Transitive meaning "to burn, set on fire" is from 1580s. Meaning "break out in violence of passion" is from 1540s; the sense of "unleash invective on a computer network" is from 1980s. Related: Flamed; flaming. To flame out, in reference to jet engines, is from 1950.
Middle English flaume, also flaumbe, flambe, flame, flamme, mid-14c., "a flame;" late 14c., "a flaming mass, a fire; fire in general, fire as an element;" also figurative, in reference to the "heat" or "fire" of emotions, from Anglo-French flaume, flaumbe "a flame" (Old French flambe, 10c.), from Latin flammula "small flame," diminutive of flamma "flame, blazing fire," from PIE *bhleg- "to shine, flash," from root *bhel- (1) "to shine, flash, burn."
The meaning "a sweetheart, object of one's passion" is attested from 1640s; the figurative sense of "burning passion" was in Middle English, and the nouns in Old French and Latin also meant "fire of love, flame of passion," and, in Latin "beloved object." The Australian flame-tree is from 1857, so called for its red flowers.
The literal sense of "to cause to burn" first recorded in English late 14c. Meaning "to heat, make hot, cause inflammation" is from 1520s. Formerly also enflame, but since 16c. the spelling with in- has predominated. Related: Inflamed; Inflaming.
1832, originally in reference to a 15c.-16c. architectural style with wavy, flame-like curves, from French flamboyant "flaming, wavy," present participle of flamboyer "to flame," from Old French flamboiier "to flame, flare, blaze, glow, shine" (12c.), from flambe "a flame, flame of love," from flamble, variant of flamme, from Latin flammula "little flame," diminutive of flamma "flame, blazing fire" (from PIE root *bhel- (1) "to shine, flash, burn"). Extended sense of "showy, ornate" is from 1879. Related: Flamboyantly.