Instead of dividing by zero as Newton and Leibniz did, modern mathematicians divide by a number that they let approach zero. They do the division—perfectly legally, since there are no zeros—then they take the limit. The dirty tricks of making squared infinitesimals disappear and then dividing by zero to get a derivative were no longer necessary (see appendix C). This logic may seem like splitting hairs, like an argument as mystical as Newton’s “ghosts,” but in reality it’s not. It satisfies the mathematician’s strict requirement of logical rigor. There is a very firm, consistent basis for the
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