ABSTRACT
The culmination of Euclid's first book of Elements is the proof of Pythagoras' theorem in I, 47. But it would have been better if Euclid had not proved Pythagoras but assumed it, and, taking Pythagoras' proposition as a postulate, proved the parallel postulate as a theorem. Rational agents who were not human and did not depend on eyes and hands might not share our preferences. But even disembodied spirits can recognize other, Platonic, merits in Euclidean geometry. Pythagoras' proposition is a more fundamental one, and a more distinctive feature of Euclidean geometry regarded as an axiomatic system. It connects the concept of distance with that of a right angle— orthogonality— and it does so in the simplest possible way. More fundamentally, we could defend the Pythagorean rule as being the simplest case of Parseval's theorem.
