Cute. And there are other issues with what I wrote. For a more careful analysis, would have to do, say, a 'probabilistic' analysis, e.g., where in principle ca

Cute. And there are other issues with what I wrote. For a more careful analysis, would have to do, say, a 'probabilistic' analysis, e.g., where in principle can walk back to the beginning and start over but actually, e.g., with some common assumptions, in a space of dimension 3 or greater, probabilistically keep getting farther from the starting point. Your point should come out in the wash in estimating where there is a common language and how far back if use more than one word, etc. My guess, especially from many of the other comments on this thread, is that the basic data is so bad, noisy, etc. and pronunciation changes so large that the math I outlined, while basically correct, would mostly give such large distances that we wouldn't learn much. Maybe a better solution would be just to assume that the tree of languages would follow the tree of genetic inheritance and, then, via genetic analysis trace that tree. But the article wanted to ask what one of those 'common' words would 'sound' like 15,000 years ago, so I gave a way to get an approximation. The cute part is while 'father' in English sounds a bit different from 'vater' in German, the common ancestor should sound closer to either 'father' or 'vater' than those to two each other, that is, closer rather than farther away. That is, our intuition would be that the sound from 15,000 years ago should be much different from either the current 'father' or 'vater', but with the math model I gave that should be wrong.