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The ancient Babylonians understood squares:


     

Plimpton 322

The ancient Athenians understood cubes, if we can take doubling the cube, i.e., the Delian problem, as evidence.

My question is:

Q. When were 4th, 5th, $\ldots$, $n$-th powers contemplated/understood/used?

I am wondering how tied was the understanding of powers/exponentiation to geometry, to spatial dimensions. Did the ancients generalize their explorations to arbitrary integer exponents?

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    $\begingroup$ In geometry, you might look at the history of curves like the parabola, cycloids and variations, tractrix, and so on. If they had any geometric understanding of higher exponents, it would most likely be through such curves. $\endgroup$
    – The Masked Avenger
    Commented Sep 19, 2014 at 1:37
  • $\begingroup$ I am not a historian, but if the people from early millenia had any concept of four or higher dimensions, I suspect it would be a near-theological conception more than an algebraic or geometric conception. $\endgroup$
    – The Masked Avenger
    Commented Sep 19, 2014 at 1:43
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    $\begingroup$ I read somewhere (maybe someone has a reference) that in the classical Greek geometry powers 4 and up never appear. Second power is area, third power is volume, and geometry is a theory to describe the real world, so higher powers are nonsense. Or something to that effect. $\endgroup$
    – Gerald Edgar
    Commented Sep 19, 2014 at 13:32
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    $\begingroup$ Maybe this is a bit naive, but after all the Babylonians you mention did use a positional numeral system with base 60. When they ever needed a number larger then 215,999, they will have implicitly used cubes. (And fourth powers if they had to count until 12,960,000.) $\endgroup$
    – Moritz Firsching
    Commented Sep 19, 2014 at 22:19
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    $\begingroup$ Not a fourth power but a product of four factors occurs in Heron's formula for the area of a triangle. $\endgroup$
    – bof
    Commented Sep 20, 2021 at 1:37

4 Answers 4

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Wikipedia's article on the Mother Goose rhyme "As I was going to St. Ives" suggests that curiosity about higher powers has existed a long time.

The Rhind Mathematical Papyrus (Problem 79), dated to around 1650 BC, could be described with the following text:

There are seven houses; In each house there are seven cats; Each cat catches seven mice; Each mouse would have eaten seven ears of corn; If sown, each ear of corn would have produced seven hekat of grain. How many things are mentioned altogether?

Edit: Interestingly, the $7^5$ hekat of grain here are counterfactual (imagined, not actually existing); as is perhaps the very notion of a number like $7^5$, to the author of the papyrus.

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  • $\begingroup$ That is a nice example of repeating an operation; it would also be interesting, when powers were perceived as "the next" operation in the sequence of addition as repeated increments by 1 and, multiplication as repeated additions of the same number. $\endgroup$
    – Manfred Weis
    Commented Sep 19, 2014 at 3:58
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    $\begingroup$ $7^5$---Nice catch! $\endgroup$
    – Joseph O'Rourke
    Commented Sep 19, 2014 at 11:05
  • $\begingroup$ This may not be the definitive answer, but it is quite suggestive and informative. And so I accept it. Thanks! $\endgroup$
    – Joseph O'Rourke
    Commented Sep 19, 2014 at 22:58
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    $\begingroup$ This is a great question and a great answer, although I wonder if there's a better translation for "ear of corn" which, presumably, were unknown to the ancient Egyptians. I've heard "grain of barley..." $\endgroup$
    – Mark S
    Commented Oct 27, 2018 at 2:47
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The cuneiform tablet MS 2351 from the 19th century BC contains the 15-digit sexagesimal number 13 22 50 54 59 09 29 58 26 43 17 31 51 06 40, which happens to equal $20^{20}$. I also seem to remember they constructed a table of reciprocals for numbers of the form $125 \cdot 2^n$ for exponents up to $19$.

Edit. Last year some fragments of a large table have been identified: Ossendrijver discovered that the complete table contained the powers of 9 up to $9^{46}$.

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    $\begingroup$ That's amazing! And in A Remarkable Collection of Babylonian Mathematical Texts by Jöran Friberg I read that tablet MS 2205 gives $20^{12}$ and $20^{16}$. Friberg thinks these numbers may represent exercises in computing square roots, although to some extent of course that remains guesswork. $\endgroup$
    – R.P.
    Commented Sep 26, 2015 at 16:28
  • $\begingroup$ It should be mentioned that the tablet containing the list of powers of $9$ is of Late Babylonian origin (which is the era from approx. 400 BC to 100 BC). In this era we also encounter mathematicians like Euclid, who of course uses powers implicitly throughout the Elements (just think of the summing of the geometric series). $\endgroup$
    – R.P.
    Commented Oct 1, 2015 at 17:21
  • $\begingroup$ So they were just one step away from the Cunningham Project, homes.cerias.purdue.edu/~ssw/cun/intro $\endgroup$
    – Gerry Myerson
    Commented Sep 20, 2021 at 1:54
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    $\begingroup$ @Gerry: Not really. They used factorization techniques for computing square roots and reciprocals, but only regular factors (those dividing powers of 60). There are a couple of examples of division with remainder by 7, 11 or 13, but no prime factorizations involving primes $> 5$. $\endgroup$
    – Franz Lemmermeyer
    Commented Sep 20, 2021 at 12:03
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The first definition of the powers beyond the third is probably in Diophantus' Arithmetica (written somewhere from 100 BC to 300 AD).

In the introduction to that work, he defines fourth powers (δυναμοδύναμεις "dynamodynameis", lit. "square-squares"), fifth powers (δυναμόκυβοι "dynamokuboi", lit. "square-cubes"), and sixth powers (κυβόκυβοι "kubokuboi", lit. "cube-cubes"). Throughout the course of the work, he uses all of these powers. Moreover, he assigns them no geometrical significance whatsoever, just as he doesn't think of squares and cubes themselves in a geometric way; e.g., he sees no problem in adding the square of the unknown to a constant, etc.

Edit: After writing the above, I checked up on Sir Thomas Heath's great treatise on Diophantus. Apparently, Heron used the same term for the fourth power as Diophantus. Since Diophantus' dates are somewhat uncertain, he may or may not have been anticipated by Heron.

A screenshot from Heath's book:

Diophantus' definition of the first few powers

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Just for the record, I thought this passage from Omar Khayyam's algebra book (p.49) should be here. In particular, it shows how hard it was to to tie the understanding of powers to geometry

I say: what algebraists call square-square is an imaginary concept in continuous quantities. It has no existence in any way in materialistic objects. For continuous quantities, the terms square-square, square-cube and cube-cube are used to denote the number (coefficient) of the object (variable)... The things that algebraists use to denote objects and quantities are: number, root, square and cube. The number has to be taken as an abstract concept. It has no existence unless it is individuated by things... Square-square, which, to the algebraists, is the product of the square by itself, has no meaning in continuous objects. This is because how can one multiply a square, which is a surface, by itself? Since the square is a two-dimensional object (geometrical figure), and two-dimensional by two-dimensional is a four dimensional object. But solids cannot have more than three dimensions. All objects in algebra are generated from these four genera. And anyone who says that algebra is a trick to determine unknown numbers is wrong. So don’t pay attention to these people. It is true that algebra and equations are geometrical things...

Edit. René's post and Joël's comment gave me some new insight about Khayyam's understanding of powers higher than three. Of course, he was aware of them as he explains how a certain equation of power 4 can be solved:

Now, whoever said: square-square plus three squares equals twenty-eight; he halved the squares then multiplied it by itself and then added the number; and took the root of the result to equal five and a half; then subtracted half the squares to get four which the square, and the square of the square is sixteen...

But, for him algebra and equations were attached to geometry. Apart from number that "has to be taken as an abstract concept", $x$ , $x^2$, and $x^3$ had geometrical meaning, side, square , and cube, respectively. Thus, immediately after mentioning the solution of the equation above, he warns the reader as follows:

...and he thought that he deduced the square of the square using algebra: is very feeble in his thinking. This is because he did not deduce the square of the square but rather he deduced the square.It is exactly as if he said: square plus three roots equals twenty-eight, then he determined the root using the second reduction, and concluded that the square of this root is the square of the square, which is a secret from which you will come to know other secrets.

All in all, it is a good example of how a "philosophical" belief could impede the advance of knowledge even for such an intelligent mind.

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    $\begingroup$ Interesting. Especially in comparison with René's post, which shows that Diophantus' point of view is closer to the modern one. $\endgroup$
    – Joël
    Commented Sep 26, 2015 at 3:22
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    $\begingroup$ Indeed interesting. It is intriguing to see how long it took even the greatest minds to treat algebra as independent of geometry. Even in as late a mathematician as François Viète (late 16th century), we find a very determined attempt to reconcile a sophisticated algebraic theory with a "dimensionalized" number system. I guess that, for a long time, what we call the number line was the most powerful source for producing numbers, and then of course the numbers did come with their own specific dimension. $\endgroup$
    – R.P.
    Commented Sep 26, 2015 at 13:21
  • $\begingroup$ I do not necessarily agree with the concluding sentence of this answer nor with the comment of @Joël. Instead, Khayyam's intelligence led him to pose and to seriously ponder an extremely important mathematical/geometric question which is quite close to issues we discuss in modern geometry and topology: What is the product of a surface with itself? It is rather ahistorical to suggest that his philosophical beliefs impeded the solution. What future historians might laugh at our "philosophical" inability to solve the P=NP conjecture? $\endgroup$
    – Lee Mosher
    Commented Sep 26, 2015 at 16:33
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    $\begingroup$ I think we must not dismiss beforehand the possibility that philosophical beliefs can stand in the way of mathematical progress. It is also not really a forceful argument that Khayyam's way of thinking somehow lines up with modern-day topology: after all, Khayyam wasn't doing topology, he was doing algebra. But other than that I agree with you: the above does not convince me that Khayyam's meta-mathematical thinking prevented him from making progress that he might "otherwise" have made. $\endgroup$
    – R.P.
    Commented Sep 26, 2015 at 17:27
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    $\begingroup$ @René Indeed, my conclusion was not just based on a few paragraphs I mentioned above. It was based on my previous reading of Khayyam. Tying algebra to geometry costed him and progress of mathematics a lot (about five hundred years for the latter). $\endgroup$
    – Amir Asghari
    Commented Sep 26, 2015 at 18:08

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