Why does the American form of Government work?

Essay 10: "No man is allowed to be a judge in his own cause, because his interest would certainly bias his judgment, and, not improbably, corrupt his integrity. With equal, nay with greater reason, a body of men are unfit to be both judges and parties at the same time; yet what are many of the most important acts of legislation, but so many judicial determinations, not indeed concerning the rights of single persons, but concerning the rights of large bodies of citizens? And what are the different classes of legislators but advocates and parties to the causes which they determine? Is a law proposed concerning private debts? It is a question to which the creditors are parties on one side and the debtors on the other. Justice ought to hold the balance between them. Yet the parties are, and must be, themselves the judges; and the most numerous party, or, in other words, the most powerful faction must be expected to prevail. Shall domestic manufactures be encouraged, and in what degree, by restrictions on foreign manufactures? are questions which would be differently decided by the landed and the manufacturing classes, and probably by neither with a sole regard to justice and the public good. The apportionment of taxes on the various descriptions of property is an act which seems to require the most exact impartiality; yet there is, perhaps, no legislative act in which greater opportunity and temptation are given to a predominant party to trample on the rules of justice."

Madison advocates for a republic based on the fact that if each man had an equal say in the government of the country, nothing would get done because people would only be out for their own gain. He states that  "No man is allowed to be a judge in his own cause, because his interest would certainly bias his judgment, and, not improbably, corrupt his integrity," implying that even if progress could be made in terms of decisions many people would be treated unjustly as a result of the selfish actions of the individuals. In the republic system proposed by Madison in this essay the representatives from each faction would be held in check by the members of their parties, stating that "parties are, and must be, themselves the judges," allowing them to make decisions that benefit the overall good of the party and not just themselves. 

Essay 51: “In order to lay a due foundation for that separate and distinct exercise of the different powers of government, which to a certain extent is admitted on all hands to be essential to the preservation of liberty, it is evident that each department should have a will of its own; and consequently should be so constituted that the members of each should have as little agency as possible in the appointment of the members of the others. Were this principle rigorously adhered to, it would require that all the appointments for the supreme executive, legislative, and judiciary magistracies should be drawn from the same fountain of authority, the people, through channels having no communication whatever with one another.”

Madison's proposal for a system that maintains a balance of power shows how he is attempting to create a governmental system free from corruption or lopsided power. The creation of the three branches and the rules surrounding them would help to prevent one branch from becoming more powerful than the others and dominating the government. By implementing this system Madison clearly hopes to prevent the United States from being governed in a hierarchical fashion, similar to those still present in Europe at the time.