I have been reading about Eddington and his measurement of 1.75 arc seconds in 1919. What I am wondering is why was the measurement lateral? I picture a beam of star light entering a curvature of spacetime. The beam has no sense of up, down or left or right. So as it enters this curvature it seems arbitrary as to which curved path to follow. It could go up, down or sideways or anywhere in between. All paths seem equal. What determines which path the light will take? It would be very strange if it always took a path that considered the earth’s location.
The starlight was slightly bent by the gravity of the Sun, in the direction of the Sun, as it passed near the Sun. The Earth had nothing to do with it.
There is nothing arbitrary about it. You can calculate the curvature of spacetime near the Sun, and you can calculate a null (light-like) geodesic through that curved spacetime. A geodesic grazing the Sun bends by 1.75 arc seconds.