Well, I was lucky enough to be on holiday in Sicily recently, where I visited a number of hellenistic archeological sites. Possibly the best preserved was Morgantina, especially it's baths, a model of which was displayed at the musem of the site in nearby Aidone.
What particularily struck me was the heating system which seemed very similar to that I've seen in later Roman sites. Morgantina was occupied by Romans though, so I can't be sure of the dating e.g. what went first -there were a series of improvements of the baths throughout the city's occupation.
I don't know about the baths you are reffering to, but the greek Gymnasium did have things that Roman Bath Houses did not, details but it isn't important (and Romans built baths in all of their cities). The point I was making is if the Romans didn't come up with the concept themselves they have taken it from the Etruscans or other cultures as easily as the Greeks. The Etruscans probably gave the Romans the concept because Rome did start out as an Etruscan ruled city. That said Greek architecture is still incredible and very worth studying.
True, the baths in Morgantina still had the hot/cold element, but they were adminsitered differently. I think they actually had cermic "bath tubs" where the hot water was poured over the person sitting in the tub.
As to who came up with the idea first, I've no idea. Its interesting how such social practices evolved but are in some ways still present in modern societies -e.g the Hammam.
Bookmarks