Beauty

To know what beauty is, shouldn't one observe examples of it? But if one doesn't know what beauty is in the first place, how can one tell if one is observing examples of it?

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Nice question! Here's a quick

Nice question! Here's a quick reconstruction of your reasoning:
(1) In order to know what X is, you need to observe instances of X.
(2) But one could only know one is observing instances of X if one already knew what X is.
(3) Hence, one can neither know what X is nor know whether one is observing instances of X.

(3) looks like a pretty powerful skeptical conclusion: It would seem like we can't know whether a certain thing is beautiful unless we know what beauty is, but we can't know what beauty is unless we know which things are beautiful. We certainly seem stuck -- and the same reasoning could be used to generate skeptical conclusions about other important philosophical concepts (such as goodness, virtue, and so on).

Is there a way out of this conundrum? There are a lot of complex issues here, so let me just mention some possible ways out and leave it to you to assess their plausibility:

Testimony Others might be in a credible position to tell us which things are beautiful or what beauty is. Accepting the testimony of such experts would allow us to break out of this circle. With testimonial knowledge of which things are beautiful, we could then figure out what beauty is; with testimonial knowledge of what beauty is, we could then correctly classify which things are beautiful. (Granted, others' testimony may not seem like a very promising solution. After all, we may reasonably wonder how those experts whose testimony we accept came to their knowledge -- how, in other words, did they break out of the skeptical circle with which we started?)

Reject (1): Platonic rationalism Plato (and many other philosophers dubbed 'rationalists') seem to hold that while we may need to have our attention drawn to something in order to understand its nature, we do not come to understand its nature through observing instances of it by through an intellectual grasp of its essence. Upon seeing a triangle, one perceives that it must also be trilateral. This inference, according to rationalists, involves an intellectual grasp of the nature of triangles. One comes to understand its essence (and one is thereby able to classify any future geometrical figures one encounters as triangles) not by forming a hypothesis (every triangle is trilateral) based on perceiving this triangle and then 'confirming' this hypothesis. Rather, perceiving the triangle led to an immediate acquisition of a priori (i.e., non-empirical) knowledge.

Reject (2): Empiricism: We might be able to learn what X is by observing instances of things we believe to be X and trying to suss out the essential features of X. After observing a number of seemingly beautiful things, we might frame the hypothesis that 'because all beautiful things are F, beauty = F'. We might then observe further seemingly beautiful things to determine they have property F. If so, then our hypothesis is confirmed; if not, it is disconfirmed. With a sufficiently large number of observations, we might come to be satisfied that we have hit upon the real essence of beauty.

Nominalism: Deny real essences: Your question seemed to assume that there is something real about beauty -- that beauty is or is based on properties all beautiful things share. But one might reject that assumption. 'Nominalism' about some notion holds that there is no real essence to it. Nominalism about 'beauty' would amount to saying that there is no 'universal' beauty -- only many things that we judge to be beautiful based on convention.

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