By the dictionary definition as a verb, I can see that essay and assay have the same meaning, i.e. "make an effort or attempt".
I'm wondering if they are totally interchangeable, or is there a difference in degree associated with each?
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Both have the same root, the French essai (trial). As verbs with the meaning of “attempt or try”, the New Oxford American Dictionary marks essay as “formal”, and assay as “archaic”. Apart from that, they are synonymous (and both antiquated).
Edit: regarding modern synonyms of this meaning, one would now say attempt or try. Other synonyms are endeavour, seek, undertake or have a go at.