In geometry, the **parallel postulate**, also called ** Euclid's fifth postulate** because it is the fifth postulate in Euclid's *Elements*, is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry:

- Equivalent properties
- History
- Converse of Euclid's parallel postulate
- Criticism
- Decomposition of the parallel postulate
- See also
- Notes
- References
- External links

If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.

This postulate does not specifically talk about parallel lines;^{ [1] } it is only a postulate related to parallelism. Euclid gave the definition of parallel lines in Book I, Definition 23^{ [2] } just before the five postulates.^{ [3] }

*Euclidean geometry* is the study of geometry that satisfies all of Euclid's axioms, *including* the parallel postulate.

The postulate was long considered to be obvious or inevitable, but proofs were elusive. Eventually it was discovered that inverting the postulate gave valid, albeit different geometries. A geometry where the parallel postulate does not hold is known as a non-Euclidean geometry. Geometry that is independent of Euclid's fifth postulate (i.e., only assumes the modern equivalent of the first four postulates) is known as absolute geometry (or sometimes "neutral geometry").

Probably the best known equivalent of Euclid's parallel postulate, contingent on his other postulates, is Playfair's axiom, named after the Scottish mathematician John Playfair, which states:

In a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point.^{ [4] }

This axiom by itself is not logically equivalent to the Euclidean parallel postulate since there are geometries in which one is true and the other is not. However, in the presence of the remaining axioms which give Euclidean geometry, each of these can be used to prove the other, so they are equivalent in the context of absolute geometry.^{ [5] }

Many other statements equivalent to the parallel postulate have been suggested, some of them appearing at first to be unrelated to parallelism, and some seeming so self-evident that they were unconsciously assumed by people who claimed to have proven the parallel postulate from Euclid's other postulates. These equivalent statements include:

- There is at most one line that can be drawn parallel to another given one through an external point. (Playfair's axiom)
- The sum of the angles in every triangle is 180° (triangle postulate).
- There exists a triangle whose angles add up to 180°.
- The sum of the angles is the same for every triangle.
- There exists a pair of similar, but not congruent, triangles.
- Every triangle can be circumscribed.
- If three angles of a quadrilateral are right angles, then the fourth angle is also a right angle.
- There exists a quadrilateral in which all angles are right angles, that is, a rectangle.
- There exists a pair of straight lines that are at constant distance from each other.
- Two lines that are parallel to the same line are also parallel to each other.
- In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides (Pythagoras' Theorem).
^{ [6] }^{ [7] } - The Law of cosines, a generalization of Pythagoras' Theorem.
- There is no upper limit to the area of a triangle. (Wallis axiom)
^{ [8] } - The summit angles of the Saccheri quadrilateral are 90°.
- If a line intersects one of two parallel lines, both of which are coplanar with the original line, then it also intersects the other. (Proclus' axiom)
^{ [9] }

However, the alternatives which employ the word "parallel" cease appearing so simple when one is obliged to explain which of the four common definitions of "parallel" is meant – constant separation, never meeting, same angles where crossed by *some* third line, or same angles where crossed by *any* third line – since the equivalence of these four is itself one of the unconsciously obvious assumptions equivalent to Euclid's fifth postulate. In the list above, it is always taken to refer to non-intersecting lines. For example, if the word "parallel" in Playfair's axiom is taken to mean 'constant separation' or 'same angles where crossed by any third line', then it is no longer equivalent to Euclid's fifth postulate, and is provable from the first four (the axiom says 'There is at most one line...', which is consistent with there being no such lines). However, if the definition is taken so that parallel lines are lines that do not intersect, or that have some line intersecting them in the same angles, Playfair's axiom is contextually equivalent to Euclid's fifth postulate and is thus logically independent of the first four postulates. Note that the latter two definitions are not equivalent, because in hyperbolic geometry the second definition holds only for ultraparallel lines.

For two thousand years, many attempts were made to prove 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 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.^{ [10] } 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.

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,^{ [11] } in the course of which he introduced the concept of motion and transformation into geometry.^{ [12] } He formulated the Lambert quadrilateral, which Boris Abramovich Rozenfeld names the "Ibn al-Haytham–Lambert quadrilateral",^{ [13] } and his attempted proof contains elements similar to those found in Lambert quadrilaterals and Playfair's axiom.^{ [14] }

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."^{ [15] } He derived some of the earlier results belonging to elliptical geometry and hyperbolic geometry, though his postulate excluded the latter possibility.^{ [16] } 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*.^{ [13] } Unlike many commentators on Euclid before and after him (including Giovanni Girolamo Saccheri), Khayyám was not trying to prove the parallel postulate as such but to derive it from his equivalent postulate. He recognized that three possibilities arose from omitting Euclid's fifth postulate; if two perpendiculars to one line cross another line, judicious choice of the last can make the internal angles where it meets the two perpendiculars equal (it is then parallel to the first line). If those equal internal angles are right angles, we get Euclid's fifth postulate, otherwise, they must be either acute or obtuse. He showed that the acute and obtuse cases led to contradictions using his postulate, but his postulate is now known to be equivalent to the fifth postulate.

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. Nasir al-Din attempted to derive a proof by contradiction of the parallel postulate.^{ [17] } He also considered the cases of what are now known as elliptical and hyperbolic geometry, though he ruled out both of them.^{ [16] }

Nasir al-Din's son, Sadr al-Din (sometimes known as "Pseudo-Tusi"), wrote a book on the subject in 1298, based on his father's later thoughts, which presented one of the earliest arguments for a non-Euclidean hypothesis equivalent to the parallel postulate. "He essentially revised both the Euclidean system of axioms and postulates and the proofs of many propositions from the *Elements*."^{ [17] }^{ [18] } 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^{ [17] } which opened with a criticism of Sadr al-Din's work and the work of Wallis.^{ [19] }

Giordano Vitale (1633-1711), in his book *Euclide restituo* (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.^{ [20] }

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 which 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."

^{ [21] }

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.

Euclid did not postulate the converse of his fifth postulate, which is one way to distinguish Euclidean geometry from elliptic geometry. The Elements contains the proof of an equivalent statement (Book I, Proposition 27): *If a straight line falling on two straight lines make the alternate angles equal to one another, the straight lines will be parallel to one another.* As De Morgan ^{ [22] } pointed out, this is logically equivalent to (Book I, Proposition 16). These results do not depend upon the fifth postulate, but they do require the second postulate^{ [23] } which is violated in elliptic geometry.

Attempts to logically prove the parallel postulate, rather than the eighth axiom,^{ [24] } were criticized by Arthur Schopenhauer in * The World as Will and Idea *. However, the argument used by Schopenhauer was that the postulate is evident by perception, not that it was not a logical consequence of the other axioms.^{ [25] }

The parallel postulate is equivalent, as shown in,^{ [26] } to the conjunction of the Lotschnittaxiom and of Aristotle's axiom. The former states that the perpendiculars to the sides of a right angle intersect, while the latter states that there is no upper bound for the lengths of the distances from the leg of an angle to the other leg. As shown in,^{ [27] } the parallel postulate is equivalent to the conjunction of the following incidence-geometric forms of the Lotschnittaxiom and of Aristotle's axiom:

Given three parallel lines, there is a line that intersects all three of them.

Given a line a and two distinct intersecting lines m and n, each different from a, there exists a line g which intersects a and m, but not n.

- ↑ non-Euclidean geometries, by Dr. Katrina Piatek-Jimenez
- ↑ Euclid's Elements, Book I, Definition 23
- ↑ Euclid's Elements, Book I
- ↑ Euclid's Parallel Postulate and Playfair's Axiom
- ↑ Henderson & Taimiņa 2005 , pg. 139
- ↑ Eric W. Weisstein (2003),
*CRC concise encyclopedia of mathematics*(2nd ed.), p. 2147, ISBN 1-58488-347-2,The parallel postulate is equivalent to the

*Equidistance postulate*,*Playfair axiom*,*Proclus axiom*, the*Triangle postulate*and the*Pythagorean theorem*. - ↑ Alexander R. Pruss (2006),
*The principle of sufficient reason: a reassessment*, Cambridge University Press, p. 11, ISBN 0-521-85959-X,We could include...the parallel postulate and derive the Pythagorean theorem. Or we could instead make the Pythagorean theorem among the other axioms and derive the parallel postulate.

- ↑ Bogomolny, Alexander. "Euclid's Fifth Postulate".
*Cut The Knot*. Retrieved 30 September 2011. - ↑ Weisstein, Eric W. "Proclus' Axiom – MathWorld" . Retrieved 2009-09-05.
- ↑ Florence P. Lewis (Jan 1920), "History of the Parallel Postulate",
*The American Mathematical Monthly*, The American Mathematical Monthly, Vol. 27, No. 1,**27**(1): 16–23, doi:10.2307/2973238, JSTOR 2973238. - ↑ Katz 1998 , pg. 269
- ↑ Katz 1998 , p. 269:
In effect, this method characterized parallel lines as lines always equidistant from one another and also introduced the concept of motion into geometry.

- 1 2 Rozenfeld 1988 , p. 65
- ↑ Smith 1992
- ↑ Boris A Rosenfeld and Adolf P Youschkevitch (1996),
*Geometry*, p.467 in Roshdi Rashed, Régis Morelon (1996),*Encyclopedia of the history of Arabic science*, Routledge, ISBN 0-415-12411-5. - 1 2 Boris A. Rosenfeld and Adolf P. Youschkevitch (1996), "Geometry", in Roshdi Rashed, ed.,
*Encyclopedia of the History of Arabic Science*, Vol. 2, p. 447-494 [469], Routledge, London and New York:"Khayyam's postulate had excluded the case of the hyperbolic geometry whereas al-Tusi's postulate ruled out both the hyperbolic and elliptic geometries."

- 1 2 3 Katz 1998 , pg.271:
"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. In particular, it became the starting point for the work of Saccheri and ultimately for the discovery of non-Euclidean geometry."

- ↑ Boris A. Rosenfeld and Adolf P. Youschkevitch (1996), "Geometry", in Roshdi Rashed, ed.,
*Encyclopedia of the History of Arabic Science*, Vol. 2, p. 447-494 [469], Routledge, London and New York:"In

*Pseudo-Tusi's Exposition of Euclid*, [...] another statement is used instead of a postulate. It was independent of the Euclidean postulate V and easy to prove. [...] He essentially revised both the Euclidean system of axioms and postulates and the proofs of many propositions from the*Elements*." - ↑ MacTutor's Giovanni Girolamo Saccheri
- ↑ O'Connor, J.J.; Robertson, E.F. "Johann Heinrich Lambert" . Retrieved 16 September 2011.
- ↑ Faber 1983 , pg. 161
- ↑ Heath, T.L.,
*The thirteen books of Euclid's Elements*, Vol.1, Dover, 1956, pg.309. - ↑ Coxeter, H.S.M.,
*Non-Euclidean Geometry*, 6th Ed., MAA 1998, pg.3 - ↑ Schopenhauer is referring to Euclid's Common Notion 4: Figures coinciding with one another are equal to one another.
- ↑ http://www.gutenberg.org/files/40097/40097-pdf.pdf
- ↑ Pambuccian, Victor (1994), "Zum Stufenaufbau des Parallelenaxioms",
*Journal of Geometry*,**51**(1–2): 79–88, doi:10.1007/BF01226859, hdl: 2027.42/43033 , S2CID 28056805 - ↑ Pambuccian, Victor; Schacht, Celia (2021), "The ubiquitous axiom",
*Results in Mathematics*,**76**(3): 1--39, doi:10.1007/s00025-021-01424-3

**Euclidean geometry** is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the *Elements*. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system. The *Elements* begins with **plane geometry**, still taught in secondary school as the first axiomatic system and the first examples of mathematical proofs. It goes on to the solid geometry of three dimensions. Much of the *Elements* states results of what are now called algebra and number theory, explained in geometrical language.

In mathematics, **non-Euclidean geometry** consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises by either relaxing the metric requirement, or replacing the parallel postulate with an alternative. In the latter case one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries. When the metric requirement is relaxed, then there are affine planes associated with the planar algebras, which give rise to kinematic geometries that have also been called non-Euclidean geometry.

In geometry and trigonometry, a **right angle** is an angle of exactly 90° (degrees), corresponding to a quarter turn. If a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles. The term is a calque of Latin *angulus rectus*; here *rectus* means "upright", referring to the vertical perpendicular to a horizontal base line.

In Euclidean geometry, a convex quadrilateral with at least one pair of parallel sides is referred to as a **trapezium** in English outside North America, but as a **trapezoid** in American and Canadian English. The parallel sides are called the *bases* of the trapezoid and the other two sides are called the *legs* or the lateral sides. A *scalene trapezoid* is a trapezoid with no sides of equal measure, in contrast with the special cases below.

In mathematics, **hyperbolic geometry** is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:

**Synthetic geometry** is the study of geometry without the use of coordinates or formulae. It relies on the axiomatic method and the tools directly related to them, that is, compass and straightedge, to draw conclusions and solve problems.

In geometry, **parallel** lines are lines in a plane which do not meet; that is, two straight lines in a plane that do not intersect at any point are said to be parallel. Colloquially, curves that do not touch each other or intersect and keep a fixed minimum distance are said to be parallel. A line and a plane, or two planes, in three-dimensional Euclidean space that do not share a point are also said to be parallel. However, two lines in three-dimensional space which do not meet must be in a common plane to be considered parallel; otherwise they are called skew lines. Parallel planes are planes in the same three-dimensional space that never meet.

**Absolute geometry** is a geometry based on an axiom system for Euclidean geometry without the parallel postulate or any of its alternatives. Traditionally, this has meant using only the first four of Euclid's postulates, but since these are not sufficient as a basis of Euclidean geometry, other systems, such as Hilbert's axioms without the parallel axiom, are used. The term was introduced by János Bolyai in 1832. It is sometimes referred to as **neutral geometry**, as it is neutral with respect to the parallel postulate.

A **Saccheri quadrilateral** is a quadrilateral with two equal sides perpendicular to the base. It is named after Giovanni Gerolamo Saccheri, who used it extensively in his book *Euclides ab omni naevo vindicatus* first published in 1733, an attempt to prove the parallel postulate using the method Reductio ad absurdum.

In a Euclidean space, the **sum of angles of a triangle** equals the straight angle . A triangle has three angles, one at each vertex, bounded by a pair of adjacent sides.

In geometry, **Pasch's axiom** is a statement in plane geometry, used implicitly by Euclid, which cannot be derived from the postulates as Euclid gave them. Its essential role was discovered by Moritz Pasch in 1882.

In geometry, a **transversal** is a line that passes through two lines in the same plane at two distinct points. Transversals play a role in establishing whether two or more other lines in the Euclidean plane are parallel. The intersections of a transversal with two lines create various types of pairs of angles: **consecutive interior angles**, **consecutive exterior angles**, **corresponding angles**, and **alternate angles**. As a consequence of Euclid's parallel postulate, if the two lines are parallel, consecutive interior angles are supplementary, corresponding angles are equal, and alternate angles are equal.

**Giordano Vitale** or **Vitale Giordano** was an Italian mathematician. He is best known for his theorem on Saccheri quadrilaterals. He may also be referred to as **Vitale Giordani**, **Vitale Giordano da Bitonto**, and simply **Giordano**.

**Geometry** is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a geometer.

The **exterior angle theorem** is Proposition 1.16 in Euclid's Elements, which states that the measure of an exterior angle of a triangle is greater than either of the measures of the remote interior angles. This is a fundamental result in absolute geometry because its proof does not depend upon the parallel postulate.

**Foundations of geometry** is the study of geometries as axiomatic systems. There are several sets of axioms which give rise to Euclidean geometry or to non-Euclidean geometries. These are fundamental to the study and of historical importance, but there are a great many modern geometries that are not Euclidean which can be studied from this viewpoint. The term **axiomatic geometry** can be applied to any geometry that is developed from an axiom system, but is often used to mean Euclidean geometry studied from this point of view. The completeness and independence of general axiomatic systems are important mathematical considerations, but there are also issues to do with the teaching of geometry which come into play.

In absolute geometry, the **Saccheri–Legendre theorem** states that the sum of the angles in a triangle is at most 180°. Absolute geometry is the geometry obtained from assuming all the axioms that lead to Euclidean geometry with the exception of the axiom that is equivalent to the parallel postulate of Euclid.

**Giovanni Girolamo Saccheri** was an Italian Jesuit priest, scholastic philosopher, and mathematician.

In geometry, **Playfair's axiom** is an axiom that can be used instead of the fifth postulate of Euclid :

In a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point.

**Aristotle's axiom** is an axiom in the foundations of geometry, proposed by Aristotle in *On the Heavens*. It states:

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*Experiencing Geometry: Euclidean and Non-Euclidean with History*(3rd ed.), Upper Saddle River, NJ: Pearson Prentice Hall, ISBN 0-13-143748-8 - Katz, Victor J. (1998),
*History of Mathematics: An Introduction*, Addison-Wesley, ISBN 0-321-01618-1, OCLC 38199387 - Rozenfeld, Boris A. (1988),
*A History of Non-Euclidean Geometry: Evolution of the Concept of a Geometric Space*, Springer Science+Business Media, ISBN 0-387-96458-4, OCLC 15550634 - Smith, John D. (1992), "The Remarkable Ibn al-Haytham",
*The Mathematical Gazette*, Mathematical Association,**76**(475): 189–198, doi:10.2307/3620392, JSTOR 3620392 - Boutry, Pierre; Gries, Charly; Narboux, Julien; Schreck, Pascal (2019), "Parallel postulates and continuity axioms: a mechanized study in intuitionistic logic using Coq",
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