Thanks for the comments about my last reply, and for the photo you posted,
William!
The barometer depicted by Tolkien appears to be an aneroid one, meaning it uses a non-liquid way of measuring air pressure. A metallic cell or capsule, from which the air has been removed, expands or contracts depending on the air pressure. It is a nineteenth century invention, by a French scientist called Lucien Vidi, in 1844. Since late in that century, barometers, along with wind observations, have been used to make short-term weather forecasts.
Despite all this sophistication, we don't see hobbit meteorologists around Middle-earth in the late Third Age...