My understanding of a mercury barometer is that the force exerted by air is equal to the force exerted by mercury in the emerged portion of the tube. The forces here are weight of air and weight of mercury.
But in the picture above, there are more air molecules involved in the left side experience than on the right side one. Still the balance is not changed and the same height of mercury is actually balanced by a variable weight of air.
I don't understand why air-mercury interface area (noted S1) and tube section area (S2) don't influence the height of the mercury column.
I suspect the answer is: "because the weight by unit of area (definition of the pressure) is the same in both cases". So what I don't understand is actually why pressures are taken into account rather than weights.
I'm looking for a didactic explanation.