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The ancient romans had implivium, https://en.m.wikipedia.org/wiki/Impluvium, to do the same sort of job. The water in the pool would cool the surrounding air and air drawn in from the opening in the roof above would circulate it through the surrounding rooms.

If western architecture means northern europe, then I think they were rather more concerned with heating than cooling. Southern europe took a lot more care (fountains, shutters, huge thick walls...). But it's still a long way from the desert climates where they did really serious wind & water cooling.

You do see it in the US southwest to some degree (e.g. Santa Fe) although there is of course lots of architecture built around air conditioning as well.