Most aestheticians ignore Plato's Phaedo. It does mention the eternal form of Beauty, but it seems that Plato has more interesting things to say about Beauty in the Symposium and the Greater Hippias. The Phaedo is generally treated as mainly about the Forms, of which Beauty is just one example. But it is the first example he always gives, and so maybe it plays a more important role here than the other Forms mentioned. We learn early on that Beauty cannot be apprehended by any of the sense organs but only by the mind alone. But the most interesting discussion begins at 99c 137 (I'll use both the Stephanus numbers and the page number from the Grube translation Five Dialogues, Hackett, 2002) where we learn about the early life of Socrates, his rejection of the materialist view of causality, and his similar rejection of the philosophy of Anaxagoras, who began by talking about Mind as a causal principle, but then ended by seeming more like a traditional materialist anyway. This disappointed the young Socrates. He then came up (according to this autobiographical note) with a new theory of change, and a new methodology. The new method is to take refuge from the blinding quality of apprehension of things through the senses in "discussions," by which he means investigating "the truth of things by means of words." He interestingly sees words as not like images but more like facts in reliability. The new method is to start with a hypothesis, i.e. "the theory that seemed to me the most compelling," and assume that whatever agreed with this was true. So he turns to his hypothesis of the Forms and proceeds from this, assuming the existence e.g. of "a Beautiful itself." The end product of this new method, he predicts, will be to show that the soul is immortal (which is the goal of the Phaedo.) His claim is that if there is anything beautiful besides the Beautiful itself it is so because it "shares" in the Beautiful.
Now the interesting thing comes in the next passage (100d, 138) where he rejects "other sophisticated causes" for example the claim that "a thing is beautiful because it has a bright color or shape or any such thing." That is, he rejects any definition of beauty in terms that would seem natural in that they would refer to some combination or mixture of things in the sensible world. Perhaps he rejects any definition of beauty at all! But whether that is so is not made clear here. In any case, he would clearly reject a number of theories of beauty. He would, for example, reject formalism where the emphasis is placed on relations between lines, planes and colors. Instead, he holds ("naively, and perhaps foolishly" - recognizing the apparently outlandish nature of his claim) "all beautiful things are beautiful by the Beautiful." And then, rather wildly, he calls this "the safest answer" in the sense of being safe from falling into error (as though this sort of safety was a sufficient condition for truth). Another way of putting the thesis is then offered: that it is "through Beauty that beautiful things are made beautiful." Similarly, he argues that a thing is big because it shares in Bigness, or is bigger "by Bigness." On the face of it, all of this seems "safe" only because it is tautological: he seems to be saying that the beautiful is beautiful because it is categorized under the concept of Beauty, unless, of course, the form of Beauty (or Bigness), really does something here. I suspect that the main point is that we cannot give an account of the beauty of a thing, cannot define it or break it down into parts, but that simply see an object as beautiful if we see it as partaking in this very different thing (because eternal and unchanging) called Beauty itself, i.e. we see it as partaking in the infinite or the divine, or the divine aspect of itself. In secular terms this might be translated into a plausible hypothesis that a thing has beauty if it has an aura of the infinite, the divine or the eternal. The existence of the aura does not require that there actually be something divine or eternal. This then, is a kind of metaphysical "seeing as."