Two Geometrical Arguments Reconstructed

Euler on the tide-raising force · the equivalence of eccentric and epicycle in Swerdlow’s exposition of Ptolemy

Gerd Graßhoff

Humboldt-Universität zu Berlin · Deutsches Museum, München

2026-11-10

One method, two cases

The model written out — every object with construction and meaning
The argument as steps — premises and rule for each
A program checks: grounded · true · justified · conclusive

	Euler 1740	Ptolemy’s two forms
about	forces on a particle	two kinematic models
conclusion	approximate, to a stated order	exact
the check found	an editor’s emendation that is wrong	two cases the figure leaves out

Euler: the force that raises the tides

“si a vi acceleratrice illam particulam sollicitante auferatur vis acceleratrix, quam centrum Terrae patitur”

E57 §22

Centre C, luminary S at distance a, particle M at CP = x, PM = y. One angle varies: ACM.

The quantities argued about: force on M less force on C, along MC and along MR.

Euler’s argument

An emendation, tested

Opera omnia II 31, p. 37, note 5: “Editio princeps et Manuscriptus: (4xx − yy)/2a loco (4xx + yy)/2a. Correxit EJA.”

Which sign is right?

Relative difference from Euler’s own exact component, at ρ/a = 10⁻², 10⁻³, 10⁻⁴:

reading				order
4xx − yy (first edition, manuscript)	1.8 · 10⁻⁴	1.8 · 10⁻⁶	1.8 · 10⁻⁸	second
4xx + yy (Opera omnia, emended)	2.0 · 10⁻²	2.0 · 10⁻³	2.0 · 10⁻⁴	first only

Euler’s sign was right. The emendation introduces an error. His results use the first term only and are untouched.

The first edition and the manuscript are known to us through the editor’s note.

Eccentric and epicycle

“To construct the epicyclic model, complete parallelogram OESC and let C be the center of an epicycle of radius r = e … on a concentric circle about O of radius R′ = R”

Swerdlow 2026, p. 19

The model is Ptolemy’s. The diagram is a superposition of two models. That the two suns are one point is not drawn in — it is argued.

The argument

Three cases, one figure

Off the apsidal line: the parallelogram (Elements I.28, I.33). At the apogee and at the perigee: lengths added and subtracted along one line.

What the checks say

All four checks pass. 26 steps: 16 instances of rules or Common Notion 1, 8 read off the constructions, 2 calculations
Both hypotheses are needed — R′ = R and r = e
None is used tacitly
The equivalence is in position; with R′/r = R/e only in direction
Equivalent models are not the same model