Origin and history of squarely

early 14c., "having four equal sides and right angles," from square (n.), or from Old French esquarre, past participle of esquarrer. The meaning "honest, fair, equitable, just" is attested by 1560s and accounts for many figurative sense; the notion seems to be "accurately adjusted, as by a square," hence "true, fitting, proper."

In measurements of square area, from late 14c. Of body parts, "sturdy, strongly built," late 14c. The meaning "straight, direct" is from 1804. Square meal, one that is solid and substantial, is by 1868; OED reports it "Orig. U.S.; common from about 1880." Of accounts, etc., "even, leaving no balance," by 1859.

The sense of "old-fashioned" is by 1944 in U.S. jazz slang, said to be from shape of a conductor's hand gestures in a regular four-beat rhythm. Square-toes (1752) meant nearly the same thing in late 18c.: "precise, formal, old-fashioned person (especially one's own father)," from the style of men's shoes worn early 18c. and then fallen from fashion. Squaresville, the Limbo of the L-7, is attested from 1951.

Square dance (n.) is attested by 1831; originally one in which the couples faced inward from four sides; later of country dances generally. Square-dancing (n.) is by 1867, American English (Boston Evening Transcript).

[T]he old square dance is an abortive attempt at conversation while engaged in walking certain mathematical figures over a limited area. [The Mask, March 1868]

Square-sail is attested by c. 1600. The nautical square-rigger is by 1829; square-rigged is from 1769. Square wheel as figurative of something that doesn't work as needed is by 1920.

common adverbial suffix, forming, from adjectives, adverbs signifying "in a manner denoted by" the adjective; Middle English -li, from Old English -lice, from Proto-Germanic *-liko- (source also of Old Frisian -like, Old Saxon -liko, Dutch -lijk, Old High German -licho, German -lich, Old Norse -liga, Gothic -leiko). See -ly (1). It is 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.