Representation, model, description

Three things are easily run together: the model, the texts and pictures that represent it, and the prose that describes and uses it.

The model

The model is not a text. It is the geometry: the function from parameter values and states to configurations, with what holds of it. Ptolemy’s eccentric solar model is the same model whether Ptolemy, Regiomontanus, Swerdlow or an astra text states it. That is why an astra model text carries three different names: whose model it is (attributed "Ptolemy"), the text whose use of the model is analysed (witness fr:kepler-sec/Swerdlow2026/…), and who wrote the description (author "Gerd Graßhoff"). Swerdlow uses and explains the model; he is not its author, and neither is the author of the astra text.

Representations

A representation is anything that determines the model: Ptolemy’s prose and figure; Swerdlow’s prose and his figures; the astra model text; the compiled list of operations; the sheet. Two representations represent the same model when they agree in every configuration — what diff tests as the same geometry up to renaming.

A figure represents less and more than the model: less, because it shows one state under one set of values; more, because it chooses a view and often other values than the attested ones. Swerdlow’s Fig. 1.4 and his Fig. 1.2 draw the same eccentric model in two states, two orientations and with an exaggerated eccentricity. In astra/1 they are two figure texts of one model.

What a diagram of two models is

Fig. 1.2 is the figure of a superposition: two models in one plane. It is drawn so that a parallelogram can be seen, and the parallelogram is the place of an argument. The diagram does not state the argument and does not state its hypotheses; the prose beside it does, in part. Three things are therefore distinguished:

• the superposition: the two models with one earth, one apsidal direction, one mean anomaly;
• the hypotheses under which something holds across them (R′ = R, r = e);
• the derivation that reaches the equivalence from the hypotheses, step by step.

The printed figure shows the first, is drawn under the second, and exhibits the scene of the third.

Equivalence is not identity

The eccentric and the epicyclic form are different models. Under hypotheses they are equivalent in what is compared — the position of the sun, or only its direction — and they stay different in everything else: in the one the sun moves uniformly on a circle about E, in the other its motion is compounded of two. Swerdlow notes that only Kepler drew a consequence from that difference (p. 20).

The describing text

Swerdlow’s prose refers to the model through names: “complete parallelogram OESC”, “r is always parallel to the apsidal line AB”. The astra texts make those names resolvable (fr:astra/solar-eccentric-epicyclic/Epi.C) and make what the prose asserts checkable, each assertion as what it is:

the prose	is checked as
states a property of the construction	a proposition of the model
says that something follows	a step of a derivation, with its rule
gives a number that follows from other numbers	arithmetic, or a quantity computed under a parametrisation
gives parameters found from observations	a determination through the model

The deep reading of Swerdlow’s pages does this sentence by sentence.

What one can do with models

operation	what it is	in astra/1
parametrise	give the symbols values	parametrisation
instantiate	fix a state	at <variable> = <angle>; the result is a configuration
depict	draw a configuration, or run through them	figure
superpose	place models in one plane, identify their inputs	superposition
derive	reason from hypotheses to a conclusion, by named rules	derivation
relate	state an equivalence under hypotheses	conclude equivalent(…)
compute, determine	a quantity from the model; parameters from observations	compute, determine
compare	identity of geometry; a table of values	diff
cite	refer to a text or any part of it	fr:astra/<text>/<object>