After a lifetime of work by Newton using his gravitational theory to describe the motion of the Moon in Earth's sky, many important details remained unresolved. Newton himself declared that considering the intricate details of lunar motion "makes my head hurt and keeps me awake so often that I will think of it no more".
After Newton's death other scientists went beyond his geometrical-based efforts and began describing lunar motion with mathematical equations, often motivated by prizes offered by various scientific societies and the British government. Various problems regarding changes over time in the Earth-Moon apsis were worked out by Clairaut, d'Alembert and Euler by the mid-1750s. Self-taught mathematician Tobias Mayer then cracked the riddle of libration, which makes it possible to see more than 50% of the moon's surface over time even though one side of the Moon is tidally locked to always face towards Earth. The libration mathematics in Mayer's book Theoria lunae juxta systema Newtonianum (1767) resulted in the first lunar almanac accurate enough for use in ship navigation and won the hefty sum of £3000 from the British Admiralty for his widow. Efforts by Lagrange to describe remaining errors in lunar motion theory were addressed by Laplace in his encyclopedic Celestial Mechanics (1802), where he correctly accounted for tidal acceleration of the Moon's mean motion.
Tables of the Moon 1857
George William Hill
Tables of the Motion of the Moon 1919