https://famulus.lettre.ai/astra/lettre/2026-11-10T00:00:00Z https://famulus.lettre.ai/astra/lettre/2/2026-11-10T00:00:00ZGerd Graßhoff

A historical astronomical model is a geometrical construction with a kinematic variable. Texts about such models do four things at once: they describe a geometry, give it numbers, show it in one position, and reason about it. We present astra/1, a small language that writes each of the four as a text of its own kind. A model is the pure geometry: parameter symbols, a kinematic variable, a construction without coordinates, observables; it holds no values. A parametrisation gives the parameters values, each with the page it comes from; one model has many. A state is a value of the kinematic variable; model, parametrisation and state together give a configuration. A superposition places several models in one plane and identifies their inputs. A derivation is the operation that follows construction: hypotheses, auxiliary constructions, propositions derived from named premises by named rules, quantities computed or determined, a conclusion. Four questions are asked of a derivation: whether it is grounded, whether every proposition is true, whether every step is an instance of the rule it names with its configuration conditions holding, and whether the conclusion rests on what is cited. Every text names three roles that must not be confused: whose model it is, which text's use of it is analysed, and who wrote the description. We describe Ptolemy's solar model, in its eccentric and its epicyclic form, as Noel Swerdlow uses it in The Renaissance of Astronomy (2026). The diagram that shows both forms is a superposition, and their equivalence is the conclusion of a derivation under hypotheses on the radii: equivalence in position when the radii are equal, by Elements I.28 and I.33, and in direction when they are proportional, by VI.6 and I.29. The parallelogram argument does not apply at the apsides, where the parallelogram collapses; the derivation therefore argues by three cases. Ptolemy's reported eccentricity and apogee follow from his rounded arcs of mean motion, not from the day counts of the seasons.

https://famulus.lettre.ai/astra/lettre/1/2026-11-10T00:00:00ZGerd Graßhoff

Astronomical texts from Ptolemy to Euler argue geometrically about models: a figure, a few sentences, and inferences marked by little more than "therefore". This paper gathers what is needed to reconstruct such arguments and to check them. It distinguishes what an argument is about (a model, as pure geometry; values for its parameters; a state) from what it says (propositions) and what it does (operations: to suppose, to construct, to derive, to distinguish cases, to compute, to conclude). It says what licenses a step: an instance of a named proposition, an equality by a Common Notion, a fact read off a construction, a calculation, an expansion in a small ratio. It treats what a figure supplies without saying so — that points are distinct, lie on one side of a line, form a triangle — and what follows when the figure degenerates: cases. It treats hypotheses, how to find out whether an argument needs them, and how to detect one used tacitly; equivalence and identity of models; exact and approximate steps; quantities computed from a model and parameters determined from observations. It fixes a vocabulary for soundness, gives a format in which a reconstructed argument can be read at three depths, a procedure in six steps, and a table of rules with Euclid's wording and its place. Every notion is shown on one of three cases: Ptolemy's solar model in Swerdlow's exposition, Kepler on the equant, and Euler on the tides.