[pash]

I did not mean to imply that Wikipedia was my source for the etymology of even and odd as classifications of functions. (I hadn't really meant to make an etymological point at all, and I added the link only so that readers unfamiliar with the concept of Taylor series might be enticed to click through and learn something.)

I don't have any references for you right now, but I would be surprised if the usage of the terms doesn't pre-date your citation of Euler by at least several decades. The basic technique for deriving Taylor series is known to date to at least 1671, to the correspondence between James Gregory and John Collins; it also shows up in letters from de Moivre to Johann Bernoulli in 1694 and from Leibniz to Bernoulli in 1708. Newton had a geometric means of deriving the coefficients of power series when he was writing the Principa (published in 1687), as demonstrated by the proof of his tenth proposition, and he included a description of the algebraic technique in an early draft of his Quadrature of Curves (but removed it before publication in 1706). So Taylor series were broadly known to Europe's prominent corresponding mathematicians in the first decades of the eighteenth century, and to some of them decades earlier.

And the basic concept of power series, as well as the power-series representations of many common functions, were already widely known in the latter decades of the seventeenth century. Like I said, I don't have a reference for you at the moment, but I have always heard that the terms odd and even derive from the power-series representations of functions, and it's difficult to imagine that no one before Euler had noted that the power series of some functions involve only odd or only even powers—although one may more readily imagine that someone noted it while neglecting to name it.

Anyway, Taylor series and the more basic concept of power series certainly were known to Euler when he wrote the paper you cited. Since the power series of polynomial functions are the polynomials themselves, you're right that Euler would not have had power series in mind in the context of this paper. (But, no, Euler would not have been thinking in terms of smoothness—mathematicians up until the analytic enlightenment of nineteenth century played fairly fast and loose with derivatives. The point you made about ratios of functions is off-base, I think: the paper is about reciprocal solutions to polynominal equations.)

The question is simply whether your citation is evidence that Euler first coined the terms for the concept in the context of this paper (i.e., at that time and referring specifically to polynomials), or whether the concept and terminology already existed. If not, when was it generalized to refer to functions other than polynomials of finite order? I don't know. Perhaps nobody bothered to give the concept a name until Euler in 1727, and perhaps nobody even considered the concept with regard to a more general class of functions until even later. (I have a source book around here somewhere that might say something about the matter. ...)

In any event, if you asked a sample of mathematicians today why functions are called odd or even, I'd bet that many of them would point to power-series representations. That's what I meant to point out in my earlier comment.

[thaumasiotes]

> I would be surprised if the usage of the terms doesn't pre-date your citation of Euler by at least several decades

The citation is pretty suggestive, in that the text goes "functions, which I call even, which have this property...". That doesn't mean the terminology is original to Euler, but it does mean it can't have been established at the time he wrote.

> if you asked a sample of mathematicians today why functions are called odd or even, I'd bet that many of them would point to power-series representations

Eh. The terms apply just as much to nondifferentiable functions. I would have just said that it comes from the properties of polynomials with terms of all even or all odd degree. I've never thought of infinite polynomials as having any special place in the categorization of functions as even or odd; the concept is generally introduced with finite ones like f(x) = x^2 . Years later, when you learn about Taylor series, it gets pointed out that sine and cosine form polynomials with terms of only odd or only even degree, and -- look at that -- they conform to the definition of odd and even functions. (I'm not making a claim about the origin of the concept, but I am making a claim about how it's viewed today.)

[pash]

Yes, that's more or less what I meant, though from what I understand of pre-modern mathematics, I still think it's likely that power-series representations were the inspiration for applying the terms odd and even to functions. ...

But is it even clear that Euler's usage of "functiones pares" etymologically corresponds with our "even functions"? In studying classical Latin for four years, I never encountered a usage of "par" that would be translated as "even" in the numerical sense, but admittedly I know next to nothing about mathematical Latin of the eighteenth century. Reading Euler's words, if I were not aware of the modern English usage of even to describe the functions he named, I imagine I might translate his phrase as "equal functions". That translation seems to capture the idea that such functions have the property that f(x) equals f(-x), or that the functions' values are the same on both sides of the y-axis.

Particularly since Euler did not name what we now call odd functions, it does not seem clear to me that his Latin usage is etymologically related to our modern terms at all. ... Do you know whether par and impar were the Latin terms used in Euler's era to refer to even and odd integers?

(To counter my own objection, yes, the English even, of Germanic origin, does have several meanings that are close to the main meaning of Latin par as equal, e.g., in "even odds" or "an even split". Presumably that's the origin of the mathematical meaning of even in English: an even number is one that can be split into two equal numbers whose sum is the original.)

[thaumasiotes]

> But is it even clear that Euler's usage of "functiones pares" etymologically corresponds with our "even functions"?

Well, the etymology of "even" doesn't trace to the Latin word "par", or any cognate or ancestor of it. Is it clear that Euler's usage of "pares" is the same sense as our "even functions"? Pretty clear.

The "words" Latin dictionary ( http://archives.nd.edu/whitaker/words.htm ) includes the following gloss of 'par': "s:even, divisible by two". Perseus ( http://www.perseus.tufts.edu/hopper/morph?l=par&la=la#lexico... ) seems to cite this sense back to Horace, glossing the phrase "ludere par impar" as "play 'even and odd'".

I'll also note that par and impar are the Spanish equivalents today of even and odd.

All that, together with the fact that it was translated into English as "even", seems like a pretty strong case that that's what Euler meant.

[pash]

Yep, I looked it up too, and indeed par and impar seem to have been the standard Latin terms for even and odd in the mathematics of the era. See, e.g., Clavius's 1574 translation of Euclid's Elements [0].

0. https://books.google.com/books?id=4Ks7AAAAcAAJ&pg=PA8&lpg=PA...