square (n.)

mid-13c., "mason's tool for measuring right angles, carpenter's square," from Old French esquire "a square, squareness," from Vulgar Latin *exquadra, a back-formation from *exquadrare "to square," from Latin ex "out" (see ex-) + quadrare "make square, set in order, complete," from quadrus "a square" (from PIE root *kwetwer- "four").

The meaning "square shape or area" is recorded by late 14c. (Old English used feower-scyte). The geometric sense of "four-sided rectilinear figure" is from 1550s. The mathematical sense of "number or quantity derived from a number multiplied by itself" is also from 1550s.

As "square piece; any object in roughly square form" by c. 1600. As a body of troops, 1590s. As "old-fashioned person" by 1944 (see square adj.). As short for square meal, from 1882. Square one "the very beginning" (often what one must go back to) is from 1960, probably a figure from board games.

The sense of "quadrilateral open space in a town or park" is from 1680s; that of "area bounded by four streets in a city" is from c. 1700; in England this was often an individual building but in the U.S. commonly "a block of buildings bounded by four streets" (by 1867), which made it formerly noted as one of the words used differently in the two countries. 

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 meant nearly the same thing late 18c.: "precise, formal, old-fashioned person," 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.

late 14c., squaren, of stones, "make square in shape," from Old French esquarrer, variant of escarrer "to cut square," from Vulgar Latin *exquadrare "to square," from Latin ex "out" (see ex-) + quadrare "make square; set in order, complete," from quadrus "a square" (from PIE root *kwetwer- "four"). Also probably from or influenced by the noun.

The meaning "regulate according to any given standard" is from 1530s; the sense of "accord with" is from 1590s. With reference to accounts, "balance, make even," by 1815. In 15c.-17c. the verb also could mean "deviate, vary, digress, fall out of order." Related: Squared; squaring.

1570s, "fairly, honestly," from square (adj.). The notion seems to be of rule, regularity, exact proportion, hence integrity of conduct, honest dealing. It is attested from 1630s as "directly, in line." The sense of "completely" is American-English, colloquial, by 1862.