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?



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".

  • $\begingroup$ Great comments, thank you! But does the word "power" or any other word with the same root have this kind of meaning for potentiality? $\endgroup$ – მამუკა ჯიბლაძე Dec 23 '15 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. $\endgroup$ – მამუკა ჯიბლაძე Dec 28 '15 at 1:40
  • 2
    $\begingroup$ The English word "power" is in fact cognate with Latin "potentia". $\endgroup$ – fdb Feb 20 '16 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? $\endgroup$ – მამუკა ჯიბლაძე Jun 19 '18 at 7:06

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