c.1300, "tool for measuring right angles," from Old French esquire "a square, squareness," from Vulgar Latin *exquadra, from *exquadrare "to square," from Latin ex- "out" (see ex-) + quadrare "make square," from quadrus "a square," from quattuor "four" (see quart). Meaning "rectangular shape or area" is recorded by late 14c.; replaced Old English feower-scyte. Sense of "open space in a town or park" is from 1680s. The mathematical sense of "a number multiplied by itself" is first recorded 1550s. The verb is first attested late 14c.; with reference to accounts, from 1815.
c.1300, "containing four equal sides and right angles," from square (n.). Meaning "honest, fair," is first attested 1560s; that of "straight, direct" is from 1804. Sense of "old-fashioned" is 1944, 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 in 1771, from a style of shoes then fallen from fashion.) Squaresville is attested from 1956. Square one "the beginning" is first recorded 1960, probably from board games; square dance first attested 1870.