The texts
All texts describe Ptolemy’s solar model as Swerdlow 2026, The Renaissance of Astronomy, chapter 1, uses it (kepler-sec, Swerdlow2026, scan pages 47–53). Every value points at the page that prints it. Every text is a reference object, fr:astra/<id>.
| text | kind | check |
|---|---|---|
fr:astra/euler-fig-3 |
figure | ✓ labelled |
fr:astra/euler-tide-force |
model | ✓ documented |
fr:astra/euler-tide-force-emendation |
derivation | ✗ grounded |
fr:astra/euler-tide-force-resolution |
derivation | ✓ conclusive |
fr:astra/kepler-equant |
model | ✓ documented |
fr:astra/kepler-equant-unequal-motion |
derivation | ✓ conclusive |
fr:astra/kepler-fig-ch2-eccentric |
figure | ✓ labelled |
fr:astra/kepler-fig-ch2-epicycle |
figure | ✓ labelled |
fr:astra/kepler-fig-ch4 |
figure | ✓ labelled |
fr:astra/solar-eccentric |
model | ✓ documented |
fr:astra/solar-eccentric-epicyclic |
superposition | ✓ consistent |
fr:astra/solar-eccentric-epicyclic.equal-radii |
parametrisation | ✓ attested |
fr:astra/solar-eccentric-epicyclic.proportional |
parametrisation | ✓ attested |
fr:astra/solar-eccentric.battani |
parametrisation | ✗ well-formed |
fr:astra/solar-eccentric.ptolemy |
parametrisation | ✓ attested |
fr:astra/solar-epicyclic |
model | ✓ documented |
fr:astra/solar-equation-of-centre |
derivation | ✓ conclusive |
fr:astra/solar-equivalence-equal-radii |
derivation | ✓ conclusive |
fr:astra/solar-equivalence-proportional |
derivation | ✓ conclusive |
fr:astra/solar-fig-1-2 |
figure | ✓ labelled |
fr:astra/solar-fig-1-3 |
figure | ✓ labelled |
fr:astra/solar-fig-1-4 |
figure | ✓ labelled |
fr:astra/solar-fig-eccentric |
figure | ✓ labelled |
fr:astra/solar-fig-epicyclic |
figure | ✓ labelled |
fr:astra/solar-fig-equal-radii-steps |
figure | ✓ labelled |
fr:astra/solar-fig-equation |
figure | ✓ labelled |
fr:astra/solar-parameters-from-seasons |
derivation | ✓ conclusive |
solar-eccentric.battani is incomplete on purpose: the page gives al-Battānī no year and no epoch.
Models
fr:astra/solar-eccentric, drawn by fr:astra/solar-fig-eccentric.
fr:astra/solar-epicyclic, drawn by fr:astra/solar-fig-epicyclic with drawing values only: the pages give this form no values of its own.
Ptolemy’s solar model, epicyclic form
fr:astra/solar-epicyclic · model · astra/1
| the model of | Ptolemy |
| witness | fr:kepler-sec/Swerdlow2026/0049/p04 — the construction, p. 19 |
| witness | fr:kepler-sec/Swerdlow2026/0050/fig01 — Fig. 1.2 |
| this description by | Gerd Graßhoff |
| tradition | ptolemaic |
| frame | geocentric |
| check | documented (6/6: ✓ well-formed · ✓ constructible · ✓ astronomical · ✓ kinematic · ✓ consistent · ✓ documented) |
Parameters (symbols; values stand in parametrisations)
| symbol | unit | meaning | admissible |
|---|---|---|---|
R’ R' |
length | radius of the deferent, the circle about O | (1, 1000) |
r r |
length | radius of the epicycle | (0, R’ / 2) |
λA lamA |
angle | longitude of the apogee | any |
T T |
time | tropical year | (300, 400) |
κ̄0 kappa_0 |
angle | mean anomaly at t = 0 | any |
Kinematic variables (their values are the states)
| symbol | meaning | law |
|---|---|---|
κ̄ kappa_bar |
mean anomaly: the motion of C about O, and of the sun about C in the opposite sense | uniform, period T, epoch kappa_0 |
Construction
| object | kind | defined by | meaning |
|---|---|---|---|
O |
point | observer() |
earth |
def |
circle | circle(O, R') |
the deferent |
A' |
point | O + polar(R', lamA) |
apogee of the deferent |
B' |
point | O - polar(R', lamA) |
perigee of the deferent |
C |
point | O + polar(R', lamA + kappa_bar) |
centre of the epicycle |
epi |
circle | circle(C, r) |
the epicycle |
S' |
point | C + polar(r, lamA) |
sun |
Observables
| symbol | defined by | meaning |
|---|---|---|
λS lam |
direction(O, S') |
true longitude |
Stated propositions (hold in every sampled configuration; not derived)
| proposition | meaning | residual | |
|---|---|---|---|
| p1 | parallel(line(C, S'), line(O, A')) |
the radius of the epicycle stays parallel to the apsidal line | holds, 2e-15 over 3456 |
| p2 | collinear(A', O, B') |
the apsidal line passes through O | holds, 0e+00 over 3456 |
Figures: fr:astra/kepler-fig-ch2-epicycle, fr:astra/solar-fig-epicyclic
Model text
# Ptolemy's solar model in its epicyclic form: the pure geometry.
# Swerdlow 2026, p. 19 (scan page 49, paragraph 3): the centre C of the epicycle moves uniformly through κ̄ on a
# circle about the earth O; the sun moves on the epicycle "in the opposite direction through κ̄ about C from OC
# extended", so that the radius of the epicycle keeps the direction of the apsidal line.
model solar-epicyclic "Ptolemy's solar model, epicyclic form"
attributed "Ptolemy"
witness fr:kepler-sec/Swerdlow2026/0049/p04 "the construction, p. 19"
witness fr:kepler-sec/Swerdlow2026/0050/fig01 "Fig. 1.2"
author "Gerd Graßhoff"
tradition ptolemaic
frame geocentric
param R' : length "radius of the deferent, the circle about O" in (1, 1000)
param r : length "radius of the epicycle" in (0, R' / 2)
param lamA : angle "longitude of the apogee"
param T : time "tropical year" in (300, 400)
param kappa_0 : angle "mean anomaly at t = 0"
display lamA "λ_A"
display kappa_0 "κ̄_0"
variable kappa_bar : angle "mean anomaly: the motion of C about O, and of the sun about C in the opposite sense"
law kappa_bar = uniform(period: T, epoch: kappa_0) "C moves uniformly about O"
display kappa_bar "κ̄"
O = observer "earth"
def = circle(O, R') "the deferent"
A' = O + polar(R', lamA) "apogee of the deferent"
B' = O - polar(R', lamA) "perigee of the deferent"
C = O + polar(R', lamA + kappa_bar) "centre of the epicycle"
epi = circle(C, r) "the epicycle"
S' = C + polar(r, lamA) "sun"
body sun at S'
observe lam = direction(O, S') "true longitude"
display lam "λ_S"
proposition p1: parallel(line(C, S'), line(O, A')) "the radius of the epicycle stays parallel to the apsidal line"
proposition p2: collinear(A', O, B') "the apsidal line passes through O"Fig. 1.4: the same model, another figure
Fig. 1.4 is not another model. It is the eccentric form under Ptolemy’s values in another state (κ̄ = 108°), seen with the solstitial colure upright, with the longitudes counted from the vernal equinox.
Figure text
# Swerdlow 2026, Fig. 1.4 (p. 21, scan page 51): "Complete solar model for the annual inequality in its eccentric
# form." The same model as in Fig. 1.2, in another state, seen with the solstitial colure upright, with the
# longitudes counted from the vernal equinox. The eccentricity is exaggerated as in the printed figure.
figure solar-fig-1-4 of solar-eccentric "Ptolemy's solar model, eccentric form, with longitudes, anomalies and the equation"
witness fr:kepler-sec/Swerdlow2026/0051/fig02
author "Gerd Graßhoff"
with solar-eccentric.ptolemy
drawn with e = 17;30
at kappa_bar = 108°
up 90°
fit [ecc]
speed 30 s
label angle lamA at O from aries to A arrow
label angle kappa_bar at E from A to S arrow
label angle kappa at O from A to S arrow
label angle c at S from O to E
label angle c at O from lam to lam_bar
label length R on E S
label length e on O E
label segment line(A, B)
label arrow S on ecc
label direction lam from O dashed
label direction lam_bar from O dashed
label zodiac at O on eccGaps
- Almagest III 3–4 is not in PDAI on this machine; the attribution to Ptolemy and the loci “3.4; 3.13” are as Swerdlow’s page gives them.
- No epoch is given on the pages;
kappa_0 = 0°is a stated convention. - Models with an equant, the Maragha couples, the Copernican and Keplerian models are not written: their source pages are not located on this machine.
