Geometry is a form of rhetoric. Nobody knows for certain, but the anecdotes keep repeating that above the entrance to Plato's Academy there was a sign that says: “Here nobody is admitted who does not know his geometry.” The message is that the rules of geometry and the rules of thought are one and the same. But how do we know whether a given proposition of logic is true or false and how do we decide whether a particular proof of geometry is to be rejected or embraced? Aristotle and Euclid pointed the way to an answer, the former by legislating what can be truthfully said, the latter by discovering what can be convincingly shown. Thus, Euclidean geometry builds its acceptable forms of demonstration on a foundation of well-defined axioms and rules of inference. What must now be remembered is that Euclid did not crown all his proofs with the halo of quod erat demonstrandum (“which was to be demonstrated”). Sometimes he instead used the expression quod erat faciendum (“which was to be shown”).
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