Shakespeare does use the phrase.
Romeo and Juliet, Act 2, Scene 2
ROMEO: Lady, by yonder blessed moon I swear
That tips with silver all these fruit-tree tops--
JULIET: O, swear not by the moon, the inconstant moon,
That monthly changes in her circled orb,
Lest that thy love prove likewise variable.
31256: Inconstant Moon
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