The full theory consists of the four Maxwell equations in either integral or differential form. A coordinate transformation is the set of equations for converting the space and time coordinates of an event as measured in one reference frame to the coordinates measured in another frame. In special relativity we restrict ourselves to frames moving with constant velocities with respect to each other. The Galilean transformation is implied by the writings of Galileo, although he did not write the equations.
Velocities add, etc.
Consistency Newton 's theory of mechanics is consistent with the Principle of Relativity if the Galilean transformation is used to convert from one frame to another, but not if the Lorentz transformation is used. Maxwell 's theory is consistent with the Principle of Relativity if the Lorentz transformation is used to convert from one frame to another, but not if the Galilean transformation is used. Einstein , in 1905, declared that the Principle of Relativity and the Maxwell equations must be right. Therefore the Lorentz transformation must be the correct one to use, and Newtonian mechanics must be replaced by a new mechanics which keeps the same form under a Lorentz transformation and also yields essentially the same results as the old theory in realms where it was known to work well, i.e., at speeds small compared with the speed of light. He invented the new mechanics.