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Deep reading

A deep reading takes a few pages of a source and checks every assertion on them: the quotation is verified against the transcription, every number is recomputed, every geometrical statement is a proposition of a model or a step of a derivation, and every result comes with its evidence — the crop of the page. The record is a YAML file (readings/<name>.yaml), the report is reproducible (famulus-astra reading readings/<name>.yaml), and every item points back to its paragraph.

An item:

- id: battani-advance
  locus: fr:kepler-sec/Swerdlow2026/0052/p06
  quote: "λA = 82;17°, a notable advance of 16;37° in 743 years"
  kind: computation
  formal: "82;17 - 65;30"
  expect: "16;37"
  places: 1
  check_of: print
  evidence: {page: 52, box: [0.10, 0.855, 0.92, 0.895]}

formal is sexagesimal arithmetic or one of

form what is checked
proposition <model>: <name> a proposition stated in a model holds over its parameter space and states
step <derivation>: <name> a step (or conclusion) of a derivation is true and justified
computation <derivation>: <name> a computed or determined quantity agrees with the printed one
sign <model>[<parametrisation>]: <observable> a sign rule, read off the model
solve <model>[<parametrisation>]: <obs> at <obs2> = <angle> a value in the state where another observable has a value

check_of distinguishes the author’s own statements (print) from historical figures the author reports (reported): a discrepancy in the first is about the page, in the second about the source reported. A quotation that is not in the paragraph’s transcription fails the run.

Swerdlow’s pages on Ptolemy’s solar model

Deep reading: Swerdlow on Ptolemy’s solar model, pp. 17 and 19–23

fr:kepler-sec/Swerdlow2026 · 45 assertions: 35 confirmed, 9 discrepancies, 1 not checkable, 0 quotations not found.

# locus quotation kind of check printed recomputed verdict
1 0047/p03 365 1/4 - 1/300 days = 365;14,48^d computation print 365 + 1/4 - 1/300 365;14,48 365;14,48 confirmed
2 0047/p03 365;14,48^d = 365^d 5;55,12^h computation print (365;14,48 - 365) * 24 5;55,12 5;55,12 confirmed
3 0047/p03 is 0;6,26^h too long computation print (365;14,48 - 365.2422) * 24 0;6,26 0;6,26 confirmed
4 0047/p03 3760/304 · 29;31,50,8,20^d ≈ 111,035^d/304 computation print 3760 * 29;31,50,8,20 111035 111035 confirmed
5 0047/p03 111,035^d/304 = 365;14,48,9…^d computation print 111035 / 304 365;14,48,9 365;14,48,9 confirmed
6 0047/p03 365;15,25^d = 365^d 6;10^h computation print (365;15,25 - 365) * 24 6;10 6;10 confirmed
7 0047/p03 1° per century or 36’’ per year computation print 3600 / 100 36 36 confirmed
8 0049/p04 on which the sun S moves uniformly about E through the mean anomaly κ̄ measured from th… construction print proposition solar-eccentric: p1 holds equal(lam_bar, lamA + kappa_bar): max residual 6e-14 over 3456 configurations confirmed
9 0049/p04 complete parallelogram OESC construction print step solar-equivalence-equal-radii: d6 holds parallel(line(O, Ecc.E), line(Epi.C, Ecc.S)): justified confirmed
10 0049/p04 so that r is always parallel to the apsidal line AB construction print proposition solar-epicyclic: p1 holds parallel(line(C, S’), line(O, A’)): max residual 2e-15 over 3456 configurations confirmed
11 0050/p02 the direction OS from the earth to the sun, the diagonal of the parallelogram, is the s… construction print step solar-equivalence-equal-radii: conclusion holds coincident(Ecc.S, Epi.S’): justified confirmed
12 0050/p02 the apparent nonuniform motion of the sun is identical construction print step solar-equivalence-equal-radii: d12 holds coincident(Ecc.S, Epi.S’): justified confirmed
13 0050/p02 in both models the directions OS and OS′ are the same construction print step solar-equivalence-proportional: conclusion holds equal(direction(O, Ecc.S), direction(O, Epi.S’)): justified confirmed
14 0050/p02 There is, however, a distinction between the models, although it is recognized only by … historical-claim print — — — not checkable
15 0052/p02 The mean anomaly κ̄ = λ̄s − λA definition print proposition solar-eccentric: p1 holds equal(lam_bar, lamA + kappa_bar): max residual 6e-14 over 3456 configurations confirmed
16 0052/p02 the true anomaly κ = κ̄ ± c about O definition print proposition solar-eccentric: p2 holds equal(kappa, kappa_bar - c): max residual 1e-13 over 3456 configurations confirmed
17 0052/p03 sin c = e sin κ̄/(R^2 + e^2 + 2Re cos κ̄)^1/2 formula print step solar-equation-of-centre: conclusion2 holds equal(sin(abs(norm180(c))), e * abs(sin(kappa_bar)) / sqrt(R * R + e * e + 2 * R * e * cos(kappa_bar))): by calculation confirmed
18 0052/p03 tan c = e sin κ̄/R + e cos κ̄ formula print step solar-equation-of-centre: conclusion holds equal(tan(abs(norm180(c))), e * abs(sin(kappa_bar)) / (R + e * cos(kappa_bar))): by calculation confirmed
19 0052/p04 c < 0° for κ̄ > 180°, c > 0° for κ̄ > 180° sign-rule print sign solar-eccentric[solar-eccentric.ptolemy]: c c < 0° for κ̄ > 180°, c > 0° for κ̄ > 180° c > 0° for κ̄ < 180°, c < 0° for κ̄ > 180° discrepancy
20 0052/p04 lies on a line parallel to ES in the direction Oλ̄s, thus λs = λ̄s ± c definition print proposition solar-eccentric: p5 holds equal(lam, lam_bar - c): max residual 6e-14 over 3456 configurations confirmed
21 0052/p05 vernal equinox to summer solstice 94½d with mean motion 93;9° computation reported 94.5 * 360 / 365;14,48 93;9 93;9 confirmed
22 0052/p05 summer solstice to autumnal equinox 92½d with mean motion 91;11° computation reported 92.5 * 360 / 365;14,48 91;11 91;10 discrepancy
23 0052/p05 a total of 187d and mean motion of 184;20° computation reported 187 * 360 / 365;14,48 184;20 184;19 discrepancy
24 0052/p05 a total of 187d computation print 94.5 + 92.5 187 187 confirmed
25 0052/p05 e = 2;29,30 ≈ 2;30, thus e/R = 1/24 computation print 2;30 / 60 0;2,30 0;2,30 confirmed
26 0052/p05 from which he finds that where R = 60, e = 2;29,30 derivation reported computation solar-parameters-from-seasons: arcs.e 2;29,30 2;29,28 discrepancy
27 0052/p05 and λA = 65;30° derivation reported computation solar-parameters-from-seasons: arcs.lamA 65;30 65;35° discrepancy
28 0052/p05 vernal equinox to summer solstice 94½d derivation reported computation solar-parameters-from-seasons: days.e 2;29,30 2;28,58 discrepancy
29 0052/p05 summer solstice to autumnal equinox 92½d derivation reported computation solar-parameters-from-seasons: days.lamA 65;30 65;26° discrepancy
30 0052/p05 the maximum correction cm = sin−1e/R = 2;23° computation print asin(2;30 / 60) 2;23 2;23 confirmed
31 0052/p05 e is in error by about +0;24 computation print 2;30 - 2;6,22 0;24 0;24 confirmed
32 0052/p05 +0;24/2;6 = +19% computation print 0;24 / 2;6 * 100 19 19 confirmed
33 0052/p05 λA by −0;46° in −145 computation print 65;30 - 66;16 -0;46 -0;46 confirmed
34 0052/p05 −5;36° in 140 computation print 65;30 - 71;6 -5;36 -5;36 confirmed
35 0052/p05 its longitude has increased by +4;50° in 285 years computation print 71;6 - 66;16 4;50 4;50 confirmed
36 0052/p05 of which about 4° is the precession of the equinoxes computation print 285 / 72 4 4 confirmed
37 0052/p05 0;50° the proper or sidereal motion of the apsidal line computation print 4;50 - 4 0;50 0;50 confirmed
38 0052/p06 al-Battānī found e = 2;4,45 and cm = 1;59,10° computation reported asin(2;4,45 / 60) 1;59,10 1;59,9 discrepancy
39 0052/p06 λA = 82;17°, a notable advance of 16;37° in 743 years computation print 82;17 - 65;30 16;37 16;47 discrepancy
40 0053/p02 λA = 77;50°, a regression of 4;27° computation print 82;17 - 77;50 4;27 4;27 confirmed
41 0053/p02 cm = 2;10°, corresponding to e ≈ 2;16 computation print 60 * sin(2;10) 2;16 2;16 confirmed
42 0053/p03 λ̄s = λs + c = 180° + 2;10° = 182;10° computation print solve solar-eccentric[solar-eccentric.ptolemy]: c at lam … 2;10 2;10 confirmed
43 0053/p03 κ̄ = 182;10° −65;30° = 116;40° computation print 182;10 - 65;30 116;40 116;40 confirmed
44 0053/p03 λ̄s ≈ 183;15°, so Ptolemy’s mean longitude is in error by −1;5° computation print 182;10 - 183;15 -1;5 -1;5 confirmed
45 0053/p03 the error increases by −0;26,25° per century computation print -(0;6,26 / 24) * 100 * 360 / 365;14,48 -0;26,25 -0;26,25 confirmed
  • sign-rule (fr:kepler-sec/Swerdlow2026/0052/p04): the model gives «c > 0° for κ̄ < 180°, c < 0° for κ̄ > 180°», printed «c < 0° for κ̄ > 180°, c > 0° for κ̄ > 180°». the two clauses name the same condition; as printed the rule is contradictory — one clause must read κ̄ < 180°
  • interval-2 (fr:kepler-sec/Swerdlow2026/0052/p05): recomputed 91;10, printed 91;11, difference -0;0,43. Ptolemy’s figure; exact arithmetic on his year gives 91;10,17, so 91;11 is Ptolemy’s rounding, not a slip of the print
  • interval-sum (fr:kepler-sec/Swerdlow2026/0052/p05): recomputed 184;19, printed 184;20, difference -0;1,10. exact arithmetic gives 184;18,50; 184;20 is the sum of Ptolemy’s two rounded arcs 93;9 + 91;11
  • eccentricity-from-arcs (fr:kepler-sec/Swerdlow2026/0052/p05): computation fr:astra/solar-parameters-from-seasons/arcs.e with solar-eccentric.ptolemy: 2;29,28, reported 2;29,30, difference −0;0,1,51 (the only solution found; the intervals taken as arcs of mean motion). the model inverted exactly for the two reported arcs 93;9° and 91;11° gives 2;29,28: Ptolemy’s 2;29,30 within 0;0,2, the size of a rounding in his chord computation
  • apogee-from-arcs (fr:kepler-sec/Swerdlow2026/0052/p05): computation fr:astra/solar-parameters-from-seasons/arcs.lamA with solar-eccentric.ptolemy: 65;35°, reported 65;30°, difference +0;5,7 (the only solution found; the intervals taken as arcs of mean motion). the same exact inversion gives 65;35°; the reported 65;30° differs by 0;5°. The paragraph itself calls the derivation sensitive to errors in the intervals
  • eccentricity-from-days (fr:kepler-sec/Swerdlow2026/0052/p05): computation fr:astra/solar-parameters-from-seasons/days.e with solar-eccentric.ptolemy: 2;28,58, reported 2;29,30, difference −0;0,31,44 (the only solution found; days turned into arcs of mean motion with T = 365;14,48 d). from the day counts 94½ and 92½ with the year 365;14,48 d and exact arithmetic the model gives 2;28,58: the reported parameters follow from the rounded arcs, not from the day counts
  • apogee-from-days (fr:kepler-sec/Swerdlow2026/0052/p05): computation fr:astra/solar-parameters-from-seasons/days.lamA with solar-eccentric.ptolemy: 65;26°, reported 65;30°, difference −0;3,47 (the only solution found; days turned into arcs of mean motion with T = 365;14,48 d). from the day counts with exact arithmetic the model gives 65;26°
  • battani-maximum (fr:kepler-sec/Swerdlow2026/0052/p06): recomputed 1;59,9, printed 1;59,10, difference -0;0,0,55. arcsin(e/R) for the reported e gives 1;59,9,5; the reported 1;59,10 is al-Battānī’s own figure as Regiomontanus gives it (not checked against al-Battānī)
  • battani-advance (fr:kepler-sec/Swerdlow2026/0052/p06): recomputed 16;47, printed 16;37, difference +0;10. the advance from Ptolemy’s 65;30° to 82;17° is 16;47°; the printed 16;37° differs by exactly 0;10°

The two discrepancies of the print are misprints: the sign rule for the equation of centre names the same condition in both clauses, and the advance of al-Battānī’s apogee is given as 16;37° where the two longitudes on the page differ by 16;47°.

The seven discrepancies in reported figures are facts about the sources, not about Swerdlow’s page. Two are Ptolemy’s roundings of his mean motions. One is al-Battānī’s maximum equation as Regiomontanus transmits it, 1;59,10° against 1;59,9° from his eccentricity. Four belong to the determination of the parameters from the seasons: inverted exactly, the model gives e = 2;29,28 and λ_A = 65;35° from the two reported arcs, and e = 2;28,58 and λ_A = 65;26° from the day counts. Ptolemy’s 2;29,30 and 65;30° follow from his rounded arcs within the rounding of a chord computation, not from the day counts. The paragraph itself calls the derivation sensitive to errors in the intervals.

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