More than 2,000 years ago, a philosopher in Athens looked at the young people around him and wrote down a complaint that could have been posted this morning. They "fall in love and quickly fall out of love," he noted, "changing often within a single day."

The philosopher was Aristotle, the observation is in the eighth book of the Nicomachean Ethics, and it is not really about romance. It is an example in a longer argument about what friendship is, and about why most of what goes by that name does not last.

Aristotle took the subject seriously. He opens the book by saying that "without friends no one would choose to live, though he had all other goods".

The part that has lasted is a sorting exercise. "There are therefore three kinds of friendship," he writes, "equal in number to the things that are lovable." People love what is useful, what is pleasant, and what is good. Each produces a different kind of friend, and the three behave very differently when something changes.

The friend who is useful

The first kind is built on what each person gets. In W. D. Ross's translation: "those who love each other for their utility do not love each other for themselves but in virtue of some good which they get from each other."

Perhaps, this might be the colleague who covers your shifts and whose shifts you cover. The neighbor with the ladder. The contact who sends work your way.

Aristotle is precise about their failure mode. "Such friendships, then, are easily dissolved, if the parties do not remain like themselves," he writes, and then, more bluntly: "when the motive of the friendship is done away, the friendship is dissolved, inasmuch as it existed only for the ends in question."

Change jobs, move house, stop needing the ladder, and the friendship goes with the need. He adds an observation that is less flattering to his elders than to the young: this kind "seems to exist chiefly between old people (for at that age people pursue not the pleasant but the useful)."

The friend who is fun

The second kind is built on enjoyment. These are the people you want at the table, whose company is its own reward, and in Aristotle's account they are the characteristic friendships of youth: "the friendship of young people seems to aim at pleasure." The line about falling in and out of love within a day comes here, as a symptom. What is pleasant changes as people change, and a friendship based on it changes at the same speed.

Both of the first two kinds love the other person for something the other person provides. The friend is loved not as who they are but as a source of something, and when the source dries up there is nothing left to be loved.

The friend who is good

The third kind is the one Aristotle calls perfect, or complete. "Perfect friendship is the friendship of men who are good, and alike in virtue; for these wish well alike to each other qua good."

Each wishes the other well for the other's own sake, not for anything that comes back. These friendships, he notes, happen to be useful and pleasant as well, because good people are useful and pleasant to each other, but that is a side effect rather than the basis.

Two things are what make this kind rare. The first is time. "Such friendship requires time and familiarity," he writes, and quotes a proverb: "men cannot know each other till they have 'eaten salt together.'" As he puts it, "a wish for friendship may arise quickly, but friendship does not." The second is a limit on numbers. "One cannot be a friend to many people in the sense of having friendship of the perfect type with them," because the time cannot be spread that far. Aristotle expected such friendships to be infrequent for a simple reason: "for such men are rare."

How to tell which kind you have

Aristotle's sorting gives a sort of practical test, and it is not about how the friendship feels. It is about what would happen if the thing that brought you together disappeared. If you left the job, moved away, lost the thing that made you useful or the circumstances that made you fun, would the friendship survive the change?

Utility and pleasure friendships, on his account, mostly do not. The friendship of the good survives it, because what was loved was the person, and the person is still there.

The other half of the test is what the friendship has cost in time, and here a modern study puts numbers to Aristotle's idea. In 2018 Jeffrey Hall, a communication studies professor at the University of Kansas, published two studies in the Journal of Social and Personal Relationships on how many hours of contact it takes to move through the stages of friendship. The first surveyed 355 adults who had moved in the previous six months and were looking for new friends. The second followed 112 first-year students over their first weeks at university. Combined, as the university's account of the research reports, it took "between 40 and 60 hours to form a casual friendship, 80-100 hours to transition to being a friend and more than 200 hours together to become good friends."

Not all hours counted equally. Time spent hanging out, joking around and doing things for their own sake moved a friendship along; hours spent working side by side counted for much less.

"We have to put that time in," Hall said. "You can't snap your fingers and make a friend." And the time was not evenly spread: "When people transition between stages, they'll double or triple the amount of time they spend with that other person in three weeks' time." Some of his first-year students, he said, had spent a third of all their waking hours in a month with one good friend.

The two-hundred-hour mark is a rough average from two studies, not a threshold anyone should count toward, and Hall's own advice was simpler than the numbers. "You can't make people spend time with you, but you can invite them," he said. "Make it a priority to spend time with potential friends."

The university's article opened with a line it attributed to Aristotle, in a looser translation than Ross's: "Wishing to be friends is quick work, but friendship is a slow ripening fruit." The three kinds, in the end, come down to that. The first two ripen fast and fall. The third takes salt, and time.