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Schrijver explains Monge's reasoning in On the history of combinatorial optimization (till 1960):

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 Appell [1928]: "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 The strange case of Paul Appell’s last memoir on Monge’s problem: “sur les déblais et remblais”

Schrijver explains Monge's reasoning in On the history of combinatorial optimization (till 1960):

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 Appell [1928]: "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."

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 Appell [1928]: "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."

source: Combinatorial Optimization: Polyhedra and Efficiency, Volume 1, pages 292-293.

See also The strange case of Paul Appell’s last memoir on Monge’s problem: “sur les déblais et remblais”

Schrijver explains Monge's reasoning in On the history of combinatorial optimization (till 1960):

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 Appell [1928]: "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 The strange case of Paul Appell’s last memoir on Monge’s problem: “sur les déblais et remblais”

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 Appell [1928]: "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."

source: Combinatorial Optimization: Polyhedra and Efficiency, Volume 1, pages 292-293.

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 Appell [1928]: "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."

source: Combinatorial Optimization: Polyhedra and Efficiency, Volume 1, pages 292-293.

See also The strange case of Paul Appell’s last memoir on Monge’s problem: “sur les déblais et remblais”