In
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, the parallel postulate is the fifth postulate in
Euclid's ''Elements'' and 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
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...
. It states that, in two-dimensional geometry:
If a line segment
In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct endpoints (its extreme points), and contains every Point (geometry), point on the line that is between its endpoints. It is a special c ...
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 Degree (angle), degrees or radians corresponding to a quarter turn (geometry), turn. If a Line (mathematics)#Ray, ray is placed so that its endpoint is on a line and the ad ...
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
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 ge ...
. 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 Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
of Euclid's fifth postulate (i.e., only assumes the modern equivalent of the first four postulates) is known as
absolute geometry
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. The term was introduced by ...
(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
In geometry, Playfair's axiom is an axiom that can be used instead of the fifth postulate of Euclid (the parallel postulate
In geometry, the parallel postulate is the fifth postulate in Euclid's ''Elements'' and a distinctive axiom in Euclidea ...
, 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, mathematical structure, structure, space, Mathematica ...
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
In logic and mathematics, statements p and q are said to be logically equivalent if they have the same truth value in every model. The logical equivalence of p and q is sometimes expressed as p \equiv q, p :: q, \textsfpq, or p \iff q, depending on ...
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, one can be used to prove the other, so they are equivalent in the context of
absolute geometry
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. The term was introduced by ...
.
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
In geometry, Playfair's axiom is an axiom that can be used instead of the fifth postulate of Euclid (the parallel postulate
In geometry, the parallel postulate is the fifth postulate in Euclid's ''Elements'' and a distinctive axiom in Euclidea ...
)
# The sum of the
angle
In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...
s in every
triangle
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...
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
Congruence may refer to:
Mathematics
* Congruence (geometry), being the same size and shape
* Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure
* In modu ...
, triangles.
# Every triangle can be
circumscribe In geometry, a circumscribed circle for a set of points is a circle passing through each of them. Such a circle is said to ''circumscribe'' the points or a polygon formed from them; such a polygon is said to be ''inscribed'' in the circle.
* Circum ...
d.
# If three angles of a
quadrilateral
In Euclidean geometry, geometry a quadrilateral is a four-sided polygon, having four Edge (geometry), edges (sides) and four Vertex (geometry), corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''l ...
are
right angle
In geometry and trigonometry, a right angle is an angle of exactly 90 Degree (angle), degrees or radians corresponding to a quarter turn (geometry), turn. If a Line (mathematics)#Ray, ray is placed so that its endpoint is on a line and the ad ...
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 geometry, Euclidean plane geometry, a rectangle is a Rectilinear polygon, rectilinear convex polygon or a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that a ...
.
# 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, points, people, or ideas are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two co ...
from each other.
# Two lines that are parallel to the same line are also parallel to each other.
# In a
right-angled triangle
A right triangle or right-angled triangle, sometimes called an orthogonal triangle or rectangular triangle, is a triangle in which two sides are perpendicular, forming a right angle ( turn or 90 degrees).
The side opposite to the right angle i ...
, the square of the hypotenuse equals the sum of the squares of the other two sides (
Pythagoras' theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite ...
).
# The
law of cosines
In trigonometry, the law of cosines (also known as the cosine formula or cosine rule) relates the lengths of the sides of a triangle to the cosine of one of its angles. For a triangle with sides , , and , opposite respective angles , , and (see ...
, a generalization of Pythagoras' theorem.
# There is no upper limit to the
area
Area is the measure of a region's size on a 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 open surface or the boundary of a three-di ...
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 (, ''Próklos ho Diádokhos''), was a Greek Neoplatonist philosopher, one of the last major classical philosophers of late antiquity. He set forth one of th ...
' 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
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P' ...
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
In geometry, Playfair's axiom is an axiom that can be used instead of the fifth postulate of Euclid (the parallel postulate
In geometry, the parallel postulate is the fifth postulate in Euclid's ''Elements'' and a distinctive axiom in Euclidea ...
). 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 (, ''Próklos ho Diádokhos''), was a Greek Neoplatonist philosopher, one of the last major classical philosophers of late antiquity. He set forth one of th ...
(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 (; , ; ; – 160s/170s AD) was a Greco-Roman mathematician, astronomer, astrologer, geographer, and music theorist who wrote about a dozen scientific treatises, three of which were important to later Byzantine science, Byzant ...
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 (Latinization of names, Latinized as Alhazen; ; full name ; ) was a medieval Mathematics in medieval Islam, mathematician, Astronomy in the medieval Islamic world, astronomer, and Physics in the medieval Islamic world, p ...
(Alhazen) (965–1039), an
Arab mathematician, made an attempt at proving the parallel postulate using a
proof by contradiction
In logic, 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.
Although it is quite freely used in mathematical pr ...
, in the course of which he introduced the concept of
motion
In physics, motion is when an object changes its position with respect to a reference point in a given time. Motion is mathematically described in terms of displacement, distance, velocity, acceleration, speed, and frame of reference to an o ...
and
transformation into geometry. He formulated the
Lambert quadrilateral
In geometry, a Lambert quadrilateral (also known as Ibn al-Haytham–Lambert quadrilateral), is a quadrilateral in which three of its angles are right angles. Historically, the fourth angle of a Lambert quadrilateral was of considerable interest s ...
, which Boris Abramovich Rozenfeld names the "Ibn al-Haytham–Lambert quadrilateral",
and his attempted proof contains elements similar to those found in
Lambert quadrilateral
In geometry, a Lambert quadrilateral (also known as Ibn al-Haytham–Lambert quadrilateral), is a quadrilateral in which three of its angles are right angles. Historically, the fourth angle of a Lambert quadrilateral was of considerable interest s ...
s and
Playfair's axiom
In geometry, Playfair's axiom is an axiom that can be used instead of the fifth postulate of Euclid (the parallel postulate
In geometry, the parallel postulate is the fifth postulate in Euclid's ''Elements'' and a distinctive axiom in Euclidea ...
.
The Persian mathematician, astronomer, philosopher, and poet
Omar Khayyám
Ghiyāth al-Dīn Abū al-Fatḥ ʿUmar ibn Ibrāhīm Nīshābūrī (18 May 1048 – 4 December 1131) ( Persian: غیاث الدین ابوالفتح عمر بن ابراهیم خیام نیشابورﻯ), commonly known as Omar Khayyam (), was ...
(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 (; 384–322 BC) was an Ancient Greek philosophy, Ancient Greek philosopher and polymath. His writings cover a broad range of subjects spanning the natural sciences, philosophy, linguistics, economics, politics, psychology, a ...
), 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
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For a ...
, 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, pp. 447–494 69 ]Routledge
Routledge ( ) is a British multinational corporation, multinational publisher. It was founded in 1836 by George Routledge, and specialises in providing academic books, academic journals, journals and online resources in the fields of the humanit ...
, 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
Giovanni Girolamo Saccheri (; 5 September 1667 – 25 October 1733) was an Italian Jesuit priest, scholastic philosopher, and mathematician. He is considered the forerunner of non-Euclidean geometry.
Biography
The son of a lawyer, Saccheri w ...
), 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
Muḥammad ibn Muḥammad ibn al-Ḥasan al-Ṭūsī (1201 – 1274), also known as Naṣīr al-Dīn al-Ṭūsī (; ) or simply as (al-)Tusi, was a Persians, Persian polymath, architect, Early Islamic philosophy, philosopher, Islamic medicine, phy ...
(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
Rome (Italian language, Italian and , ) is the capital city and most populated (municipality) of Italy. It is also the administrative centre of the Lazio Regions of Italy, region and of the Metropolitan City of Rome. A special named with 2, ...
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 (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
In geometry, a Lambert quadrilateral (also known as Ibn al-Haytham–Lambert quadrilateral), is a quadrilateral in which three of its angles are right angles. Historically, the fourth angle of a Lambert quadrilateral was of considerable interest s ...
'', 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 deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences o ...
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
Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observatory and ...
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
Poincaré is a French surname. Notable people with the surname include:
* Henri Poincaré
Jules Henri Poincaré (, ; ; 29 April 185417 July 1912) was a French mathematician, Theoretical physics, theoretical physicist, engineer, and philos ...
into hyperbolic geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For a ...
(the acute case) and elliptic geometry
Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines ...
(the obtuse case). The independence
Independence is a condition of a 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 status of ...
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
Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines ...
. 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 known for his 1818 work ''The World as Will and Representation'' (expanded in 1844), which characterizes the Phenomenon, phenomenal world as ...
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.
Decomposition of the parallel postulate
The parallel postulate is equivalent 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''.
The splitting of the parallel postulate into the conjunction of these incidence-geometric axioms is possible only in the presence of absolute geometry
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. The term was introduced by ...
.
See also
* Line at infinity
In geometry and topology, the line at infinity is a projective line that is added to the affine plane in order to give closure to, and remove the exceptional cases from, the incidence properties of the resulting projective plane. The line at ...
* Non-Euclidean geometry
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 ge ...
Notes
References
* Carroll, Lewis, ''Euclid and His Modern Rivals'', Dover,
*
*
*
*
*
*
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*
*
External links
On Gauss' Mountains
{{Authority control
Elementary geometry
Foundations of geometry
Non-Euclidean geometry
History of geometry
Greek mathematics
tr:Paralel#Paralel aksiyomu