At the entrance to a large city, Zarathustra encounters a foaming fool called "Zarathustra's ape," who has learned to copy much of what Zarathustra says. He warns Zarathustra not to enter the city because it is full of small people and small minds. Zarathustra stops his tirade, saying that the ape despises these people for all the wrong reasons. He despises because he resents the people for not flattering him enough, whereas Zarathustra despises out of love for what these people could be. Zarathustra suggests that this fool should leave the city if he hates it so much: "where one can no longer love, there one should pass by."
Zarathustra finds to his dismay that many of his disciples have turned to God. They found it more comforting to have faith than to struggle forward alone. Zarathustra suggests that when the old gods died, they died from laughter at the God who said, "There is one God. Thou shalt have no other God before me!"
Zarathustra returns to his home in the mountains and delights in his solitude. He remarks on how peculiar humans are, that they talk but say nothing, and that the "good" among them are the most spiteful.
We first encounter the eternal recurrence in the chapter "On the Vision and the Riddle," and the rest of Part III deals with Zarathustra's failed attempt to come to terms with the full consequences of this doctrine. On a very basic level, the doctrine of the eternal recurrence is simple. Events do not happen once; they recur an infinite number of times, so that every event in the present has already happened an infinite number of times in the past and will recur an infinite number of times in the future. However, placing this doctrine in the context of Nietzsche's philosophy and explaining its importance might prove a little trickier.
First, we might want to question the scientific validity of this doctrine, and also ask whether scientific validity has anything to do with it. As presented in "On the Vision and the Riddle," the doctrine of the eternal recurrence seems to be based on the claim that if time is infinite, all events must recur at some time. This claim is mathematically unsound. To follow an example by Georg Simmel, we can imagine three wheels lined up together on an axis, with a mark at the top of each wheel to show that they are lined up. If the three wheels start rotating, the first at one revolution per second, the second at two revolutions per second, and the third at one/¹ revolutions per second, the three marks on the three wheels will never again line up with one another, even given an infinite amount of time.
While Walter Kaufmann provides evidence that Nietzsche sought some kind of scientific validation for this doctrine, Gilles Deleuze suggests that the eternal recurrence goes deeper than a simple mathematical assertion. One of Nietzsche's fundamental claims is that the universe is in a state of flux and that there are no absolutes, no constants. Nothing is permanent. If everything is governed by the will to power, and the will to power drives everything to change itself and to overcome itself, nothing will remain fixed. According to Deleuze, the eternal recurrence is the full expression of what he calls "the being of becoming." Everything, in Deleuze's reading, is in a state of becoming, and it always has been. If there were a final state of being that things were moving toward, they would have reached it long ago, and if there were an initial state of being that things were moving from, they would never have left it. The only constant in the universe is becoming, or change.