Schrijver explains Monge's reasoning in <A HREF="https://homepages.cwi.nl/~lex/files/histco.pdf">On the history of combinatorial optimization (till 1960)</A>: 

> Monge [1784] described an interesting geometric method to solve this problem.
> Consider a line that is tangent to both areas, and move the molecule
> $m$ touched in the first area to the position $x$ touched in the
> second area, and repeat, until all earth has been transported. Monge's
> argument that this would be optimum is simple: if molecule $m$ would
> be moved to another position, then another molecule should be moved to
> position $x$, implying that the two routes traversed by these
> molecules cross, and that therefore a shorter assignment exists.
> 
> Although geometrically intuitive, the method is however not fully
> correct, as noted by <A HREF="http://www.numdam.org/article/MSM_1928__27__1_0.pdf">Appell [1928]:</A> "It is very easy to make the
> figure in such a way that the routes followed by the two particles of
> which Monge speaks, do not cross each other."

See also <A HREF="https://www.sciencedirect.com/science/article/pii/S0315086016300209">The strange case of Paul Appell’s last memoir on Monge’s problem: “sur les déblais et remblais”</A>