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Having done some searches on the internet, seems like the term "power" is a mistranslation. The Wikipedia article links to an article in the MacTutor History of Mathematics archive where it is written

The notation and terminology for powers and exponents is interesting. Power is first used for the square. Euclid uses the phrase in power , for example he says that magnitudes are commensurable in power when their squares are commensurable. Of course Euclid thought geometrically and the square to him was the geometrical square not a new number formed by multiplying the number by itself. Henry Billingsley, the first English translator of Euclid in 1570, makes the definition precise in his translation of Euclid's Second book:- The power of a line is the square of the same line.

Are these the intended meanings? For the first instance (concerning commensurability) -

εὐθεῖαι δυνάμει σύμμετροί εἰσιν, ὅταν τὰ ἀπ᾽ αὐτῶν τετράγωνα τῷ αὐτῷ χωρίῳ μετρῆται

I could not find the corresponding place in the Book 2 for Billingsley's translation.

I would translate "δυνάμει" in the above context as "potentially", which must be closely related to "δύνατον" ("possible") frequently used across Euclid's books. Cf. German "Potenz" for "power".

Thus although "potence" is sort of a synonym for "power", that another meaning of potentiality is lost when using "power" I think.

Is my interpretation correct? Are there any authoritative opinions on this?

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    $\begingroup$ Note that the syntax of δυνάμει in that definition is dative of respect, so a literal translation of δυνάμει σύμμετροι would be "commensurable in respect to power". $\endgroup$ Commented Oct 6, 2023 at 5:46

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Comments

The correct source is Euclid : Book X, Def.2.

For δυνάμει, it is a word used in Aristotle : see Potentiality and actuality and Aristotle's Metaphysics : Actuality and Potentiality :

the Aristotelian distinction between potentiality (dunamis) and actuality (entelecheia or energeia). This distinction is the main topic of Book $\Theta$.

Thus, from the Greek dunamis to the Latin potentia and finally to power.

We can transalte dunamei summetros as "potential commensurability" : two segments of lenght $1$ and $\sqrt 2$ are incommensurable but "commensurable in square".


See this post on Aristotelian terminology, dynamis and entelechy for some comments on the multiplicity of meaning of dunamis.

Thus, if dunamis stands for potentiality and capacity, it has some "nuances" that also power has : "ability to act or do".

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    $\begingroup$ Great comments, thank you! But does the word "power" or any other word with the same root have this kind of meaning for potentiality? $\endgroup$ Commented Dec 23, 2015 at 19:54
  • $\begingroup$ It just occurred to me that Aristotle's notions cited by you became reflected in physics in a still different form, - cf. force (rather than power) and energy. $\endgroup$ Commented Dec 28, 2015 at 1:40
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    $\begingroup$ The English word "power" is in fact cognate with Latin "potentia". $\endgroup$
    – fdb
    Commented Feb 20, 2016 at 19:29
  • $\begingroup$ @fdb Sorry only noticed your comment now. So can the word "power" be used to express potentiality? For example, could "in power" mean "potentially" in some context? $\endgroup$ Commented Jun 19, 2018 at 7:06
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    $\begingroup$ @მამუკაჯიბლაძე, Not sure. One English version of the term omnipotent used about a god (or about God), is "all-powerful". But another one is "almighty". He is able to (potentially) do anything. $\endgroup$ Commented Apr 17, 2023 at 12:31
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The Greek word δύναμις is, I believe, an ordinary (Ancient) Greek word meaning power, ability, strength, etc. The first entry in Liddell/Scott/Jones (click on LSJ) cites Homer:

ἦ τ᾽ἂν ἀμυναίμην, εἴ μοι δύναμίς γε παρείη. (Homer, Od. 2.62)
I would defend myself if the power were in me. (transl. Lattimore)

Liddell/Scott/Jones in entry V gives the mathematical meaning as "power...usu. second power" and cites a passage in Aristotle's Metaphysics (1019b) that gives a derivation. Plato also uses δύναμίς to mean squaring in the Republic (587d) and Timaeus (31c/32a), but something else in Theatetus (147e et seq.) (see below). I do not know of older references that might suggest how far back the mathematical meaning goes.

Aristotle connects the mathematical meaning of δύναμις to the fact that "it is impossible [ἀδύνατον] that the diagonal of a square should be commensurable with the sides" (transl. Tredennick). The word ἀδύνατον has the same root as δύναμις, of course, and Aristotle claims δύναμις derives its meaning as "square" as an extension of the geometric fact. While that may have been clear to his contemporary readers, I have to guess at what is meant. My guess, in somewhat modern terms, is that a number is δύναμις if a square with an area equal to a (whole) number of unit squares has a root (side) equal to a number of units, that is, we have the power or ability to construct such a square from multiple-unit lengths. My explanation does not fit quite as neatly with the incommensurability of the diagonal as I would like, but it seems plausible to me.

The passage in Theatetus seems relevant and opposes this. There δύναμις is ultimately defined to be the sides of squares that are not integers (i.e., in modern terms, irrational square roots of integers). I cannot resolve this conflict. Nonetheless, the passage clearly connects "power," squares, and irrational roots. One translator (sorry, I've lost the reference) suggested that the mathematical meaning of δύναμις might have been in flux in Plato's time.

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  • $\begingroup$ But does not ἀδύνατον in the meaning of impossibility appear in completely different contexts, in texts not having to do anything with squaring, geometry, numbers, or in general power in any of its meanings? I mean, you similarly may interpret "it cannot be" as "it is unable to be", "has no power to be" but... $\endgroup$ Commented Oct 5, 2023 at 19:05
  • $\begingroup$ @მამუკაჯიბლაძე Don't both ἀδύνατον and δύνατον appear in many contexts? I'm afraid I don't really understand what you're getting at. I'm suggesting that Aristotle means that to describe numbers for which it is possible to find an integer square root, such as 4, 9, 16, etc., the term δύναμίς was settled on by, say, mathematicians, because one has the power to expresse the square root of such numbers. I don't see how any other contexts are relevant. (I'm probably misunderstanding your point.) $\endgroup$
    – Michael E2
    Commented Oct 5, 2023 at 19:36
  • $\begingroup$ For instance mathematicians settled on "derivative" in the 18th cent., even though its meaning as something derived appears in many different contexts not having to do with calculus or analysis. $\endgroup$
    – Michael E2
    Commented Oct 5, 2023 at 19:38
  • $\begingroup$ N.B. The normal nominative form δύναμις only has the one accent on the antepenult; in the line of Homer that you quote, the second accent occurs because the word is followed by the enclitic γε. $\endgroup$ Commented Oct 6, 2023 at 5:55
  • $\begingroup$ @AlexanderCampbell Thanks, and thanks for the correct form I could paste. $\endgroup$
    – Michael E2
    Commented Oct 6, 2023 at 10:46

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