Euler’s Argument for the Tide-Raising Force, Reconstructed
One Segment of a Source, Passage by Passage, beside Its Formal Reconstruction
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.
history of astronomy, Euler, tides, geometrical reasoning, argument reconstruction, source edition
1 The segment
In chapter II of his prize essay on the tides (Euler 1996) Euler asks by what force the sun or the moon tends to move a particle of the earth relative to the earth’s centre. He answers on seven pages, 33 to 39 of the volume of the Opera omnia: a principle at the end of §22, a figure in §24, the force resolved exactly in §§25 and 26, expanded in a small ratio in §27, and read for its consequences in §§28 and 29. An earlier paper treated this argument briefly as one of two cases (Graßhoff 2026c). Here it is the only subject, and the aim is different: to set Euler’s own words beside the reconstruction, passage by passage, so that a reader can see for every sentence what has been made of it.
The method is that of the series (Graßhoff 2026b, 2026a). The model is written out: the points of Euler’s figure with their constructions, the lengths as symbols, the forces as quantities defined on them. The argument is written as steps, one proposition each, with the premises it is drawn from and the rule that licenses it. A program checks whether every premise is there, whether every proposition holds in the model for every admissible value and position, whether every step is an instance of its rule, and whether the conclusion rests only on what is cited.
The figure is drawn here from the model, after Euler’s Fig. 3: the earth as a circle about C, the luminary S, the particle M with its perpendicular MP on CS. It moves: the angle φ at C, which fixes the place of M, runs through a full turn. The distance of S is a drawing value, as it is in the printed figure.
fr:astra/euler-fig-3 draws fr:astra/euler-tide-force-resolution · astra/12 How to read the passages
The segment is divided into twelve passages. A passage is a stretch of Euler’s text that does one thing: states the principle, sets up the figure, gives a formula, draws a consequence. Each passage has three parts.
The source. Euler’s text, in Latin as the edition transcribes it, or in English. A mark such as |35| shows where a page of the Opera omnia begins. The English is the edition’s stored page translation, a machine translation; it was compared with the Latin for this paper and is a working translation, not a published one. Three wordings were corrected, all three at a page break, where a translation made page by page had broken a sentence in two.
The reconstruction. The step or steps of the argument that say what the passage says. A step has a number, a claim in words, the proposition in the notation of the figure, the steps or constructions it is drawn from, the rule, and a mark: ✓ an instance of the rule named, ◦ read off the construction, ≈ by calculation or by expansion. Where a passage is not a step, the reconstruction says what it is instead: a definition in the model, or a part of the text that is not reconstructed.
The notes. What the reconstruction supplies that Euler leaves unsaid, where it departs from his route, and what the check found.
In this version the passages are interactive. Latin · English switches the source. Overview · Steps · Full sets how much of the reconstruction is shown; every step also opens by itself. Notes adds the third column. Selecting a step shows in the figure what it speaks of. Open in the edition opens the page of the Opera omnia in the edition’s viewer, in a separate window.
3 The passages
4 What the reconstruction shows
The principle is a definition. The sentence at the end of §22 is not a premise among others. It fixes what the argument is about: the force on the particle less the force on the centre. In the model it is the definition of the two quantities observed, along MC and along MR, and every later step is a statement about them.
The exact components are confirmed, by another route than Euler’s. §26 reaches the two components through a perpendicular QV and the similar triangles QVM and MPC. The reconstruction does not rebuild this. It states that Euler’s closed expressions equal the quantities defined, and the check confirms it in every sampled configuration. This is the widest gap between the source and the reconstruction, and the notes say so at the passage.
The expansion has an order, and the order can be tested. §27 keeps first two terms and then one. A step of this kind is not an equality. It is checked by making the ratio of the particle’s distance to the luminary’s distance small, a hundredth, a thousandth, a ten-thousandth, and requiring the relative difference to fall like the stated power of the ratio.
An emendation of the editor is wrong. For the component along MR the Opera omnia print 4xx + yy in the term of second order, with the note “Editio princeps et Manuscriptus: (4xx − yy)/2a loco (4xx + yy)/2a. Correxit EJA.” (Figure 1). The reconstruction follows the first edition and the manuscript. Their reading follows from Euler’s own substitution three lines above, and it is of second order: its relative difference from the exact component is 1.8 · 10⁻⁴, 1.8 · 10⁻⁶ and 1.8 · 10⁻⁸ at the three ratios. The emended reading is of first order only: 2.0 · 10⁻², 2.0 · 10⁻³ and 2.0 · 10⁻⁴. The emendation does not touch Euler’s results, which use the first term. We have not seen the first edition or the manuscript; their reading is known to us from the editor’s note.
fr:enestroem/E57/0018): the component along MR with the emended sign, the first-order formulae, and note 5.
A rounded angle. §28 gives the angle whose tangent is √2 as 54°45′ “circiter”. To the nearest minute it is 54°44′.
What is not reconstructed. The first resolution of §25, along CS and along MP with the diagonal MO, is left out: §26 replaces it, and no formula of §§26 to 29 cites it. Euler’s Fig. 4 is not drawn. The checks report the positions of the particle on the line CS as not covered by the construction of the perpendicular MP.
5 The exposition as a format
The page is built from one file that names, for every passage, where it begins and where it ends on a page of the edition, which steps reconstruct it, and the notes. Nothing of the source is typed into that file. The builder takes the Latin from the page of the edition and the English from the stored translation, and it refuses to build when a passage is not found exactly once, when a step of the argument stands beside no passage, or when a passage lies in no paragraph of the reference register. A correction of the translation is listed with its reason and shown as a correction. The reference of the source comes from a checked record.
The same file gives the interactive page and the print, so the two cannot drift apart. The procedure for another source is the same: choose the segment, write the model, write the argument, divide the text into passages, attach the steps, and annotate what remains.
6 Limits
The Latin is the edition’s transcription, not a new collation; one spelling of the transcription (Praetera) is kept as it stands. The English is a machine translation compared with the Latin by the author, not a scholarly translation. The reconstruction covers what Euler argues in these sections, not what he presupposes from the sections before: that the earth may be treated as if collected in its centre (the paragraph before §22), and the form of the force. The emendation is judged on the argument alone, without the first edition or the manuscript.
Availability
The exposition is expositions/euler-e57-tide.yaml in the Famulus workspace famulus-astronomica; the model, the derivation, the tested emendation and the figure are in examples/euler/. The page element is <astra-exposition>. Euler is cited by the pages of the Famulus collection enestroem, document E57; page n of the collection is page n + 19 of Opera omnia II 31.
Publication
Famulus Lettre · Euler, No. 1. Published on 10 November 2026 by OpenScienceTechnology GmbH, Duisburg; version 1.0. Editor of the series: Gerd Graßhoff.
How to cite. Graßhoff, Gerd. Euler’s Argument for the Tide-Raising Force, Reconstructed: One Segment of a Source, Passage by Passage, beside Its Formal Reconstruction. Famulus Lettre · Euler, No. 1. Duisburg: OpenScienceTechnology, 2026. Version 1.0. https://famulus.lettre.ai/euler/lettre/1/
Licence. © 2026 Gerd Graßhoff. This issue is published under the licence Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0): it may be copied and passed on unchanged for non-commercial purposes when the author and the source are named. Reproductions from the sources cited are not covered by this licence.
Versions. 1.0 (10 November 2026): First publication. A published version is not changed; a correction appears as a new version under the same number and address.
Format. This issue is a Famulus Notebook. Open the notebook to follow every segment to its source, or read the PDF of this issue.
