1597, "short non-fiction literary composition" (first attested in writings of Francis Bacon, probably in imitation of Montaigne), from M.Fr.
essai "trial, attempt, essay," from L.L.
exagium "a weighing, weight," from L.
exigere "test," from
ex- "out" +
agere apparently meaning here "to weigh." The suggestion is of unpolished writing.
Essayist is from 1609. The more literal verb meaning "to put to proof, test the mettle of" is from 1483; this sense has mostly gone with the divergent spelling
assay (q.v.).
[n. es-ey for 1, 2; es-ey, e-sey for 3–5; v. e-sey] 
eɪ
