All 3 Uses

divine
in
The Comedy of Errors
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Man, more divine, the masters of all these,
Scene 2.1 *

divine = god-like
Sweet mistress,—what your name is else, I know not, Nor by what wonder you do hit on mine,—Less, in your knowledge and your grace, you show not Than our earth's wonder: more than earth divine.†
Scene 3.2
—To conclude: this drudge or diviner laid claim to me; called me Dromio; swore I was assured to her; told me what privy marks I had about me, as the mark of my shoulder, the mole in my neck, the great wart on my left arm, that I, amazed, ran from her as a witch: and, I think, if my breast had not been made of faith and my heart of steel, she had transformed me to a curtail-dog, and made me turn i' the wheel.†
Scene 3.2

Definitions:

(1) (divine as in: to forgive is divine)
wonderful; or god-like or coming from God
(2) (divine as in: divined from tea leaves)
to discover or predict something supernaturally (as if by magic)
(3) (divine as in: divined through intuition)
to discover or guess something -- usually through intuition or reflection
(4) (meaning too rare to warrant focus)
In the time of Shakespeare, divine was sometimes used as a noun to reference a priest or a person of the church.

Divinity typically refers to a god or to a school of religion, but on rare occasions, it refers to the name of a kind of soft white candy. To remember that sense, you might think of it as tasting divine/wonderful.