Euler E389: Research and calculations on the true elliptical orbit of the comet of 1769 and its period | English translation | OMH v1.0

English translation of “Recherches et calculs sur la vraie orbite elliptique de la Comete de l’An. 1769. et son tems periodique” (Eneström E389), titled “Research and calculations on the true elliptical orbit of the comet of 1769 and its period.” Attribution: work executed under the direction of Leonhard Euler, by the efforts of Mr. Lexell. This PDF includes the original title and blank verso, text pp. 3–159, and both plates with thirteen figures. Prepared for Open Math Heritage with OpenAI-assisted translation, mathematical transcription and LaTeX typesetting. OMH v1.0 is a machine-assisted draft that has undergone a full, fresh independent AI-assisted technical audit of source coverage, translation, formulas, tables and figures where present, and rendered output. This is not a claim of human scholarly review or certification of the historical mathematics. Version 2 is reserved for a fully reviewed edition. Original publication: Saint Petersburg: Press of the Imperial Academy of Sciences, 1770, pp. [1]–159 and Plates I–II. Sources: https://scholarlycommons.pacific.edu/euler-works/389/; https://doi.org/10.3931/e-rara-2112. The ETH-Bibliothek Zürich witness, Rar 4270, explicitly carries the Public Domain Mark. Terminal digits of log a and log b on p. 88 are clipped; the edition marks visible readings with ellipses and cites complete values on p. 90 separately. The Euler Archive copy is the same clipped scan, not an independent corroboration. The historical original is public domain by age; no publication license is assigned here to the new translation or editorial notes.