![]() ![]() Although known from the time of Proclus, this became known as Playfair's Axiom after John Playfair wrote a famous commentary on Euclid in 1795 in which he proposed replacing Euclid's fifth postulate by this axiom. Playfair's Axiom:- Given a line and a point not on the line, it is possible to draw exactly one line through the given point parallel to the line. However he did give the following postulate which is equivalent to the fifth postulate. Proclus then goes on to give a false proof of his own. Proclus (410- 485) wrote a commentary on The Elements where he comments on attempted proofs to deduce the fifth postulate from the other four, in particular he notes that Ptolemy had produced a false 'proof'. Another comment worth making at this point is that Euclid, and many that were to follow him, assumed that straight lines were infinite. It did not satisfy Euclid and he tried to avoid its use as long as possible - in fact the first 28 propositions of The Elements are proved without using it. It is clear that the fifth postulate is different from the other four. ![]() That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, if produced indefinitely, meet on that side on which are the angles less than the two right angles.That all right angles are equal to each other.To describe a circle with any centre and distance.To produce a finite straight line continuously in a straight line. ![]()
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