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Galileo's secretary and scientific successor, Evangelista Torricelli, helped prove the existence of air pressure through a famous experiment that he did with a mercury-filled glass tube. Some of Galileo's contemporaries had built simple barometers, using water in long tubes, to try to prove the existence of a vacuum. But in 1643, Torricelli took the barometer idea a step further. He realized that air--contrary to contemporary thought--was not weightless and that the rise and fall of liquid in a barometric tube was related to air pressure.
To show this, he developed an experiment using liquid mercury or 'quicksilver.' Since mercury is about 14 times as heavy as water, Torricelli was able to use a much shorter tube than the earlier water barometers required. He filled the tube with mercury and then inverted it into a dish of mercury. Some of the mercury drained from the tube, but then it stopped at a certain level--14 times less than the level water stopped at, to be exact.
Torricelli postulated that air exerted pressure on the mercury in the dish, pushing hard enough on the surface area to keep most of the mercury inside the tube. Thus, it was not the attraction of the vacuum at the top of the tube which held the mercury inside (as other scientists thought), but rather, the phenomenon was the result of the pressure exerted by air. Thus, the barometer could be used to measure air pressure. Later generations of scientists, most notably Blaise Pascal, developed the barometer further.
You can demonstrate air pressure's effects through an experiment similar to Torricelli's. Stick a plastic straw in a glass of juice or other colored drink, and suck enough liquid into the straw to fill it about halfway up. When you suck on the straw, a partial vacuum is created in the top of the straw. Air pressure on the liquid in the glass forces the juice up the straw and into your mouth. Now, hold your finger over the top of the straw and slowly pull the straw out of the glass. While your finger is pressed over the top, a partial vacuum is maintained in the top of the straw. The air pressure is greatest underneath the straw and will keep the liquid from dripping out.