’s
Elements, originally Euclid’s 5th
Postulate, here reclassified and presented as
Axiom XII by the editor. From the beginning, the postulate came under attack as being provable, and therefore not a postulate, and for more than two thousand years, many attempts were made to prove (derive) the parallel postulate using Euclid's first four postulates. The main reason that such a proof was so highly sought after was that, unlike the first four postulates, the parallel postulate is not self-evident. If the order in which the postulates were listed in the Elements is significant, it indicates that Euclid included this postulate only when he realised he could not prove it or proceed without it. 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). 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 his own axiom. Today, over two thousand two hundred years later, Euclid's fifth postulate remains a postulate.
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'. Proclus then goes on to give a false proof of his own. However, he did give a postulate which is equivalent to the fifth postulate.
Ibn al-Haytham (Alhazen) (965–1039), an
Arab mathematician, made an attempt at proving the parallel postulate using a
proof by contradiction, in the course of which he introduced the concept of
motion and
transformation into geometry. He formulated the
Lambert quadrilateral, which Boris Abramovich Rozenfeld names the "Ibn al-Haytham–Lambert quadrilateral", and his attempted proof contains elements similar to those found in
Lambert quadrilaterals and
Playfair's axiom. The Persian mathematician, astronomer, philosopher, and poet
Omar Khayyám (1050–1123), attempted to prove the fifth postulate from another explicitly given postulate (based on the fourth of the five
principles due to the Philosopher (
Aristotle), namely, "Two convergent straight lines intersect and it is impossible for two convergent straight lines to diverge in the direction in which they converge." He derived some of the earlier results belonging to
elliptical geometry and
hyperbolic geometry, though his postulate excluded the latter possibility. 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. His work was published in
Rome in 1594 and was studied by European geometers. This work marked the starting point for Saccheri's work on the subject which opened with a criticism of Sadr al-Din's work and the work of Wallis.
Giordano Vitale (1633–1711), in his book
Euclide restituto (1680, 1686), used the Khayyam-Saccheri quadrilateral to prove that if three points are equidistant on the base AB and the summit CD, then AB and CD are everywhere equidistant.
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. He worked with a figure that today we call a
Lambert quadrilateral, a quadrilateral with three right angles (can be considered half of a Saccheri quadrilateral). He quickly eliminated the possibility that the fourth angle is obtuse, as had Saccheri and Khayyám, and then proceeded to prove many theorems under the assumption of an acute angle. Unlike Saccheri, he never felt that he had reached a contradiction with this assumption. He had proved the non-Euclidean result that the sum of the angles in a triangle increases as the area of the triangle decreases, and this led him to speculate on the possibility of a model of the acute case on a sphere of imaginary radius. He did not carry this idea any further. Where Khayyám and Saccheri had attempted to prove Euclid's fifth by disproving the only possible alternatives, the nineteenth century finally saw mathematicians exploring those alternatives and discovering the
logically consistent geometries that result. 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.
Carl Friedrich Gauss had also studied the problem, but he did not publish any of his results. Upon hearing of Bolyai's results in a letter from Bolyai's father,
Farkas Bolyai, Gauss stated: If I commenced by saying that I am unable to praise this work, you would certainly be surprised for a moment. But I cannot say otherwise. To praise it would be to praise myself. 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 resulting geometries were later developed by
Lobachevsky,
Riemann and
Poincaré into
hyperbolic geometry (the acute case) and
elliptic geometry (the obtuse case). The
independence of the parallel postulate from Euclid's other axioms was finally demonstrated by
Eugenio Beltrami in 1868. ==Converse of Euclid's parallel postulate==