navstokgap

Euler, Methodus inveniendi, Additamentum II (1744)

Companion source

Source: Internet Archive, Smithsonian Libraries scan methodusinvenie00eule; local OCR excerpt .txt, lines 23944–24585 of the archive’s _djvu.txt, retrieved 2026-09-08. Metadata: Leonhard Euler, Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici latissimo sensu accepti (Lausanne and Geneva: Marc-Michel Bousquet, 1744), Eneström E065; Additamentum II, “De motu projectorum in medio non resistente, per methodum maximorum ac minimorum determinando”, pp. 311–320. Extraction: archive.org OCR of the 1744 printing. Long s is read as f, ligatures and Greek are lost, and displayed formulae are fragmentary. The Euler Archive’s per-section PDF E065h.pdf at scholarlycommons.pacific.edu returned an interstitial page on 2026-09-08 and was not retrieved. Rights: public domain.

Source digest

Section 1 opens with the claim that all effects of nature follow a law of maximum or minimum, so that the curves described by projected bodies must have such a property, and that the aggregate of all motions in the body should be the least. Section 2 states the quantity: the integral of mass times speed times element of arc, \(\int M v\,ds\), is a minimum on the actual trajectory. The first application is uniform gravity, where \(v^2=a+gx\) and the condition becomes \(\int ds\sqrt{a+gx}\) minimum, which yields the parabola.

For this project, this is the earliest published variational characterization of the constant-force trajectory of C001. Euler’s functional is Hamilton’s characteristic function at fixed energy; C001’s \(\Delta S\) is a fixed-duration comparison of the principal function. Maupertuis’s 1746 Berlin memoir acknowledges this Additamentum.

Passage anchors

Coverage and limits

Reading level passage for sections 1–2 and the uniform-gravity application, on OCR text; target for verification against the scan for every formula. The scan itself, about three hundred pages, was not stored.