After studying a number of real and theoretical city-states' constitutions, Aristotle classified them according to various criteria. On one side stand the true (or good) constitutions, which are considered such because they aim for the common good, and on the other side the perverted (or deviant) ones, considered such because they aim for the well being of only a part of the city. The constitutions are then sorted according to the "number" of those who participate to the magistracies: one, a few, or many. Aristotle's sixfold classification is slightly different from the one found in The Statesman by Plato. The diagram above illustrates Aristotle's classification. Moreover, following Plato's vague ideas, he developed a coherent theory of integrating various forms of power into a so-called mixed state:
It is … constitutional to take … from oligarchy that offices are to be elected, and from democracy that this is not to be on a property-qualification. This then is the mode of the mixture; and the mark of a good mixture of democracy and oligarchy is when it is possible to speak of the same constitution as a democracy and as an oligarchy.— Aristotle. Politics, Book 4, 1294b.10–18
To illustrate this approach, Aristotle proposed a first-of-its-kind mathematical model of voting, albeit textually described, where the democratic principle of "one voter–one vote" is combined with the oligarchic "merit-weighted voting"; for relevant quotes and their translation into mathematical formulas see (Tangian 2020).