In mathematics, non-
Euclidean geometry consists of two geometries
based on axioms closely related to those specifying Euclidean
Euclidean geometry lies at the intersection of metric
geometry and affine geometry, non-
Euclidean geometry arises when
either the metric requirement is relaxed, or the parallel postulate is
replaced with an alternative one. 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
The essential difference between the metric geometries is the nature
of parallel lines. Euclid's fifth postulate, the parallel postulate,
is equivalent to Playfair's postulate, which states that, within a
two-dimensional plane, for any given line ℓ and a point A, which is
not on ℓ, there is exactly one line through A that does not
intersect ℓ. In hyperbolic geometry, by contrast, there are
infinitely many lines through A not intersecting ℓ, while in
elliptic geometry, any line through A intersects ℓ.
Another way to describe the differences between these geometries is to
consider two straight lines indefinitely extended in a two-dimensional
plane that are both perpendicular to a third line:
In Euclidean geometry, the lines remain at a constant distance from
each other (meaning that a line drawn perpendicular to one line at any
point will intersect the other line and the length of the line segment
joining the points of intersection remains constant) and are known as
In hyperbolic geometry, they "curve away" from each other, increasing
in distance as one moves further from the points of intersection with
the common perpendicular; these lines are often called ultraparallels.
In elliptic geometry, the lines "curve toward" each other and
1.2 Discovery of non-Euclidean geometry
2 Axiomatic basis of non-Euclidean geometry
3 Models of non-Euclidean geometry
3.1 Elliptic geometry
3.2 Hyperbolic geometry
3.3 Three-dimensional non-Euclidean geometry
4 Uncommon properties
6 Planar algebras
7 Kinematic geometries
9 See also
12 External links
Hyperbolic geometry § History
Euclidean geometry, named after the Greek mathematician Euclid,
includes some of the oldest known mathematics, and geometries that
deviated from this were not widely accepted as legitimate until the
The debate that eventually led to the discovery of the non-Euclidean
geometries began almost as soon as Euclid's work Elements was written.
In the Elements,
Euclid began with a limited number of assumptions (23
definitions, five common notions, and five postulates) and sought to
prove all the other results (propositions) in the work. The most
notorious of the postulates is often referred to as "Euclid's Fifth
Postulate," or simply the "parallel postulate", which in Euclid's
original formulation is:
If a straight line falls on two straight lines in such a manner that
the interior angles on the same side are together less than two right
angles, then the straight lines, if produced indefinitely, meet on
that side on which are the angles less than the two right angles.
Other mathematicians have devised simpler forms of this property.
Regardless of the form of the postulate, however, it consistently
appears to be more complicated than Euclid's other postulates:
1. To draw a straight line from any point to any point.
2. To produce [extend] a finite straight line continuously in a
3. To describe a circle with any centre and distance [radius].
4. That all right angles are equal to one another.
For at least a thousand years, geometers were troubled by the
disparate complexity of the fifth postulate, and believed it could be
proved as a theorem from the other four. Many attempted to find a
proof by contradiction, including
Ibn al-Haytham (Alhazen, 11th
Omar Khayyám (12th century), Nasīr al-Dīn al-Tūsī
(13th century), and
Giovanni Girolamo Saccheri
Giovanni Girolamo Saccheri (18th century).
The theorems of Ibn al-Haytham, Khayyam and al-Tusi on quadrilaterals,
Lambert quadrilateral and
Saccheri quadrilateral, were
"the first few theorems of the hyperbolic and the elliptic
geometries." These theorems along with their alternative postulates,
such as Playfair's axiom, played an important role in the later
development of non-Euclidean geometry. These early attempts at
challenging the fifth postulate had a considerable influence on its
development among later European geometers, including Witelo, Levi ben
John Wallis and Saccheri. All of these early
attempts made at trying to formulate non-
Euclidean geometry however
provided flawed proofs of the parallel postulate, containing
assumptions that were essentially equivalent to the parallel
postulate. These early attempts did, however, provide some early
properties of the hyperbolic and elliptic geometries.
Khayyam, for example, tried to derive it from an equivalent postulate
he formulated from "the principles of the Philosopher" (Aristotle):
"Two convergent straight lines intersect and it is impossible for two
convergent straight lines to diverge in the direction in which they
converge." Khayyam then considered the three cases right, obtuse,
and acute that the summit angles of a
Saccheri quadrilateral can take
and after proving a number of theorems about them, he correctly
refuted the obtuse and acute cases based on his postulate and hence
derived the classic postulate of
Euclid which he didn't realize was
equivalent to his own postulate. Another example is al-Tusi's son,
Sadr al-Din (sometimes known as "Pseudo-Tusi"), who wrote a book on
the subject in 1298, based on al-Tusi's later thoughts, which
presented another 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 in 1594 and was studied by European geometers,
including Saccheri who criticised this work as well as that of
Giordano Vitale, in his book Euclide restituo (1680, 1686), used the
Saccheri quadrilateral to prove that if three points are equidistant
on the base AB and the summit CD, then AB and CD are everywhere
In a work titled Euclides ab Omni Naevo Vindicatus (
Euclid Freed from
All Flaws), published in 1733,
Saccheri quickly discarded elliptic
geometry as a possibility (some others of Euclid's axioms must be
modified for elliptic geometry to work) and set to work proving a
great number of results in hyperbolic geometry.
He finally reached a point where he believed that his results
demonstrated the impossibility of hyperbolic geometry. His claim seems
to have been based on Euclidean presuppositions, because no logical
contradiction was present. In this attempt to prove Euclidean geometry
he instead unintentionally discovered a new viable geometry, but did
not realize it.
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
Khayyam, 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.
At this time it was widely believed that the universe worked according
to the principles of Euclidean geometry.
Discovery of non-Euclidean geometry
The beginning of the 19th century would finally witness decisive steps
in the creation of non-Euclidean geometry. Circa 1813, Carl Friedrich
Gauss and independently around 1818, the German professor of law
Ferdinand Karl Schweikart had the germinal ideas of non-Euclidean
geometry worked out, but neither published any results. Then, around
1830, the Hungarian mathematician
János Bolyai and the Russian
Nikolai Ivanovich Lobachevsky
Nikolai Ivanovich Lobachevsky separately published
treatises on hyperbolic geometry. Consequently, hyperbolic geometry is
called Bolyai-Lobachevskian geometry, as both mathematicians,
independent of each other, are the basic authors of non-Euclidean
geometry. Gauss mentioned to Bolyai's father, when shown the younger
Bolyai's work, that he had developed such a geometry several years
before, though he did not publish. While
Lobachevsky created a
Euclidean geometry by negating the parallel postulate, Bolyai
worked out a geometry where both the Euclidean and the hyperbolic
geometry are possible depending on a parameter k. Bolyai ends his
work by mentioning that it is not possible to decide through
mathematical reasoning alone if the geometry of the physical universe
is Euclidean or non-Euclidean; this is a task for the physical
Bernhard Riemann, in a famous lecture in 1854, founded the field of
Riemannian geometry, discussing in particular the ideas now called
manifolds, Riemannian metric, and curvature. He constructed an
infinite family of geometries which are not Euclidean by giving a
formula for a family of Riemannian metrics on the unit ball in
Euclidean space. The simplest of these is called elliptic geometry and
it is considered to be a non-
Euclidean geometry due to its lack of
By formulating the geometry in terms of a curvature tensor, Riemann
Euclidean geometry to be applied to higher dimensions.
It was Gauss who coined the term "non-Euclidean geometry". He was
referring to his own work which today we call hyperbolic geometry.
Several modern authors still consider "non-Euclidean geometry" and
"hyperbolic geometry" to be synonyms.
Arthur Cayley noted that distance between points inside a conic could
be defined in terms of logarithm and the projective cross-ratio
function. The method has become called the
Cayley-Klein metric because
Felix Klein exploited it to describe the non-euclidean geometries in
articles in 1871 and 73 and later in book form. The Cayley-Klein
metrics provided working models of hyperbolic and elliptic metric
geometries, as well as Euclidean geometry.
Klein is responsible for the terms "hyperbolic" and "elliptic" (in his
system he called
Euclidean geometry "parabolic", a term which
generally fell out of use). His influence has led to the current
usage of the term "non-Euclidean geometry" to mean either "hyperbolic"
or "elliptic" geometry.
There are some mathematicians who would extend the list of geometries
that should be called "non-Euclidean" in various ways.
Axiomatic basis of non-Euclidean geometry
Euclidean geometry can be axiomatically described in several ways.
Unfortunately, Euclid's original system of five postulates (axioms) is
not one of these as his proofs relied on several unstated assumptions
which should also have been taken as axioms. Hilbert's system
consisting of 20 axioms most closely follows the approach of
Euclid and provides the justification for all of Euclid's proofs.
Other systems, using different sets of undefined terms obtain the same
geometry by different paths. In all approaches, however, there is an
axiom which is logically equivalent to Euclid's fifth postulate, the
parallel postulate. Hilbert uses the Playfair axiom form, while
Birkhoff, for instance, uses the axiom which says that "there exists a
pair of similar but not congruent triangles." In any of these systems,
removal of the one axiom which is equivalent to the parallel
postulate, in whatever form it takes, and leaving all the other axioms
intact, produces absolute geometry. As the first 28 propositions of
Euclid (in The Elements) do not require the use of the parallel
postulate or anything equivalent to it, they are all true statements
in absolute geometry.
To obtain a non-Euclidean geometry, the parallel postulate (or its
equivalent) must be replaced by its negation. Negating the Playfair's
axiom form, since it is a compound statement (... there exists one and
only one ...), can be done in two ways:
Either there will exist more than one line through the point parallel
to the given line or there will exist no lines through the point
parallel to the given line. In the first case, replacing the parallel
postulate (or its equivalent) with the statement "In a plane, given a
point P and a line ℓ not passing through P, there exist two lines
through P which do not meet ℓ" and keeping all the other axioms,
yields hyperbolic geometry.
The second case is not dealt with as easily. Simply replacing the
parallel postulate with the statement, "In a plane, given a point P
and a line ℓ not passing through P, all the lines through P meet
ℓ", does not give a consistent set of axioms. This follows since
parallel lines exist in absolute geometry, but this statement says
that there are no parallel lines. This problem was known (in a
different guise) to Khayyam,
Saccheri and Lambert and was the basis
for their rejecting what was known as the "obtuse angle case". In
order to obtain a consistent set of axioms which includes this axiom
about having no parallel lines, some of the other axioms must be
tweaked. The adjustments to be made depend upon the axiom system being
used. Among others these tweaks will have the effect of modifying
Euclid's second postulate from the statement that line segments can be
extended indefinitely to the statement that lines are unbounded.
Riemann's elliptic geometry emerges as the most natural geometry
satisfying this axiom.
Models of non-Euclidean geometry
Further information: Models of non-Euclidean geometry
On a sphere, the sum of the angles of a triangle is not equal to
180°. The surface of a sphere is not a Euclidean space, but locally
the laws of the
Euclidean geometry are good approximations. In a small
triangle on the face of the earth, the sum of the angles is very
Euclidean geometry is modelled by our notion of a
Main article: Elliptic geometry
The simplest model for elliptic geometry is a sphere, where lines are
"great circles" (such as the equator or the meridians on a globe), and
points opposite each other (called antipodal points) are identified
(considered to be the same). This is also one of the standard models
of the real projective plane. The difference is that as a model of
elliptic geometry a metric is introduced permitting the measurement of
lengths and angles, while as a model of the projective plane there is
no such metric.
In the elliptic model, for any given line ℓ and a point A, which is
not on ℓ, all lines through A will intersect ℓ.
Main article: Hyperbolic geometry
Even after the work of Lobachevsky, Gauss, and Bolyai, the question
remained: "Does such a model exist for hyperbolic geometry?". The
model for hyperbolic geometry was answered by Eugenio Beltrami, in
1868, who first showed that a surface called the pseudosphere has the
appropriate curvature to model a portion of hyperbolic space and in a
second paper in the same year, defined the
Klein model which models
the entirety of hyperbolic space, and used this to show that Euclidean
geometry and hyperbolic geometry were equiconsistent so that
hyperbolic geometry was logically consistent if and only if Euclidean
geometry was. (The reverse implication follows from the horosphere
model of Euclidean geometry.)
In the hyperbolic model, within a two-dimensional plane, for any given
line ℓ and a point A, which is not on ℓ, there are infinitely many
lines through A that do not intersect ℓ.
In these models the concepts of non-Euclidean geometries are being
represented by Euclidean objects in a Euclidean setting. This
introduces a perceptual distortion wherein the straight lines of the
Euclidean geometry are being represented by Euclidean curves which
visually bend. This "bending" is not a property of the non-Euclidean
lines, only an artifice of the way they are being represented.
Three-dimensional non-Euclidean geometry
Main article: Thurston geometry
In three dimensions, there are eight models of geometries. There
are Euclidean, elliptic, and hyperbolic geometries, as in the
two-dimensional case; mixed geometries that are partially Euclidean
and partially hyperbolic or spherical; twisted versions of the mixed
geometries; and one unusual geometry that is completely anisotropic
(i.e. every direction behaves differently).
Lambert quadrilateral in hyperbolic geometry
Saccheri quadrilaterals in the three geometries
Euclidean and non-Euclidean geometries naturally have many similar
properties, namely those which do not depend upon the nature of
parallelism. This commonality is the subject of absolute geometry
(also called neutral geometry). However, the properties which
distinguish one geometry from the others are the ones which have
historically received the most attention.
Besides the behavior of lines with respect to a common perpendicular,
mentioned in the introduction, we also have the following:
Lambert quadrilateral is a quadrilateral which has three right
angles. The fourth angle of a
Lambert quadrilateral is acute if the
geometry is hyperbolic, a right angle if the geometry is Euclidean or
obtuse if the geometry is elliptic. Consequently, rectangles exist (a
statement equivalent to the parallel postulate) only in Euclidean
Saccheri quadrilateral is a quadrilateral which has two sides of
equal length, both perpendicular to a side called the base. The other
two angles of a
Saccheri quadrilateral are called the summit angles
and they have equal measure. The summit angles of a Saccheri
quadrilateral are acute if the geometry is hyperbolic, right angles if
the geometry is Euclidean and obtuse angles if the geometry is
The sum of the measures of the angles of any triangle is less than
180° if the geometry is hyperbolic, equal to 180° if the geometry is
Euclidean, and greater than 180° if the geometry is elliptic. The
defect of a triangle is the numerical value (180° - sum of the
measures of the angles of the triangle). This result may also be
stated as: the defect of triangles in hyperbolic geometry is positive,
the defect of triangles in
Euclidean geometry is zero, and the defect
of triangles in elliptic geometry is negative.
Before the models of a non-Euclidean plane were presented by Beltrami,
Klein, and Poincaré,
Euclidean geometry stood unchallenged as the
mathematical model of space. Furthermore, since the substance of the
subject in synthetic geometry was a chief exhibit of rationality, the
Euclidean point of view represented absolute authority.
The discovery of the non-Euclidean geometries had a ripple effect
which went far beyond the boundaries of mathematics and science. The
philosopher Immanuel Kant's treatment of human knowledge had a special
role for geometry. It was his prime example of synthetic a priori
knowledge; not derived from the senses nor deduced through
logic — our knowledge of space was a truth that we were born
with. Unfortunately for Kant, his concept of this unalterably true
geometry was Euclidean. Theology was also affected by the change from
absolute truth to relative truth in the way that mathematics is
related to the world around it, that was a result of this paradigm
Euclidean geometry is an example of a scientific revolution in the
history of science, in which mathematicians and scientists changed the
way they viewed their subjects. Some geometers called Lobachevsky
Copernicus of Geometry" due to the revolutionary character of his
The existence of non-Euclidean geometries impacted the intellectual
Victorian England in many ways and in particular was one
of the leading factors that caused a re-examination of the teaching of
geometry based on Euclid's Elements. This curriculum issue was hotly
debated at the time and was even the subject of a book,
Euclid and his
Modern Rivals, written by Charles Lutwidge Dodgson (1832–1898)
better known as Lewis Carroll, the author of Alice in Wonderland.
In analytic geometry a plane is described with Cartesian
coordinates : C = (x,y) : x, y ∈ ℝ . The points are
sometimes identified with complex numbers z = x + y ε where ε2 ∈
–1, 0, 1 .
The Euclidean plane corresponds to the case ε2 = −1 since the
modulus of z is given by
displaystyle zz^ ast =(x+yepsilon )(x-yepsilon )=x^ 2 +y^ 2
and this quantity is the square of the
Euclidean distance between z
and the origin. For instance, z z z* = 1 is the unit circle.
For planar algebra, non-
Euclidean geometry arises in the other cases.
When ε2 = +1, then z is a split-complex number and conventionally j
replaces epsilon. Then
displaystyle zz^ ast =(x+ymathbf j )(x-ymathbf j )=x^ 2 -y^
and z z z* = 1 is the unit hyperbola.
When ε2 = 0, then z is a dual number.
This approach to non-
Euclidean geometry explains the non-Euclidean
angles: the parameters of slope in the dual number plane and
hyperbolic angle in the split-complex plane correspond to angle in
Euclidean geometry. Indeed, they each arise in polar decomposition of
a complex number z.
Hyperbolic geometry found an application in kinematics with the
physical cosmology introduced by
Hermann Minkowski in 1908. Minkowski
introduced terms like worldline and proper time into mathematical
physics. He realized that the submanifold, of events one moment of
proper time into the future, could be considered a hyperbolic space of
three dimensions. Already in the 1890s Alexander Macfarlane
was charting this submanifold through his Algebra of Physics and
hyperbolic quaternions, though Macfarlane did not use cosmological
language as Minkowski did in 1908. The relevant structure is now
called the hyperboloid model of hyperbolic geometry.
The non-Euclidean planar algebras support kinematic geometries in the
plane. For instance, the split-complex number z = eaj can represent a
spacetime event one moment into the future of a frame of reference of
rapidity a. Furthermore, multiplication by z amounts to a Lorentz
boost mapping the frame with rapidity zero to that with rapidity a.
Kinematic study makes use of the dual numbers
displaystyle z=x+yepsilon ,quad epsilon ^ 2 =0,
to represent the classical description of motion in absolute time and
space: The equations
displaystyle x^ prime =x+vt,quad t^ prime =t
are equivalent to a shear mapping in linear algebra:
displaystyle begin pmatrix x'\t'end pmatrix = begin pmatrix
1&v\0&1end pmatrix begin pmatrix x\tend pmatrix .
With dual numbers the mapping is
displaystyle t^ prime +x^ prime epsilon =(1+vepsilon
)(t+xepsilon )=t+(x+vt)epsilon .
Another view of special relativity as a non-
Euclidean geometry was
advanced by E. B. Wilson and Gilbert Lewis in Proceedings of the
American Academy of Arts and Sciences
American Academy of Arts and Sciences in 1912. They revamped the
analytic geometry implicit in the split-complex number algebra into
synthetic geometry of premises and deductions.
Euclidean geometry often makes appearances in works of science
fiction and fantasy.
H. G. Wells
H. G. Wells published the short story "The Remarkable Case of
Davidson’s Eyes". To appreciate this story one should know how
antipodal points on a sphere are identified in a model of the elliptic
plane. In the story, in the midst of a thunderstorm, Sidney Davidson
sees "Waves and a remarkably neat schooner" while working in an
electrical laboratory at Harlow Technical College. At the story’s
close Davidson proves to have witnessed H.M.S. Fulmar off Antipodes
Euclidean geometry is sometimes connected with the influence of
the 20th century horror fiction writer H. P. Lovecraft. In his works,
many unnatural things follow their own unique laws of geometry: In
Lovecraft's Cthulhu Mythos, the sunken city of
R'lyeh is characterized
by its non-Euclidean geometry. It is heavily implied this is achieved
as a side effect of not following the natural laws of this universe
rather than simply using an alternate geometric model, as the sheer
innate wrongness of it is said to be capable of driving those who look
upon it insane.
The main character in Robert Pirsig's Zen and the Art of Motorcycle
Maintenance mentioned Riemannian
Geometry on multiple occasions.
In The Brothers Karamazov, Dostoevsky discusses non-Euclidean geometry
through his main character Ivan.
Christopher Priest's novel
Inverted World describes the struggle of
living on a planet with the form of a rotating pseudosphere.
Robert Heinlein's The Number of the Beast utilizes non-Euclidean
geometry to explain instantaneous transport through space and time and
between parallel and fictional universes.
Antichamber uses non-
Euclidean geometry to create a
minimal, Escher-like world, where geometry and space follow unfamiliar
HyperRogue is a roguelike game set on the hyperbolic
plane, allowing the player to experience many properties of this
geometry. Many mechanics, quests, and locations are strongly dependent
on the features of hyperbolic geometry.
Renegade Legion science fiction setting for FASA's wargame,
role-playing-game and fiction, faster-than-light travel and
communications is possible through the use of Hsieh Ho's
Polydimensional Non-Euclidean Geometry, published sometime in the
middle of the 22nd century.
In Ian Stewart's
Flatterland the protagonist Victoria Line visit all
kinds of non-Euclidean worlds.
In Jean-Pierre Petit's Here's looking at
Euclid (and not looking at
Archibald Higgins stumbles upon spherical geometry
^ Eder, Michelle (2000), Views of Euclid's Parallel Postulate in
Ancient Greece and in Medieval Islam, Rutgers University, retrieved
^ Boris A. Rosenfeld & Adolf P. Youschkevitch, "Geometry", p. 470,
in Roshdi Rashed & Régis Morelon (1996), Encyclopedia of the
History of Arabic Science, Vol. 2, pp. 447–494, Routledge, London
and New York:
"Three scientists, Ibn al-Haytham, Khayyam and al-Tusi, had made the
most considerable contribution to this branch of geometry whose
importance came to be completely recognized only in the nineteenth
century. In essence their propositions concerning the properties of
quadrangles which they considered assuming that some of the angles of
these figures were acute of obtuse, embodied the first few theorems of
the hyperbolic and the elliptic geometries. Their other proposals
showed that various geometric statements were equivalent to the
Euclidean postulate V. It is extremely important that these scholars
established the mutual connection between this postulate and the sum
of the angles of a triangle and a quadrangle. By their works on the
theory of parallel lines Arab mathematicians directly influenced the
relevant investigations of their European counterparts. The first
European attempt to prove the postulate on parallel lines –
made by Witelo, the Polish scientists of the thirteenth century, while
revising Ibn al-Haytham's
Book of Optics
Book of Optics (Kitab al-Manazir) –
was undoubtedly prompted by Arabic sources. The proofs put forward in
the fourteenth century by the Jewish scholar Levi ben Gerson, who
lived in southern France, and by the above-mentioned
Spain directly border on Ibn al-Haytham's demonstration. Above, we
have demonstrated that Pseudo-Tusi's Exposition of
stimulated borth J. Wallis's and G. Saccheri's studies of the theory
of parallel lines."
^ Boris A. Rosenfeld & Adolf P. Youschkevitch (1996), "Geometry",
p. 467, in Roshdi Rashed & Régis Morelon (1996), Encyclopedia of
the History of Arabic Science, Vol. 2, pp. 447–494, Routledge,
^ a b Victor J. Katz (1998), History of Mathematics: An Introduction,
p. 270–271, Addison–Wesley, ISBN 0-321-01618-1:
"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 , 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.
^ A notable exception is David Hume, who as early as 1739 seriously
entertained the possibility that our universe was non-Euclidean; see
David Hume (1739/1978) A Treatise of Human Nature, L.A. Selby-Bigge,
ed. (Oxford: Oxford University Press), pp. 51-52.
^ In a letter of December 1818, Ferdinand Karl Schweikart (1780-1859)
sketched a few insights into non-Euclidean geometry. The letter was
forwarded to Gauss in 1819 by Gauss's former student Gerling. In his
reply to Gerling, Gauss praised Schweikart and mentioned his own,
earlier research into non-Euclidean geometry. See:
Carl Friedrich Gauss, Werke (Leipzig, Germany: B. G. Teubner, 1900),
volume 8, pages 180-182.
English translations of Schweikart's letter and Gauss's reply to
Gerling appear in: Course notes: "Gauss and non-Euclidean geometry",
University of Waterloo, Ontario, Canada; see especially pages 10 and
Letters by Schweikart and the writings of his nephew Franz Adolph
Taurinus (1794-1874), who also was interested in non-Euclidean
geometry and who in 1825 published a brief book on the parallel axiom,
appear in: Paul Stäckel and Friedrich Engel, Die theorie der
Parallellinien von Euklid bis auf Gauss, eine Urkundensammlung der
nichteuklidischen Geometrie (The theory of parallel lines from Euclid
to Gauss, an archive of non-Euclidean geometry), (Leipzig, Germany: B.
G. Teubner, 1895), pages 243 ff.
^ In the letter to Wolfgang (Farkas) Bolyai of March 6, 1832 Gauss
claims to have worked on the problem for thirty or thirty-five years
(Faber 1983, pg. 162). In his 1824 letter to Taurinus (Faber 1983, pg.
158) he claimed that he had been working on the problem for over 30
years and provided enough detail to show that he actually had worked
out the details. According to Faber (1983, pg. 156) it wasn't until
around 1813 that Gauss had come to accept the existence of a new
^ However, other axioms besides the parallel postulate must be changed
in order to make this a feasible geometry.
^ Felix Klein, Elementary
Mathematics from an Advanced Standpoint:
Geometry, Dover, 1948 (reprint of English translation of 3rd Edition,
1940. First edition in German, 1908) pg. 176
^ F. Klein, Über die sogenannte nichteuklidische Geometrie,
Mathematische Annalen, 4(1871).
^ The Euclidean plane is still referred to as "parabolic" in the
context of conformal geometry: see Uniformization theorem.
^ for instance, Manning 1963 and Yaglom 1968
^ a 21st axiom appeared in the French translation of Hilbert's
Grundlagen der Geometrie according to Smart 1997, pg. 416
^ (Smart 1997, pg.366)
^ while only two lines are postulated, it is easily shown that there
must be an infinite number of such lines.
^ Book I
Proposition 27 of Euclid's Elements
^ * William Thurston. Three-dimensional geometry and topology. Vol. 1.
Edited by Silvio Levy. Princeton Mathematical Series, 35. Princeton
University Press, Princeton, NJ, 1997. x+311 pp.
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^ Imre Toth, "Gott und Geometrie: Eine viktorianische Kontroverse,"
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Vesalius was to
Copernicus was to
Ptolemy that was
Lobachevsky to Euclid."
— W. K. Clifford
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^ Richard C. Tolman (2004) Theory of Relativity of Motion, page 194,
§180 Non-Euclidean angle, §181 Kinematical interpretation of angle
in terms of velocity
Hermann Minkowski (1908–9). "
Space and Time" (Wikisource).
^ Scott Walter (1999) Non-Euclidean Style of
Isaak Yaglom (1979) A simple non-
Euclidean geometry and its physical
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Edwin B. Wilson &
Gilbert N. Lewis
Gilbert N. Lewis (1912) "The Space-time
Manifold of Relativity. The Non-Euclidean
Geometry of Mechanics and
Electromagnetics" Proceedings of the American Academy of Arts and
^ Synthetic Spacetime, a digest of the axioms used, and theorems
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^ "The Call of Cthulhu".
^ Jean-Pierre Petit. "Here's Looking at Euclid".
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Greenberg, Marvin Jay
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Morris Kline (1972) Mathematical Thought from Ancient to Modern Times,
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Bernard H. Lavenda, (2012) " A New Perspective on Relativity : An
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Wikiquote has quotations related to: Non-Euclidean geometry
Roberto Bonola (1912) Non-Euclidean Geometry, Open Court, Chicago.
MacTutor Archive article on non-Euclidean geometry
Non-euclidean geometry at PlanetMath.org.
Non-Euclidean geometries from Encyclopedia of Math of European
Mathematical Society and Springer Science+Business Media
Synthetic Spacetime, a digest of the axioms used, and theorems proved,
by Wilson and Lewis. Archived by WebCite.
BNF: cb119798569 (d