Explain the formula for representation in the small and large republics ...

Prior to the creation of the US Constitution, the general view of representative government is that it only worked in very small-scale environments - essentially a city-state or smaller. Otherwise, the actual reflection of popular will declined in the representatives (ie, the difference between 1 representative per 100 citizens and 1 representative per 10,000 citizens), and generally the whole process became inefficient and self-destructive.

James Madison, however, was like 'no beezy'. He remarked that small republics were prone to faction and thus collapsed quickly. A large republic, he said, had so many interests and factions that none could rule, and thus they had to moderate and compromise.