In mathematics, powers of 2 and 3 are often referred to using "square" and "cube" terminology: a "number squared/cubed", the "square/cube root of a number". But then you have "quadratic and cubic" Beziér curves. Why is the confusing term "quadratic" used when "squaric" would be more consistent? Alternatively, why isn't the term "hexatic" used instead of "cubic"?

• Now there's a can of worms. Why "hexatic"? If we're looking at edges, wouldn't that be a pyramid and a cube "dodecic"? Vertices? Then you get a quadratic pyramid and an octic cube. And would a square be two or three dimensional? "Two faced squares" sounds a bit Kerouac to me... Commented Feb 15, 2023 at 12:19
• The Latin for square was "quadra" and later "quadratus", and for cube was "cubus" so quadratic and cubic are reasonably consistent in their English etymology (there are steps through Old French) Commented Feb 16, 2023 at 0:47
• Fun fact: in German, equations/polynomials can be "quadratisch" or "kubisch". so, the same as English. But while a square is a "Quadrat", and there isn't really an alternative word, noone outside math would call a cube a "Kubus", everyone uses "Würfel" ("dice"). Commented Feb 16, 2023 at 6:53
• In a lot of modern (and dead!) European languages square is some form of "quadrat" Commented Feb 16, 2023 at 7:27
• By the way, to answer my own hypothetical question (`why does the word "square" exist when quadrilateral already covers it?`): It is from the Old French `esquarrer`, which was a stone cutting term to mean "to cut a block into an equilateral quadrilateral" Commented Feb 16, 2023 at 21:44

It is an old usage:

1650s, "square," with -ic + obsolete quadrate "a square; a group of four things" (late 14c.), from Latin quadratum, noun use of neuter adjective quadratus "square, squared," past participle of quadrare "to square, make square.

(Etymonline)

Terms of Latin origin were often adopted in the 16th/17th c.

“During the English Renaissance, from around 1500–1650, some 10,000 to 12,000 words entered the English lexicon, including the word lexicon. Many of these words were borrowed directly from Latin, both in its classical and medieval forms. In turn, Late Latin also included borrowings from Greek.

(Wikipedia)

Note that also squaric is used but with the following sense:

Squaric acid, also called quadratic acid because its four carbon atoms approximately form a square, is a diprotic organic acid with the chemical formula C4O2(OH)2.

(Wikipedia)

• So mathematics actually went from consistent terminology to inconsistent terminology? lol
– Jez
Commented Feb 15, 2023 at 11:33
• @Jez Not math, the English language adopted this terminology. In the 17th century Latin terms were sort of fashionable in English. Commented Feb 15, 2023 at 11:35
• Lots and lots of terms in maths are nonsensical unless you know the definition (or to put it positively, they only make any kind of sense along with their definition). Take all terms based on names of their inventors/proponents ("Abelian Group", "Hilbert Space" etc.). Totally inconsistent and nonsensical. This is nothing particularly fancy or lol-worthy, it's just the way it is. @Jez
– AnoE
Commented Feb 16, 2023 at 8:15
• @Jez Inconsistent mixing of terms from Latin/French and Saxon origin is a pervading universal trait of English. Why do you say "eye" but "ocular"? Commented Feb 16, 2023 at 8:54
• @AnoE: There's also the fact that some mathematicians were rather prolific and consequently have rather a lot of things named after them. Commented Feb 16, 2023 at 9:08

Simple answer: quadra- is a Latin root meaning "square". (And cub- is a Latin root meaning... "cube".)

So it is consistent—give or take using Latin roots.

• @Jez: "Quadratic" refers to the shape, and "cubic" refers to the shape. They're both the Latin names for the shapes. I'm not quite sure what's inconsistent? Commented Feb 15, 2023 at 11:36
• @Jez: No, it just happens that the Latin word for "square" derives from the word for "four", but the Latin word for "cube" doesn't happen to derive from a number. I'm not sure where "hexatic" comes in - that would presumably derive from the Greek word for "six", but having a Latin word for the 2 case and a Greek one for the 3 case doesn't seem any more consistent to me! Commented Feb 15, 2023 at 12:23
• @Jez: "sextic" is a word, and it's used for equations with x^6. The equivalent for x^4 is "quartic". Using an "English" word in one context, and a Latin/Greek-derived word in another--but referring to the same thing--is very common (eg we speak with a "voice" but the study of that is "phonology" (phon- being Greek for "voice")). (I put "English" in quotes because, in fact, both "square" and "voice" derive, indirectly, from their Latin equivalents...) Commented Feb 15, 2023 at 13:14
• @Jez: I'm not sure which terminologies you're talking about? As I've said, "quadratic" and "cubic" are derived exactly the same - both come from the Latin words for the corresponding shapes (square and cube, respectively). So that's consistent, right? Commented Feb 15, 2023 at 14:32
• @Jez:You seem to keep changing what you're complaining about. Yes, we use the "English" words (square/cube) in one context--when raising to a power--and the Latinate words (quadratic/cubic) in another context--talking about the degree of an equation. Yes, it happens that (a) the "English" words both derive from the Latin ones, and (b) the ones for cube happen to be the same. But as I've already mentioned--it's very common to use an "English" word in one context and a Latin-based one in another; this is just an example of that. It's no more inconsistent than the rest of the English language :) Commented Feb 15, 2023 at 16:42

I will suggest that your question is backwards. Rather than asking "why is the word quadratic used", you might astutely inquire "why is the word square in use?"

In Old French,`esquarrer` was a term meaning "to cut a block of stone into an equilateral quadrilateral."

As others have suggested, `quadra` is from the Latin word for "four" (`quattuor`) but the French invaded the British Island from Normandy because they caught wind that Harolð Harðráði had taken wine to a dinner party.

Then they brought their Frenchy-French Frenchiness into the mix and now we have the word "square" because they wanted to cut stones into cubes.

• Random, useless knowledge to add since we're talking about Latin numbers: if you ever wonder why some people pronounce the number 101 in word form as "one hundred and one" I point you to the Latin for the number: `centum et unus` ...God bless the Romans; terrific. Commented Feb 16, 2023 at 22:02
• Wait, is there another widely used formal way to say 101?! I'm pretty sure my dialect only has "a hundred and one" and "one hundred and one," which is also found in titles like One Hundred and One Dalmatians and One Thousand and One Nights. Commented Feb 18, 2023 at 1:53
• @Vectornaut in American English, teachers instruct that introducing the word "and" into the pronunciation of a number is improper. "One hundred and twenty seven" is instructed to be read "One hundred twenty seven." The special distinction I mentioned exists for the number "One hundred and one" being that it is `centum et unus` in Latin, but English teachers tend towards overenforcement of the "inappropriate 'and' rule" teaching it instead as "one hundred one." Commented Jul 18, 2023 at 13:13
• Weird—I don't recall ever hearing a teacher say that! I'd love to know where and when you've encountered it. I went to school in the Midwest in the 1990s–2000s. Commented Jul 21, 2023 at 21:54
• I never thought to investigate how I use "and" in three-digit numbers; it's more complicated than I expected! I've found some examples (101, 105, 112, 117, 180) where the "and" feels almost obligatory to me, although it sometimes seems to drop if I speak very quickly. I've also found examples (365, 542) where the "and" seems optional, and tends to be absent or very clipped by default. There are even a few (156, 256, 756) where the "and" seems distinctly disfavored. I'm guessing that prosody drives a lot of the variation. Commented Jul 21, 2023 at 22:23