Superposition

Swerdlow’s Fig. 1.2 does not show one model. It shows two — the eccentric and the epicyclic form of Ptolemy’s solar model — in one plane. That is a superposition:

# The two forms of Ptolemy's solar model in one plane: what Swerdlow's Figs. 1.2 and 1.3 draw.
# The superposition says only that both models are about the same earth, the same apsidal direction and the
# same mean motion. That the two suns coincide (Fig. 1.2) or lie on one ray (Fig. 1.3) is not said here: it is
# derived, under hypotheses on the radii, in solar-equivalence-equal-radii and solar-equivalence-proportional.
superposition solar-eccentric-epicyclic "Eccentric and epicyclic form of Ptolemy's solar model, superposed"
witness fr:kepler-sec/Swerdlow2026/0050/fig01 "Fig. 1.2"
witness fr:kepler-sec/Swerdlow2026/0051/fig01 "Fig. 1.3"
author "Gerd Graßhoff"
place solar-eccentric as Ecc
place solar-epicyclic as Epi
identify Ecc.O = Epi.O                 "one earth"
identify Ecc.lamA = Epi.lamA           "one apsidal direction"
identify Ecc.T = Epi.T                 "one year"
identify Ecc.kappa_0 = Epi.kappa_0     "one epoch"
identify Ecc.kappa_bar = Epi.kappa_bar "one mean anomaly"
display Epi.lam "λ'_S"
display Ecc.lam "λ_S"

Every name of a member gets the member’s prefix (Ecc.S, Epi.S'); the names the text identifies become one (O, lamA, kappa_bar). Identification says that the two models are about the same earth, the same apsidal direction, the same mean motion. It adds no geometry. Only inputs — parameters of one unit, kinematic variables — and the observer can be identified; a text that identifies two constructed points is refused:

identify Ecc.S = Epi.S': only parameters of one unit, kinematic variables and the observer can be identified;
that two constructed points coincide is a proposition to derive

The superposition is again a model: pure geometry over the parameters of both members. In it the two suns are two points. They coincide for some parameter values and not for others; nothing in the superposition says which.

Eccentric and epicyclic form of Ptolemy’s solar model, superposed

fr:astra/solar-eccentric-epicyclic · superposition · astra/1

witness	fr:kepler-sec/Swerdlow2026/0050/fig01 — Fig. 1.2
witness	fr:kepler-sec/Swerdlow2026/0051/fig01 — Fig. 1.3
this description by	Gerd Graßhoff
frame	geocentric
check	consistent (4/4: ✓ well-formed · ✓ constructible · ✓ superposed · ✓ consistent)

Members

prefix	model	whose
Ecc	fr:astra/solar-eccentric — Ptolemy’s solar model, eccentric form	Ptolemy
Epi	fr:astra/solar-epicyclic — Ptolemy’s solar model, epicyclic form	Ptolemy

Identified

• Ecc.O = Epi.O as O — one earth
• Ecc.lamA = Epi.lamA as lamA — one apsidal direction
• Ecc.T = Epi.T as T — one year
• Ecc.kappa_0 = Epi.kappa_0 as kappa_0 — one epoch
• Ecc.kappa_bar = Epi.kappa_bar as kappa_bar — one mean anomaly

Parameters (symbols; values stand in parametrisations)

symbol	unit	meaning	admissible
R Ecc.R	length	radius of the eccentric	(1, 1000)
e Ecc.e	length	eccentricity, the distance OE	(0, Ecc.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
R’ Epi.R'	length	radius of the deferent, the circle about O	(1, 1000)
r Epi.r	length	radius of the epicycle	(0, Epi.R’ / 2)

Kinematic variables (their values are the states)

symbol	meaning	law
κ̄ kappa_bar	mean anomaly about E, counted from the apogee	uniform, period T, epoch kappa_0

Construction

object	kind	defined by	meaning
O	point	observer()	earth
Ecc.E	point	O + polar(Ecc.e, lamA)	centre of the eccentric
Ecc.ecc	circle	circle(Ecc.E, Ecc.R)	the eccentric
Ecc.A	point	Ecc.E + polar(Ecc.R, lamA)	apogee
Ecc.B	point	Ecc.E - polar(Ecc.R, lamA)	perigee
Ecc.S	point	Ecc.E + polar(Ecc.R, lamA + kappa_bar)	sun
Epi.def	circle	circle(O, Epi.R')	the deferent
Epi.A'	point	O + polar(Epi.R', lamA)	apogee of the deferent
Epi.B'	point	O - polar(Epi.R', lamA)	perigee of the deferent
Epi.C	point	O + polar(Epi.R', lamA + kappa_bar)	centre of the epicycle
Epi.epi	circle	circle(Epi.C, Epi.r)	the epicycle
Epi.S'	point	Epi.C + polar(Epi.r, lamA)	sun

Observables

symbol	defined by	meaning
λ̄S Ecc.lam_bar	direction(Ecc.E, Ecc.S)	mean longitude
λS Ecc.lam	direction(O, Ecc.S)	true longitude
κ Ecc.kappa	Ecc.lam - lamA	true anomaly about O
c Ecc.c	Ecc.lam_bar - Ecc.lam	equation of centre, mean minus true longitude
λ’S Epi.lam	direction(O, Epi.S')	true longitude

Stated propositions (hold in every sampled configuration; not derived)

	proposition	meaning	residual
Ecc.p1	equal(Ecc.lam_bar, lamA + kappa_bar)	the mean longitude is the longitude of the apogee plus the mean anomaly	holds, 6e-14 over 3456
Ecc.p2	equal(Ecc.kappa, kappa_bar - Ecc.c)	the true anomaly is the mean anomaly less the equation	holds, 1e-13 over 3456
Ecc.p5	equal(Ecc.lam, Ecc.lam_bar - Ecc.c)	the true longitude is the mean longitude less the equation	holds, 6e-14 over 3456
Ecc.p3	collinear(Ecc.A, Ecc.E, Ecc.B)	the apsidal line passes through E	holds, 2e-16 over 3456
Ecc.p4	collinear(Ecc.A, O, Ecc.B)	and through O	holds, 2e-16 over 3456
Epi.p1	parallel(line(Epi.C, Epi.S'), line(O, Epi.A'))	the radius of the epicycle stays parallel to the apsidal line	holds, 3e-15 over 3456
Epi.p2	collinear(Epi.A', O, Epi.B')	the apsidal line passes through O	holds, 0e+00 over 3456

Parametrisations: fr:astra/solar-eccentric-epicyclic.equal-radii, fr:astra/solar-eccentric-epicyclic.proportional

Derivations: fr:astra/solar-equivalence-equal-radii, fr:astra/solar-equivalence-proportional

Figures: fr:astra/solar-fig-1-2, fr:astra/solar-fig-1-3

Notes and gaps

• parameters of one member only (free until a hypothesis or a parametrisation relates them): Ecc.R, Ecc.e, Epi.R’, Epi.r

Equivalence is a relation, derived

That the two forms are equivalent is a relation between the two models. It holds relative to hypotheses and to what is compared:

derivation	hypotheses	the models are equivalent in
solar-equivalence-equal-radii (Fig. 1.2)	R′ = R, r = e	position: the two suns are one point in every state
solar-equivalence-proportional (Fig. 1.3)	R′/r = R/e	direction: the sun is seen from the earth in the same direction; the distances differ

Each is the conclusion of a derivation in the superposition (Derivations), not something the figure asserts by drawing one point.

Identity is something else

Two texts are the same model when they are the same geometry up to renaming. diff tests that:

$ famulus-astra diff solar-eccentric solar-epicyclic
solar-eccentric ↔ solar-epicyclic: different geometries
  parameters
    − R
    − e
    + R'
    + r
  construction
    − E (point) = O + polar(e, lamA)
    − ecc (circle) = circle(E, R)
    …

The eccentric and the epicyclic form are different models that are equivalent under hypotheses. What stays different under those hypotheses is everything that is not compared: in the one the sun moves uniformly on a circle about E, in the other its motion is compounded of two.

Values and figures of a superposition

A parametrisation of a superposition takes over the values of a member’s parametrisation and adds its own:

# Ptolemy's values in the superposition, with the radii of Fig. 1.2: R' = R and r = e (p. 19).
parametrisation solar-eccentric-epicyclic.equal-radii of solar-eccentric-epicyclic "Ptolemy's values, R' = R and r = e"
attributed "Ptolemy"
witness fr:kepler-sec/Swerdlow2026/0049/p04
author "Gerd Graßhoff"
use Ecc = solar-eccentric.ptolemy
Epi.R' = Ecc.R   "R' = R"   source fr:kepler-sec/Swerdlow2026/0049/p04
Epi.r  = Ecc.e   "r = e"    source fr:kepler-sec/Swerdlow2026/0049/p04

Fig. 1.3: the same superposition under another parametrisation, R′/r = R/e. The two suns are two points on one ray from the earth.