What does "which heaven to gaudy day denies" mean?

"Which heaven to gaudy day denies" means that the woman's beauty is like a starlit night, possessing a heavenly light which the garish day could never match.

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Throughout "She Walks in Beauty," the poet uses images of light to describe a particular beautiful woman. He writes, for example, that the woman's beauty is like a "tender light / Which heaven to gaudy day denies." The implication here is that the light which the woman seems to emit is a heavenly light, which heaven does not grant to ordinary, earthly days. The light of those earthly days, provided by the sun, is, the poet writes, "gaudy" in comparison to the radiant and "tender" light of the woman's beauty. The implication is that the woman's beauty is heavenly and celestial and transcends the earthly, corporeal realm.

The idea that the woman's beauty is heavenly and celestial is emphasized elsewhere in the poem too. In the third stanza, for example, the poet writes that the woman's beauty is so pure as to suggest that she has lived a life "in goodness spent." Her beauty also seems to indicate that she has a "mind at peace with all below," suggesting that the woman is like an angel looking down upon the earth. In the second stanza, the woman's beauty is also described as "the nameless grace," the word "grace" again connoting something spiritual and heavenly.

The title of the poem, and thus also the opening line of the poem, further suggest that the woman possesses an angelic beauty. Saying that the woman walks "in" beauty suggests that the woman is surrounded by her beauty, and thus surrounded by the light of her beauty, much as in many depictions of angels surrounded by the light cast by their haloes.

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