My first thought was Shakespeare's Globe - the London theatre where many of Shakespeare's plays were performed. Built in 1599, the name generally comes from the idea that the whole world is a stage and we its actors. But why globe and not world or earth or some other name for this place we find ourselves in.
We can see how science gets reflected in culture by looking at things today. Take a TV show like The Big Bang Theory for example - the name only makes sense in a post-big bang theory world. It reflects what we believe now. It couldn't have been created before we had this idea. It no doubt will sound out of date in a few centuries time too. Likewise I'm guessing the same for the Globe. So for the time being I'm going to choose 1599, the year of Shakespeare's Globe, as the high water mark of the ball hysteria. I'm going to suggest that the global model was new enough and hip enough at that time to have been considered very much à la mode. I'll take it as a marker for further investigation.
1599 - the year of the ball.