Etymology
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square (n.)

mid-13c., "tool for measuring right angles, carpenter's square," from Old French esquire "a square, squareness," from Vulgar Latin *exquadra, 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").

Meaning "square shape or area" is recorded by late 14c. (Old English used feower-scyte). Geometric sense "four-sided rectilinear figure" is from 1550s; mathematical sense of "a number multiplied by itself" is first recorded 1550s. Sense of "open space in a town or park" is from 1680s; that of "area bounded by four streets in a city" is from c. 1700. 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.

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square (adj.)

early 14c., "containing four equal sides and right angles," from square (n.), or from Old French esquarre, past participle of esquarrer. Meaning "honest, fair," is first attested 1560s; that of "straight, direct" is from 1804. Of meals, from 1868.

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 late 18c.: "precise, formal, old-fashioned person," from the style of men's shoes worn early 18c. and then fallen from fashion. Squaresville is attested from 1956. Square dance attested by 1831; originally one in which the couples faced inward from four sides; later of country dances generally.

[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]
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square (v.)
late 14c. of stones, from Old French esquarrer, 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"). Meaning "regulate according to standard" is from 1530s; sense of "to accord with" is from 1590s. With reference to accounts from 1815. In 15c.-17c. the verb also could mean "to deviate, vary, digress, fall out of order." Related: Squared; squaring.
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square (adv.)
1570s, "fairly, honestly," from square (adj.). From 1630s as "directly, in line." Sense of "completely" is American-English, colloquial, by 1862.
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four-square (adj.)
also foursquare, c. 1300, "having four equal sides," from four + square (adj.). As an adverb, in figurative use, "forthrightly, honestly" from 1845.
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squared (adj.)
late 14c., "made square," past-participle adjective from square (v.). Meaning "drawn up in squares" is from 1660s. Of numbers, "multiplied by itself," from 1550s.
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squarely (adv.)
1540s, of multiplication, from square (adj.) + -ly (2). From 1560s as "in a straightforward manner;" meaning "firmly, solidly" is from 1860.
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Sloane Square 
neighborhood near Chelsea in London, named for Sir Hans Sloane who purchased the manor of Chelsea in 1712 and whose collections contributed to the British Museum. Previous to development the place was known as Great Bloody Field ["Oxford Dictionary of London Place Names"]. Sloane Ranger attested from 1975, with a play on Lone Ranger.
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rectangle (n.)

in geometry, "quadrilateral plane figure having all its angles right and all its opposite sides equal," 1570s, from French rectangle (16c.), from rect-, combining form of Latin rectus "right" (from PIE root *reg- "move in a straight line," with derivatives meaning "to direct in a straight line") + Old French angle (see angle (n.)). Late/Medieval Latin rectiangulum meant "a triangle having a right angle," noun use of neuter of rectiangulus "having a right angle." When the adjacent sides are equal, it is a square, but rectangle usually is limited to figures where adjacent sides are unequal.

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*kwetwer- 
Proto-Indo-European root meaning "four."

It forms all or part of: cadre; cahier; carillon; carrefour; catty-cornered; diatessaron; escadrille; farthing; firkin; fortnight; forty; four; fourteen; fourth; quadrant; quadraphonic; quadratic; quadri-; quadrilateral; quadriliteral; quadrille; quadriplegia; quadrivium; quadroon; quadru-; quadruped; quadruple; quadruplicate; quarantine; quarrel (n.2) "square-headed bolt for a crossbow;" quarry (n.2) "open place where rocks are excavated;" quart; quarter; quarterback; quartermaster; quarters; quartet; quarto; quaternary; quatrain; quattrocento; quire (n.1) "set of four folded pages for a book;" squad; square; tessellated; tetra-; tetracycline; tetrad; tetragrammaton; tetrameter; tetrarch; trapezium.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit catvarah, Avestan čathwaro, Persian čatvar, Greek tessares, Latin quattuor, Oscan petora, Old Church Slavonic četyre, Lithuanian keturi, Old Irish cethir, Welsh pedwar.
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