Thus liberty, when out of all order and reason, passes into the worst form of servitude...
In the previous books Plato has described the ideal State; now he returns to the perverted or declining forms, on which he had lightly touched at the end of Book IV.

These he describes in a succession of parallels between the individuals and the States, tracing the origin of either in the State or individual which has preceded them.

He begins by asking the point at which he digressed; and is thus led shortly to recapitulate the substance of the three former books, which also contain a parallel of the philosopher and the State.
Of the first decline he gives no intelligible account; he would not have liked to admit the most probable causes of the fall of his ideal State, which to us would appear to be the impracticability of communism or the natural antagonism of the ruling and subject classes.

He throws a veil of mystery over the origin of the decline, which he attributes to ignorance of the law of population.

Of this law the famous geometrical figure or number is the expression.