"free from special defect or injury," c. 1200, from Old English gesund "sound, safe, having the organs and faculties complete and in perfect action," from Proto-Germanic *sunda-, from Germanic root *swen-to- "healthy, strong" (source also of Old Saxon gisund, Old Frisian sund, Dutch gezond, Old High German gisunt, German gesund "healthy," as in the post-sneezing interjection gesundheit; also Old English swið "strong," Gothic swinþs "strong," German geschwind "fast, quick"), with connections in Indo-Iranian and Balto-Slavic. Meaning "right, correct, free from error" is from mid-15c. Meaning "financially solid or safe" is attested from c. 1600; of sleep, "undisturbed," from 1540s. Sense of "holding accepted opinions" is from 1520s.
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 soundly from WordNet
deeply or completely;
the baby is sleeping soundly
slept soundly through the storm
completely and absolutely (`good' is sometimes used informally for `thoroughly');