A translation of L. Euler's "An easy method for calculating the motion of celestial bodies perturbed in any manner avoiding astronomical computations"

TL;DR

This paper translates Euler's work on a geometrical and mechanical method to simplify celestial motion calculations, particularly for the Moon, aiming to reduce complex astronomical computations in three-body problems.

Contribution

It provides a translation and clarification of Euler's original method, highlighting its analytical approach and limitations in solving lunar motion equations.

Findings

Euler's method involves eight coupled differential equations.

The approach reduces astronomical computations for the Moon.

Euler's method lacks practical guidance for solving the equations.

Abstract

This is a translation from Latin of E348 'Methodus facilis motus corporum coelestium utcunque perturbatos ad rationem calculi astronomici revocandi', in which Euler develops a method to alleviate the astronomical computations in a typical celestial three-body problem represented by Sun, Earth and Moon. In this work, Euler's approach consists of two parts: geometrical and mechanical. The geometrical part contains most of the analytical developments, in which Euler makes use of Cartesian and spherical trigonometry as well - the latter not always in a clear enough way. With few sketches to show the geometrical constructions envisaged by Euler - represented by several geometrical variables -, it is a hard to follow publication. The Translator, on trying to clear the way to the non-specialized reader, used the best of his abilities to add his own figures to the translation. In the latter part of the work, Euler particularizes his developments to the Moon, ending up with eight coupled differential equations for resolving the perturbed motion of this celestial body, which makes his claim of an "easy method" as being rather fallacious. Despite showing great analytical skills, Euler did not give indications on how this system of equations could be solved, which renders his efforts practically useless in the determination of the variations of the nodal line and inclination of the Moon's orbit.