The distinction, if there is one, between a democracy and a republic is contentious. One common way of thinking about this topic is to contrast democracy as direct rule by the people, unconstrained by any external or constitutional factors, with a republic as rule by representatives (so-called), whether elected or appointed, whose actions are subject to some form of constitutional constraint. If these representatives were to be elected by the people, then you would seem to have a representative democracy which is simultaneously a republic. In my Libertarian Anarchy, I’ve argued that so-called representative democracies are neither representative nor democratic being, in fact, merely elective oligarchies. The signers of the Constitution feared democracy as a form of political chaos, where government would be conducted at the whim of a volatile and fickle electorate.
In my Freedom’s Progress?, I wrote: “What will seem strange to modern eyes is that Athenian democracy was largely non-elective. The Council (βουλή) with 500 members, the Supreme Court (Ήλιαία) with six thousand members and roughly six hundred of the seven hundred magistracies (to be held only once by any one citizen) were filled by casting lots, the remaining magistracies, including the ten strategoi (the generals or diplomats) and the officials responsible for financial matters, were filled by election. The allotment of the non-technical responsibilities was intended to prevent the domination of governance roles by the rich or the well-born which, it was feared (with some justification) would happen under an elective system. As Aristotle remarked, ‘the appointment of magistrates by lot is thought to be democratical, and the election of them oligarchical…’ Since not every citizen participated in every aspect of government at all times, it would perhaps be something of an exaggeration to describe Athenian democracy as direct. David Van Reybrouck suggests that it might be described as a ‘non-electoral representative democracy’ or, indeed, given the use of lots, as an ‘aleatoric representational democracy.’” [see his Against Elections] As can be seen from Aristotle’s comment, appointment of officials by lot was thought to be democratic, whilst their appointment by election was thought to be undemocratic! Today, we tend to think of democracy and elections as being intrinsically connected so Aristotle’s comment seems counter-intuitive!]