NavList:

Antoine C., you wrote:
"with one exception being the [rhumbline] course initially published by Martins Caminos at 53°18.6' unless it is actually published for WGS84, and not for a spherical Earth as is the case for the immediately preceeding [great circle] course."

Well, that's the thing. It doesn't take WGS84 or Vincenty code or any of the other bullshit to incorporate the key feature of the ellipsoidal form of the globe. Navigators don't get this... And that's why I launched this discussion. The key issue here is the scale of the spacing of the latitude and longitude lines on the globe (primarily latitude; the convergence of longitude lines towards the poles is obvious and well-known and swamps other concerns). Parallels of latitude near the equator are closer together by just about 1 part in 150 compared to the spacing of the meridians of longitude. This is not due to some complicated geodesic analysis (e.g. Vincenty). It is a simple result of the way we define latitude and longitude. 

Over a century ago, the calculations and the tables for rhumblines were adjusted to incorporate thie ellipsoidal correction. Arguably (and I would argue), this was completely un-necessary given the practical purpose of those tables, but if you check modern "meridional parts" tables which show the relative scales of degrees of latitude to degrees of longitude, you will find that they match up with the ellipsoidal definition of latitude. And note that this is not something dependent on the exact, perfect specification of an ellipsoi. Whether it's the old Clarke ellipsoid of 1866 or some other antique set of ellipsoid parameters or the now universal WGS84 ellipsoid is immaterial. 

The strange result in this story is that the rhumbline course from A to C in the original scenario yields the correct sextant angle at the southern corner, point C. This is strange because photons do not carry compasses and surely are not bound by rhumblines! It works because we're dealing with a very small triangle. It's really just damn close to a pair of straight lines, A-to-B and A-to-C. But the rhumbline formulae and tables were "cleaned up" (as I say, over a century ago) to incorporate the geodetic latitude --the definition of latitude which respects the fundamental concept of astronomical latitude. 

 Antoine, you continued:
"I had never figured this out, because that is almost the first time I have been carefully comparing results on WGS84 against their spherical results counterparts for surface navigation, while I have always used WGS84 for Astronomical computations"

But think about it... You have always used WGS84 coordinates. Well, ok not always, but for the entirety of the 21st century at least. Our coordinates are always founded on the WGS84 ellipsoid, unless we are looking at data or publications from decades ago. As I say, this scenario is something that you could construct, on the ground (in the real world). And it doesn't just apply to that specific island setup at the mouth of the Amazon. Find yourself some targets for observation at ranges similar to the scenario in my original post at any latitude, calculate the sextant angles between targets using the plain latitudes and longitudes from GPS devices (or mapping intended to be used with GPS devices --meaning anything in the modern world), and you will discover comparably large errors in the angles [except at one latitude... anyone?].

You added:
"If we are to address this drill more in depth, then we need to work on the Geoid. EGM 08 gives us our [...]"

This really isn't relevant at all. Yes, it's true, you should assess the further significance of the three-dimensional aspect of this problem, but it's a tiny correction far below the primary issue here. So you should assess that significance, recognize that is negligible, and move on! 

Something to consider (just to get you thinking): what is the actual path followed by those photons in this scenario? They follow straight lines, and they would be unmeasurably different from Euclidean straight lines without an atmosphere. But what about refraction? The photon trajectories will "droop" downward by about one minute of arc in those six miles from A to B. Would that be worth worrying about? Or is that negligible, too?

Frank Reed