However he did not fall into the trap that Saccheri fell into and investigated the hypothesis of the acute angle without obtaining a contradiction. In 1766 Lambert followed a similar line to Saccheri. However he eventually 'proved' that the hypothesis of the acute angle led to a contradiction by assuming that there is a 'point at infinity' which lies on a plane. Saccheri then studied the hypothesis of the acute angle and derived many theorems of non-Euclidean geometry without realising what he was doing.
Saccheri proved that the hypothesis of the obtuse angle implied the fifth postulate, so obtaining a contradiction. Saccheri has shown:Ī ) The summit angles are > 90° (hypothesis of the obtuse angle ).ī ) The summit angles are < 90° (hypothesis of the acute angle ).Ĭ ) The summit angles are = 90° (hypothesis of the right angle ).Įuclid's fifth postulate is c ). Here is the Saccheri quadrilateral In this figure Saccheri proved that the summit angles at D D D and C C C were equal.The proof uses properties of congruent triangles which Euclid proved in Propositions 4 and 8 which are proved before the fifth postulate is used. The importance of Saccheri's work was that he assumed the fifth postulate false and attempted to derive a contradiction. It was produced in 1697 by Girolamo Saccheri. One of the attempted proofs turned out to be more important than most others.
To each triangle, there exists a similar triangle of arbitrary magnitude. One such 'proof' was given by Wallis in 1663 when he thought he had deduced the fifth postulate, but he had actually shown it to be equivalent to:. Invariably the mistake was assuming some 'obvious' property which turned out to be equivalent to the fifth postulate. Many attempts were made to prove the fifth postulate from the other four, many of them being accepted as proofs for long periods of time until the mistake was found. 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.
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