Damage
Middle English: from Old French, from dam, damne ‘loss or damage’, from Latin damnum ‘loss or hurt’; compare with damn.
wiktionary
From Middle English damage, from Old French damage (Modern French dommage), from Vulgar Latin *damnaticum from Classical Latin damnum.
etymonline
damage (n.)
c. 1300, "harm, injury; hurt or loss to person, character, or estate," from Old French damage, domage "loss caused by injury" (12c., Modern French dommage), from dam "damage," from Latin damnum "loss, hurt, damage" (see damn). In law (as damages) "the value in money of what was lost or withheld, that which is given to repair a cost," from c. 1400. Colloquial sense of "cost, expense" is by 1755. Damage control "action taken to limit the effect of an accident or error" is attested by 1933 in U.S. Navy jargon.
damage (v.)
"cause damage to, hurt, injure, harm," early 14c., from Old French damagier, from damage "loss caused by injury" (see damage (n.)). Related: Damaged; damaging.
