In Tabulae astronomicae solis et lunae (Solar and lunar astronomical tables) (E87), published in 1746, Euler made a first attempt to correct his astronomical tables for the gravitational attraction of the Earth toward the Moon, simply by accounting for the difference between the position of the Earth and the center of gravity of the Earth-Moon system. In this article Euler revisited the problem, noting that this center of gravity would not itself take an elliptical path around the Sun. To model the motion more accurately, Euler considered the separate gravitational attractions of the Sun and Moon acting on the Earth. With this approach, Euler found expressions dependent on the angle between the Sun and Moon as seen from Earth that can be used to correct the direction of the Sun as seen from Earth as well as the distance from the Earth to the Sun. Unfortunately for Euler, the accuracy of these corrections depends on accurate estimates of the density of the Moon and the mean distance from the Earth to the Sun, neither of which were available.
Patrick T Headley (Sun,) studied this question.