https://famulus.lettre.ai/kepler/lettre/2026-11-10T00:00:00Z 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.