Political scientist here. If this answer is deemed inappropriate by mods, please remove it.

It's important to have in mind when reading Madison that before the US Constitution, democracy was commonly viewed in a negative light by political theorists and philosophers, and the term democracy was exclusively used to mean what today we may call direct democracy, or democracy in terms of Greek city-states like Athens. He is not thinking of any particular example, but instead is in dialogue with current ideas about democracy at the time

There's a lot to unpack in this text. At the time many thinkers feared democracy because of how succeptible it is to falling prey to demagogues and degenerating into mob rule. Some instances such as the trial of the Athenian admirals after the battle of Arginusae serve as illustrations to the potential for injustice in this form of government.

But at least since Aristotle, the philosophical discourse about democracies had largely been about this risk of devolving into chaos and lynching. Madison's main worry is with the "many" oppressing the minorities, which of course means the poor uprooting the propertied and well to do, but also means the majority shutting down the minorities. Here is where Madison introduces several arguments in favor of representation that are game changers in political theory.

You see, Madison in particular is heavily influenced by Montesquieu (who cited Aristotle's Politics in his L'esprit des lois) and the Federalist n. 10 is a direct confrontation with this thinker's ideas. Montesquieu innovated by considering democracy a form of republic, which for him meant that democracy depends on the Civic virtue as put forth for Aristotle (the virtue of governing and being governed well, i. e. respecting the rule of law and moderation). Montesquieu famously argued that republics cannot be very big because of their internal dynamics, but they could perhaps federate to cover large territories.

Madison on the other hand uses the term "Republic" to mean the opposite of "Democracy". This republic is not supposed to be democratic, but it is also not supposed to be an aristocracy (rule of the "best"). It's supposed to be a mixed system of these two. This mixture is something Aristotle himself advocates, but Madison proposes it in very novel terms for the time: the English idea of representation and Montesquieu's idea of branches of government. He wants a system that does not have the problems of a democracy, which are to him the emergence of sectarian factions and majority tyranny. He also believes republics can indeed be very big.

And to make all of this work he introduces three things: that the people elect Representatives (different from aristocratic elections, where you choose the best to take decisions and not someone to represent anyone else); that the people vote based on their interest (not virtue), having the vote act as a filter of candidates who are able to better deliberate according to the interest of electors; and multiple dimensions of government that hinder the expansion of any faction, which means the representative republic must be big so it has many many factions, many different Interests, and never one of them would be appealing to an overwhelming majority. This also includes the separation of three powers checking and balancing each other, as a way to control abuses based on interest, not depending on the virtues of the current titularies of each branch.

I'm sorry if this wasn't historical enough of an answer. But I think the "history of ideas" approach is a bit more useful this time. I can propose reading Aristotle's Politics, especially accompanied by Richard Kraut's excellent commentaries, that make the aristotelic work very comprehensible (it's just called Aristotle and published by Oxford U. Press.) Montesquieu's relevant chapters are the very first ones of Spirit of the Laws.

Others have mentioned cases like classical Greece or medieval Florence, just want to add a much more recent (to Madison) case of the Corsican Republic. This was surely on the founders' minds as well. For example, Alexander Hamilton (another Federalist author) was a member of the militia The Hearts of Oak which originally was named The Corsicans.

Although the failure of the Corsican Republic had more to do with regional geopolitics and less to do with democracy so it may not be the best example, but it was certainly short lived and violent in death.