My question refers to p.307-335 of volume 8 of Gauss's werke, which are part of a section of volume 8 which deals with Gauss's applications of complex numbers for geometry. The pages mentioned are entitled "Pothenot's task and the square", and deal in a comprehensive way with the "Snellius–Pothenot problem". First of all, the length of his treatment suggests there is something interesting here; the fact that almost 30 pages of this volume are dedicated to this problem seems to indicate that Gauss considered this problem to be very important.
The "Snellius–Pothenot problem" is an early example of a problem of resection - determination of a observer position by observations of angles between lines of sight to known points. Like the triangulation method, this kind of methods is very important, for example, in surveying (geodesy) and astronomy (in satellite navigation, for example), so that partially answers the question why Gauss was occupied with this problem.
Therefore, my questions are:
- why the solution of this problem involves such complications? If i recall correctly, triangulation-like problems can be solved very easily by graphical method and also algebrically with a bit more effort. what am i missing here?
- Why Gauss dedicated lengthy computations to it? why did he attach importance to it?
- how exactly did he use complex numbers to solve this problem? what advantages do they offer in this case?
- From a more modern point of view, do those computations represent a significant idea?
Any usefull comment/good reference/answer, will be blessed!