Euler E320: Investigations into the construction of the new five- and six-lens telescopes and their further improvement | English translation | OMH v1.0

English translation of “Recherches sur la construction des nouvelles lunettes à 5 & 6 verres & leur perfection ultérieure” by Leonhard Euler (Eneström E320), titled “Investigations into the construction of the new five- and six-lens telescopes and their further improvement.” The historical accompanying notice and formula extract are distinguished from Euler’s memoir. 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 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: Mélanges de philosophie et de mathématique de la Société royale de Turin, vol. 3 (1766), second pagination, pp. 92–151; the accompanying historical editorial notice and formula extract, pp. 152–155, are separately translated. Plate 3 contains figures 1–3 and a starred diagram. Sources: https://scholarlycommons.pacific.edu/euler-works/320/; https://archive.org/details/mlangesdephilo03soci. The NHM/BHL volume is described as public domain; that source status is separate from rights in the new translation. The original historical text is public domain by age; no publication license is assigned here to the new translation or editorial notes.