“Natural” coincidence is otherwise known as a spacetime point. Einstein has already spent twenty-odd pages of this very brief book laying out the assumptions which underlie the train experiment. He is very careful about being consistent with them, and he is a devoted and very strict Euclidean. But Einstein was not, it appears, quite careful enough. We know that he is assuming, along with Euclid, that the definition of the coincidence of two points is a point. However, we have never gotten (and never get, in any of Einstein’s writings) a definition of a “natural” coincidence of two points. This alone prevents us from going on and this argument, which defined the twentieth century, abruptly ends. We also have a problem if we try to resolve the issue ourselves.

** If we simply drop the term “naturally” we run into a situation in which Einstein has told us to assume two Cartesian coordinate systems, but now leaves us with one, since, following from the definition of the coincidence of two points, if two parallel coordinate systems coincide at one point, they coincide at all points and are one coordinate system, not two.** We have been led to a contradiction.