Fifth postulate

Euclid's axiom of parallelism

One can draw just one straight line through a point $P$ not on a straight line $AA_1$ that does not intersect $AA_1$ and lies in the plane containing $P$ and $AA_1$. In Euclid's Elements the fifth postulate is given in the following equivalent form: "If a straight line incident to two straight lines has interior angles on the same side of less than two right angles, then the extension of these two lines meets on that side where the angles are less than two right angles" (see [1]). Among the commentators of Euclid there arose the view that a proof of this statement could be found based on the remaining axioms. Attempts at proofs occurred as long ago as in Ancient Greece. These attempts continued in the East in the Middle Ages and then in Western Europe. If direct logical mistakes are overlooked, then usually an implicit (and sometimes also a clearly understood) assumption was made which was not deducible from the remaining axioms and which turned out to be equivalent to the fifth postulate. For example, the distance between parallels is bounded, the space admits a "simple" motion (all trajectories are straight lines), two converging straight lines always intersect, there exist similar but unequal figures, the sum of the angles in a triangle is equal to two right angles, etc. G. Saccheri (1733) considered a quadrangle with right angles at the base and with equal lateral sides. Omar Khayyam (11th–12th century) had considered such a quadrangle earlier. Of the three possible hypotheses about the remaining two equal angles (they are obtuse, they are acute, they are right angles) he tried to reject the first two since the third implied the fifth postulate. Saccheri succeeded in deriving a contradiction from the first hypothesis, but he made a logical mistake in refuting the hypothesis on the acute angle. J. Lambert (1766, published 1786) with a similar approach refuted the hypothesis on the acute angle but also made a serious mistake. He assumed that such a geometry is realized only on an imaginary sphere. A. Legendre (1800), in the first edition of the textbook Eléments de la géométrie, started from the sum $S$ of the angles of a triangle. Having rejected the hypothesis $S>\pi$, he made a mistake in deriving the consequences of the hypothesis $S<\pi$, namely, he implicitly introduced the axiom that for any point inside an acutely-angled sector there exists a straight line passing through this point and intersecting both sides of the sector. The solution to the problem of the fifth postulate (more precisely its removal) was obtained by a geometry created by N.I. Lobachevskii (1826) in which the fifth postulate does not hold. From the fact that Lobachevskii geometry is consistent, it follows that the fifth postulate is independent of the other axioms in Euclidean geometry.

References

[1]	Th.L. Heath, "The thirteen books of Euclid's elements" , 1–3 , Cambridge Univ. Press (1926) ((Translated from the Greek))
[2]	V.F. Kagan, "Foundations of geometry" , 1 , Moscow-Leningrad (1949) (In Russian)
[3]	N.V. Efimov, "Höhere Geometrie" , Deutsch. Verlag Wissenschaft. (1960) (Translated from Russian)
[4]	, On the foundations of geometry. A collection of classical papers on Lobachevskii geometry , Moscow (1956) (In Russian) (Collection of translations)
[5]	B.A. [B.A. Rozenfel'd] Rosenfel'd, "A history of non-euclidean geometry" , Springer (1988) (Translated from Russian)

Comments

See also Elements of Euclid. The fifth postulate is also called the parallel postulate. A nice account of the history briefly described in the article above is [a2].

References

[a1]	H.S.M. Coxeter, "Non-Euclidean geometry" , Univ. Toronto Press (1957)
[a2]	R. Bonola, "Non-Euclidean geometry" , Dover, reprint (1955) (Translated from Italian)