This book sets forth the authors' Neo-Newtonian mechanics with extension to relativistic velocities but without recourse to Einstein's relativity theories. It is motivated by Eric Laithwaite's infamous lecture to the Royal Society that (more or less) resulted in him being run out of town on a rail. The authors have set themselves the task of giving a rigorous and elementary theory to explain his amazing demonstrations, but using only as minimal an modification of Newtonian mechanics as both possible and necessary. And then the authors extend this alteration to relativistic velocities. Finally, examples are given of this theory, and related topics are discussed. Radiative effects are (God Willing) to be treated in Part 2. This is the second edition.