In geometry, the parallel postulate, also called Euclid's fifth postulate since it is the fifth postulate in Euclid's Elements, is a distinctive axiom in what is now called Euclidean geometry. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry Euclid's Elements ( Greek:) is a mathematical and geometric Treatise consisting of 13 books written by the Greek In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. It states that:

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. In Geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its end points In Geometry and Trigonometry, a right angle is an angle of 90 degrees corresponding to a quarter turn (that is a quarter of a full circle

Euclidean geometry is the study of geometry that satisfies all of Euclid's axioms, including the parallel postulate. A geometry where the parallel postulate cannot hold is known as a non-euclidean geometry. In mathematics non-Euclidean geometry describes how this all works--> hyperbolic and Elliptic geometry, which are contrasted with Euclidean geometry Geometry that is independent of Euclid's fifth postulate (i. e. , only assumes the first four postulates) is known as absolute geometry (or, in some places, neutral geometry). Absolute geometry is a Geometry based on an Axiom system that does not assume the Parallel postulate or any of its alternatives Absolute geometry is a Geometry based on an Axiom system that does not assume the Parallel postulate or any of its alternatives

Converse of Euclid's parallel postulate

If the sum of the two interior angles equals 180°, the lines are parallel and will never intersect.

Euclid did not postulate the converse of his fifth postulate, which is one way to distinguish Euclidean geometry from elliptic geometry. Conversion is a concept in Traditional logic referring to a "type of immediate Inference in which from a given Proposition another proposition is inferred Elliptic geometry (sometimes known as Riemannian geometry) is a Non-Euclidean geometry, in which given a line L and a point The Elements contains the proof of an equivalent statement (Book I, Proposition 17): Any two angles of a triangle are together less than two right angles. In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line In Geometry and Trigonometry, a right angle is an angle of 90 degrees corresponding to a quarter turn (that is a quarter of a full circle The proof depends on an earlier proposition: In a triangle ABC, the exterior angle at C is greater than either of the interior angles A or B. Geometry, an interior angle (or internal angle) is an Angle formed by two sides of a Simple polygon that share an endpoint namely the angle Geometry, an interior angle (or internal angle) is an Angle formed by two sides of a Simple polygon that share an endpoint namely the angle This in turn depends on Euclid's unstated assumption that two straight lines meet in only one point, a statement not true of elliptic geometry.

In other words, the converse of the fifth postulate follows from Euclid's axioms minus the fifth postulate, plus an axiom stating that two distinct non-parallel straight lines meet in only one point.

Logically equivalent properties

It is often and erroneously asserted that Euclid's parallel postulate is equivalent to Playfair's axiom, named after the Scottish mathematician John Playfair. A mathematician is a person whose primary area of study and research is the field of Mathematics. Professor John Playfair FRSE ( March 10, 1748 &ndash July 20, 1819) was a Scottish Scientist. It states:

Exactly one line can be drawn through any point not on a given line parallel to the given line.

This axiom is actually more powerful than Euclid's parallel postulate, as it assumes that a single parallel line exists. This does not follow from Euclid's postulate. In fact, it is possible to develop spherical geometry without contradicting the parallel postulate, as it does not assert that the lines will not meet again on the side of the obtuse interior angles. Euclid himself believed he had shown in his Proposition 1. 27 that parallel lines exist independently of the parallel postulate, which would have ruled out spherical geometry. However this proof depends on an implicit assumption made in Proposition 1. 16 which Euclid does not appear to have recognized. This assumption along with the parallel postulate are together equivalent to Playfair's axiom.

Some of the other statements that are equivalent to the parallel postulate or to Playfair's axiom appear at first to be unrelated to parallelism. Some even seem so self-evident that they were unconsciously assumed by people who claimed to have proven the parallel postulate from Euclid's other postulates. In Epistemology (theory of knowledge a self-evident proposition is one that is known to be true by understanding its meaning without proof. Many observers throughout history have argued that there are influences on Consciousness from other parts of the Mind. Here are some of these results:

1. The sum of the angles in every triangle is 180°. In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line
2. There exists a triangle whose angles add up to 180°.
3. The sum of the angles is the same for every triangle.
4. There exists a pair of similar, but not congruent, triangles. Geometry Two geometrical objects are called similar if one is congruent to the result of a uniform scaling (enlarging or shrinking of the other In Geometry, two sets of points are called congruent if one can be transformed into the other by an Isometry, i
5. Every triangle can be circumscribed.
6. If three angles of a quadrilateral are right angles, then the fourth angle is also a right angle. In Geometry, a quadrilateral is a Polygon with four sides or edges and four vertices or corners. In Geometry and Trigonometry, a right angle is an angle of 90 degrees corresponding to a quarter turn (that is a quarter of a full circle
7. There exists a quadrilateral of which all angles are right angles.
8. There exists a pair of straight lines that are at constant distance from each other. Distance is a numerical description of how far apart objects are
9. Two lines that are parallel to the same line are also parallel to each other.
10. Given two parallel lines, any line that intersects one of them also intersects the other.
11. In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides (Pythagoras' Theorem). Two types of special right triangles appear commonly in geometry the "angle based" and the "side based" (or Pythagorean Triangles The former are characterised In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry
12. There is no upper limit to the area of a triangle. [1]

However, the alternatives which employ the word "parallel" cease appearing so simple when one is obliged to explain which of the three common definitions of "parallel" is meant - constant separation, never meeting or same angles where crossed by a third line - since the equivalence of these three is itself one of the unconsciously obvious assumptions equivalent to Euclid's fifth postulate.

History

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 the fifth postulate isn't self-evident unlike the other postulates. 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^[1].

Omar Khayyám (1050-1123) recognized that three possibilities arose from omitting Euclid's Fifth; 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). For the Thoroughbred racehorse see Omar Khayyam (horse Ghiyās od-Dīn Abol-Fath Omār ibn Ebrāhīm Khayyām Neyshābūri (غیاث الدین If those equal internal angles are right angles, we get Euclid's Fifth; otherwise, they must be either acute or obtuse. He persuaded himself that the acute and obtuse cases lead to contradiction, but had made a tacit assumption equivalent to the fifth to get there.

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 debunk the acute case (although he managed to wrongly persuade himself that he had). Giovanni Girolamo Saccheri ( September 5, 1667 - October 25, 1733) was an Italian Jesuit priest and mathematician

Where Saccheri and Khayyám 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 Lobachevski published an account of acute geometry in an obscure Russian journal (later re-published in 1840 in German). Nikolai Ivanovich Lobachevsky (Никола́й Ива́нович Лобаче́вский ( December 1 1792 &ndash February 24 1856 ( N

In 1831, János Bolyai included, in a book by his father, an appendix describing acute geometry, which, doubtlessly, he had developed independently of Lobachevski. János Bolyai ( December 15, 1802 – January 27, 1860) was a Hungarian Mathematician, known for his work in Non-Euclidean

It is probable that Carl Friedrich Gauss had actually studied the problem before that, but if so, he didn't publish any of his results. Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German

The resulting geometries were later developed by Lobachevsky, Riemann and Poincaré into hyperbolic geometry (the acute case) and spherical geometry (the obtuse case). Nikolai Ivanovich Lobachevsky (Никола́й Ива́нович Лобаче́вский ( December 1 1792 &ndash February 24 1856 ( N Jules Henri Poincaré ( 29 April 1854 &ndash 17 July 1912) (ˈʒyl ɑ̃ˈʁi pwɛ̃kaˈʁe was a French Mathematician In Spherical geometry is the Geometry of the two- Dimensional surface of a Sphere.

The independence of the parallel postulate from Euclid's other axioms was finally demonstrated by Eugenio Beltrami in 1868. In Mathematical logic, a sentence &sigma is called independent of a given first-order theory T if T neither proves nor Eugenio Beltrami ( 16 November, 1835 - 4 June, 1899) was an Italian mathematician notable for his work on Non-Euclidean geometry Year 1868 ( MDCCCLXVIII) was a Leap year starting on Wednesday (link will display the full calendar of the Gregorian Calendar (or a Leap

Criticism

Attempts to logically prove this postulate, rather than the eighth axiom, were criticized by Schopenhauer, as described in Schopenhauer's criticism of the proofs of the Parallel Postulate. Arthur Schopenhauer criticized mathematicians' attempts to prove Euclid 's Parallel Postulate because they try to prove from indirect

See also

• For more information, see the history of non-Euclidean geometry. In mathematics non-Euclidean geometry describes how this all works--> hyperbolic and Elliptic geometry, which are contrasted with Euclidean geometry

References

1. ^ Florence P. Lewis (Jan 1920). "History of the Parallel Postulate". The American Mathematical Monthly 27 (1): 16-23.  

Further reading

• Lewis Carroll, Euclid and His Modern Rivals, Dover, ISBN 0-486-22968-8