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Good people!

I'm presently in the process of putting something together on Euler and Gauss and cyclotomy and modular arithmetic, and I noticed that when it comes to the terminology primitive root modulo $n$, that particular phrase seems to come from Euler's Demonstrationes circa residua ex divisione potestatum per numeros primos resultantia from 1774. Euler there isn't actually concerned with the algebraic roots of any equations, rather he is looking at geometric progressions, and that which we moderns call the ratio of a geometric progression, Euler dubs the root of the same.

Now this made me go, "Hang on, haven't I read somewhere in some textbook on geometric progressions or sequences someone write something along the lines of 'and the $r$ in the expression $ar^k$ stands for the common ratio or root'?" But looking as much as I have, I have yet to see anyone use the term root as opposed to ratio when discussing geometric sequences.

If this was something that was common terminology in the 18th century, I have been unable to verify it. Indeed, in the German language edition of Euler's Elements of Algebra, he calls it the Nenner, literally, the denominator of the sequence. Still, I cannot shake the feeling that I have read the term root used for the ratio somewhere at some point.

Do anyone of you know of any instance in which the term root (Latin radix, radicis, radici, radicem, radice depending on declension) in the 18th century or at any other point, or does this appear to just be terminology that Euler used for this one particular paper in 1774?

Look forward to your responses!

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