c. 1300, "stretched out (on a surface), prostrate, lying the whole length on the ground;" mid-14c., "level, all in one plane; even, smooth;" of a roof, "low-pitched," from Old Norse flatr "flat," from Proto-Germanic *flata- (source also of Old Saxon flat "flat, shallow," Old High German flaz "flat, level," Old High German flezzi "floor"), from PIE root *plat- "to spread."
From c. 1400 as "without curvature or projection." Sense of "prosaic, dull" is from 1570s, on the notion of "featureless, lacking contrast." Used of drink from c. 1600; of women's bosoms by 1864. Of musical notes from 1590s, because the tone is "lower" than a given or intended pitch. As the B of the modern diatonic scale was the first tone to be so modified, the "flat" sign as well as the "natural" sign in music notation are modified forms of the letter b (rounded or square).
Flat tire or flat tyre is from 1908. Flat-screen (adj.) in reference to television is from 1969 as a potential technology. Flat-earth (adj.) in reference to refusal to accept evidence of a global earth, is from 1876.
Old English stæf (plural stafas), "walking stick, strong pole used for carrying, rod used as a weapon, pastoral staff," probably originally *stæb, from Proto-Germanic *stab- (source also of Old Saxon staf, Old Norse stafr, Danish stav, Old Frisian stef, Middle Low German and Middle Dutch staf, Old High German stab, German Stab, Gothic *stafs "element;" Middle Dutch stapel "pillar, foundation"), from PIE root *stebh- "post, stem, to support, place firmly on, fasten" (source also of Old Lithuanian stabas "idol," Lithuanian stiebas "staff, pillar;" Old Church Slavonic stoboru "pillar;" Sanskrit stabhnati "supports;" Greek stephein "to tie around, encircle, wreathe," staphyle "grapevine, bunch of grapes;" Old English stapol "post, pillar").
As "pole from which a flag is flown," 1610s. In musical notation from 1660s. Sense of "group of military officers that assists a commander" is attested from 1702, apparently from German, from the notion of the "baton" that is a badge of office or authority (a sense attested in English from 1530s); hence staff officer (1702), staff-sergeant (1811). Meaning "group of employees (as at an office or hospital)" is first found 1837. Staff of life "bread" is from the Biblical phrase break the staff of bread meaning "cut off the supply of food" (Leviticus xxvi.26), translating Hebrew matteh lekhem.
The Old English word, in plural, was the common one used for "letter of the alphabet, character," hence "writing, literature," and many compounds having to do with writing, such as stæfcræft "grammar," stæfcræftig "lettered," stæflic "literary," stæfleahtor "grammatical error," with leahtor "vice, sin, offense."