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

One segment of a historical source is taken through its argument: the end of §22 and §§24 to 29 of Euler's Inquisitio physica in causam fluxus ac refluxus maris (Eneström 57), seven pages of Opera omnia II 31, in which Euler derives the force by which the sun or the moon tends to raise the tides. The segment is divided into twelve passages. Each passage is quoted in Latin and in English and set beside the step of the reconstruction that says the same thing explicitly, with its premises, the rule that licenses it, and the result of a check. Annotations stand in a third column. The texts are not typed: the Latin is taken from the page of the edition, the English from the edition's stored translation, and every passage links to its page in the edition. The reconstruction confirms Euler's exact components and his expansion. It follows the first edition and the manuscript against an emendation of the editor, which a test shows to be wrong. It marks where it does not follow Euler's route: the resolution by similar triangles in §26 is compared with the result, not rebuilt. The format, called an exposition, is general.

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

Five pages of Noel Swerdlow's posthumous The Renaissance of Astronomy (2026) set out Ptolemy's solar model: the length of the year, the eccentric and the epicyclic form and their equivalence, the equation of centre, Ptolemy's derivation of the parameters from the seasons, and the later values of al-Battānī, az-Zarqāl and the Alfonsine Tables. We read these pages deeply. Every assertion is quoted and the quotation verified against the transcription. Every assertion is then checked as what it is: a property of the construction as a proposition of a model written in the language astra/1, a statement that something follows as a step of a derivation with its rule, a number as sexagesimal arithmetic or as a quantity computed from the model, parameters found from observations as a determination through the model. Every result is shown with the crop of the page it checks. Of 45 assertions, 35 are confirmed, one is a historical judgement that arithmetic cannot reach, and nine differ from recomputation. Seven of the nine concern figures that Swerdlow reports from his sources and are facts about those sources: Ptolemy's rounded arcs of mean motion, al-Battānī's maximum equation, and Ptolemy's eccentricity and apogee, which follow from his rounded arcs and not from the day counts of the seasons. Two concern the page itself and are misprints: the sign rule for the equation of centre names the same condition in both of its clauses, and the advance of al-Battānī's apogee is printed as 16;37° where the two longitudes on the page differ by 16;47°. The derivation of the equivalence, written out and checked step by step, shows further that the parallelogram argument does not apply at the apsides. The reading is a record that runs, every item of which points back to its paragraph.

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

In chapter 4 of the first part of the Astronomia nova Kepler gives, in eight lines and one figure, the argument by which Copernicus charged Ptolemy's equant with making the heavens move unequally: about the centre of its circle the planet passes a lesser arc and a greater arc in equal times. We reconstruct this argument. First the model is made explicit: every point of Kepler's figure with its construction and its astronomical meaning, the one angle that varies, and what is assumed of the lengths. Then the argument is written as steps, each with the premises it uses and the proposition of Euclid that licenses it, none of which Kepler names. The reconstruction is checked by a program: every step is tested over the whole parameter space of the model and all its states, matched against the rule it cites, and tested for the hypotheses it needs. Three things come out that the passage does not say. Kepler argues for the position in which the angle at the equant is right; the central inequality holds in every position of the half revolution. The hypothesis that the equant lies as far above the centre as the eye lies below it, which Kepler states, is not needed for the conclusion. And the argument rests on a triangle that vanishes when the planet is at the apogee or the perigee. A coda treats chapter 2, where Kepler states the equivalence of the eccentric and the concentric with epicycle and leaves the proof to the figure: "Schema loquitur Geometrae". The paper is meant as a model for doing the same with other passages.

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.