What was the first astronomical measurement which demonstrated that "the Earth is surrounded by vacuum"? The question Who was the first to realize that the Earth is surrounded by vacuum? was closed because some users felt it was answered by answers to a different question in an different SE site: Who was the first to postulate that space was a vacuum?

To me the distance between realize and postulate is astronomical!

That aside, I've asked a new question: What was the first astronomical measurement which demonstrated that "the Earth is surrounded by vacuum"?

By demonstrated I'm talking about a measurement that shows it is, or at least is likely to be the case that Earth is surrounded by vacuum. It should be something such that it could be presented as scientifically convincing evidence to other scientists of the time, unencumbered by non-scientific predispositions (e.g. religion, rivalry, philosophy...)
Torricelli, the inventor of the Mercury Barometer (~1644) argued that the height of the column of mercury was governed by atmospheric pressure (the "weight of the atmosphere" as he would have put it).   He asserted that the space above the mercury in his tube was a vacuum, a totally anti-Aristotlean concept at that time.
To test this he enlisted the help of Blaise Pascal and Florin Perier in carrying one of two barometers up a mountain, the Puy de Dome in central France. His prediction was that the pressure at the top of the mountain would be less than at the base because of a lesser weight of air.
The experiment showed a drop in height (ie of pressure) and Torriceli himself concluded that the height of the atmosphere would be about 20km, above which there would be vacuum - the same vacuum as in the top of the barometer tube. 
The experiment was IMHO one of the greatest experiments in the  history of physics.  See for example  https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3768090/