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Enough of analysing Mathis's argument. Simply by examining his conclusion you can see that it is wrong without having to waste time looking for the fallacy. He fails to do what any self respecting mathematician would do and that is to subject his results to a sanity check--that is, to ask "Is the result I have supposedly proved a reasonable one?" If you look at the attached diagram, you can see that Mathis (in effect) claims that the perimeter of a square has the same length as the circumference of the inscribed circle. Since this is clearly bollocks, we can state, without even bothering to look at his "proof", that his reasoning must be faulty.
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