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by pash
3612 days ago
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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.) |
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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.