What's the difference between i.e. and e.g.?
Old English belyfan "to believe," earlier geleafa (Mercian), gelefa (Northumbrian), gelyfan (West Saxon) "believe," from Proto-Germanic *ga-laubjan "to believe," perhaps literally "hold dear, love" (cf. Old Saxon gilobian "believe," Dutch geloven, Old High German gilouben, German glauben), ultimately a compound based on PIE *leubh- "to care, desire, love" (see belief).
Spelling beleeve is common till 17c.; then altered, perhaps by influence of relieve, etc. To believe on instead of in was more common in 16c. but now is a peculiarity of theology; believe of also sometimes was used in 17c. Related: Believed (formerly occasionally beleft); believing. Expression believe it or not attested by 1874; Robert Ripley's newspaper cartoon of the same name is from 1918. Emphatic you better believe attested from 1854.