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geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, the parallel postulate, also called
Euclid Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
's fifth postulate because it is the fifth postulate in Euclid's ''Elements'', is a distinctive
axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...
in Euclidean geometry. It states that, in two-dimensional geometry:
''If a
line segment In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between i ...
intersects two straight lines forming two interior angles on the same side that are less than two
right angle In geometry and trigonometry, a right angle is an angle of exactly 90 degrees or radians 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. Th ...
s, 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; it is only a postulate related to parallelism. Euclid gave the definition of parallel lines in Book I, Definition 23 just before the five postulates. ''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 Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s * Independe ...
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").


Equivalent properties

Probably the best-known equivalent of Euclid's parallel postulate, contingent on his other postulates, is Playfair's axiom, named after the Scottish
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...
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.''
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. 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
angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles ...
s in every
triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...
is 180° (
triangle postulate A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non-collin ...
). # 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 angle In geometry and trigonometry, a right angle is an angle of exactly 90 degrees or radians 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. Th ...
s, then the fourth angle is also a right angle. # There exists a quadrilateral in which all angles are right angles, that is, a
rectangle In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram contain ...
. # There exists a pair of straight lines that are at constant
distance Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
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). #The law of cosines, a generalization of Pythagoras' theorem. # There is no upper limit to the
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an op ...
of a triangle. ( Wallis axiom) # 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 Proclus Lycius (; 8 February 412 – 17 April 485), called Proclus the Successor ( grc-gre, Πρόκλος ὁ Διάδοχος, ''Próklos ho Diádokhos''), was a Greek Neoplatonist philosopher, one of the last major classical philosophe ...
' axiom) 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.


History

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 Proclus Lycius (; 8 February 412 – 17 April 485), called Proclus the Successor ( grc-gre, Πρόκλος ὁ Διάδοχος, ''Próklos ho Diádokhos''), was a Greek Neoplatonist philosopher, one of the last major classical philosophe ...
(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 Claudius Ptolemy (; grc-gre, Πτολεμαῖος, ; la, Claudius Ptolemaeus; AD) was a mathematician, astronomer, astrologer, geographer, and music theorist, who wrote about a dozen scientific treatises, three of which were of importanc ...
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 Ḥasan Ibn al-Haytham, Latinized as Alhazen (; full name ; ), was a medieval mathematician, astronomer, and physicist of the Islamic Golden Age from present-day Iraq.For the description of his main fields, see e.g. ("He is one of the pr ...
(Alhazen) (965-1039), an Arab mathematician, made an attempt at proving the parallel postulate using a
proof by contradiction In logic and mathematics, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition, by showing that assuming the proposition to be false leads to a contradiction. Proof by contradiction is also known ...
, in the course of which he introduced the concept of
motion In physics, motion is the phenomenon in which an object changes its position with respect to time. Motion is mathematically described in terms of displacement, distance, velocity, acceleration, speed and frame of reference to an observer and m ...
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 Ghiyāth al-Dīn Abū al-Fatḥ ʿUmar ibn Ibrāhīm Nīsābūrī (18 May 1048 – 4 December 1131), commonly known as Omar Khayyam ( fa, عمر خیّام), was a polymath, known for his contributions to mathematics, astronomy, philosophy, an ...
(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 Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of ...
), 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.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 69
Routledge Routledge () is a British multinational publisher. It was founded in 1836 by George Routledge, and specialises in providing academic books, journals and online resources in the fields of the humanities, behavioural science, education, law ...
, London and New York:
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''. 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. He also considered the cases of what are now known as elliptical and hyperbolic geometry, though he ruled out both of them. 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''." His work was published in
Rome , established_title = Founded , established_date = 753 BC , founder = King Romulus ( legendary) , image_map = Map of comune of Rome (metropolitan city of Capital Rome, region Lazio, Italy).svg , map_caption ...
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 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 Giovanni Girolamo Saccheri (; 5 September 1667 – 25 October 1733) was an Italian Jesuit priest, scholastic philosopher, and mathematician. Saccheri was born in Sanremo. He entered the Jesuit order in 1685 and was ordained as a priest in 169 ...
(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 In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consiste ...
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 János Bolyai (; 15 December 1802 – 27 January 1860) or Johann Bolyai, was a Hungarian mathematician, who developed absolute geometry—a geometry that includes both Euclidean geometry and hyperbolic geometry. The discovery of a consis ...
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 Farkas Bolyai (; 9 February 1775 – 20 November 1856; also known as Wolfgang Bolyai in Germany) was a Hungarian mathematician, mainly known for his work in geometry. Biography Bolyai was born in Bolya, a village near Hermannstadt, Grand P ...
, 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 Independence is a condition of a person, nation, country, or state in which residents and population, or some portion thereof, exercise self-government, and usually sovereignty, over its territory. The opposite of independence is the stat ...
of the parallel postulate from Euclid's other axioms was finally demonstrated by Eugenio Beltrami in 1868.


Converse of Euclid's parallel postulate

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 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 which is violated in elliptic geometry.


Criticism

Attempts to logically prove the parallel postulate, rather than the eighth axiom, were criticized by
Arthur Schopenhauer Arthur Schopenhauer ( , ; 22 February 1788 – 21 September 1860) was a German philosopher. He is best known for his 1818 work ''The World as Will and Representation'' (expanded in 1844), which characterizes the phenomenal world as the prod ...
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.http://www.gutenberg.org/files/40097/40097-pdf.pdf


Decomposition of the parallel postulate

The parallel postulate is equivalent, as shown in, 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, 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''. As shown in, the splitting of the parallel postulate into the conjunction of these incidence-geometric axioms is possible only in the presence of absolute geometry.


See also

* Line at infinity * Non-Euclidean geometry


Notes


References

*
Carroll, Lewis Charles Lutwidge Dodgson (; 27 January 1832 – 14 January 1898), better known by his pen name Lewis Carroll, was an English author, poet and mathematician. His most notable works are ''Alice's Adventures in Wonderland'' (1865) and its seque ...
, ''Euclid and His Modern Rivals'', Dover, * * * * * * * *


External links


On Gauss' Mountains
{{Authority control Elementary geometry Foundations of geometry Non-Euclidean geometry History of geometry tr:Paralel#Paralel aksiyomu