The dividing line between Book VIII and Book IX of Aristotle's Nicomachean Ethics seems to be arbitrary, since it occurs right in the middle of what one can think of as discussion of friendship and dispute. Some commentators have noted, though, that whereas Book VIII seems mostly concerned with friendship as a social phenomenon, Book IX seems mostly concerned with friendship insofar as we find it in the individual.
I have already looked at Book VIII. You can read Nicomachean Ethics Book IX online in English at the Perseus Project.
Where A and B are friends, but do not have the same aim in the friendship, preservation of the equality required for friendship depends on being able to maintain some kind of proportion in what people get in the friendship. People fight and break up when they are not getting what they think is important to the friendship. This is why friendships of utility, however pedestrian they may seem in comparison to pleasure-friendships like erotic relationships, tend to work reasonably well despite dissimilarity in aims. A shoemaker and a person buying shoes have dissimilar aims in the friendship (remember, all our social interactions are friendships of one kind or another on Aristotle's account), but there is an objective way to measure whether both sides are getting what they want (namely, contract and money), and, whatever its limits, this often works quite well: I pay you, you give me shoes, and as long as we're both trying to make it work, we both get what we want and neither feels put upon. Contrast this with erotic relationships, in which proportion is extremely difficult to maintain, thus often leading people to complain that they were promised everything and then got nothing, and this can be the case even if both parties are genuinely trying to make things work (i.e., haven't already started dissolving the friendship). This is especially like to happen, Aristotle notes, when friends love each other for their qualities rather than for themselves, which is another reason why virtue-friendships are the most stable friendships of all, since such friendships exist for their own sake.
Aristotle considers the question of whose judgment matters more when deciding the value of a benefit. He uses the example of teaching. Protagoras is said to have taught, and then left it to his students to decide what they thought the teaching worth; but others think it better if everyone pays the same. Sophists in general, on the other hand, take money beforehand on extravagant promises of success and then don't do what they were supposed to do, leading to complaints. But we also see that there are lots of cases where there's no definite contract, and in such cases we tend to think that it's unreasonable to complain about having received a benefit from a person who did not have to give it in the first place, even if it's not the benefit you would have preferred. In cases of piety -- which we have with the gods, or our parents, or people who teach us philosophy -- we are given an extraordinary benefit we cannot repay, and so maintaining equality with your philosophy teacher has to be understood in terms of just returning what you rationally can. In cases of clear contract or conditional benefit, on the other hand, what obviously makes the most sense is for each to give what seems fair to both, which is the standard at which we aim at friendships involving buying and selling. In each case, we have to take into account what is appropriate to the friendship itself.
Breaking Off Friendships
When is it a good idea to break off friendships? It seems clear enough with utility-friendships and pleasure-friendships that they should break up when the grounds for them are gone. But things get a clear complicated when we are dealing with utility-friendships or pleasure-friendships that people treat as if they were virtue-friendships; the single most dangerous thing to a friendship is for people not really to be friends in the way they think they are.
With virtue-friendships, we have situations in which someone actually turns out not to be virtuous at all. We might hold that they can be rehabilitated, and so try to maintain the friendship on those grounds; but we can also see that if this didn't seem possible, a person might reasonably give up on the friendship. What if, however, the difference arises not because of any vice, but because one of the friends turns out to be much more virtuous than the other? This is a trickier issue. Obviously if the gap were small, it wouldn't matter, but the larger the gap, the more difficult it is to maintain the equality of the friendship. Aristotle's example is of a virtue-friendship that begins with childhood. What if one of the friends grows up and the other remains childish in mind? Despite both being virtuous, they will not approve and disapprove the same things, nor be able easily to share in each other's joys and sorrows, nor live together as equals. But Aristotle does not that even when a friendship like this must be dissolved, it makes sense to govern our relations with someone by our prior friendship with them.
Loving One's Friend as Oneself
The section of Aristotle's discussion of friendship that has had the most influence through the ages is that which begins at IX.4, because of its interesting relation to Christian ethics. In this part of the discussion, he argues that in loving a friend we are in some way loving them as we love ourselves, particularly if we are virtuous. In particular:
(1) A virtuous person is not divided of mind but in agreement with himself.
(2) A virtuous person wishes real good for himself.
(3) A virtuous person wishes his life and mind to be preserved.
(4) A virtuous person spends time with himself, in memory and anticipation of the future, and enjoys doing so because he has nothing to regret.
These are the kinds of things we recognize as appropriate to friendship, so even if we want to rule out the idea that you can be friends with yourself, it is still the case that a virtuous person stands to his friend in such a way that his friend is a sort of 'other self'. This, Aristotle notes, emphasizes the importance of virtue for friendship, since the more wicked we are, the less able we are to be genuine friends, not even being able to treat ourselves in a friendly way.
This recognition of the importance of virtue continues as Aristotle goes through several things that in some way pertain to friendship: benevolence, concord, and beneficence. Benevolence or goodwill is not itself friendship because it need not be mutual, loving, or close. But we can think of it as a sort of latent friendship, since friendship can originate from it. However, what kind of friendship most naturally arises from benevolence? Virtue-friendship, because genuine goodwill is more concerned with what is actually good than with being useful or pleasant. Concord has to do with agreement about what is beneficial, and Aristotle argues that it is a kind of civic friendship. But concord is again something that is more properly found rooted in virtue than in anything else, since the virtuous person has a sort of inner concord, and virtue-friendships are therefore more naturally a kind of concord than any other friendship. Beneficence is a bit trickier, because the relationship between benefactor and beneficiary is asymmetrical, thus making it relatively easy for a situation to arise in which the benefactor loves the beneficiary as a sort of extension of himself, while the beneficiary loves the benefactor primarily as a choice of benefits. (As Aristotle often does, he uses the motherhood as a primary example of asymmetric friendship.)
Aristotle passes on to the question of whether you should love yourself or others more. On the one hand, we do not regard in a favorable light people who love themselves more than anything, and we often find that bad people in some way are selfish, whereas good people are often neglecting themselves in favor of friends. But, he argues, this is all highly misleading. If we should love most the person who is most our friend, well, in a sense we are our own best friend, especially if we are virtuous. If the virtuous person treats a friend as 'another self', it follows from this that all the characteristic governing how much you should love a friend are found most in the virtuous person's relation with himself or herself. All the proverbial things we say about friendship have the implication that we should actually love ourselves most of all.
The obvious issue here is what is meant by loving oneself. When we say someone is selfish, we are generally saying that they try to get money, pleasure, and recognition for themselves; we aren't saying that they are trying to get genuine good, like virtue, for themselves. But (as we've already seen) the virtuous person is the person who can most properly be said to love himself or herself. So there is an unselfish self-love, and in this kind of love of self is the root of the kind of love of others that makes for the best kinds of friendships. This kind of person will sacrifice money, honor, and pleasure for a friend. But in doing so he is, in a sense, wishing a greater good for himself than for his friend, since he is giving himself the benefit of being noble, while only giving the friend some other, lesser kind of advantage. Or to put it in other words: the best kind of friends love themselves more than they love their friends, and it is because their self-love is of a noble kind that they are able to love their friends so well. We might also say that for the virtuous person, love of a friend is already a part of -- and therefore in that sense less than -- love of self.
Happiness and Friendship
The aim of all human action, on Aristotle's view, is eudaimonia, the complete good of a rational life. The best translation for this used to be happiness, although the term has shifted enough that it's a difficult fit now. When Thomas Jefferson in the Declaration of Independence says that one of our inalienable rights is "pursuit of happiness", he did not mean that we had an inalienable right to try to have fun (which would be a stupid thing to have a bloody revolution over), but that we had an inalienable right to try to achieve a life that was genuinely appropriate to human beings. Unfortunately, we don't really have any words that fit this older idea of happiness well, so translators often flounder a bit on the question, trying things like 'flourishing'. I think I will stick with happiness, as long as it's clear that it is to be understood as 'life we can call blessed' rather than 'feeling of happiness'.
Happy people are self-sufficient on their own, so it might seem that they don't need friends; but, on the other side, it seems absurd to think that the culmination of human life would be having every important kind of good except friendship, or that human beings, who are civic animals, would be fulfilled by becoming solitary. Aristotle suggests that if you think of happiness as being something you can have without friends, it is primarily because you are thinking in terms of utility-friendships; the happy person, qua happy person, would not need these kinds of friendships. And the same is the case if we are thinking in terms of pleasure-friendships, since the happy person is satisfied without having to find pleasures in other people. With virtue-friendships, though, it is entirely different; everything that we can identify as being something suitable to a happy life is something we can identify as being even more appropriate to a happy life with virtue-friendships. For instance, someone with a genuinely happy life will take joy in virtue; but this is more easily done if you have virtuous friends. It likewise seems clear that a virtuous friend is by nature the sort of thing that is worthy of being part of a virtuous life, since a virtuous person loves a virtuous friend as another self.
One might think from this that we should get as many friends as possible. With utility-friendships this is often not true; being as useful to others as they are to you is often hard work. It makes sense that we should have only those utility-friendships that are adequate for living our lives. And, Aristotle says, we should see pleasure-friendships is a similar way, as a sort of seasoning. But virtue-friendships are so good in themselves that it's not obvious that we should restrict ourselves. But Aristotle notes that we can't have room in our lives for endless numbers of friends, because there are only so many people you can genuinely share your life with. Thus in reality the best route is what we naturally tend to do anyway: we keep our profound friendships few, because those who try to be friends with everyone are really not much different from people who are friends with no one, and are not generally regarded as admirable people. (It should perhaps be noted here that Aristotle is not talking about being friendly. Friendliness in Aristotle is the virtue concerned with keeping open the possibility of friendship, and is an admirable quality; this is different from actively trying to be friends with lots and lots of people, and is something people in practice tend to treat with contempt.)
Aristotle then goes on to argue that we need friends more when we are dealing with bad fortune, but that it is more splendid to have them in good fortune. This is partly due to the fact that bad fortune can be alleviated by receiving benefits from friends, but good fortune gives us the means to do good to our friends. Even if our friends can't help us in our bad fortune, however, it is pleasant just to have friends, although this is complicated by the possibility that our being in bad fortune might cause them pain. In particular, Aristotle says, this is true of those who are manly by nature; they will go out of their way to avoid causing their friends distress, even if they themselves are in distress, whereas women and effeminate men in misery get pleasure from the commiseration of others. (There is no question about which Aristotle considers the better way.) There is also the point that good fortune is by nature the kind of thing to be shared with friends, whereas bad fortune is obviously not the kind of thing we are eager to share with friends; this means that in a healthy friendship, we try to make sure that, when in bad fortune, we call on our friends to help only in ways that would do us great good with as little trouble to them as possible. On the other side, though, it's obviously the part of a friend to benefit friends in trouble, even if they haven't demanded it; so in a healthy friendship, one tries to do good to one's friends in such a way that even when in trouble they don't have to demand anything of oneself, because one has already done it. The overall implication is that friendships is something that is appropriate to all conditions of human life.
Aristotle ends his discussion by talking about friendship-activities. All friendship activities in some way have to do with the sharing of life. Because of that, what we want to share with our friends is the kind of thing we want for ourselves -- we want to be able to share with our friends an answer to the question, "Why is it good to be alive?" This is why we tend to associate friendship with certain kinds of social activity that make life good in some way -- drinking, gambling, athletics, hunting, and philosophy are the examples Aristotle gives. We share with friends, and we share the things we like. This is why friendship with bad people is degrading and friendship with good people is improving: bad people are likely to share bad activities, while good people are likely to share good activities.
And that's essentially the end of Aristotle's discussion of friendship, which is one of the most important and influential philosophical discussions in history.