Reddit mentions of Problem Book in Relativity and Gravitation

The original work is apparantly in Volume 4 of Laplace's Treatise on Celestial Mechanics, but I couldn't find it online. Laplace shows that the speed of gravity would need to be at least 10^6 times the speed of light (this was in the 1830s, we probably have better bounds now). An excellent discussion of how GR deals with this issue with all the relevant references can be found in this paper by Carlip. He cites this paper (which I do not have access to) and a problem in this textbook for how the Newtonian calculation fails.

It's not so hard to see how the calculation goes. If gravity pointed to where the sun used to be, it would be off from where the sun is by an angle of about v/c radians. So the force would be F(total) = F(central) + F(c) where F(central) = F(newton)(1-v/c) is almost exactly the Newton term and F(c) points along the direction of Earth's motion with magnitude GMm(v/c)/r^(2). Unlike a central force, this second term should generate a calculable violation of angular momentum conservation from which you can compute how fast the radius of Earth's orbit varies, and you can show the for c = the speed of light, the orbit varies way too quickly.