Proper being related to private of course got me thinking about that word, so here we go.
Proper first showed up in the fourteenth century from the Old French propre and classical Latin proprius, which can mean proper or individual. That word is actually taken from a phrase, pro privo, which could mean things like to deprive or private, because yes that privo is from privus, the origin word for private. Interestingly enough, while proper showed up in English meaning apt, it morphed into meaning “pertaining to oneself; individual” and then separate or distinct, which as we all know is very appropriate for a word related to private. Even though we don’t generally use proper that way anymore, that individual definition is where we get proper name from and probably proper noun, too.
Next, property also showed up in the fourteenth century as properte and it meant a quality before it meant something that was owned. It comes from the Old French propriete, individuality or property, and classical Latin proprietatem, property. That word comes from the above mentioned proprius, which means that it’s also related to private, so at one point the phrase “private property” would have been redundant.
Appropriate showed up in the early fifteenth century, first meaning to take possession of (like to appropriate something) before meaning suitable or apt. It comes from the Late Latin appropriatus, the past participle of appropriare, to make one’s own. The a comes from ad-, to, and the rest is from proprius. Appropriate is… to proper? To individual? I guess appropriating something is taking it to an individual. As to why appropriate now means proper, I guess that’s just because the word’s just intertwined in there, even in English.