"near in position, close to," 1640s, from Late Latin approximatus, past participle of approximare "to come near to," from Latin ad "to" (see ad-) + proximare "come near," from proximus "nearest," superlative of prope "near" (see propinquity). Meaning "near in accuracy or correctness" is by 1816. Also used in Middle English in a sense "similar" (early 15c.).
common adverbial suffix, forming from adjectives adverbs signifying "in a manner denoted by" the adjective, Middle English, from Old English -lice, from Proto-Germanic *-liko- (cognates: Old Frisian -like, Old Saxon -liko, Dutch -lijk, Old High German -licho, German -lich, Old Norse -liga, Gothic -leiko); see -ly (1). Cognate with lich, and identical with like (adj.).
Weekley notes as "curious" that Germanic uses a word essentially meaning "body" for the adverbial formation, while Romanic uses one meaning "mind" (as in French constamment from Latin constanti mente). The modern English form emerged in late Middle English, probably from influence of Old Norse -liga.
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Definitions of approximately from WordNet
(of quantities) imprecise but fairly close to correct;