By The People

There is a pair of signs in the men's changing room at the gym, one in English, the other French, which say approximately the same thing. I didn't take a picture of them, because nobody wants to see a guy running around a locker room with a camera (and because there's another pair of signs which say you can't use cell phone in the changing room, for the obvious reason that all cell phones nowadays have cameras in them), but they're asking you to refrain from fuming up the place with colognes and such, not that anybody pays attention to it--the place stank of Axe or the like when I got in this morning.

The English sign reads

Please respect our scent free policy

and the French reads

Veuillez respecter notre politique sans parfum

so "policy" in English is "politique" in French, and damned if that doesn't look just like "politic" in English, and therefore they must be related, right?

Right. And what a chain of evolution it is, too.

"Policy", meaning "an official course of action", is descended from
French "policie" (unsurprisingly), "civil administration", from
Latin "politia", "the state", from
Greek "politea", "the state", from
Greek "polites", "citizen", from
Greek "polis", "city", from
Indo-European "pelh-", "a fortified citadel", from
Proto-Indo-European "p(o)lh-", "an enclosed space".

"Politics", "the science of governance", is from French "politique", originally "political", which is from Latin "politicus", "of the state", which then follows the previous line of descent to and beyond "polis", which, you will be unsurprised to learn, is also the source of "metropolis" (literally "mother-city") and "police".

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Now, if "politique" and "policy" are related, and the aim of the gym's policy is to avoid incommoding other members--in other words, to be polite--then surely "politique/policy" and "polite" are related, yes?

No, believe it or not. "Polite" is from an entirely different IE root, "pel-" (one of many), which meant "to beat". Eventually one of its meanings was "to beat woollen cloth to full it", which has the effect of making it smoother, and this gave rise to the Latin verb "polire", "to polish, to make smooth", which eventually made it into French in the form "polir", with the same meaning (where it remains today): one of its verb forms, "poliss-", gave English "polish". The adjective "politus", "polished", led directly to English "polite"; exhibiting polished, refined behaviour. No relation to policy of any sort.

However, it may interest you to know that "interpolate" is related to "polish": literally meaning "to polish up", it's from Latin "interpolare", "to touch up", and means "to alter [and presumably improve] a text by inserting material into it".