compliment (n.)
"An act, or expression of civility, usually understood to include some hypocrisy, and to mean less than it declares" [Johnson], 1570s, complement, via French compliment (17c.), from Italian complimento "expression of respect and civility," from Vulgar Latin *complire, for Latin complere "to complete" (see complete (adj.)), via notion of "complete the obligations of politeness." Same word as complement but by a different etymological route; differentiated by spelling after 1650.
compliment (v.)
1610s, from French complimenter, from compliment (see compliment (n.)). Related: Complimented; complimenting.