I have studied now spherical trigonometry although I do not consider myself at all an expert yet. I am wondering is the first calculation the proof of some theorem? I am wondering how Vincenty arrived at this conclusion. Is it arc length that is being measured? Unfortunately I am inquiring on the details of this equation and how it functions. I am very curious about this function and it does not make sense to me yet.

The greek letter in Tod Hunters book for geodetic calculations looks a little different but I believe it's still sigma with Lambda being a distance in this case. Although it isn't really clear to me how lambda is supposed to fluctuate in Vincenty's calculations. It says in wikipedia to iterate from Lambda but it doesn't say how the next iteration should look like.