The Lorentz transformation (LT) has been shown to be self-contradictory (Clock Puzzle). The reason is because it predicts both proportional time dilation and the remote non-simultaneity of events. Consequently, the LT erroneously requires that when a null time difference is multiplied with a constant time-dilation factor, the resulting time difference in another rest frame can be different from zero. This possibility stands in direct conflict with a fundamental axiom of elementary algebra. The precursor of the LT was introduced by Voigt in 1887. He pointed out that one can make a change in the Galilean transformation that makes it compatible with experimental data which indicate that the speed of light in free space is the same for all observers, regardless of their state of motion. The history of how the Voigt transformation ultimately evolved into the LT over the next 20 years is reviewed. In the process, it is shown how a different approach can be taken which eliminates the above contradiction inherent in the LT, while at the same time retaining the essential characteristics of light-speed constancy and consistency with Galileo's Relativity Principle (RP). The resulting space-time transformation eschews the space-time mixing characteristic of both the LT and the original Voigt transformation. In this way, it removes the possibility of remote non-simultaneity, and instead insists on the proportionality of time dilation, which in turn is shown to follow directly from a straightforward extension of Newton's Law of Inertia. The resulting set of equations [Newton-Voigt Transformation (NVT)] is found to agree with all experimental timing data obtained to the present day.