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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.

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]

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.

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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Definitions of square from WordNet
1
square (v.)
make square;
square the wood with a file
Synonyms: square up
square (v.)
raise to the second power;
square (v.)
cause to match, as of ideas or acts;
square (v.)
position so as to be square;
He squared his shoulders
square (v.)
be compatible with;
one idea squares with another
square (v.)
pay someone and settle a debt;
I squared with him
square (v.)
turn the paddle; in canoeing;
Synonyms: feather
square (v.)
turn the oar, while rowing;
Synonyms: feather
2
square (n.)
(geometry) a plane rectangle with four equal sides and four right angles; a four-sided regular polygon;
you can compute the area of a square if you know the length of its sides
Synonyms: foursquare
square (n.)
the product of two equal terms;
gravity is inversely proportional to the square of the distance
Synonyms: second power
square (n.)
an open area at the meeting of two or more streets;
Synonyms: public square
square (n.)
something approximating the shape of a square;
square (n.)
someone who doesn't understand what is going on;
Synonyms: lame
square (n.)
a formal and conservative person with old-fashioned views;
Synonyms: square toes
square (n.)
any artifact having a shape similar to a plane geometric figure with four equal sides and four right angles;
a checkerboard has 64 squares
square (n.)
a hand tool consisting of two straight arms at right angles; used to construct or test right angles;
the carpenter who built this room must have lost his square
3
square (adj.)
without evasion or compromise;
a square contradiction
square (adj.)
leaving no balance;
my account with you is now all square
square (adj.)
having four equal sides and four right angles or forming a right angle;
a square peg in a round hole
a square corner
square (adj.)
providing abundant nourishment;
four square meals a day
square (adj.)
rigidly conventional or old-fashioned;
Synonyms: straight
square (adj.)
characterized by honesty and fairness;
a square deal
Synonyms: straight
4
square (adv.)
in a straight direct way;
ran square into me
Synonyms: squarely
square (adv.)
in a square shape;
folded the sheet of paper square
Synonyms: squarely
square (adv.)
firmly and solidly;
planted his great bulk square before his enemy
Synonyms: squarely
From wordnet.princeton.edu