NavList:

Yvon, you wrote:
"For the little story, I came across the method of lunar distances while trying to locate the time when navigation was done with a chronometer without using the Sumner line"

And just for general background, it may be worth knowing that the significance of the "Sumner line" is almost always re-told in a completely anachronistic fashion. Celestial lines of position (known for a few decades as "Sumner lines") did not become popular until the early 20th century and did not really dominate marine navigation until the 1940s. Most short histories [the sort of thing found in introductions to navigation manuals and in brief popular accounts, the type that I call "cartoon histories"] would have us believe that Sumner lines were instantly successful and revolutionized navigation immediately after Sumner's little book was published. This is emphatically not true. The proof is in the primary sources: the logbooks, the navigation notes scratched in the margins of other books, the actual work of navigators. The Sumner Line seems to have intrigued some navigators and nautical astronomers, but among practical navigators? It was greeted with indifference.

You wrote:
"To come back to John L.'s formula, I fully agree with you: it is normal that it run in connection with Chauvenet's method and this proves nothing more. It is however sure that John knew and used the Chauvenet method which, it seems to me, must have been quite easy to use at the time when calculations were made with logarithms."

Actually Chauvenet's methods for lunars were quite esoteric and not at all easy to use. Most importantly, they were too little, too late. There were numerous methods and tables which were genuinely easy to use, and the series methods, of which Letcher's little equation is an example, were among the most popular in the era when lunars were widely used at sea. To counter your contention that "it is... sure" that Letcher used Chauvenet's method, I would instead suggest that Letcher developed his clever little version of the standard series method because he found Chauvenet's method so arcane. I'm just positing one speculation against another here. We do know that Letcher was at least aware of Chauvenet's work on lunars, and he has a very brief, parenthetical reference to it in his book. 

You concluded:
"Thank you for the text of Mendoza that I will read with interest, and all the more easily as it is in French, my native language."

You're welcome! Have you gotten into it? It's quite an antique approach to spherical trigonometry, but there's a lot of fun it, of course. It displays very nicely, also, the fascination for the so-called "Douwes problem" for calculating latitude from two Sun altitudes and the time interval between them (that's the first half of the Menodza Rios article) among mathematicians at the time. It's unclear whether this was used more than rarely by real navigators at sea, but it certainly inspired the math people!

Frank Reed
Clockwork Mapping / ReedNavigation.com
Conanicut Island USA