Translation and exegesis of Leonhard Euler’s “De momentis virium respectu axis cuiuscunque inveniendis”

In this article, we carried out the English translation and the exegesis of the Leonhard Eulero’s work titled “De momentis virium respectu axis cuiuscunque inveniendis,” submitted in 1780 to the journal “Nova acta Academiae scientiarum imperialis petropolitanae” of Saint Petersburg, but appeared posthumously in 1793 (Eneström index E658). The original text, written in neo-Latin (the lingua franca of science and culture throughout Europe up to 18th century) and enriched by a few illustrations, is reported in the Appendix. We recognize that, in the work translated here, Euler was the first to represent, in Cartesian coordinates, the moment of any force with respect to an arbitrary axis: this is maybe the most important step needed to recognize the (pseudo-)vectorial nature of the moment of forces. Indeed, Euler proved that the moments can be composed exactly as the usual forces are, according to the parallelogram rule. The proof was developed following two parallel tracks: the geometric problem and the mechanical problem. In fact, as a preliminary result, Euler needs to obtain an analytical expression to compute, in the three-dimensional ambient space, the minimum distance between two skew lines, as the length of a straight segment resulting orthogonal to both these lines. The rigor and clarity of Euler are remarkable, endowed by an insatiable quest for mathematical expressions resulting in a perfect synthesis of simplicity and significance, and by a deep yearning, that sometimes becomes awareness, for statements having a lasting impact “in universal mechanics.”