When referring to dictionaries, there seems to be no such meaning as "quarrel" under the word "square", only "in agreement".

But in II 1 of A Midsummer Night's Dream, "square" in the following text means "quarrel":

And now they never meet in grove, or green,
By fountain clear, or spangled starlight sheen,
But they do square, that all their elves for fear
Creep into acorn cups, and hide them there.

Where does this meaning come from?

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    A question that probably would also be well-suited for Literature SE – CowperKettle Feb 18 at 11:07
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    it's simply as in "square off for a fight". – Fattie Feb 18 at 13:37
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    Just BTW, I certainly wouldn't translate that as "quarrel". That's far too soft and pansy-assed. The sense is an all out brawl, fist-fight, war, battle, slaughterfest. Nothing to do with "quarrel" – Fattie Feb 18 at 13:43

The sense of to "fall out, to be at variance or discord, to disagree or quarrel" (OED, sense 8b of square, v.; paywalled) is obsolete. It appears in historical dictionaries. OED attests the sense with quotes from 1542 to 1608.

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    Thank you. I also check the meaning of quarrel and found out that one of its meanings as a noun is "A short, heavy, square-headed arrow or bolt used in a crossbow or arbalest." And its origin is "Middle English: from Old French, based on late Latin quadrus ‘square’." I think there might be some kind of relation generated from this sense. And this kind of arrow is used around middle age, so it makes sense I guess. – kimXU Feb 18 at 2:39
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    @kimXU The derivation of "square" (to argue) from "quarrel" (square-headed arrow) feels like a folk etymology to me. Unless you have evidence of some intermediate forms I wouldn't believe it. – Martin Bonner Feb 18 at 7:52
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    @Sarriesfan More like High or Late Renaissance since it started in the 13thC and turned into The Enlightenment in the early18thC. Shakespeare was writing in the late 16th and early 17th centuries. – BoldBen Feb 18 at 11:24
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    @kimXU , there is utterly no connection to quarrel in the sense of archery. you are piling confusion on confusion. – Fattie Feb 18 at 13:39
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    Even if the use was somewhat old-fashioned in Shakespeare's time, you need to remember that he needed to find a word that would fit both the meaning and the meter, since "A Midsummer Night's Dream" is in verse. – jamesqf Feb 18 at 18:03

We actually still have versions of this usage in modern English: to square up to or against1 someone means (from Cambridge Dictionaries)

to prepare to fight, compete, or argue with someone:
The players squared up to each other and started shouting.

And to square off means (also Cambridge)

to oppose someone in a competition or prepare to fight someone:
Bradley is expected to square off with Cook in the next election.

And a square go is a Scottish term for (Collins Dictionary)

a fair fight between two individuals

This "fighting" usage of square seems to be related to the idea of facing one's opponent squarely (still Cambridge) and squaring one's shoulders (Merriam-Webster) when readying oneself for an unpleasant task, as well as other square-related concepts that have to do with forthrightness and not shirking, such as foursquare (M-W), on the square, and the other ways in which one can square up (both Collins). It may also be related to the idea of two individuals being at right angles to one another.2

The Oxford English Dictionary1 dates most of these positive senses of the word to the late 16th through the 17th century. They seem to have arisen metaphorically from the square's geometric traits: a square is upright (as opposed to any shape with non-right-angles), shows its full face (since a square seen at an angle doesn't look like a square), and is equitably arranged (since all its sides and angles are equal).

So Oberon and Titania were squaring in the sense that they were coming face-to-face with one another, and not edging away or sloping off from the fight.

1 My impression is that the to version is more British (square up to is actually listed as a UK usage in several dictionaries) and against is more American (I hear squared up against especially in US sports contexts).

2 Compare the adjective cross and the verb thwart (both OED, see note below), which both evolved from the idea of being at a right angle (to something) to, respectively, being generally angry and actively opposing someone.

3 Note that the OED Online is unfortunately a prescription service (behind a paywall). However, many folks, especially in the UK, will have access through their local public library, and most students should be able to access through their institution's library. More info available at About the OED, including, as of February 2019, a special subscription price that is available for one more month in celebration of the OED's 90th anniversary.

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    Just a note that for UK based people access to the OED is likely to be free if you are a member of your local authority/university/college library. Just use your library number at Sign-in. – Spagirl Feb 18 at 10:50
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    Also worth pointing out that a "Square go" is a common modern Scots expression for a fight. Can also be used as an expression to bait an opponent. See the following quote from Trainspotting (1996) : "When this hard cunt comes in. Obviously fuckin' fancies himself, like. Starts staring at me. Lookin' at me, right fuckin' at me, as if to say, "Come ahead, square go." " imdb.com/title/tt0117951/characters/nm0001015 – Smeato Feb 18 at 11:09
  • Is it possible "square" is related to the arena a fight or duel would take place in? – usul Feb 18 at 15:56
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    @usul given that the arenas used to be circular; eg boxing "ring", I wouldn't think it likely ... but anything is possible when it comes to language. I'd have thought it more likely to be related to the formations of armies that would have their troops in box formation - thus if they squared off they were preparing to fight – UKMonkey Feb 18 at 16:56
  • My speculation would be that it has to do with raising the fists up, as preparation for a fight (to block or to punch), creating a roughly square shape between fists and shoulders. – KRyan Feb 19 at 1:11

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