"make complete," 1640s, from complement (n.). Earlier in a now-obsolete sense of "exchange courtesies" (1610s), from complement (n.) in a 16c. sense "that which is added, not as necessary, but as ornamental," now going with compliment. Related: Complemented; complementing.
late 14c., "means of completing; that which completes; what is needed to complete or fill up," from Old French compliement "accomplishment, fulfillment" (14c., Modern French complément), from Latin complementum "that which fills up or completes," from complere "fill up," from com-, here probably as an intensive prefix (see com-), + plere "to fill" (from PIE root *pele- (1) "to fill").
From c. 1600 as "full quality or number, full amount;" musical sense of "simple interval that completes an octave from another simple interval" is from 1873. In 16c. also having senses which were taken up c. 1650-1725 by compliment.
Old English full "containing all that can be received; having eaten or drunk to repletion; filled; perfect, entire, utter," from Proto-Germanic *fullaz "full" (source also of Old Saxon full, Old Frisian ful, Dutch vol, Old High German fol, German voll, Old Norse fullr, Gothic fulls), from PIE root *pele- (1) "to fill." Related: Fuller; fullest.
The adverb is Old English ful "very, fully, entirely, completely" and was common in Middle English (full well, full many, etc.); sense of "quite, exactly, precisely" is from 1580s. Full moon, one with its whole disc illuminated, was Old English fulles monan; first record of full-blood in reference to racial purity is from 1812. Full house is 1710 in the theatrical sense, 1887 in the poker sense (three of a kind and a pair, earlier full-hand, 1850). Full-dress (adj.) "appropriate to a formal occasion" is from 1761, from the noun phrase.