Adeimantus interrupts Socrates to point out that being a ruler sounds unpleasant. Since the ruler has no private wealth, he can never take a trip, keep a mistress, or do the things that people think make them happy. Socrates responds by reminding his friends that their goal in building this city is not to make any one group happy at the expense of any other group, but to make the city as a whole as happy as it can be. We cannot provide the guardians with the sort of happiness that would make them something other than guardians. He compares this case to the building of a statue. The most beautiful color in the world, he states matter-of-factly, is purple. So if our intention were to make the statue’s eyes as beautiful as possible, we would paint them purple. Since no human being actually has purple eyes this would detract from the beauty of the statue as a whole, so we do not paint the eyes purple. On the statue, as in the city, we must deal with each part appropriately, in order to make the situation best for the whole.
Socrates proceeds to address several topics regarding the lifestyle of the guardians. He tells the money-loving Adeimantus that there will be no wealth or poverty at all in the city since there will be no money. Adeimantus objects that a city without money cannot defend itself against invaders, but Socrates reminds Adeimantus that our city will have the best warriors and points out that any neighboring city would be happy to come to our aid if we promised them all the spoils of war. Socrates limits the size of the city, warning against it becoming so large that it can no longer be governed well under the current system. He suggests that guardians guard their own elementary education above all else, and that they share everything in common among them, including wives and children. He declares that the just city has no use for laws. If the education of guardians proceeds as planned, then guardians will be in a position to decide any points of policy that arise. Everything we think of as a matter of law can be left to the judgement of the properly educated rulers.
Socrates declares the just city complete. Since this city has been created to be the best city possible, we can be sure that it has all the virtues. In order to define these virtues, all we need to do is look into our city and identify them. So we will now look for each of the four virtues: wisdom, courage, moderation, and justice.
We find wisdom first. Wisdom lies with the guardians because of their knowledge of how the city should be run. If the guardians were not ruling, if it were a democracy, say, their virtue would not translate into the virtue of the city. But since they are in charge, their wisdom becomes the city’s virtue. Courage lies with the auxiliaries. It is only their courage that counts as a virtue of the city because they are the ones who must fight for the city. A courageous farmer, or even ruler, would do the city no good. Moderation and justice, in contrast to wisdom and courage, are spread out over the whole city. Moderation is identified with the agreement over who should rule the city, and justice, finally, is its complement—the principle of specialization, the law that all do the job to which they are best suited.
So now we have reached one of our two aims, at least partially. We have identified justice on a city-wide level. Our next task is to see if there is an analogous virtue in the case of the individual.
Socrates has at last provided a definition of justice. This definition bears strong resemblance to the two definitions of justice put forward in Book I. Cephalus ventured that justice was the honoring of legal obligations, while his son Polemarchus suggested that justice amounts to helping one’s friends and harming one’s enemies. These two definitions are linked by the imperative of rendering what is due, or giving to each what is appropriate. This same imperative finds variant expression in Plato’s definition of justice—justice as a political arrangement in which each person plays the appropriate role. What is due to each person is rendered all at once. Each is assigned the role in society that best suits their nature and that best serves society as a whole.
The definitions on this list are mostly helpful, but the term "Understanding" is incorrect and misleading. The highest grade of cognitive activity in Plato's fourfold epistemological scheme is, in Greek, "noesis." In English this should be called Intellection, Higher Reason, or simply Noesis. To call it Understanding badly confuses things because of the myriad unrelated meanings and senses of "understanding" in English.
Similarly, what the list above calls "Reason" would be better termed ratiocination, calculation, reasoning, or lower reason. To simply call this faculty Reason confounds two distinct faculties: ratiocination (dianoia) and Higher Reason (nous or Nous). Ratiocination is somewhat like the ability that animals have to think and plan. It is the Higher Reason that is associated with mans immortal soul, and on which basis, according to Plato, man may attain "likeness to God insofar as possible."
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Spouses in the republic are not drawn by lot, as is asserted in the summary for Book V. Rather, they are chosen so that 'bronze-souled' individuals will be paired off with other bronze-souled individuals, and gold-souled with other gold-souled. The idea of the lottery is a lie fabricated by the framers of the city to placate citizens.