What semantic notions underlie "a weighing, a weight" and 'trial, attempt'?
1590s, "trial, attempt, endeavor," also "short, discursive literary composition" (first attested in writings of Francis Bacon, probably in imitation of Montaigne),
from Middle French essai "trial, attempt, essay" (in Old French from 12c.),
from Late Latin exagium "a weighing, a weight"
from Latin exigere "drive out; require, exact; examine, try, test,"
from ex "out" (see ex-) + agere "to set in motion, drive"
(from PIE root *ag- "to drive, draw out or forth, move") apparently meaning here "to weigh." The suggestion is of unpolished writing. Compare assay, also examine.