"uncertain as to specifics," 1540s, from French vague "empty, vacant; wild, uncultivated; wandering" (13c.), from Latin vagus "strolling, wandering, rambling," figuratively "vacillating, uncertain," perhaps from PIE *Huog-o- and cognate with Old Norse vakka "to stray, hover," Old High German wankon "to totter, stagger," Old High German winkan "to waver, stagger, wink," Old English wincian "to nod" [de Vaan]. Related: Vagueness.
late 14c., in constituciouns extravagaunt, a term in Canon Law for papal decrees not originally included or codified in the Decretals, from Medieval Latin extravagantem (nominative extravagans), present participle of extravagari "wander outside or beyond," from Latin extra "outside of" (see extra-) + vagari "wander, roam" (see vague).
In 15c. it also could mean "rambling, irrelevant; extraordinary, unusual." Extended sense of "excessive, extreme, exceeding reasonable limits" first recorded 1590s, probably via French; that of "wasteful, lavish, exceeding prudence in expenditure" is from 1711. Related: Extravagantly. Wordsworth ("Prelude") used extravagate (v.).