The implicit function theorem from Newton to the present day

In 1676, I. Newton, in a letter to the Secretary of the Royal London Society, G. Oldenburg, described an iterative procedure for finding the root of a nonlinear equation, which is known today as Newton's method. Inverse or implicit function theorems are theorems on the existence of a solution to a non-linear equation, and their proofs, based on Newton's method, are natural and transparent. Moreover, this approach allows us to get more: the solution exists not only for this equation, but also for a close (in a certain sense) to it, which is important for applications when we are dealing with an approximate description of an object.