A more accurate treatment of the problem of drawing the shortest line on a surface

TL;DR

This paper revisits Euler's 1779 work on deriving equations for geodesics on a surface, aiming to clarify the calculus of variations steps involved in finding shortest lines.

Contribution

It provides a detailed analysis of Euler's original approach to geodesics, clarifying the calculus of variations methods used in deriving shortest paths on surfaces.

Findings

Euler's equations for geodesics are derived from the surface equation

Clarification of calculus of variations steps in geodesic problem

Historical insight into 18th-century differential geometry

Abstract

E727 in the Enestrom index. This is a translation from the Latin original "Accuratior evolutio problematis de linea brevissima in superficie quacunque ducenda" (1779). Given a surface $pdx+qdy+rdz=0$, Euler wants to develop equations that give the geodesics on this surface. I am new to the calculus of variations, so it is not clear to me what steps follow from results that are previously known (like the Euler-Lagrange equation in the calculations) and what steps follow from earlier in this paper. I would appreciate comments from any readers who are familiar with calculus of variations.