Galilean moons century when other inner moons were discovered, and Marius' names became widely used. Galileo was able to develop a method of determining longitude based on the timing of the orbits of the Galilean moons. The times of the eclipses of the moons could be precisely calculated in advance, and compared with local observations on land or on ship to determine the local time and hence longitude. The main problem with the technique was that it was difficult to observe the Galilean moons through a telescope on a moving ship; a problem that Galileo tried to solve with the invention of
Newtonian dynamics a separate symbol and then treated as independent variables. The quantities are used as internal coordinates of a point of the phase space formula_20 of the constrained Newtonian dynamical system. Geometrically, the vector-function () implements an embedding of the configuration space formula_18 of the constrained Newtonian dynamical system into the formula_29-dimensional flat comfiguration space of the unconstrained Newtonian dynamical system (). Due to this embedding the Euclidean structure of the ambient space induces the Riemannian metric onto the manifold formula_18. The components of the metric tensor of this induced metric are given by the formula where formula_31 is the scalar