In article <email@example.com>, firstname.lastname@example.org (Robert Israel) writes: |> Euler never used that word. J.J Sylvester coined it around 1883, and |> he was writing in English. He doesn't say where he took it from, but it could |> just as plausibly be from the letter "t" (Sylvester used \tau n for \phi(n)) |> + the ending of "quotient".
Correction. Sylvester first used the word in "On Certain Ternary Cubic-Form Equations", Amer. J. Math 2 (1879) 280-285, 357-393, in Sylvester's Collected Mathematical Papers vol. III p. 321. He writes: "The so-called \phi function of any number I shall here and hereafter designate as its \tau function and call its Totient".
My suggestion that the "t" came from "tau" was rather silly, and in fact Sylvester says (in vol. IV p. 589 of his Collected Mathematical Papers) "I am in the habit of representing the totient of n by the symbol \tau n, \tau (taken from the initial of the word it denotes) being a less hackneyed letter than Euler's \phi, which has no claim to preference over any other letter of the Greek alphabet, but rather the reverse".
BTW, according to Dickson's "Theory of Numbers", it seems Euler didn't use \phi either, but rather \pi, and Gauss introduced \phi (Disq. Arith., article 38).
Robert Israel email@example.com Department of Mathematics (604) 822-3629 University of British Columbia fax 822-6074 Vancouver, BC, Canada V6T 1Y4