Proof that the Earth revolves around the sun.
It wasn't until 1838 that stellar parallax was first measured. Astronomers were excited because this was the first time that the distance to a star could be directly measured. It also happened to be the first proof that the earth went around the sun, but by that time, science and society had so thoroughly accepted Newton's mechanics, that no one cared. (Bessel measured the parallax of 61 Cygni to be about 1/3 of a second, which meant that it was about 11 light years away.)
There are two other effects caused by the movement of the earth around the sun. There is a measurable Doppler shift when observing stars that are in the direction of the earth's motion. (Towards the stars causes a little blue shift, and away from the star causes a little red shift.) In addition, there is an effect called stellar aberation, in which the positions of stars appear to shift slightly due the motion of the earth. (I do not know when these were first observed.)
Proof that the Earth rotates on its axis.
In 1851, the French physicist Foucault demonstrated the rotation of the earth with a massive pendulum. The pendulum was so big and massive that it maintained its swing for hours. Even though there was no way to change the swing of the pendulum, the plane of its swing slowly shifted. (There is a Foucault Pendulum in the lobby of the Museum of Science.)
To see how this would happen, imagine putting a pendulum on the north pole. As the pendulum swings, it would maintain a constant plane of motion. However, as the earth rotates on its axis, it would be rotating underneath the pendulum. From the earth's point of view, it would appear that the pendulum was slowly rotating in the opposite direction.
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