I’m starting a Classical Latin course and I noticed that proelium, meaning battle, sounds very much like English broil/embroil, and could plausibly have some etymological relationship.
Is it just a coincidence, or am I onto something here?
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We don't know, but it's unlikely!
Both of these words have uncertain etymologies. But there are reasons to think that the resemblances in form between broil and proelium are coincidental, which would mean they're no more likely to be related than any other pair of words of uncertain origin.
Proelium looks like it probably starts with the prefix prō-/pro-. This comes from a Proto-Indo-European source starting with *p, which regularly turns into Latin /p/ and Germanic *f (as in from, for, first).
There isn't an easy way for the start of broil to come from that same source. The word (em)broil in the sense "struggle" seems to be from French brouiller, of uncertain origin, but often thought to be taken from a Germanic language. If broil is ultimately from a Germanic word, then the b would come from a PIE source starting in *bʰ. And PIE *bʰ at the start of a word regularly turned into Latin f-.* (If broil somehow had a Celtic source, which apparently has sometimes been proposed, we would also expect the initial /b/ to go back to *bʰ.)
As for the end of the words, according to one hypothesis the /l/ in French brouiller comes from a suffix rather than from the original root of the word (the hypothesis is Proto-Indo-European *bʰrewh₁- > *brod- in a West Germanic language > *brodicare in Late Latin/Romance > *brodiculare > French brouiller; see CNRTL and Wiktionary).
*Some examples of words thought to come from *bʰrewh₁- in Latin: ferveo "burn, boil, seethe", fermentum "fermentation, yeast", dēfrutum "boiled-down grape juice/must".