We must, however, not only describe virtue as a state of character, but also say what sort of state it is. We may remark, then, that every virtue or excellence both brings into good condition the thing of which it is the excellence and makes the work of that thing be done well; e.g. the excellence of the eye makes both the eye and its work good; for it is by the excellence of the eye that we see well. Similarly the excellence of the horse makes a horse both good in itself and good at running and at carrying its rider and at awaiting the attack of the enemy. Therefore, if this is true in every case, the virtue of man also will be the state of character which makes a man good and which makes him do his own work well.
This is a surprising section because the examples Aristotle chooses aren't to states of character, or even any of the elements of the soul he mentioned before: they appear to be not examples, but analogies to organs and animals.
Now as we frequently discuss here, all analogies always break. Because you are making comparisons between two things that are not alike, there will always come a point of difference. This insight is known to Aristotle, so we have to question that apparent use of analogy. When he says, "...if this is true in every case..." we should ask if he is making a claim about something that all three kinds of things have, something that isn't an analogy after all because they are really the same kind of thing.
The claim involves not merely states of character in people, but also the telos of organs and the telos of an animal.* We might assume, then, that this is a claim about teloi in general. The telos is the end of the thing, whatever kind of thing it is; and it is, coincidentally, the telos against which the thing is to be judged. You can tell good eyes from bad ones by how well they see. You can tell a good horse from a bad one by how well it performs.
So this seems to be the right kind of answer: you can tell a good from a bad man by how virtuous he is. Why? Because of the standards established in I.3: the EN is highly pragmatic. A virtuous man will perform better ('always or for the most part' given the existence of chance) when tested against objective reality. That's also how Aristotle closes this paragraph: "the virtue of man will also be the state of character which makes a man good and which makes him do his own work well."
The next paragraph requires a bit of explanation about Aristotle's math.
How this is to happen we have stated already, but it will be made plain also by the following consideration of the specific nature of virtue. In everything that is continuous and divisible it is possible to take more, less, or an equal amount, and that either in terms of the thing itself or relatively to us; and the equal is an intermediate between excess and defect. By the intermediate in the object I mean that which is equidistant from each of the extremes, which is one and the same for all men; by the intermediate relatively to us that which is neither too much nor too little- and this is not one, nor the same for all. For instance, if ten is many and two is few, six is the intermediate, taken in terms of the object; for it exceeds and is exceeded by an equal amount; this is intermediate according to arithmetical proportion. But the intermediate relatively to us is not to be taken so; if ten pounds are too much for a particular person to eat and two too little, it does not follow that the trainer will order six pounds; for this also is perhaps too much for the person who is to take it, or too little- too little for Milo, too much for the beginner in athletic exercises. The same is true of running and wrestling. Thus a master of any art avoids excess and defect, but seeks the intermediate and chooses this- the intermediate not in the object but relatively to us.
When talking about 'everything that is continuous and divisible' the contemporary mind probably turns to something like the number line, or the set of Real Numbers. This is not what Aristotle meant. There were no negative numbers in Ancient Greece, nor in fact even zero. Aristotle spends quite a lot of time worrying about the status of the infinite in both the Physics and the Metaphysics (especially the former, where he treats Zeno's problems). His solution is the infinite conceptual divisibility of any unity. If you have one, you can divide it in half; and then the half into halves; etc. Even if you can't physically do this to, say, an apple, you can do it conceptually forever. Thus, his 'real potential infinity' problem is solved; he rejects actually real infinities.
Here we get a clear and unambiguous statement that the 'intermediate' or 'mean between extremes' is not just the middle position. He distinguishes the true middle of a division of a unity from the kind of division that is appropriate to ethics. The true middle of the thing is "the intermediate in the object," which is the same for all (math being objective) and not what he is talking about. The intermediate we should be aiming at is "the intermediate relative to us." He gives some examples to show how that will differ from person to person: Milo of Croton is going to need more food than the novice who is just starting out on building muscle. The 'intermediate' amount of food, which is neither excessive nor insufficient, will therefore not be the same depending on which of them is being considered.**
That should clarify what is meant by 'the mean between extremes' or 'the intermediate' once and for all for students of Aristotle's ethics. If you run into anyone who thinks he's arguing for simply avoiding the extreme in all cases, refer them here: Milo might really need those extra calories.
*We can note that the horse might see its own telos differently from the one Aristotle gives -- it might not think that 'awaiting the attack of the enemy' is any part of its good life, and that even running is only in the service of defending a good long life of peacefully eating grass. Aristotle has an answer to that kind of objection, which is in Physics II.8. That section requires quite a bit of interpretation that might be distracting here, but the upshot is that he accepts that reality admits of different teloi from different perspectives, as the grass might consider the purpose of the rain the good of the corn, but things look different from other perspectives. There is a way of sorting out the priority laid out there for those who are interested in it. There's also in this section Aristotle's rejection of the contemporary scientific assumption that telos isn't required because things just happen due to necessary natural forces.
** This mathematical proof of an ethical concept is, by the way, exactly the sort of thing he warned us to avoid in I.3. We'll skip over that.
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