), yet still "white and loud" because the discharge now passes a larger charge per spark than the bare gap. This pattern (large C: short, fat spark; small series C: long, white spark) is exactly what the model predicts, in sign and rough size. The same applies to Morton 1867 (a single jar gave 1 in, six jars in cascade 4 in, [read by Investigator]), which is the series-capacitance rule: n equal jars in series have C/n.

Houston's two-conductor result in this model. With one conductor on each pole and neither earthed, the capacitance seen across the gap is the two in series, C_a C_b / (C_a + C_b). When one conductor is made small, that series value approaches the small one's capacitance (about 10 pF at 1 sq ft, more in practice). So the spark length returns toward the free-air 6 in: Houston got "about five inches". This is the model's prediction, and Houston's observation agrees in direction. His step "diminishing one and increasing the other" is the same as moving C_series from about C/4 (equal halves) to about C_b. His observation that length came back "without sensible diminution in quantity" needs care: in regime 2, Q x length is constant, so a spark of 5 in carries about 6/5 = 1.2 times the charge of a 6 in spark, a 20% change, well inside what an eye can judge [computed, model]. The statement is therefore not contradicted, but also could not distinguish between competing views. A thick, silvery spark at 5 in was presumably a real brightening, since the series C is not zero.

Spark energy per break. If a loaded coil sparks at about 200 kV into about 100 pF, the spark energy is ½ C V² = 2 J [computed; V from 21 in at about 3-5 kV/cm gives 160-270 kV, [memory, low] for the rule, consistent with the web range above for positive long gaps]. A primary of 200 ft of 0.1655 in wire on a 14 lb iron core carrying some tens of amps stores order a few joules at break [computed; turns and permeability assumed, so only the order of magnitude is meaningful]. So Wahl's 21 in claim is energetically plausible. I cannot verify the 29 in London coil failure or the "sixty cells" independently.

Houston's "maximum" prediction. He wrote the maximum increase "will be obtained" with several hundred sq ft on one pole and earth on the other. In the model there is no maximum in charge at a fixed gap setting until regime 2 is reached, and in regime 2 charge keeps growing as sqrt(C) while length shrinks to nothing; the spark ceases when V_max falls below the smallest gap that can be set (no spark at all). So "maximum" depends on what the experimenter wants to keep (length, charge or brightness). His prediction is therefore not wrong but underspecified: it names a size without naming the quantity being maximised. The dimensional conclusion that a few hundred sq ft is 100-1000 pF, roughly the same order as the coil's own capacitance, is why he expected a large effect there.

Side observations

• Shock at each break. Houston insulated the break-piece handle after "a slight shock at every break". The break lies in the primary circuit, which is coupled to the secondary through the coil and its condenser. A secondary pole tied to earth and a large conductor on the other raise the potential of the whole coil structure relative to the experimenter in step with the break, and the handle is the nearest conductor to the primary. This is my inference [computed/model, not in the text], not Houston's, and it neither supports nor contradicts the "radiant shock" reading of the Energetic Forum.
• Stray sparks at 8-9 ft and 30 ft. The system of table, gas pipes, stoves and steam engine is a big conductor tied to the secondary through a partial earth; sparks between it and a grounded or floating person need only that conductor's potential to swing by a few kV relative to the person's local potential. Capacitive and inductive coupling in the pipe network can do that without radiation. The frequency scale of the spark discharge in a loop of tens of metres of pipe: L about 10 microhenry, C about 100 pF gives f = 1/(2 pi sqrt(LC)) = 5 MHz, wavelength about 60 m [computed; L and C assumed]. A 30 ft (9 m) separation is then about 0.15 wavelength, which is neither clearly near-field nor far-field, so this arithmetic cannot decide between "quasi-static coupling in the pipes" and "radiation" (Atherton 1984, Woodbury 1944 and the free-energy readings). What would decide it: a measurement of the spark's frequency content and the field versus distance, none of which exists in the record. The coil's own secondary ringing (hundreds of hertz to a few kHz for a large secondary inductance, [computed; L_s assumed 10^2-10^3 H]) is a separate and far lower frequency than the gap-discharge oscillation. The Energetic Forum's "probably not more than 5 kHz" for the "frequency" of the system mixes the two.
• Units and vocabulary. "Quantity" in 1871 means electric charge (the amount "of electricity") and is never separated from current or energy. Wahl uses "quantity" when coils are coupled for quantity (see claim 6). The Houston 1871 article gives no equations, no units for measured quantity, and no capacitance. His "capacity" is the word-less idea of "large conductor". Surface areas are the only size measure given. The 1876 reply uses "some hundred thousand miles per second" for current velocity (Houston, [read: 1876 p.61 per Investigator]); modern value for signal speed in wires is of order 10^5-10^8 m/s depending on the line and is not the electron drift speed, so the statement is consistent with the era's loose knowledge and is not load-bearing.
• Frictional-machine analogy. Houston says the connection is "somewhat similar" to earthing one conductor of a plate machine. That analogy is loose: a frictional machine is a charge-pump (current-like source) and the induction coil's secondary is closer to a voltage source with a capacitance. Earthing one pole matters in the model because it adds the pole-to-earth capacitance C_x to the system, not because it supplies a return path for charge (a spark between the poles is already a complete circuit). Houston's phrase "instead of being lost by passing along the wires to the earth" mixes the two pictures.

Claim-by-claim review

Claims as in claims.md (claim 5 with the correction above).

Claim 1: one pole earthed, the other on a large imperfectly insulated conductor (lecture table), turns the spark thicker, shorter, silvery white, like a Leyden jar; long sparks from metal 8-9 ft away.

a. Mathematics: there is none in the article. The qualitative statement is internally consistent and matches the capacitance-loading model (regime 2: shorter, higher charge). Houston's own sentence carries a small inconsistency of language: the electricity "instead of being lost ... jumped" mixes a loss picture with a charge-storage picture (above). b. Experimental consistency: consistent with the known behaviour of a Leyden jar or condenser in a coil's secondary circuit: Morton 1867 (single jar 1 in; the investigator's note), Wahl 1871 (2 in with a 14 sq ft jar vs 21 in bare) [read]. The whitening and shortening of the spark by a parallel condenser is an established coil-and-jar effect [memory, high]. c. Tested: Houston observed it himself in 1871 (once, no controls) and reported repeating it in Dec 1875 with Elihu Thomson [read: 1876 p.60]; Woodbury 1944 (a Thomson-side biography) reports the same set-up, with a "vivid blue" jar spark instead of "silvery white" [read by Investigator]. No neutral measured test is on record; no one replicated with instruments. d. If true: nothing in mainstream physics changes. Verdict: Established (qualitative effect). The quantitative size (new spark length, charge) was never measured.

Claim 2: sparks drawn from objects 8-9 ft away, a steam engine ~30 ft away and from a wire in a room below.

a. Not a mathematical claim. Distances and the missing path to the engine are an incomplete description, not an inconsistency. b. Consistent with induced potentials on a connected, grounded or floating conductor network driven by a high-voltage coil. Houston's 1876 paper adds that the engine and boiler were "in metallic connection with the water-pipes" [read: 1876 p.59-60]. That fills the gap in the 1871 description: the engine was part of the pipe network, not free in space. This matters: it makes the 30 ft observation a statement about a pipe network. c. Houston and Thomson repeated it in 1875 [read: 1876]; Edison's November 1875 "etheric force" sparks are of the same kind (Edison Papers chronology, [web] https://edison.rutgers.edu/life-of-edison/chronology/1871-1880), and Houston and Thomson reproduced them as induction effects [read by Investigator]. Woodbury and Snyder's rooftop account (90 ft, 1875) is a single published Thomson-side source; I weigh it as documented testimony, not as measurement. No spark lengths or field values were ever recorded. d. Nothing, if quasi-static or standard EM; the radiated-wave reading by Atherton and Susskind would make this an early observation of EM waves (a historical priority question, not new physics). Verdict: Established (that the sparks occur, qualitatively). Interpretation (induction vs radiation) is Not testable as stated from the record, because no frequency or field data exist; the 5 MHz arithmetic above cannot decide it.

Claim 3: a large insulated conductor raises the "quantity"; shrinking one of two conductors below ~1 sq ft restores a ~5 in spark "without sensible diminution in quantity".

a. Consistent with the series-capacitance model: C_series tends to the small conductor as it shrinks; length returns toward 6 in. In the energy-limited regime, length x charge is fixed, so "no sensible loss" of charge is a 20% effect at 5 in (above). The statement is internally consistent but too weakly stated to be falsified. b. Direction matches known loading effects; Houston's conductors are small in capacitance (10-170 pF isolated), so the effect on a 6 in coil is large only in the tens-to-hundreds of pF range, as seen. c. One experimenter, one run, "quantity" judged by eye. No electrometer, galvanometer, perforation or heating test. Houston 1876 later ran electroscope and galvanometer tests on the stray sparks and found no deflection (a different effect) [read by Investigator]. d. Nothing. Verdict: Plausible, untested for the charge increase (the model says it should happen); Never properly tested for the specific "5 in at under 1 sq ft with no loss of quantity" result: no neutral controlled test or measurement exists, and nothing suggests it was dismissed; it was never followed up.

Claim 4: mechanism: the large conductor allows rapid discharge and hence rapid induction in the secondary; an insulated conductor of several hundred sq ft on one pole, other earthed, gives the maximum.

a. The mechanism as written ("rapid induction ... by its rapid discharge") is not a derivation and does not match the modern account (added capacitance, energy transfer and charge storage, above). The 1876 restatement ("condensation of the spark by connection with extended surfaces, similar to a Leyden battery", [read: 1876 p.59]) is closer to the capacitance picture and is correct in outline. The "maximum" is underspecified (see worked example). No units or numbers carry the argument. b. Consistent with all coil-and-condenser observations, in a loose verbal form. c. The several-hundred-sq-ft arrangement was never built (Houston: "would seem", untested). I found no later test. d. Nothing. Verdict (split): the "extended surfaces act like a Leyden coating" mechanism is Established in outline (1876 form); the 1871 phrase "rapid induction by rapid discharge" is Not testable as stated; the "several hundred sq ft gives the maximum" prediction is Never properly tested (and, by the model, underspecified: no single maximum exists without naming what is maximised).

Claim 5: Ritchie's coil for Morton gives 21 in sparks from three bichromate cells; a larger London coil failed (Wahl, a co-editor writing on his colleague's coil).

a. After correction (44⅓ miles, 44½ lb wire, 250 lb coil, 325 sq ft condenser), the specification is internally consistent: 234,100 ft = 44.3 miles; the primary weight check passes (16.7 vs 17 lb); the secondary weight is 27% off the bare-wire figure (loose end, coverings). The claim that the 3 in glass block resists as 21 in of air is plausible (160-270 kV across 7.6 cm is 21-35 kV/cm, within the range for thick glass with embedded electrodes [memory, low]). b. Consistent with Spottiswoode's and Apps' large coils of the period (29 in; [read: Wahl]) and with long-gap breakdown fields (above). c. Tested by the maker and Morton's own use (months of lectures); reported by Wahl, who is the journal co-editor, a colleague of Morton and sympathetic to Ritchie (stake: the Institute's own editor and the coil's owner both edit the journal in which it is praised). No independent measurement. Brenni (2015) is an instrument-historical account, paywalled [web] https://www.academia.edu/13825373/Very_large_induction_coils , not read. d. Nothing. Verdict: Plausible, untested (independently); the specifications are consistent after correction.

Claim 6: with a Leyden jar or Morton's secondary condenser the coil gives short, white, loud sparks (2 in with a 14 sq ft jar; 14 in with a 20-pane condenser); coupling the halves "for quantity" shortens the spark to 12 in.

a. Consistent with the capacitance model: 29 nF (jar, assumed) vs 41 pF (20 panes in cascade) [computed] ordered correctly with length (2 in vs 14 in). A fit with a single energy-limited law is only rough: V ∝ 1/sqrt(C_total) would give C_s of 0.27 nF from the jar and 33 pF from the cascade, a factor of 8 apart, so the quantitative law is not tested by these two points (short gaps need more volts per inch, and the glass parameters are assumed). Coupling for quantity: if the two coil sections are put in parallel, each section has half the turns, so about half the voltage: 21/2 = 10.5 in against the reported 12 in [computed; assumes "for quantity" means parallel, [memory, medium]]. Agreement within the rough proportionality of breakdown to length. b. Consistent with Morton 1867 and with ordinary coil practice (jar: short fat spark; cascade condensers: long white spark). c. Reported by Wahl (same stake as above); Morton's earlier 1867 numbers are consistent [read by Investigator]. No replication on record. d. Nothing. Verdict: Established (the qualitative jar/cascade/parallel trade-offs); the specific figures are one report.

Investigator's findings I keep in the record (with my weighing)

• Houston 1876 restatement (own work): adds the pneumatic trough, half-horsepower engine and boiler on the pipe system; that makes the "30 ft engine" a pipe-network effect. Gives the 1875 repetition's battery (ten-cell electro-poion). The ambiguity noted in Correction 4 stays.
• Houston and Thomson April 1876 (balanced-battery and paired-magnet tests): the make-and-vanish tests make the effect a polarity or balance dependent one, as an induced, capacitively coupled effect should be; I treat it as stronger evidence than anything in 1871.
• Morton 1867: supplies the missing condenser result; I used it for the cascade rule (n jars in series, C/n).
• Authorship dispute (Woodbury 1944, Thomson side; stake: Thomson's credit): it says the 1871 observations were Thomson's and Houston "took sole credit." Single published source, quoted by the Investigator; Houston's own note names no assistant; Thomson's 1902 recollection does not mention the coil work (silence). I cannot settle this; Thomson's 1871 notebooks (APS Mss.Ms.Coll.74) would. It does not change the mathematics.
• Wave and antenna readings (Atherton 1984, Susskind 1964, Woodbury, the Energetic Forum "radiant energy" reading): each is that author's characterisation, the free-energy reading having no stake beyond advocacy. My arithmetic shows the geometry cannot decide between near-field and radiated coupling (0.15 wavelength at 5 MHz). The forum's "table top was an antenna, frequency 5 kHz" is a guess, and the "1872" date and Thomson attribution differ from the June 1871 Houston note (the SciAm reprint is dated 10 June 1871).
• Absence of follow-up (no Wahl sequel on the Morton coil; no Houston note on the connection until Dec 1875): competing explanations as in the Investigator's list (work not written up, published elsewhere, not done); the record cannot decide yet.
• Wahl-Houston link (Wahl substituted for Houston 1872-73): noted; no mathematical consequence.

Web verification (budget 6; used 1)

• Long-gap breakdown field, about 4.5-5 kV/cm for a positive streamer, lower slope for leaders above 0.5-1 m, range 3.4-11.2 kV/cm in numerical studies [web: https://archive.nptel.ac.in/content/storage2/courses/108104048/lecture8/slide2.htm, a result of a web search that I did not open in full]. Checked.
• Not checked and not load-bearing: glass block breakdown, coil self-capacitance, the meaning of "coupled for quantity" (all marked [memory] above).

Verdict summary

1. Thicker, shorter, white spark with earth plus large conductor: Established (qualitative); size never measured.
2. Sparks at 8-9 ft, 30 ft: Established as a phenomenon; interpretation (induction vs radiation) Not testable as stated from the record.
3. Quantity increase and the 5 in, under 1 sq ft result: charge increase Plausible, untested; specific result Never properly tested.
4. Mechanism and "several hundred sq ft maximum": capacitor mechanism (1876 form) Established in outline; 1871 wording Not testable as stated; maximum prediction Never properly tested and underspecified.
5. Morton coil 21 in from three cells: Plausible, untested independently (figures corrected to 44⅓ miles).
6. Jar, condenser and quantity coupling effects: Established qualitatively; figures from one interested-colleague report.