involved voluminous and highly laborious amounts of infinitesimal calculus and trigonometry. It also remained necessary, for completing the theories of this period, to refer to observational measurements. Results of the theories The lunar theorists used (and invented) many different mathematical approaches to analyse the gravitational problem. Not surprisingly, their results tended to converge. From the time of the earliest gravitational analysts among Newton's successors, Euler, Clairaut and d'Alembert, it was recognized that nearly all of the main lunar perturbations could be expressed in terms of just a few angular arguments and coefficients. These can be represented by: the mean motions or positions of the Moon and the Sun, together with three coefficients and three angular positions, which together define the shape and location of their apparent orbits: the two eccentricities (, about 0.0549, and , about 0.01675) of the ellipses that approximate to the apparent orbits of the Moon and the Sun; the angular direction of the perigees ( and ) (or their opposite points the apogees) of the two orbits; and the angle of inclination (, mean value about 18523") between the planes of the two orbits, together with the direction () of the line of nodes in which those two planes intersect. The ascending node () is the node passed by the Moon when it is tending northwards relative to the ecliptic. From these basic parameters, just four basic differential angular arguments are enough to express, in their different combinations, nearly all of the most significant perturbations of the lunar motions. They are given here with their conventional symbols due to Delaunay; they are sometimes known as the Delaunay arguments: the Moon's mean anomaly (angular distance of the mean longitude of the Moon from the mean longitude of its perigee ); the Sun's mean anomaly (angular distance of the mean longitude of the Sun from the mean longitude of its perigee ); the Moon's mean argument of latitude (angular distance of the mean longitude of the Moon from the mean longitude of its ascending (northward-bound) node ); the Moon's mean (solar) elongation (angular distance of the mean longitude of the Moon from the mean longitude of the Sun). This work culminated into Brown's lunar theory (1897–1908) and Tables of the Motion of the Moon (1919). These were used in the American Ephemeris and Nautical Almanac until 1968, and in a modified form until 1984. Largest or named lunar