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Geometric axiom
In geometry, the parallel postulate is the fifth postulate in Euclid's Elements and a distinctive axiom in Euclidean geometry. It states that, in two-dimensional
Parallel_postulate
German philosopher (1788–1860)
evident in his criticism of contemporaneous attempts to prove the parallel postulate in Euclidean geometry. Writing shortly before the discovery of hyperbolic
Arthur_Schopenhauer
Statement that is taken to be true
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
Axiom
Mathematical model of the physical space
axioms (postulates) and deducing many other propositions (theorems) from these. One of those is the parallel postulate which relates to parallel lines on
Euclidean_geometry
Modern formulation of Euclid's parallel postulate
instead of the fifth postulate of Euclid (the parallel postulate): In a plane, given a line and a point not on it, at most one line parallel to the given line
Playfair's_axiom
Two geometries based on axioms closely related to those specifying Euclidean geometry
affine geometry, non-Euclidean geometry arises by either replacing the parallel postulate with an alternative, or consideration of quadratic forms other than
Non-Euclidean_geometry
Fundamental result in geometry
triangle postulate states that the sum of the angles of a triangle is two right angles. This postulate is equivalent to the parallel postulate. In the
Sum_of_angles_of_a_triangle
Study of geometries as axiomatic systems
seemed so intuitively obvious (with the possible exception of the parallel postulate) that any theorem proved from them was deemed true in an absolute
Foundations_of_geometry
Relation used in geometry
direction (not necessarily the same length). Parallel lines are the subject of Euclid's parallel postulate. Parallelism is primarily a property of affine
Parallel_(geometry)
Type of non-Euclidean 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
Hyperbolic_geometry
Relation between sides of a right triangle
W. (1999). "Parallel Postulate". CRC Concise Encyclopedia of Mathematics. CRC Press. p. 1313. ISBN 0-8493-9640-9. The parallel postulate is equivalent
Pythagorean_theorem
Geometry without the parallel postulate
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
Absolute_geometry
Mathematical treatise by Euclid
numbers. While the first four postulates are relatively straightforward, the fifth is not. It is known as the parallel postulate, and the question of its independence
Euclid's_Elements
Geometry of the surface of a sphere
geometry sufficient to resolve the ancient problem of whether the parallel postulate is a logical consequence of the rest of Euclid's axioms of plane geometry
Spherical_geometry
Quadrilateral with two pairs of parallel sides
consequence of the Euclidean parallel postulate and neither condition can be proven without appealing to the Euclidean parallel postulate or one of its equivalent
Parallelogram
Line intersecting 2 coplanar lines at 2 points
angles, and linear pairs. As a consequence of Euclid's parallel postulate, if the two lines are parallel, consecutive angles and linear pairs are supplementary
Transversal_(geometry)
Fundamental space of geometry
impossible to prove (parallel postulate). After the introduction at the end of the 19th century of non-Euclidean geometries, the old postulates were re-formalized
Euclidean_space
Geometric axiom
the Parallel Postulate, as shown by Max Dehn. In the presence of the Archimedean axiom, the Lotschnittaxiom is equivalent with the Parallel Postulate. As
Lotschnittaxiom
Branch of mathematics
Later in the 19th century, it appeared that geometries without the parallel postulate (non-Euclidean geometries) can be developed without introducing any
Geometry
Category of mathematical proof
parallel postulate from Euclid's Elements is equivalent to the statement that given a straight line and a point not on that line, only one parallel to
Proof_of_impossibility
a theorem. The Parallel Postulate is the fifth postulate in Euclid's Elements and it is independent from the preceding four postulates. This was first
Axiom_independence
Framework of distances and directions
that does not include the parallel postulate, called hyperbolic geometry. In this geometry, an infinite number of parallel lines pass through the point
Space
Quadrilateral with two equal sides perpendicular to the base
vindicatus (Euclid freed of every flaw), an attempt to prove the parallel postulate using the method reductio ad absurdum. Such a quadrilateral is sometimes
Saccheri_quadrilateral
Ancient Greek mathematician (fl. 300 BC)
comparison of magnitudes. While postulates 1 through 4 are relatively straightforward, the 5th is known as the parallel postulate and particularly famous. Book
Euclid
constructibility Rank-into-rank Kripke–Platek axioms Diamond principle Parallel postulate Birkhoff's axioms (4 axioms) Hilbert's axioms (20 axioms) Tarski's
List_of_axioms
2D surface which extends indefinitely
particular the parallel postulate. A projective plane may be constructed by adding "points at infinity" where two otherwise parallel lines would intersect
Plane_(mathematics)
Basic framework of mathematics
unprovability of the parallel postulate lead to several philosophical problems, the main one being that before this discovery, the parallel postulate and all its
Foundations_of_mathematics
Non-Euclidean geometry
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
Elliptic_geometry
Historical development of geometry
very old problem of proving Euclid's Fifth Postulate, the "Parallel Postulate", from his first four postulates had never been forgotten. Beginning not long
History_of_geometry
Persian polymath and poet (1048–1131)
acute cases as self-contradictory. His elaborate attempt to prove the parallel postulate was significant for the further development of geometry, as it clearly
Omar_Khayyam
In absolute geometry, the sum of the angles in a triangle is at most 180°
geometry with the exception of the axiom that is equivalent to the parallel postulate of Euclid. The theorem is named after Giovanni Girolamo Saccheri and
Saccheri–Legendre_theorem
German polymath (1646–1716)
create a definition for a straight line while attempting to prove the parallel postulate. While most mathematicians defined a straight line as the shortest
Gottfried_Wilhelm_Leibniz
Shape with three sides
space is always 180 degrees. This fact is equivalent to Euclid's parallel postulate. This allows the determination of the measure of the third angle of
Triangle
Topics referred to by the same term
Independence (mathematical logic), unprovability of a sentence (e.g. the Parallel postulate) from other sentences (e.g. the remaining Euclidean geometry axioms)
Independent
Exterior angle of a triangle is greater than either of the remote interior angles
result in absolute geometry because its proof does not depend upon the parallel postulate. In several high school treatments of geometry, the term "exterior
Exterior_angle_theorem
no need to break into different cases, such as parallel to the x {\displaystyle x} -axis and non-parallel. The other advantage is that these homogeneous
Laguerre_transformations
Non-Euclidean geometry
Euclidean space, but such that Euclid's parallel postulate is no longer assumed to hold. Instead, the parallel postulate is replaced by the following alternative
Hyperbolic_space
Proposition in mathematics that is unproven
negation, as a new axiom in a consistent manner (much as Euclid's parallel postulate can be taken either as true or false in an axiomatic system for geometry)
Conjecture
Property of objects which are scaled or mirrored versions of each other
exists is Wallis's postulate and is logically equivalent to Euclid's parallel postulate. In hyperbolic geometry (where Wallis's postulate is false) similar
Similarity_(geometry)
Limitative results in mathematical logic
geometry without the parallel postulate is incomplete, because some statements in the language (such as the parallel postulate itself) can not be proved
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
Mathematical concept
the infinite". There is a similar controversy concerning Euclid's parallel postulate, sometimes translated: If a straight line falling across two [other]
Infinity
about what he perceived as flaws in Euclid's Elements, especially the parallel postulate. He was also the first to find the general geometric solution to cubic
History_of_mathematics
Subfield of mathematics
for geometry became known. In addition to the independence of the parallel postulate, established by Nikolai Lobachevsky in 1826, mathematicians discovered
Mathematical_logic
synthetic geometry similar to Euclidean geometry but without the parallel postulate. Abstract algebra The part of algebra devoted to the study of algebraic
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Set of points equidistant from a center
sphere fails to satisfy some of classical geometry's postulates, including the parallel postulate. In spherical trigonometry, angles are defined between
Sphere
Relationship between two lines that meet at a right angle
that are both perpendicular to a third line are parallel to each other, because of the parallel postulate. Conversely, if one line is perpendicular to a
Perpendicular
Arab physicist, mathematician and astronomer (c. 965 – c. 1040)
formula. Alhazen explored what is now known as the Euclidean parallel postulate, the fifth postulate in Euclid's Elements, using a proof by contradiction, and
Ibn_al-Haytham
Mathematical structures that allow quantum mechanics to be explained
conventionally termed a "ray". Accompanying Postulate I is the composite system postulate: Composite system postulate The Hilbert space of a composite system
Mathematical formulation of quantum mechanics
Mathematical_formulation_of_quantum_mechanics
Italian Jesuit priest, philosopher and mathematician (1667–1733)
Euclid's parallel postulate. To do so, he assumed that the parallel postulate was false and attempted to derive a contradiction. Since Euclid's postulate is
Giovanni_Girolamo_Saccheri
German polymath and scholar (1777–1855)
and known as the Gauss–Bonnet theorem. During Gauss's lifetime, the parallel postulate of Euclidean geometry was heavily discussed. Numerous efforts were
Carl_Friedrich_Gauss
Common point(s) shared by two lines in Euclidean geometry
projective space Distance between two parallel lines Distance from a point to a line Line–plane intersection Parallel postulate Triangulation (computer vision)
Line–line_intersection
geometry, the point–line–plane postulate is a collection of assumptions (axioms) that can be used in a set of postulates for Euclidean geometry in two
Point–line–plane_postulate
Topological space that locally resembles Euclidean space
important results. Non-Euclidean geometry considers spaces where Euclid's parallel postulate fails. Saccheri first studied such geometries in 1733, but sought
Manifold
Axiom in the foundations of geometry
of a right angle intersect", is equivalent to the Parallel Postulate. Without the parallel postulate, Aristotle's axiom is equivalent to each of the following
Aristotle's_axiom
English mathematician (1616–1703)
equivalent to the parallel postulate. After reading this, Wallis then wrote about his ideas as he developed his own thoughts about the postulate, trying to prove
John_Wallis
Topics referred to by the same term
Convergence of parallel lines can refer to: In everyday life, the vanishing point phenomenon Non-Euclidean geometry in which Euclid's parallel postulate does not
Convergence_of_parallel_lines
Branch of mathematics that studies sets
several paradoxes or counter-intuitive results. For example, that the parallel postulate cannot be proved, the existence of mathematical objects that cannot
Set_theory
Problem of constructing equal-area shapes
latter AP equal to AM. Through P draw PQ parallel to MN and meeting AM at Q. Join OQ and through T draw TR, parallel to OQ and meeting AQ at R. Draw AS perpendicular
Squaring_the_circle
Field of knowledge
non-Euclidean geometries, which do not follow the parallel postulate. By questioning that postulate's truth, this discovery has been viewed as joining
Mathematics
Hypothetical group of multiple universes
describe them. The different universes within the multiverse are called "parallel universes", "flat universes", "other universes", "alternate universes"
Multiverse
Research program on the symmetries of geometry
traditional Euclidean geometry had revealed the independence of the parallel postulate from the others, and non-Euclidean geometry had been born. Klein proposed
Erlangen_program
Index of articles associated with the same name
which takes values in Ω(t), gives a model of Euclidean geometry. The parallel postulate is true in this model, but if the deviation from the perpendicular
Dehn_plane
Geometry without using coordinates
appropriate to speak of "geometry" in the singular. Historically, Euclid's parallel postulate has turned out to be independent of the other axioms. Simply discarding
Synthetic_geometry
Italian mathematician (1633–1711)
Euclides Restitutus of 1658). In examining Borelli's proof of the parallel postulate, Giordano noted that it depended upon the assumption that a line everywhere
Giordano_Vitale
Scottish minister and scientist (1748–1819)
textbook Elements of Geometry made a brief expression of Euclid's parallel postulate known now as Playfair's axiom. In 1783 he was a co-founder of the
John_Playfair
Euclidean geometry without distance and angles
infinity. In affine geometry, there is no metric structure but the parallel postulate does hold. Affine geometry provides the basis for Euclidean structure
Affine_geometry
Hungarian mathematician (1802–1860)
the age of 24, a captain. Bolyai became so obsessed with Euclid's parallel postulate that his father, who had pursued the same subject for many years,
János_Bolyai
Relates the length of a median of a triangle to the lengths of its sides
Euclidean geometry Golden ratio Lune of Hippocrates Method of exhaustion Parallel postulate Platonic solid Regular polygon Straightedge and compass construction
Apollonius's_theorem
1868 – Eugenio Beltrami demonstrates the independence of Euclid’s parallel postulate from the other axioms of Euclidean geometry. 1870 – Felix Klein constructs
Timeline_of_mathematics
Reasoning for mathematical statements
axioms is called undecidable (from those axioms). One example is the parallel postulate, which is neither provable nor refutable from the remaining axioms
Mathematical_proof
Book by Gottlob Frege
incongruous and paradoxical." Frege 1884, §14: "The fact that [denying the parallel postulate] is possible shows that the axioms of geometry are independent of
The_Foundations_of_Arithmetic
development of the two forms of non-Euclidean geometry, where the parallel postulate of Euclidean geometry no longer holds. The Russian mathematician Nikolai
19th_century_in_science
Book by Jordan Ellenberg
Universe": This chapter talks about János Bolyai, and his work on the parallel postulate. Others mentioned in this chapter include David Hilbert, and Gottlob
How_Not_to_Be_Wrong
Pythagoreans were sailing because his discovery was too heretical. Euclid's parallel postulate stated that if two lines cross a third in a plane in such a way that
List_of_conjectures
Term in mathematical logic
the foundations of physics. List of statements independent of ZFC Parallel postulate for an example in geometry Paterek, T.; Kofler, J.; Prevedel, R.;
Independence (mathematical logic)
Independence_(mathematical_logic)
Impossibility of angle trisection (geometry) Independence of the parallel postulate (geometry) Inscribed angle theorem (geometry) Intercept theorem (Euclidean
List_of_theorems
Concept in geometry and topology
and Q. Elliptic geometry § Elliptic plane Hyperplane at infinity Parallel postulate Plane at infinity Point at infinity Weisstein, Eric W. "Line at Infinity"
Line_at_infinity
Basic notion of sameness in mathematics
showed a contradiction of naive set theory, it was shown that the parallel postulate cannot be proved, the existence of mathematical objects that cannot
Equality_(mathematics)
9th century Islamic geometer
additional propositions and an attempted mathematical proof of the parallel postulate.[citation needed] Described as having superb knowledge of Greek, which
Al-Abbās_ibn_Said_al-Jawharī
geometry, the AA postulate states that two triangles are similar if they have two corresponding angles congruent. The AA postulate follows from the fact
AA_postulate
Overview of and topical guide to geometry
theorem Interactive geometry software Involutes Goat grazing problem Parallel postulate Polygon Star polygon Pick's theorem Shape dissection Bolyai–Gerwien
Outline_of_geometry
Quadrilateral with only 3 right angles
since if it could be shown to be a right angle, then the Euclidean parallel postulate could be proved as a theorem. It is now known that the type of the
Lambert_quadrilateral
Mathematics of Ancient Greece and the Mediterranean, 5th BC to 6th AD
such as Carl Gauss, and attempts to prove or disprove Euclid's parallel line postulate spurred the development of non-Euclidean geometry. Ancient Greek
Ancient_Greek_mathematics
Geometry where the axiom of Archimedes is negated
in particular, there are infinitely many parallels to a straight line through a point—so the parallel postulate fails—but the sum of the angles of a triangle
Non-Archimedean_geometry
Euclidean geometry Golden ratio Lune of Hippocrates Method of exhaustion Parallel postulate Platonic solid Regular polygon Straightedge and compass construction
A History of Greek Mathematics
A_History_of_Greek_Mathematics
possibility to construct valid non-Euclidean geometries in which the parallel postulate is wrong, the Weierstrass function that is continuous but nowhere
Philosophy_of_mathematics
Axiom set used in first-order logic
the remaining Tarski's axioms, and indeed equivalent to Euclid's parallel postulate. A: ( ( B x y w ∧ x y ≡ y w ) ∧ ( B x u v ∧ x u ≡ u v ) ∧ ( B y u
Tarski's_axioms
Italian mathematician (1835–1900)
Euclidean geometry of the space, and in particular, that Euclid's parallel postulate could not be derived from the other axioms of Euclidean geometry.
Eugenio_Beltrami
was based on two postulates, as detailed below. Some formulations modify these postulates or attempt to derive the second postulate by deduction. Others
Formulations of special relativity
Formulations_of_special_relativity
Calculus using a logically rigorous notion of infinitesimal numbers
the parallel postulate is violated. For example, any line passing through the point (0, 1) on the y-axis and having infinitesimal slope is parallel to
Nonstandard_analysis
2nd-degree plane curve which is reducible
two to define one line, and the third for the parallel line to pass through, by the parallel postulate. Given two distinct points, there is a unique double
Degenerate_conic
Theory of interwoven space and time by Albert Einstein
light constancy, or the principle of light speed invariance. The first postulate was first formulated by Galileo Galilei (see Galilean invariance). Relativity
Special_relativity
Theorem in plane geometry
the fact that it has a different proof that does not presuppose the parallel postulate and is therefore valid in non-Euclidean geometry as well. By its help
Hjelmslev's_theorem
four axioms of Euclidean geometry are kept but the fifth axiom, the parallel postulate, is changed. The fifth axiom of hyperbolic geometry says that given
Constructions in hyperbolic geometry
Constructions_in_hyperbolic_geometry
American Midwest landscape depressions
structures are possibly remnants of volcanos. Rampino and Volk (1996) postulated that these structures could be the remains of a serial meteorite strike
38th_parallel_structures
Ordered field that does not satisfy the Archimedean property
ordered field, to construct non-Euclidean geometries in which the parallel postulate fails to be true but nevertheless triangles have angles summing to
Non-Archimedean_ordered_field
1992 book by Underwood Dudley
trisection, Fermat's Last Theorem, non-Euclidean geometry and the parallel postulate, the golden ratio, perfect numbers, the four color theorem, advocacy
Mathematical_Cranks
Emerging field of applied ethics
mathematics and ethics are closely parallel, a pluralist attitude should be taken to the truths of both. Just as the parallel postulate is true in Euclidean geometry
Ethics_in_mathematics
Iranian philosopher, astronomer, astrologer and mathematician
lit. Correction) of Euclid, one of which is an attempt to prove the parallel postulate, which was commented upon and criticized by Shams al-Dīn al-Samarqandī
Athir_al-Din_al-Abhari
Aug The abstract parabola fits the concrete world 1981 Oct Euclid's parallel postulate and its modern offspring 1981 Dec The Laffer curve and other laughs
List of Martin Gardner Mathematical Games columns
List_of_Martin_Gardner_Mathematical_Games_columns
PARALLEL POSTULATE
PARALLEL POSTULATE
Surname or Lastname
English
English : according to Reaney this is a nickname from an unattested Old English word cybbe meaning ‘clumsy’ or ‘thickset’. Reaney’s speculation is apparently based on taking the Middle English word kibble ‘cudgel’ as a diminutive of an unattested Old English word. Corresponding personal names have been postulated for the place names Kibworth (‘enclosure of a man called Cybba’) and Kibblesworth (‘enclosure of a man called Cybbel’); so, in theory, the surname could be a reflex of these Old English personal names.North German : nickname for a cantankerous person, from Middle Low German, Middle High German kiven ‘to quarrel’.
Girl/Female
Muslim
Example, Allegory, Parable
Boy/Male
Shakespearean
All's Well That Ends Well.' A follower of Bertram, Count of Rousillon.
Biblical
a parable; governing
Surname or Lastname
English (Lancashire)
English (Lancashire) : of uncertain origin. The most plausible suggestion is that it is a Norman nickname from Old French mort ‘dead’ (Latin mortuus), presumably referring to a person of deathly pallor or unnaturally still countenance, or possibly to someone who played the part of death in a pageant. However, it could also be the result of survival into the Middle English period of an Old English personal name, Morta, or an Old English vocabulary word mort ‘young salmon or trout’, both postulated by Ekwall to explain various place names (see for example Morcom).French : either a nickname from Old French mort ‘dead’ (see above), or an alteration, by folk etymology, of the personal name Mor(e) (see Moore 3).
Biblical
parables; governing
Girl/Female
Biblical
Parables, governing.
Surname or Lastname
Scottish
Scottish : possibly, as Black postulates, a habitational name from a place recorded in 1661 as Hantestoun.English : variant of Hampton.
Girl/Female
Biblical
A parable, governing.
Surname or Lastname
English
English : occupational name from Middle English combere, an agent derivative of Old English camb ‘comb’, referring perhaps to a maker or seller of combs, or to someone who used them to prepare wool or flax for spinning. This was an alternative process to carding, and caused the wool fibers to lie more or less parallel to one another, so that the cloth produced had a hard, smooth finish without a nap.English : variant of Coomber.Probably an Americanized spelling of German Kommer or Kammer.
Surname or Lastname
English
English : nickname for a noisy person, from Middle English lude ‘loud’ (Old English hlūd), perhaps in part preserving the Old English byname Hlūda that Ekwall postulates to explain the place names Loudham (Suffolk) and Lowdham (Nottinghamshire).English : topographic name for someone who lived by a roaring stream, Old English hlūde or hl̄de literally ‘the loud one’, or a habitational name from any of the places named from hl̄de, for example Lyde in Herefordshire and Somerset.English : variant of Louth.
Girl/Female
Arabic, Muslim
Example; Allegory; Parable
PARALLEL POSTULATE
PARALLEL POSTULATE
Girl/Female
English French
Brit. A native of England: (Britain) or France: (Brittany). In literature Lady Brett Ashley was...
Male
Scottish
Scottish Gaelic form of Latin Alexandrus, ALASTAIR means "defender of mankind."
Biblical
who overthrows or destroys a multitude
Girl/Female
Hindu
Big
Girl/Female
Tamil
Lustrous
Boy/Male
Indian, Sanskrit
Oath; Order
Girl/Female
Gujarati, Hindu, Indian
Sweet
Boy/Male
Hindu, Indian, Tamil
Joker
Girl/Female
Indian
The beautiful woman, Beautiful jewel
Girl/Female
British, English
Purest; Wind; Fair
PARALLEL POSTULATE
PARALLEL POSTULATE
PARALLEL POSTULATE
PARALLEL POSTULATE
PARALLEL POSTULATE
p. pr. & vb. n.
of Parallel
n.
A comparison made; elaborate tracing of similarity; as, Johnson's parallel between Dryden and Pope.
adv.
In a parallel manner; with parallelism.
v. t.
Fig.: To make to conform to something else in character, motive, aim, or the like.
a.
Having opposite surfaces exactly plane and parallel, as a piece of glass.
v. i.
To be parallel; to correspond; to be like.
v. t.
To equal; to match; to correspond to.
a.
Meeting and intersecting, as lines; not parallel; -- opposed to parallel.
n.
One of the imaginary circles on the surface of the earth, parallel to the equator, marking the latitude; also, the corresponding line on a globe or map.
imp. & p. p.
of Parallel
v. t.
To render parallel.
v. t.
To produce or adduce as a parallel.
n.
A comparison; a similitude; specifically, a short fictitious narrative of something which might really occur in life or nature, by means of which a moral is drawn; as, the parables of Christ.
n.
A line which, throughout its whole extent, is equidistant from another line; a parallel line, a parallel plane, etc.
a.
Continuing a resemblance through many particulars; applicable in all essential parts; like; similar; as, a parallel case; a parallel passage.
v. t.
To represent by parable.
n.
One of a series of long trenches constructed before a besieged fortress, by the besieging force, as a cover for troops supporting the attacking batteries. They are roughly parallel to the line of outer defenses of the fortress.
v. t.
To place or set so as to be parallel; to place so as to be parallel to, or to conform in direction with, something else.
n.
A character consisting of two parallel vertical lines (thus, ) used in the text to direct attention to a similarly marked note in the margin or at the foot of a page.
a.
Extended in the same direction, and in all parts equally distant; as, parallel lines; parallel planes.