Plato briefly describes and justifies the method of dialectic thusly:
Then isn’t this at last, Glaucon, the song that dialectic sings? It is intelligible,  but it is imitated by the power of sight. We said that sight tries at last to look at the animals themselves, the stars themselves, and, in the end, at the sun itself. In the same way, whenever someone tries through argument and apart from all sense perceptions to find the being itself of each thing and doesn’t give up until he grasps the good itself with [b] understanding itself, he reaches the end of the intelligible, just as the other reached the end of the visible.
And what about this journey? Don’t you call it dialectic?
Then the release from bonds and the turning around from shadows to statues and the light of the fire and, then, the way up out of the cave to the sunlight and, there, the continuing inability to look at the animals, the plants, and the light of the sun, but the newly acquired ability to look at [c] divine images in water and shadows of the things that are, rather than, as before, merely at shadows of statues thrown by another source of light that is itself a shadow in relation to the sun—all this business of the crafts we’ve mentioned has the power to awaken the best part of the soul and lead it upward to the study of the best among the things that are, just as, before, the clearest thing in the body was led to the brightest thing in the [d] bodily and visible realm.
I accept that this is so, even though it seems very hard to accept in one way and hard not to accept in another. All the same, since we’ll have to return to these things often in the future, rather than having to hear them just once now, let’s assume that what you’ve said is so and turn to the song itself, discussing it in the same way as we did the prelude. So tell us: what is the sort of power dialectic has, what forms is it divided into, and what paths does it follow? For these lead at last, it seems, towards [e] that place which is a rest from the road, so to speak, and an end of journeying for the one who reaches it.
 You won’t be able to follow me any longer, Glaucon, even though there is no lack of eagerness on my part to lead you, for you would no longer be seeing an image of what we’re describing, but the truth itself. At any rate, that’s how it seems to me. That it is really so is not worth insisting on any further. But that there is some such thing to be seen, that is something we must insist on. Isn’t that so?
And mustn’t we also insist that the power of dialectic could reveal it only to someone experienced in the subjects we’ve described and that it cannot reveal it in any other way?
That too is worth insisting on.
[b] At any rate, no one will dispute it when we say that there is no other inquiry that systematically attempts to grasp with respect to each thing itself what the being of it is, for all the other crafts are concerned with human opinions and desires, with growing or construction, or with the care of growing or constructed things. And as for the rest, I mean geometry and the subjects that follow it, we described them as to some extent grasping what is, for we saw that, while they do dream about what is, they are unable to command a waking view of it as long as they make use of hypotheses that they leave untouched and that they cannot give any account [c] of. What mechanism could possibly turn any agreement into knowledge when it begins with something unknown and puts together the conclusion and the steps in between from what is unknown?
Therefore, dialectic is the only inquiry that travels this road, doing away with hypotheses and proceeding to the first principle itself, so as to be [d] secure. And when the eye of the soul is really buried in a sort of barbaric bog, dialectic gently pulls it out and leads it upwards, using the crafts we described to help it and cooperate with it in turning the soul around. From force of habit, we’ve often called these crafts sciences or kinds of knowledge, but they need another name, clearer than opinion, darker than knowledge. We called them thought somewhere before.5 But I presume that we won’t dispute about a name when we have so many more important matters to investigate. [e]
Of course not.
It will therefore be enough to call the first section knowledge, the second thought, the third belief, and the fourth imaging, just as we did before. The last two together we call opinion, the other two, intellect. Opinion is  concerned with becoming, intellect with being. And as being is to becoming, so intellect is to opinion, and as intellect is to opinion, so knowledge is to belief and thought to imaging. But as for the ratios between the things these are set over and the division of either the opinable or the intelligible section into two, let’s pass them by, Glaucon, lest they involve us in arguments many times longer than the ones we’ve already gone through.
I agree with you about the others in any case, insofar as I’m able to follow. [b]
Then, do you call someone who is able to give an account of the being of each thing dialectical? But insofar as he’s unable to give an account of something, either to himself or to another, do you deny that he has any understanding of it?
How could I do anything else?
Then the same applies to the good. Unless someone can distinguish in an account the form of the good from everything else, can survive all refutation, as if in a battle, striving to judge things not in accordance with [c] opinion but in accordance with being, and can come through all this with his account still intact, you’ll say that he doesn’t know the good itself or any other good. And if he gets hold of some image of it, you’ll say that it’s through opinion, not knowledge, for he is dreaming and asleep throughout his present life, and, before he wakes up here, he will arrive in Hades and go to sleep forever. [d]
So we learn that apprehension of the forms is facilitated by (and, it is claimed, only by) the dialectic method. Plato himself contrasts the dialectic method with what has come to be known as the axiomatic method but was called by Plato the method of hypotheticals when he discussed it in Book VI in the course of drawing the famous divided line.
Recall our discussion Tuesday when I discussed the history of the parallel line postulate (axiom) of Euclidean Geometry: 1) how that axiom especially troubled mathematicians because, unlike any of the other axioms, it presupposes facts at the limits of infinity, well beyond finite human comprehension; 2) how disconcerted mathematicians and philosophers spent the following fifteen hundred years attempting to prove that the parallel line postulate can be proven assuming just the other Euclidean axioms (and thus that the postulate is not an axiom but a theorem); 3) how it was eventually demonstrated that there could be no proof of the parallel line postulate on the basis of the other axioms; and 4) how that shortly led to the construction of non-euclidean geometries one of which became the mathematical basis for relativity theory and the notion of curved space-time.
Bear in mind that the axiomatic method is like the dialectic method insofar as both methods lead to knowledge (our knowledge of mathematics and our knowledge of the forms or essences, respectively). Thus we are firmly above the divided line in the realm of what is knowable: the objects of mathematics via the axiomatic method versus the forms themselves via the dialectic method. Yet since the realm of what is knowable (as opposed to what is merely opinable, below the divided line) is further divided between the inferior realm of mathematics and the superior realm of the forms, the method of dialectic transcends the axiomatic method, enabling us to understand the essential nature of the eternal, perfect forms of which ordinary sensible objects are but dim reflections. See the handout I passed out Tuesday for clarification: The Divided Line (gif)
In contrast with the axiomatic method, what is the method of dialectic, and how does it permit us on Plato's view to achieve knowledge of the forms? That is, given what little Socrates asserts here, can you develop a theory of dialectic as distinct from the axiomatic method? In particular, be sure to inform your answer by a well-chosen example of the method of dialectic as it is applied in the Republic (thus, quotes are permitted, but only when they are carefully explained in developing your answer.) In the end, is Plato correct in holding that dialectic is the best and only method to achieve genuine understanding (noesis) and, thus, knowledge? Two pages is too short for this. Let us expand the limit a bit to three pages.