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

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.