Originally developed to model the physical world, geometry has applications in almost all
sciences, and also in
architecture, and other activities that are related to
graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in
Wiles's proof of
Fermat's Last Theorem, a problem that was stated in terms of
elementary arithmetic, and remained unsolved for several centuries.
European and an
Arab practicing geometry in the 15th century
The earliest recorded beginnings of geometry can be traced to ancient
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
astronomy, and various crafts. The earliest known texts on geometry are the
EgyptianRhind Papyrus (2000–1800 BC) and
Moscow Papyrus (
c. 1890 BC), and 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
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's 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 today.Archimedes (
c. 287–212 BC) of
Syracuse, Italy 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
surfaces of revolution.
Woman teaching geometry. Illustration at the beginning of a medieval translation of
Euclid's Elements, (
Indian mathematicians also made many important contributions in geometry. The Shatapatha 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 Pythagorean 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
Bakhshali manuscript, there are 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 zero."Aryabhata's Aryabhatiya (499) includes the computation of areas and volumes.
Brahmagupta wrote his astronomical work Brāhmasphuṭasiddhānta in 628. Chapter 12, containing 66
Sanskrit verses, 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).
Euclid took an abstract approach to geometry in his
Elements, one of the most influential books ever written. Euclid introduced certain
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.
Points are generally considered fundamental objects for building geometry. They may be defined by the properties that they must have, as in Euclid's definition as "that which has no part", or in
synthetic geometry. In modern mathematics, they are generally defined as
elements of a
space, which is itself
With these modern definitions, every geometric shape is defined as a set of points; this is not the case in synthetic geometry, where a line is another fundamental object that is not viewed as the set of the points through which it passes.
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
In Euclidean geometry a
plane is a flat, two-dimensional surface that extends infinitely; the definitions for other types of geometries are generalizations of that. Planes are used in many areas 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.
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.
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
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 differentiable  Algebraic geometry studies
algebraic curves, which are defined as
algebraic varieties of
In a different direction, the concepts of length, area and volume are extended by
measure theory, which studies methods of assigning a size or measure to
sets, where the measures follow rules similar to those of classical area and volume.
similarity are concepts that describe when two shapes have similar characteristics. In Euclidean geometry, similarity is used to describe objects that have the same shape, while congruence is used to describe objects that are the same in both size and shape.Hilbert, in his work on creating a more rigorous foundation for geometry, treated congruence as an undefined term whose properties are defined by
Congruence and similarity are generalized in
transformation geometry, which studies the properties of geometric objects that are preserved by different kinds of transformations.
Classical geometers paid special attention to constructing geometric objects that had been described in some other way. Classically, the only instruments used in most geometric constructions are the
straightedge.[b] 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
neusis, parabolas and other curves, or mechanical devices, were found.
Where the traditional geometry allowed dimensions 1 (a
line), 2 (a
plane) and 3 (our ambient world conceived of as
three-dimensional space), mathematicians and physicists have used
higher dimensions for nearly two centuries. One example of a mathematical use for higher dimensions is the
configuration space of a physical system, which has a dimension equal to the system's
degrees of freedom. For instance, the configuration of a screw can be described by five coordinates.
A different type of symmetry is the principle of
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 the result is an equally true theorem. A similar and closely related form of duality exists between a
vector space and its
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
Artists have long used concepts of
proportion in design.
Vitruvius developed a complicated theory of ideal proportions for the human figure. These concepts have been used and adapted by artists from
Michelangelo to modern comic book artists.
golden ratio is a particular proportion that has had a controversial role in art. Often claimed to be the most aesthetically pleasing ratio of lengths, it is frequently stated to be incorporated into famous works of art, though the most reliable and unambiguous examples were made deliberately by artists aware of this legend.
Cézanne advanced the theory that all images can be built up from the
cone, and the
cylinder. This is still used in art theory today, although the exact list of shapes varies from author to author.
The field of
astronomy, especially as it relates to mapping the positions of
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.
^Until the 19th century, geometry was dominated by the assumption that all geometric constructions were Euclidean. In the 19th century and later, this was challenged by the development of
hyperbolic geometry by
Lobachevsky and other
non-Euclidean geometries by
Gauss and others. It was then realised that implicitly non-Euclidean geometry had appeared throughout history, including the work of
Desargues in the 17th century, all the way back to the implicit use of
spherical geometry to understand the
Earth geodesy and to navigate the oceans since antiquity.
^The ancient Greeks had some constructions using other instruments.
^Pythagorean triples are triples of integers with the property: . Thus, , , etc.
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."
Boyer 1991, "The Arabic Hegemony" pp. 241–242) "Omar Khayyam (c. 1050–1123), the "tent-maker," wrote an 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."".
"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 al-Haytham's 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."
abcdeEuclid's Elements – All thirteen books in one volume, Based on Heath's translation, Green Lion Press
^Gerla, G. (1995).
"Pointless Geometries"(PDF). In Buekenhout, F.; Kantor, W. (eds.). Handbook of incidence geometry: buildings and foundations. North-Holland. pp. 1015–1031. Archived from
the original(PDF) on 17 July 2011.
Hayashi, Takao (2003). "Indian Mathematics". In Grattan-Guinness, Ivor (ed.). Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences. Vol. 1. Baltimore, MD: The
Johns Hopkins University Press. pp. 118–130.