The title of Wessel's paper directly refers to the 'analytical representation of direction'. That is, he, as a surveyor, was not interested in some 'foundational' issue, but rather with how to make complex numbers work for his profession.
Even a superficial examination of his text clearly indicates that he develops a discorse 'about lines' through which he can represent *the lines and the calculus he developed for them* using complex numbers, not the other way round.
By the time he read his memoir (1797), hardly any mathematician (with the exception of the exceptional ones) was interested in 'foundations'. Why would a surveyor be?
For many years I have been puzzled by the recurring insistence on saying that Wessel was the first one to present a work on the geometrical foundations/representation of complex numbers. He even mentions series!
The truest pearl I found in his memoir is in (his) 'paragraph' 3, where he states (for no further use!) something quite close to the 'linear independence' of length, width and depth (sorry if the words are not accordingly to the English translation; I hope the sense is still available).
all the best, Romulo
> There was no connection between Wessel and Argand that anyone knows about.