A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Position-Finding
From: Brian Walton
Date: 2018 Sep 8, 23:15 -0700
Excellent simplified layout !
Long division to four figures is not really much more difficult than the long multiplication needed in your Doniol method, and its azimuth calculation.
The usefulness of a time sight, also called Longitude by Chronometer, is that the result can seen and plotted on any chart having that longitude, directly with dividers. No azimuth is required, but another calculation at a slightly different latitude will form, as you say, a Sumner line.
The all haversine formula, available since at least 1899, works just as well with versines. The (British) Reed’s Nautical Almanac has since 1932 advocated versines and longitude by chronometer to amateur yachtsmen, and is still published.
A longitude by chronometer can be crossed directly using dividers with a noon latitude. Vasco da Gama knew how to do that !