Parallel postulate

[11] Many attempts were made to prove the fifth postulate from the other four, many of them being accepted as proofs for long periods until the mistake was found.

Invariably the mistake was assuming some 'obvious' property which turned out to be equivalent to the fifth postulate (Playfair's axiom).

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'.

Ibn al-Haytham (Alhazen) (965–1039), an Arab mathematician, made an attempt at proving the parallel postulate using a proof by contradiction,[12] in the course of which he introduced the concept of motion and transformation into geometry.

[17] The Saccheri quadrilateral was also first considered by Omar Khayyám in the late 11th century in Book I of Explanations of the Difficulties in the Postulates of Euclid.

Nasir al-Din al-Tusi (1201–1274), in his Al-risala al-shafiya'an al-shakk fi'l-khutut al-mutawaziya (Discussion Which Removes Doubt about Parallel Lines) (1250), wrote detailed critiques of the parallel postulate and on Khayyám's attempted proof a century earlier.

"He essentially revised both the Euclidean system of axioms and postulates and the proofs of many propositions from the Elements.

Girolamo Saccheri (1667–1733) pursued the same line of reasoning more thoroughly, correctly obtaining absurdity from the obtuse case (proceeding, like Euclid, from the implicit assumption that lines can be extended indefinitely and have infinite length), but failing to refute the acute case (although he managed to wrongly persuade himself that he had).

In 1766 Johann Lambert wrote, but did not publish, Theorie der Parallellinien in which he attempted, as Saccheri did, to prove the fifth postulate.

In 1829, Nikolai Ivanovich Lobachevsky published an account of acute geometry in an obscure Russian journal (later re-published in 1840 in German).

In 1831, János Bolyai included, in a book by his father, an appendix describing acute geometry, which, doubtlessly, he had developed independently of Lobachevsky.

Indeed the whole contents of the work, the path taken by your son, the results to which he is led, coincide almost entirely with my meditations, which have occupied my mind partly for the last thirty or thirty-five years.

The independence of the parallel postulate from Euclid's other axioms was finally demonstrated by Eugenio Beltrami in 1868.

Attempts to logically prove the parallel postulate, rather than the eighth axiom,[25] were criticized by Arthur Schopenhauer in The World as Will and Idea.

""But in a manuscript probably written by his son Sadr al-Din in 1298, based on Nasir al-Din's later thoughts on the subject, there is a new argument based on another hypothesis, also equivalent to Euclid's, [...] The importance of this latter work is that it was published in Rome in 1594 and was studied by European geometers.

"Eder, Michelle (2000), Views of Euclid's Parallel Postulate in Ancient Greece and in Medieval Islam, Rutgers University, retrieved 2008-01-23