What ideas does Aristotle use to support his claim that the third type of friendship is “perfect”?

Aristotle makes it clear that the number of people with whom one can sustain the kind of relationship he calls a perfect friendship is quite small (IX.10). Even if one lived in a city populated entirely by perfectly virtuous citizens, the number with whom one could carry on a friendship of the perfect type would be at most a handful. For he thinks that this kind of friendship can exist only when one spends a great deal of time with the other person, participating in joint activities and engaging in mutually beneficial behavior; and one cannot cooperate on these close terms with every member of the political community.