What's the relationship between second, the unit of time, and second, the thing after first? Well, the latter has been around much longer, but we can trace both to Latin. Back when Ptolemy was categorizing all kinds of stuff, he called minutes pars minuta prima, which meant "first small part" and implied that it was the first major subdivision of an hour. Seconds, on the other hand, were designated pars minuta secunda, which meant "second small part" and implied that it was the second major subdivision of an hour. As the phrases evolved, the first one shed the pars and prima to become English minute, and the second one lost pars and minuta to become "second". Interesting as that is, let's move on. Secunda (which, meaning "the thing after first", is obviously the etymon of the other meaning of "second") comes from another Latin word, sequor, which meant "to follow" and comes from Proto-Indo-European sek, "to follow", through Proto-Italic. Minuta, meanwhile, comes from minuere, meaning "to diminish", from PIE mey, meaning "small".
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The Etymology Nerd