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Herbert,

I am respectful with Emile Lemoine who published as a pioneer on 1873, and with Antoine Yvon-Villarceau who published more extensively on 1877.

"Nouvelle Navigation Astronomique - I Théorie" is a visionary work and I am used to teach that Yvon-Villarceau is the father of modern navigation: he was the first one to settle  the fix problem in cartesian coordinates and to show that it can be solved as a linear (Kramer) system. See Chapter I, N°5 in this book.

I know what Yvon-Villarceau wrote in N°12, N°13 and in note IX of this same book. Impressive indeed, from somebody living in XIXth century !

But I have observed that some deviation from analytics may occur when the position triangle -or more generally the polygon- is going to limit conditions. For instance, when the triangle has 2 angles growing significantly, tending to Pi/2 (or when a high DOP is anticipated). That the reason why I write "should be" in my comment.

I propose to bring more information, if possible, within a week. Now, I am busy elsewhere.

Best Regards,

Yves