3rd century BC Greek mathematician Euclid (holding calipers), as imagined by Raphael in this detail from The School of Athens (1509–1511)
Detail from Raphael's The School of Athens presumed to represent Donato Bramante as Euclid
The distribution of prime numbers is a central point of study in number theory. This Ulam spiral serves to illustrate it, hinting, in particular, at the conditional independence between being prime and being a value of certain quadratic polynomials.
Euclidis quae supersunt omnia (1704)
The quadratic formula expresses concisely the solutions of all quadratic equations
One of the oldest surviving fragments of Euclid's Elements, found at Oxyrhynchus and dated to circa AD 100 (P. Oxy. 29). The diagram accompanies Book II, Proposition 5.
Rubik's cube: the study of its possible moves is a concrete application of group theory
Euclid's construction of a regular dodecahedron.
The Babylonian mathematical tablet Plimpton 322, dated to 1800 BC.
Construction of a dodecahedron by placing faces on the edges of a cube.
Archimedes used the method of exhaustion, depicted here, to approximate the value of pi.
19th-century statue of Euclid by Joseph Durham in the Oxford University Museum of Natural History
The numerals used in the Bakhshali manuscript, dated between the 2nd century BC and the 2nd century AD.
A page from al-Khwārizmī's Algebra
Leonardo Fibonacci, the Italian mathematician who introduced the Hindu–Arabic numeral system invented between the 1st and 4th centuries by Indian mathematicians, to the Western World.
Leonhard Euler created and popularized much of the mathematical notation used today.
Carl Friedrich Gauss, known as the prince of mathematicians
The front side of the Fields Medal
Euler's identity, which American physicist Richard Feynman once called "the most remarkable formula in mathematics".

His Elements is one of the most influential works in the history of mathematics, serving as the main textbook for teaching mathematics (especially geometry) from the time of its publication until the late 19th or early 20th century.

- Euclid

A fitting example is the problem of integer factorization, which goes back to Euclid, but which had no practical application before its use in the RSA cryptosystem (for the security of computer networks).

- Mathematics
3rd century BC Greek mathematician Euclid (holding calipers), as imagined by Raphael in this detail from The School of Athens (1509–1511)

8 related topics

Alpha

An angel carrying the banner of "Truth", Roslin, Midlothian

Axiom

Statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments.

Statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments.

An angel carrying the banner of "Truth", Roslin, Midlothian

As used in mathematics, the term axiom is used in two related but distinguishable senses: "logical axioms" and "non-logical axioms".

The root meaning of the word postulate is to "demand"; for instance, Euclid demands that one agree that some things can be done (e.g., any two points can be joined by a straight line).

Title page of Sir Henry Billingsley's first English version of Euclid's Elements, 1570. Billingsley erroneously attributed the original work to Euclid of Megara.

Euclid's Elements

Title page of Sir Henry Billingsley's first English version of Euclid's Elements, 1570. Billingsley erroneously attributed the original work to Euclid of Megara.
A fragment of Euclid's Elements on part of the Oxyrhynchus papyri
Double-page from the Ishaq ibn Hunayn's Arabic Translation of Elementa. Iraq, 1270. Chester Beatty Library
An illumination from a manuscript based on Adelard of Bath's translation of the Elements, c. undefined 1309–1316; Adelard's is the oldest surviving translation of the Elements into Latin, done in the 12th-century work and translated from Arabic.
Euclidis – Elementorum libri XV Paris, Hieronymum de Marnef & Guillaume Cavelat, 1573 (second edition after the 1557 ed.); in 8:350, (2)pp. THOMAS–STANFORD, Early Editions of Euclid's Elements, n°32. Mentioned in T.L. Heath's translation. Private collection Hector Zenil.
A page with marginalia from the first printed edition of Elements, printed by Erhard Ratdolt in 1482
The different versions of the parallel postulate result in different geometries.
An animation showing how Euclid constructed a hexagon (Book IV, Proposition 15). Every two-dimensional figure in the Elements can be constructed using only a compass and straightedge.
Codex Vaticanus 190
Propositions plotted with lines connected from Axioms on the top and other preceding propositions, labelled by book.
The Italian Jesuit Matteo Ricci (left) and the Chinese mathematician Xu Guangqi (right) published the Chinese edition of Euclid's Elements (幾何原本) in 1607.
Proof of the Pythagorean theorem in Byrne's The Elements of Euclid and published in colored version in 1847.

The Elements ( Stoikheîa) is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt c. undefined 300 BC. It is a collection of definitions, postulates, propositions (theorems and constructions), and mathematical proofs of the propositions.

An illustration of Euclid's proof of the Pythagorean theorem.

Greek mathematics

An illustration of Euclid's proof of the Pythagorean theorem.
Detail of Pythagoras with a tablet of ratios, from The School of Athens by Raphael. Vatican Palace, Rome, 1509.
A fragment from Euclid's Elements (c. 300 BC), widely considered the most influential mathematics textbook of all time.
Cover of Arithmetica written by Greek Mathematician Diophantus

Greek mathematics refers to mathematics texts written during and ideas stemming from the Archaic through the Hellenistic and Roman periods, mostly extant from the 7th century BC to the 4th century AD, around the shores of the Eastern Mediterranean.

It has been customary to credit almost half of the material in Euclid's Elements to the Pythagoreans, as well as the discovery of irrationals, attributed to Hippassus (c.

An illustration of Desargues' theorem, a result in Euclidean and projective geometry

Geometry

An illustration of Desargues' theorem, a result in Euclidean and projective geometry
A European and an Arab practicing geometry in the 15th century
Woman teaching geometry. Illustration at the beginning of a medieval translation of Euclid's Elements, (c. 1310).
An illustration of Euclid's parallel postulate
Acute (a), obtuse (b), and straight (c) angles. The acute and obtuse angles are also known as oblique angles.
A sphere is a surface that can be defined parametrically (by  or implicitly (by x2 + y2 + z2 − r2 = 0.)
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.
The Koch snowflake, with fractal dimension=log4/log3 and topological dimension=1
A tiling of the hyperbolic plane
Differential geometry uses tools from calculus to study problems involving curvature.
A thickening of the trefoil knot
Quintic Calabi–Yau threefold
Discrete geometry includes the study of various sphere packings.
The Cayley graph of the free group on two generators a and b
Bou Inania Madrasa, Fes, Morocco, zellige mosaic tiles forming elaborate geometric tessellations
The Pythagoreans discovered that the sides of a triangle could have incommensurable lengths.

Geometry is, with arithmetic, one of the oldest branches of mathematics.

Euclid took an abstract approach to geometry in his Elements, one of the most influential books ever written.

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Non-Euclidean geometry

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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 nearly 180°.
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In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry.

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.

A proof from Euclid's Elements (c. 300 BC), widely considered the most influential textbook of all time.

History of mathematics

A proof from Euclid's Elements (c. 300 BC), widely considered the most influential textbook of all time.
Geometry problem on a clay tablet belonging to a school for scribes; Susa, first half of the 2nd millennium BCE
The Babylonian mathematical tablet Plimpton 322, dated to 1800 BC.
Image of Problem 14 from the Moscow Mathematical Papyrus. The problem includes a diagram indicating the dimensions of the truncated pyramid.
The Pythagorean theorem. The Pythagoreans are generally credited with the first proof of the theorem.
One of the oldest surviving fragments of Euclid's Elements, found at Oxyrhynchus and dated to circa AD 100. The diagram accompanies Book II, Proposition 5.
Archimedes used the method of exhaustion to approximate the value of pi.
Apollonius of Perga made significant advances in the study of conic sections.
Title page of the 1621 edition of Diophantus' Arithmetica, translated into Latin by Claude Gaspard Bachet de Méziriac.
The Hagia Sophia was designed by mathematicians Anthemius of Tralles and Isidore of Miletus.
Equipment used by an ancient Roman land surveyor (gromatici), found at the site of Aquincum, modern Budapest, Hungary
The Tsinghua Bamboo Slips, containing the world's earliest decimal multiplication table, dated 305 BC during the Warring States period
Counting rod numerals
The Nine Chapters on the Mathematical Art, one of the earliest surviving mathematical texts from China (2nd century AD).
The numerals used in the Bakhshali manuscript, dated between the 2nd century BCE and the 2nd century CE.
Explanation of the sine rule in Yuktibhāṣā
Page from The Compendious Book on Calculation by Completion and Balancing by Muhammad ibn Mūsā al-Khwārizmī (c. AD 820)
The Maya numerals for numbers 1 through 19, written in the Maya script
Nicole Oresme (1323–1382), shown in this contemporary illuminated manuscript with an armillary sphere in the foreground, was the first to offer a mathematical proof for the divergence of the harmonic series.
Portrait of Luca Pacioli, a painting traditionally attributed to Jacopo de' Barbari, 1495, (Museo di Capodimonte).
Gottfried Wilhelm Leibniz.
Leonhard Euler by Emanuel Handmann.
Carl Friedrich Gauss.
Behavior of lines with a common perpendicular in each of the three types of geometry
A map illustrating the Four Color Theorem
Newtonian (red) vs. Einsteinian orbit (blue) of a lone planet orbiting a star, with relativistic precession of apsides
The absolute value of the Gamma function on the complex plane.

The history of mathematics deals with the origin of discoveries in mathematics and the mathematical methods and notation of the past.

It was there that Euclid (c.

The conic sections, or two-dimensional figures formed by the intersection of a plane with a cone at different angles. The theory of these figures was developed extensively by the ancient Greek mathematicians, surviving especially in works such as those of Apollonius of Perga. The conic sections pervade modern mathematics.

Apollonius of Perga

Ancient Greek geometer and astronomer known for his work on conic sections.

Ancient Greek geometer and astronomer known for his work on conic sections.

The conic sections, or two-dimensional figures formed by the intersection of a plane with a cone at different angles. The theory of these figures was developed extensively by the ancient Greek mathematicians, surviving especially in works such as those of Apollonius of Perga. The conic sections pervade modern mathematics.
The animated figure depicts the method of "application of areas" to express the mathematical relationship that characterizes a parabola. The upper left corner of the changing rectangle on the left side and the upper right corner on the right side is "any point on the section." The animation has it following the section. The orange square at the top is "the square on the distance from the point to the diameter; i.e., a square of the ordinate. In Apollonius, the orientation is horizontal rather than the vertical shown here. Here it is the square of the abscissa. Regardless of orientation, the equation is the same, names changed. The blue rectangle on the outside is the rectangle on the other coordinate and the distance p. In algebra, x2 = py, one form of the equation for a parabola. If the outer rectangle exceeds py in area, the section must be a hyperbola; if it is less, an ellipse.
Pages from the 9th century Arabic translation of the Conics
1654 edition of Conica by Apollonius edited by Francesco Maurolico
Visual form of the Pythagorean theorem as the ancient Greeks saw it. The area of the blue square is the sum of the areas of the other two squares.
Cartesian coordinate system, standard in analytic geometry

Beginning from the contributions of Euclid and Archimedes on the topic, he brought them to the state prior to the invention of analytic geometry.

For such an important contributor to the field of mathematics, scant biographical information remains.

Types of conic sections: 1: Circle     2: Ellipse 3: Parabola  4: Hyperbola

Conic section

Types of conic sections: 1: Circle     2: Ellipse 3: Parabola  4: Hyperbola
Table of conics, Cyclopaedia, 1728
The black boundaries of the colored regions are conic sections. Not shown is the other half of the hyperbola, which is on the unshown other half of the double cone.
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Conic parameters in the case of an ellipse
Standard forms of an ellipse
Standard forms of a parabola
Standard forms of a hyperbola
Development of the conic section as the eccentricity e increases
Diagram from Apollonius' Conics, in a 9th-century Arabic translation
The paraboloid shape of Archeocyathids produces conic sections on rock faces
Definition of the Steiner generation of a conic section
Parallelogram method for constructing an ellipse
Three different types of conic sections. Focal-points corresponding to all conic sections are placed at the origin.
In this interactive SVG, move left and right over the SVG image to rotate the double cone

In mathematics, a conic section (or simply conic) is a curve obtained as the intersection of the surface of a cone with a plane.

Euclid (fl.