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Whence did the verbs 'square' and 'cube', in the sense (if there exist others) of 'to the second power' and 'to the third power' respectively, originate? There is some degree of similarity between the verbs and the nouns, but not really a great deal, seems more of an evocative relation.

While not strictly applicable to this site, more for a sense of the age of the terms, do any other Romantic or Germanic languages use the same constructions for their powers?

And finally, given the pattern, is there any precedent at all for tesseract or one of the myriad other words for a 4-cube being used to indicate the fourth power? This may be too esoteric, but it is, strictly speaking, English usage, so it probably belongs here more than the math board.

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You should split this up into 2 questions, 'cube as verb' and 'provenance of tesseract' (leave out the multilanguage one...that will be more appropriate for the linguistics.SE site (which you should commit to!)). – Mitch Aug 24 '11 at 13:34
The Greeks definitely talked about triangular numbers and square numbers, so the English may in fact be a direct translation (I don't know Greek or Latin so I can't verify this). – Peter Shor Aug 24 '11 at 13:41
@Peter Shor - that's from the different field of mathematics. In short - triangular numbers are the amount of pebbles you can use to lay out a triangle. 1 is triangular - 1 pebble is assumed to be a triangle. You add 2 pebbles to first one as a base - making 3 a triangular number. Then you add 3 more as new base and so on. Square numbers are essentially the same - you lay out squares instead of triangles. But I believe you are correct that Greek mathematics had the most influence in naming second power as square and third as cube. – Philoto Aug 24 '11 at 14:45
@Philoto: If a triangular number is the number of pebbles laid out in rows to make a triangle, and a number squared is the number of pebbles laid out in rows to make a square (with the specified size length), how's that different? – Ben Voigt Aug 26 '11 at 23:50
@Ben Voigt Triangular numbers are 1, 3, 6, 10, etc - nowhere near cubes. Square numbers happen to be all squares, that's true. But this discussion is better suited to, we've went way offtopic here. – Philoto Aug 29 '11 at 6:00

4 Answers 4

Why do you think there is "not really a great deal" of similarity between the noun concepts of 'square' and 'cube' and the verbs for raising to the 2nd and 3rd powers? The link is pretty clear. If you have a square of side N then its area is N-squared: if you have a cube of side N then its volume is N-cubed.

We also call N-squared the square of N so it seems simple enough that when going from N to N-squared we are squaring N.

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The Oxford English Dictionary gives its first use of square in the 1560s, writing:

a. To multiply (a number) by itself.

?a1560 L. Digges Geom. Pract.: Pantometria (1571) i. xxx. sig. Iiv, Now square 2400 pase, so haue you 5760000.

?a1560 L. Digges Geom. Pract.: Pantometria (1571) ii. xii. sig. Nijv, The number proceeding of the perches squared.

It says that the etymology of the word itself is even earlier, and that there are several known spellings:

Forms: Also ME squaryn, sqvare, sqware, 15 squyer.

Etymology: < Old French esquarrer (escarrer, equarrer), = Portuguese esquadrar, Spanish escuadrar, Italian squadrare < popular Latin *exquadrāre, < Latin ex out + quadra square. Old French had also esquarrir (escarrir, etc., modern French équarrir).

The first known use of the noun is in 1557:

1557 R. Record Whetstone of Witte sig. Giiiv, Twoo multiplications doe make a Cubike nomber. Likewaies .3. multiplications doe giue a square of squares.

For cube, the OED says that the etymology is

Etymology: corresponds to French cuber (1554 in Hatzfeld & Darmesteter) and probably modern Latin cubāre, < Latin cubuscube n.

As a verb, the first recorded use is given as

1588 C. Lucar tr. N. Tartaglia Three Bks. Shooting 62, I did cube those foure ynches and the Cube thereof was 64.

Note that the noun cube was used to refer to a geometric solid in the 1300s, but the use of a number to its third power dates only to 1557:

  1. Arith. and Algebra. The product formed by multiplying any quantity into its square; the third power of a quantity.

1557 R. Record Whetstone of Witte sig. Civ, When I saie twoo tymes twoo, twise, maketh 8. that number is a sounde number: and is named a Cube.

As @GEdgar mentioned, there is a word for taking a number to its fourth power. The only recorded use of this in the OED is in the 1600s, with no further examples since:

biquadrate trans. To raise (a number) to its fourth power.

1694 Philos. Trans. (Royal Soc.) 18 70 Performed by squaring, cubing, biquadrating, etc. of the terms.

The use of the associated noun has recorded use after this into the 1800s, but it is likely that it fell out of use following this:

The square of the square (power or root); the fourth power in arithmetic and algebra; = biquadratic adj. and n.

1706 Phillips's New World of Words (ed. 6) , Biquadrate,‥the fourth Power in Arithmetic and Algebra.

1806 C. Hutton Course Math. I. 171 Its‥cube (a3), or biquadrate (a4).

1806 C. Hutton Course Math. I. 203 The biquadrate root of 16a4 − 96a3x + 216a2x2 − 216ax3 + 81x4.

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Etymonline gives for "cube" :

The verb is 1580s in the mathematical sense;

For "square" :

The mathematical sense of "a number multiplied by itself" is first recorded 1550s. The verb is first attested late 14c.

I dabble a little in mathematics, but one thing's clear, there is no "special" term for a number to the power of four. It's just stated as "x to the power of 4" This is done for all numbers above three.

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Oxford English Dictionary has

biˈquadrate, v.
trans. To raise (a number) to its fourth power.

dating from 1694.

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Undoubtedly true, but as this NGram suggests, it's been virtually unheard-of since the mid 1800s. And I suspect most of the few usages recorded in the last century are simply references to earlier instances. – FumbleFingers Aug 24 '11 at 18:23
@FumbleFingers - unfamiliar to you mere 3-dimensional beings perhaps – mgb Aug 25 '11 at 4:46

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