Geometry (from the Ancient Greek: γεωμετρία; geo- "earth",
-metron "measurement") is a branch of mathematics concerned with
questions of shape, size, relative position of figures, and the
properties of space. A mathematician who works in the field of
geometry is called a geometer.
Geometry arose independently in a number of early cultures as a
practical way for dealing with lengths, areas, and volumes. Geometry
began to see elements of formal mathematical science emerging in the
West as early as the 6th century BC. By the 3rd century BC,
geometry was put into an axiomatic form by Euclid, whose treatment,
Euclid's Elements, set a standard for many centuries to follow.
Geometry arose independently in India, with texts providing rules for
geometric constructions appearing as early as the 3rd century BC.
Islamic scientists preserved Greek ideas and expanded on them during
the Middle Ages. By the early 17th century, geometry had been put
on a solid analytic footing by mathematicians such as René Descartes
and Pierre de Fermat. Since then, and into modern times, geometry has
expanded into non-
Euclidean geometry and manifolds, describing spaces
that lie beyond the normal range of human experience.
While geometry has evolved significantly throughout the years, there
are some general concepts that are more or less fundamental to
geometry. These include the concepts of points, lines, planes,
surfaces, angles, and curves, as well as the more advanced notions of
manifolds and topology or metric.
Geometry has applications to many fields, including art, architecture,
physics, as well as to other branches of mathematics.
3 Important concepts in geometry
3.9 Topologies and metrics
3.10 Compass and straightedge constructions
3.13 Non-Euclidean geometry
4 Contemporary geometry
4.1 Euclidean geometry
4.2 Differential geometry
Topology and geometry
4.4 Algebraic geometry
5.4 Other fields of mathematics
6 See also
6.2 Related topics
6.3 Other fields
9 Further reading
10 External links
Contemporary geometry has many subfields:
Euclidean geometry is geometry in its classical sense. The mandatory
educational curriculum of the majority of nations includes the study
of points, lines, planes, angles, triangles, congruence, similarity,
solid figures, circles, and analytic geometry. Euclidean geometry
also has applications in computer science, crystallography, and
various branches of modern mathematics.
Differential geometry uses techniques of calculus and linear algebra
to study problems in geometry. It has applications in physics,
including in general relativity.
Topology is the field concerned with the properties of geometric
objects that are unchanged by continuous mappings. In practice, this
often means dealing with large-scale properties of spaces, such as
connectedness and compactness.
Convex geometry investigates convex shapes in the
Euclidean space and
its more abstract analogues, often using techniques of real analysis.
It has close connections to convex analysis, optimization and
functional analysis and important applications in number theory.
Algebraic geometry studies geometry through the use of multivariate
polynomials and other algebraic techniques. It has applications in
many areas, including cryptography and string theory.
Discrete geometry is concerned mainly with questions of relative
position of simple geometric objects, such as points, lines and
circles. It shares many methods and principles with combinatorics.
Main article: History of geometry
A European and an
Arab practicing geometry in the 15th century.
The earliest recorded beginnings of geometry can be traced to ancient
Mesopotamia and Egypt in the 2nd millennium BC. Early geometry
was a collection of empirically discovered principles concerning
lengths, angles, areas, and volumes, which were developed to meet some
practical need in surveying, construction, astronomy, and various
crafts. The earliest known texts on geometry are the Egyptian Rhind
Papyrus (2000–1800 BC) and Moscow Papyrus (c. 1890 BC), the
Babylonian clay tablets such as
Plimpton 322 (1900 BC). For example,
the Moscow Papyrus gives a formula for calculating the volume of a
truncated pyramid, or frustum. Later clay tablets (350–50 BC)
demonstrate that Babylonian astronomers implemented trapezoid
procedures for computing Jupiter's position and motion within
time-velocity space. These geometric procedures anticipated the Oxford
Calculators, including the mean speed theorem, by 14 centuries.
South of Egypt the ancient Nubians established a system of geometry
including early versions of sun clocks.
In the 7th century BC, the Greek mathematician
Thales of Miletus
Thales of Miletus used
geometry to solve problems such as calculating the height of pyramids
and the distance of ships from the shore. He is credited with the
first use of deductive reasoning applied to geometry, by deriving four
corollaries to Thales' Theorem.
Pythagoras established the
Pythagorean School, which is credited with the first proof of the
Pythagorean theorem, though the statement of the theorem has a
long history. Eudoxus (408–c. 355 BC) developed the
method of exhaustion, which allowed the calculation of areas and
volumes of curvilinear figures, as well as a theory of ratios that
avoided the problem of incommensurable magnitudes, which enabled
subsequent geometers to make significant advances. Around 300 BC,
geometry was revolutionized by Euclid, whose Elements, widely
considered the most successful and influential textbook of all
time, introduced mathematical rigor through the axiomatic method
and is the earliest example of the format still used in mathematics
today, that of definition, axiom, theorem, and proof. Although most of
the contents of the Elements were already known,
Euclid arranged them
into a single, coherent logical framework. The Elements was known
to all educated people in the West until the middle of the 20th
century and its contents are still taught in geometry classes
Archimedes (c. 287–212 BC) of Syracuse used the
method of exhaustion to calculate the area under the arc of a parabola
with the summation of an infinite series, and gave remarkably accurate
approximations of Pi. He also studied the spiral bearing his name
and obtained formulas for the volumes of surfaces of revolution.
Woman teaching geometry. Illustration at the beginning of a medieval
translation of Euclid's Elements, (c. 1310)
Indian mathematicians also made many important contributions in
Satapatha Brahmana (3rd century BC) contains rules for
ritual geometric constructions that are similar to the Sulba
Sutras. According to (Hayashi 2005, p. 363), the Śulba
Sūtras contain "the earliest extant verbal expression of the
Theorem in the world, although it had already been known
to the Old Babylonians. They contain lists of Pythagorean triples,
which are particular cases of Diophantine equations. In the
Bakhshali manuscript, there is a handful of geometric problems
(including problems about volumes of irregular solids). The Bakhshali
manuscript also "employs a decimal place value system with a dot for
Aryabhatiya (499) includes the computation of
areas and volumes.
Brahmagupta wrote his astronomical work Brāhma
Sphuṭa Siddhānta in 628. Chapter 12, containing 66
was divided into two sections: "basic operations" (including cube
roots, fractions, ratio and proportion, and barter) and "practical
mathematics" (including mixture, mathematical series, plane figures,
stacking bricks, sawing of timber, and piling of grain). In the
latter section, he stated his famous theorem on the diagonals of a
cyclic quadrilateral. Chapter 12 also included a formula for the area
of a cyclic quadrilateral (a generalization of Heron's formula), as
well as a complete description of rational triangles (i.e. triangles
with rational sides and rational areas).
In the Middle Ages, mathematics in medieval Islam contributed to the
development of geometry, especially algebraic geometry.
Al-Mahani (b. 853) conceived the idea of reducing geometrical problems
such as duplicating the cube to problems in algebra. Thābit ibn
Qurra (known as Thebit in Latin) (836–901) dealt with arithmetic
operations applied to ratios of geometrical quantities, and
contributed to the development of analytic geometry. Omar Khayyám
(1048–1131) found geometric solutions to cubic equations. The
Ibn al-Haytham (Alhazen), Omar Khayyam and Nasir al-Din
al-Tusi on quadrilaterals, including the
Lambert quadrilateral and
Saccheri quadrilateral, were early results in hyperbolic geometry, and
along with their alternative postulates, such as Playfair's axiom,
these works had a considerable influence on the development of
Euclidean geometry among later European geometers, including
Witelo (c. 1230–c. 1314),
Alfonso, John Wallis, and Giovanni Girolamo Saccheri.
In the early 17th century, there were two important developments in
geometry. The first was the creation of analytic geometry, or geometry
with coordinates and equations, by
René Descartes (1596–1650) and
Pierre de Fermat
Pierre de Fermat (1601–1665). This was a necessary precursor to the
development of calculus and a precise quantitative science of physics.
The second geometric development of this period was the systematic
study of projective geometry by
Girard Desargues (1591–1661).
Projective geometry is a geometry without measurement or parallel
lines, just the study of how points are related to each other.
Two developments in geometry in the 19th century changed the way it
had been studied previously. These were the discovery of non-Euclidean
geometries by Nikolai Ivanovich Lobachevsky,
János Bolyai and Carl
Friedrich Gauss and of the formulation of symmetry as the central
consideration in the
Erlangen Programme of
Felix Klein (which
generalized the Euclidean and non-Euclidean geometries). Two of the
master geometers of the time were
Bernhard Riemann (1826–1866),
working primarily with tools from mathematical analysis, and
Riemann surface, and Henri Poincaré, the founder of
algebraic topology and the geometric theory of dynamical systems. As a
consequence of these major changes in the conception of geometry, the
concept of "space" became something rich and varied, and the natural
background for theories as different as complex analysis and classical
Important concepts in geometry
The following are some of the most important concepts in
An illustration of Euclid's parallel postulate
See also: Euclidean geometry
Euclid took an abstract approach to geometry in his Elements, one of
the most influential books ever written.
Euclid introduced certain
axioms, or postulates, expressing primary or self-evident properties
of points, lines, and planes. He proceeded to rigorously deduce other
properties by mathematical reasoning. The characteristic feature of
Euclid's approach to geometry was its rigor, and it has come to be
known as axiomatic or synthetic geometry. At the start of the 19th
century, the discovery of non-Euclidean geometries by Nikolai
Ivanovich Lobachevsky (1792–1856),
János Bolyai (1802–1860), Carl
Friedrich Gauss (1777–1855) and others led to a revival of interest
in this discipline, and in the 20th century, David Hilbert
(1862–1943) employed axiomatic reasoning in an attempt to provide a
modern foundation of geometry.
Main article: Point (geometry)
Points are considered fundamental objects in Euclidean geometry. They
have been defined in a variety of ways, including Euclid's definition
as 'that which has no part' and through the use of algebra or
nested sets. In many areas of geometry, such as analytic geometry,
differential geometry, and topology, all objects are considered to be
built up from points. However, there has been some study of geometry
without reference to points.
Main article: Line (geometry)
Euclid described a line as "breadthless length" which "lies equally
with respect to the points on itself". In modern mathematics,
given the multitude of geometries, the concept of a line is closely
tied to the way the geometry is described. For instance, in analytic
geometry, a line in the plane is often defined as the set of points
whose coordinates satisfy a given linear equation, but in a more
abstract setting, such as incidence geometry, a line may be an
independent object, distinct from the set of points which lie on
it. In differential geometry, a geodesic is a generalization of
the notion of a line to curved spaces.
Main article: Plane (geometry)
A plane is a flat, two-dimensional surface that extends infinitely
far. Planes are used in every area of geometry. For instance,
planes can be studied as a topological surface without reference to
distances or angles; it can be studied as an affine space, where
collinearity and ratios can be studied but not distances; it can
be studied as the complex plane using techniques of complex
analysis; and so on.
Main article: Angle
Euclid defines a plane angle as the inclination to each other, in a
plane, of two lines which meet each other, and do not lie straight
with respect to each other. In modern terms, 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.
Acute (a), obtuse (b), and straight (c) angles. The acute and obtuse
angles are also known as oblique angles.
In Euclidean geometry, angles are used to study polygons and
triangles, as well as forming an object of study in their own
right. The study of the angles of a triangle or of angles in a
unit circle forms the basis of trigonometry.
In differential geometry and calculus, the angles between plane curves
or space curves or surfaces can be calculated using the
A curve is a 1-dimensional object that may be straight (like a line)
or not; curves in 2-dimensional space are called plane curves and
those in 3-dimensional space are called space curves.
In topology, a curve is defined by a function from an interval of the
real numbers to another space. In differential geometry, the same
definition is used, but the defining function is required to be
Algebraic geometry studies algebraic curves, which
are defined as algebraic varieties of dimension one.
Main article: Surface (mathematics)
A sphere is a surface that can be defined parametrically (by x = r sin
θ cos φ, y = r sin θ sin φ, z = r cos θ) or implicitly (by x2 +
y2 + z2 − r2 = 0.)
A surface is a two-dimensional object, such as a sphere or
paraboloid. In differential geometry and topology,
surfaces are described by two-dimensional 'patches' (or neighborhoods)
that are assembled by diffeomorphisms or homeomorphisms, respectively.
In algebraic geometry, surfaces are described by polynomial
Main article: Manifold
A manifold is a generalization of the concepts of curve and surface.
In topology, a manifold is a topological space where every point has a
neighborhood that is homeomorphic to Euclidean space. In
differential geometry, a differentiable manifold is a space where each
neighborhood is diffeomorphic to Euclidean space.
Manifolds are used extensively in physics, including in general
relativity and string theory
Topologies and metrics
Main article: Topology
Visual checking of the
Pythagorean theorem for the (3, 4, 5) triangle
as in the
Zhoubi Suanjing 500–200 BC. The Pythagorean theorem
is a consequence of the Euclidean metric.
A topology is a mathematical structure on a set that tells how
elements of the set relate spatially to each other. The best-known
examples of topologies come from metrics, which are ways of measuring
distances between points. For instance, the Euclidean metric
measures the distance between points in the Euclidean plane, while the
hyperbolic metric measures the distance in the hyperbolic plane. Other
important examples of metrics include the
Lorentz metric of special
relativity and the semi-Riemannian metrics of general relativity.
Compass and straightedge constructions
Main article: Compass and straightedge constructions
Classical geometers paid special attention to constructing geometric
objects that had been described in some other way. Classically, the
only instruments allowed in geometric constructions are the compass
and straightedge. Also, every construction had to be complete in a
finite number of steps. However, some problems turned out to be
difficult or impossible to solve by these means alone, and ingenious
constructions using parabolas and other curves, as well as mechanical
devices, were found.
Main article: Dimension
The Koch snowflake, with fractal dimension=log4/log3 and topological
Where the traditional geometry allowed dimensions 1 (a line), 2 (a
plane) and 3 (our ambient world conceived of as three-dimensional
space), mathematicians have used higher dimensions for nearly two
Dimension has gone through stages of being any natural
number n, possibly infinite with the introduction of Hilbert space,
and any positive real number in fractal geometry.
Dimension theory is
a technical area, initially within general topology, that discusses
definitions; in common with most mathematical ideas, dimension is now
defined rather than an intuition. Connected topological manifolds have
a well-defined dimension; this is a theorem (invariance of domain)
rather than anything a priori.
The issue of dimension still matters to geometry, in the absence of
complete answers to classic questions. Dimensions 3 of space and 4 of
space-time are special cases in geometric topology.
Dimension 10 or 11
is a key number in string theory. Research may bring a satisfactory
geometric reason for the significance of 10 and 11 dimensions.
Main article: Symmetry
A tiling of the hyperbolic plane
The theme of symmetry in geometry is nearly as old as the science of
geometry itself. Symmetric shapes such as the circle, regular polygons
and platonic solids held deep significance for many ancient
philosophers and were investigated in detail before the time of
Euclid. Symmetric patterns occur in nature and were artistically
rendered in a multitude of forms, including the graphics of M. C.
Escher. Nonetheless, it was not until the second half of 19th century
that the unifying role of symmetry in foundations of geometry was
recognized. Felix Klein's
Erlangen program proclaimed that, in a very
precise sense, symmetry, expressed via the notion of a transformation
group, determines what geometry is.
Symmetry in classical Euclidean
geometry is represented by congruences and rigid motions, whereas in
projective geometry an analogous role is played by collineations,
geometric transformations that take straight lines into straight
lines. However it was in the new geometries of Bolyai and Lobachevsky,
Riemann, Clifford and Klein, and
Sophus Lie that Klein's idea to
'define a geometry via its symmetry group' proved most influential.
Both discrete and continuous symmetries play prominent roles in
geometry, the former in topology and geometric group theory, the
Lie theory and Riemannian geometry.
A different type of symmetry is the principle of duality in projective
geometry (see Duality (projective geometry)) among other fields. This
meta-phenomenon can roughly be described as follows: in any theorem,
exchange point with plane, join with meet, lies in with contains, and
you will get an equally true theorem. A similar and closely related
form of duality exists between a vector space and its dual space.
Differential geometry uses tools from calculus to study problems
In the nearly two thousand years since Euclid, while the range of
geometrical questions asked and answered inevitably expanded, the
basic understanding of space remained essentially the same. Immanuel
Kant argued that there is only one, absolute, geometry, which is known
to be true a priori by an inner faculty of mind: Euclidean geometry
was synthetic a priori. This dominant view was overturned by the
revolutionary discovery of non-
Euclidean geometry in the works of
Bolyai, Lobachevsky, and Gauss (who never published his theory). They
demonstrated that ordinary
Euclidean space is only one possibility for
development of geometry. A broad vision of the subject of geometry was
then expressed by
Riemann in his 1867 inauguration lecture Über die
Hypothesen, welche der Geometrie zu Grunde liegen (On the hypotheses
on which geometry is based), published only after his death.
Riemann's new idea of space proved crucial in Einstein's general
relativity theory, and Riemannian geometry, that considers very
general spaces in which the notion of length is defined, is a mainstay
of modern geometry.
Geometry lessons in the 20th century
Euclidean geometry has become closely connected with computational
geometry, computer graphics, convex geometry, incidence geometry,
finite geometry, discrete geometry, and some areas of combinatorics.
Attention was given to further work on
Euclidean geometry and the
Euclidean groups by crystallography and the work of H. S. M. Coxeter,
and can be seen in theories of Coxeter groups and polytopes. Geometric
group theory is an expanding area of the theory of more general
discrete groups, drawing on geometric models and algebraic techniques.
Differential geometry has been of increasing importance to
mathematical physics due to Einstein's general relativity postulation
that the universe is curved. Contemporary differential geometry is
intrinsic, meaning that the spaces it considers are smooth manifolds
whose geometric structure is governed by a Riemannian metric, which
determines how distances are measured near each point, and not a
priori parts of some ambient flat Euclidean space.
Topology and geometry
A thickening of the trefoil knot
The field of topology, which saw massive development in the 20th
century, is in a technical sense a type of transformation geometry, in
which transformations are homeomorphisms. This has often been
expressed in the form of the dictum 'topology is rubber-sheet
geometry'. Contemporary geometric topology and differential topology,
and particular subfields such as Morse theory, would be counted by
most mathematicians as part of geometry.
Algebraic topology and
general topology have gone their own ways.[dubious
Quintic Calabi–Yau threefold
The field of algebraic geometry is the modern incarnation of the
Cartesian geometry of co-ordinates. From late 1950s through mid-1970s
it had undergone major foundational development, largely due to work
Jean-Pierre Serre and Alexander Grothendieck. This led to the
introduction of schemes and greater emphasis on topological methods,
including various cohomology theories. One of seven Millennium Prize
problems, the Hodge conjecture, is a question in algebraic geometry.
The study of low-dimensional algebraic varieties, algebraic curves,
algebraic surfaces and algebraic varieties of dimension 3 ("algebraic
threefolds"), has been far advanced.
Gröbner basis theory and real
algebraic geometry are among more applied subfields of modern
Arithmetic geometry is an active field combining
algebraic geometry and number theory. Other directions of research
involve moduli spaces and complex geometry. Algebro-geometric methods
are commonly applied in string and brane theory.
Geometry has found applications in many fields, some of which are
Mathematics and art
Mathematics and art are related in a variety of ways. For instance,
the theory of perspective showed that there is more to geometry than
just the metric properties of figures: perspective is the origin of
Mathematics and architecture and Architectural geometry
Mathematics and architecture are related, since, as with other arts,
architects use mathematics for several reasons. Apart from the
mathematics needed when engineering buildings, architects use
geometry: to define the spatial form of a building; from the
Pythagoreans of the sixth century BC onwards, to create forms
considered harmonious, and thus to lay out buildings and their
surroundings according to mathematical, aesthetic and sometimes
religious principles; to decorate buildings with mathematical objects
such as tessellations; and to meet environmental goals, such as to
minimise wind speeds around the bases of tall buildings.
Main article: Mathematical physics
The 421polytope, orthogonally projected into the E8
Lie group Coxeter
plane. Lie groups have several applications in physics.
The field of astronomy, especially as it relates to mapping the
positions of stars and planets on the celestial sphere and describing
the relationship between movements of celestial bodies, have served as
an important source of geometric problems throughout history.
Modern geometry has many ties to physics as is exemplified by the
links between pseudo-
Riemannian geometry and general relativity. One
of the youngest physical theories, string theory, is also very
geometric in flavour.
Other fields of mathematics
Geometry has also had a large effect on other areas of mathematics.
For instance, the introduction of coordinates by
René Descartes and
the concurrent developments of algebra marked a new stage for
geometry, since geometric figures such as plane curves could now be
represented analytically in the form of functions and equations. This
played a key role in the emergence of infinitesimal calculus in the
17th century. The subject of geometry was further enriched by the
study of the intrinsic structure of geometric objects that originated
Euler and Gauss and led to the creation of topology and
Pythagoreans discovered that the sides of a triangle could have
An important area of application is number theory. In ancient Greece
Pythagoreans considered the role of numbers in geometry. However,
the discovery of incommensurable lengths, which contradicted their
philosophical views, made them abandon abstract numbers in favor of
concrete geometric quantities, such as length and area of figures.
Since the 19th century, geometry has been used for solving problems in
number theory, for example through the geometry of numbers or, more
recently, scheme theory, which is used in Wiles's proof of Fermat's
While the visual nature of geometry makes it initially more accessible
than other mathematical areas such as algebra or number theory,
geometric language is also used in contexts far removed from its
traditional, Euclidean provenance (for example, in fractal geometry
and algebraic geometry).
Analytic geometry applies methods of algebra to geometric questions,
typically by relating geometric curves to algebraic equations. These
ideas played a key role in the development of calculus in the 17th
century and led to the discovery of many new properties of plane
curves. Modern algebraic geometry considers similar questions on a
vastly more abstract level.
Leonhard Euler, in studying problems like the Seven Bridges of
Königsberg, considered the most fundamental properties of geometric
figures based solely on shape, independent of their metric properties.
Euler called this new branch of geometry geometria situs (geometry of
place), but it is now known as topology.
Topology grew out of
geometry, but turned into a large independent discipline. It does not
differentiate between objects that can be continuously deformed into
each other. The objects may nevertheless retain some geometry, as in
the case of hyperbolic knots.
List of geometers
List of formulas in elementary geometry
List of geometry topics
List of important publications in geometry
List of mathematics articles
Flatland, a book written by
Edwin Abbott Abbott
Edwin Abbott Abbott about two- and
three-dimensional space, to understand the concept of four dimensions
Interactive geometry software
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Qurra al-Harrani", MacTutor History of
Mathematics archive, University
of St Andrews .
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Shape-Shifting Spaces, Mathematician Wins $3-Million Prize".
Scientific American. Retrieved 2016-08-29.
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curriculum". American Educator, 26(2), 1–18.
^ J. Friberg, "Methods and traditions of Babylonian mathematics.
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Mathematics and Astronomy", pp. 71–96.
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^ (Boyer 1991, "
Euclid of Alexandria" p. 104)
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Pythagorean triples are triples of integers
with the property:
displaystyle a^ 2 +b^ 2 =c^ 2
displaystyle 3^ 2 +4^ 2 =5^ 2
displaystyle 8^ 2 +15^ 2 =17^ 2
displaystyle 12^ 2 +35^ 2 =37^ 2
^ (Cooke 2005, p. 198): "The arithmetic content of the Śulva
Sūtras consists of rules for finding
Pythagorean triples such as (3,
4, 5), (5, 12, 13), (8, 15, 17), and (12, 35, 37). It is not certain
what practical use these arithmetic rules had. The best conjecture is
that they were part of religious ritual. A Hindu home was required to
have three fires burning at three different altars. The three altars
were to be of different shapes, but all three were to have the same
area. These conditions led to certain "Diophantine" problems, a
particular case of which is the generation of Pythagorean triples, so
as to make one square integer equal to the sum of two others."
^ (Hayashi 2005, p. 371)
^ a b (Hayashi 2003, pp. 121–122)
^ R. Rashed (1994), The development of Arabic mathematics: between
arithmetic and algebra, p. 35 London
^ Boyer (1991). "The Arabic Hegemony". A History of Mathematics.
pp. 241–242. Omar Khayyam (c. 1050–1123), the "tent-maker,"
Algebra that went beyond that of al-Khwarizmi to include
equations of third degree. Like his
Arab predecessors, Omar Khayyam
provided for quadratic equations both arithmetic and geometric
solutions; for general cubic equations, he believed (mistakenly, as
the 16th century later showed), arithmetic solutions were impossible;
hence he gave only geometric solutions. The scheme of using
intersecting conics to solve cubics had been used earlier by
Menaechmus, Archimedes, and Alhazan, but Omar Khayyam took the
praiseworthy step of generalizing the method to cover all third-degree
equations (having positive roots). .. For equations of higher degree
than three, Omar Khayyam evidently did not envision similar geometric
methods, for space does not contain more than three dimensions, ...
One of the most fruitful contributions of Arabic eclecticism was the
tendency to close the gap between numerical and geometric algebra. The
decisive step in this direction came much later with Descartes, but
Omar Khayyam was moving in this direction when he wrote, "Whoever
thinks algebra is a trick in obtaining unknowns has thought it in
vain. No attention should be paid to the fact that algebra and
geometry are different in appearance. Algebras are geometric facts
which are proved."
^ O'Connor, John J.; Robertson, Edmund F., "Al-Mahani", MacTutor
Mathematics archive, University of St Andrews .
^ O'Connor, John J.; Robertson, Edmund F., "Omar Khayyam", MacTutor
Mathematics archive, University of St Andrews .
^ 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 , 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 19th 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 13th century, while revising Ibn
Book of Optics
Book of Optics (Kitab al-Manazir) – was undoubtedly
prompted by Arabic sources. The proofs put forward in the 14th century
by the Jewish scholar Levi ben Gerson, who lived in southern France,
and by the above-mentioned
Alfonso from Spain directly border on Ibn
al-Haytham's demonstration. Above, we have demonstrated that
Pseudo-Tusi's Exposition of
Euclid had stimulated both J. Wallis's and
G. Saccheri's studies of the theory of parallel lines."
^ a b c d e
Euclid's Elements – All thirteen books in one volume,
Based on Heath's translation, Green Lion Press
^ Clark, Bowman L. (Jan 1985). "Individuals and Points". Notre Dame
Journal of Formal Logic. 26 (1): 61–75.
doi:10.1305/ndjfl/1093870761. Retrieved 29 August 2016.
^ Gerla, G., 1995, "Pointless Geometries" in Buekenhout, F., Kantor,
W. eds., Handbook of incidence geometry: buildings and foundations.
^ John Casey (1885) Analytic
Geometry of the Point, Line, Circle, and
Conic Sections, link from Internet Archive.
^ Buekenhout, Francis (1995), Handbook of Incidence Geometry:
Buildings and Foundations, Elsevier B.V.
^ "geodesic – definition of geodesic in English from the Oxford
dictionary". OxfordDictionaries.com. Retrieved 2016-01-20.
^ a b c d e Munkres, James R. Topology. Vol. 2. Upper Saddle River:
Prentice Hall, 2000.
^ Szmielew, Wanda. 'From affine to Euclidean geometry: An axiomatic
approach.' Springer, 1983.
^ Ahlfors, Lars V. Complex analysis: an introduction to the theory of
analytic functions of one complex variable. New York,
^ Sidorov, L.A. (2001) , "Angle", in Hazewinkel, Michiel,
Encyclopedia of Mathematics,
Springer Science+Business Media
Springer Science+Business Media B.V. /
Kluwer Academic Publishers, ISBN 978-1-55608-010-4
^ Gelʹfand, Izrailʹ Moiseevič, and Mark Saul. "Trigonometry."
'Trigonometry'. Birkhäuser Boston, 2001. 1–20.
^ Stewart, James (2012). Calculus: Early Transcendentals, 7th ed.,
Brooks Cole Cengage Learning. ISBN 978-0-538-49790-9
^ Jost, Jürgen (2002), Riemannian
Geometry and Geometric Analysis,
Berlin: Springer-Verlag, ISBN 3-540-42627-2 .
^ Baker, Henry Frederick. Principles of geometry. Vol. 2. CUP Archive,
^ a b c Do Carmo, Manfredo Perdigao, and Manfredo Perdigao Do Carmo.
Differential geometry of curves and surfaces. Vol. 2. Englewood
Cliffs: Prentice-hall, 1976.
^ a b Mumford, David (1999). The Red Book of Varieties and Schemes
Includes the Michigan Lectures on Curves and Their Jacobians (2nd
ed.). Springer-Verlag. ISBN 3-540-63293-X.
^ Briggs, William L., and Lyle Cochran Calculus. "Early
Transcendentals." ISBN 978-0321570567.
^ Yau, Shing-Tung; Nadis, Steve (2010). The Shape of Inner Space:
String Theory and the
Geometry of the Universe's Hidden Dimensions.
Basic Books. ISBN 978-0-465-02023-2.
^ Dmitri Burago, Yu D Burago, Sergei Ivanov, A Course in Metric
Geometry, American Mathematical Society, 2001,
^ Wald, Robert M. (1984), General Relativity, University of Chicago
Press, ISBN 0-226-87033-2
^ Kline (1972) "Mathematical thought from ancient to modern times",
Oxford University Press, p. 1032. Kant did not reject the logical
(analytic a priori) possibility of non-Euclidean geometry, see Jeremy
Gray, "Ideas of
Space Euclidean, Non-Euclidean, and Relativistic",
Oxford, 1989; p. 85. Some have implied that, in light of this, Kant
had in fact predicted the development of non-Euclidean geometry, cf.
Leonard Nelson, "Philosophy and Axiomatics," Socratic Method and
Critical Philosophy, Dover, 1965, p. 164.
^ "Ueber die Hypothesen, welche der Geometrie zu Grunde liegen".
Archived from the original on 18 March 2016.
^ It is quite common in algebraic geometry to speak about geometry of
algebraic varieties over finite fields, possibly singular. From a
naïve perspective, these objects are just finite sets of points, but
by invoking powerful geometric imagery and using well developed
geometric techniques, it is possible to find structure and establish
properties that make them somewhat analogous to the ordinary spheres
Boyer, C. B. (1991) . A History of
Mathematics (Second edition,
revised by Uta C. Merzbach ed.). New York: Wiley.
Nikolai I. Lobachevsky, Pangeometry, translator and editor: A.
Papadopoulos, Heritage of European
Mathematics Series, Vol. 4,
European Mathematical Society, 2010.
Jay Kappraff, A Participatory Approach to Modern Geometry, 2014, World
Scientific Publishing, ISBN 978-981-4556-70-5.
Leonard Mlodinow, Euclid's Window – The Story of
Parallel Lines to Hyperspace, UK edn. Allen Lane, 1992.
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