Geometrical Reasoning in the History of Astronomy

Fundamentals: models, propositions, steps, rules, and what the figure supplies

Gerd Graßhoff

Humboldt-Universität zu Berlin · Deutsches Museum, München

2026-11-10

One sentence, one “ergo”

“DCE exterior aequet CBE, CEB interiores : ergo parte CEB ablata, residuus CBE vel DBE minor erit quam DCE”

Kepler, Astronomia nova, ch. 4 (KGW 3, p. 73)

Behind it: a figure with six letters, Elements I.32, a Common Notion, and the fact that C and D lie on one ray from B. None is named.

To reconstruct the argument is to say what it is about, what it says, and by what right it moves.

What an argument is about

A model is pure geometry: no values. A parametrisation gives values. A state fixes the moving variable. An argument is about the model — a figure shows one configuration.

What it says, what it does

Propositions — only they are true or false

PQ ∥ RS, PQ ⊥ RS, P = Q
a = b, a < b
a ≈ b to order n in a small ratio

Operations

given — a hypothesis on the parameters
suppose — one position
derive — from premises, by a rule
show … qed — a sub-argument
cases — covering every configuration
conclude

What licenses a step

mark	licence	example
✓	an instance of a named proposition	OE ∥ CS by Elements I.33
✓	equality, Common Notion 1	ES = R = R′ = OC
◦	read off the construction — holds without any hypothesis	ES = R
≈	calculation; expansion to a stated order	Euler’s first-order components
?	true, but not licensed: a gap

A proposition of Euclid enters the rule table only after its wording has been compared with the page of Heath.

What the figure supplies

I.33 needs a parallelogram. At the apogee and the perigee there is none: degenerate states. The remedy is a distinction of cases — the argument is longer than its figure.

Hypotheses

Is it needed? Lift it and check again.

Is it cited? If needed and not cited: a tacit premise.

Is it used at all? Kepler states BC = BA; no step of his argument needs it.

A supposition restricts the states, a hypothesis the parameters.

An argument at the depth “overview”

Kepler’s argument, every step

A vocabulary

true / false — of propositions
justified — of a step: its rule licenses it
sound — all true, all justified. A false premise makes an argument unsound, not its conclusion false
conclusive — sound, and resting only on what is cited
evidence, not proof — sampled configurations, rules matched as schemas

A procedure

Fix the passage — addressable paragraphs, quotations checked
Write the model — every object with construction and meaning
Separate hypotheses from geometry
Write the steps — premises and rule for every “therefore”
Run the checks — grounded · true · justified · conclusive
Read the report as a finding about the text