Why does the American form of government work?

In The Federalist Essay #10, I believe that Madison's main point is that factions are what holds together the government, but also what could destroy the government.  Madison argues that these factions, another way to say political parties, could destroy the government if one certain party gains too much power over the others, creating an imbalance and thus making the United States's strive for democracy useless.  However, Madison writes, "Liberty is to faction what air is to fire, an aliment without which it instantly expires. But it could not be less folly to abolish liberty, which is essential to political life, because it nourishes faction, than it would be to wish the annihilation of air, which is essential to animal life, because it imparts to fire its destructive agency".  Basically what Madison is trying to say is that without factions, the government itself would not be able to function.  This creates a problem.  If factions can destroy the government from within, but are necessary to keep democracy, what is the solution?  Madison's answer to the question is to simply have the government keep control over the factions.  Instead of having many factions, Madison believes that the government should narrow it down to just a few, and whichever faction has the majority is the one that maintains control over the government

In The Federalist Essay #51, Madison's main point is about the system of checks and balances, or separation of powers.  The purpose of separation of powers is so one branch in the government does not gain more control over the other ones, hence giving themselves more power over the government as a whole.  Madison's ideas of separation of powers comes from Montesquieu and uses his ideas when he writes, "The different governments will control each other, at the same time that each will be controlled by itself".  If all of the other branches over government control each other's power, it will prevent the other ones from gaining too much power.  However, it is possible for the branches to work together and boost one of the branches powers, right?  Madison thought this through and said that all of the branches should have, "as little agency as possible in the appointment of the members of the others".  In writing this, Madison is basically saying that all of the branches will control each other's power keeping everybody in check and maintaining democracy in the US.