3rd century BC Greek mathematician Euclid (holding calipers), as imagined by Raphael in this detail from The School of Athens (1509–1511)
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.
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.
A fragment of Euclid's Elements on part of the Oxyrhynchus papyri
The quadratic formula expresses concisely the solutions of all quadratic equations
Double-page from the Ishaq ibn Hunayn's Arabic Translation of Elementa. Iraq, 1270. Chester Beatty Library
Rubik's cube: the study of its possible moves is a concrete application of group theory
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.
The Babylonian mathematical tablet Plimpton 322, dated to 1800 BC.
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.
Archimedes used the method of exhaustion, depicted here, to approximate the value of pi.
A page with marginalia from the first printed edition of Elements, printed by Erhard Ratdolt in 1482
The numerals used in the Bakhshali manuscript, dated between the 2nd century BC and the 2nd century AD.
The different versions of the parallel postulate result in different geometries.
A page from al-Khwārizmī's Algebra
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.
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.
Codex Vaticanus 190
Leonhard Euler created and popularized much of the mathematical notation used today.
Propositions plotted with lines connected from Axioms on the top and other preceding propositions, labelled by book.
Carl Friedrich Gauss, known as the prince of mathematicians
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.
The front side of the Fields Medal
Proof of the Pythagorean theorem in Byrne's The Elements of Euclid and published in colored version in 1847.
Euler's identity, which American physicist Richard Feynman once called "the most remarkable formula in mathematics".

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.

- Euclid's Elements

In the history of mathematics, the concept of a proof and its associated mathematical rigour first appeared in Greek mathematics, most notably in Euclid's Elements.

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

7 related topics

Alpha

Detail from Raphael's The School of Athens presumed to represent Donato Bramante as Euclid

Euclid

Not to be confused with Euclid of Megara.

Not to be confused with Euclid of Megara.

Detail from Raphael's The School of Athens presumed to represent Donato Bramante as Euclid
Euclidis quae supersunt omnia (1704)
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.
Euclid's construction of a regular dodecahedron.
Construction of a dodecahedron by placing faces on the edges of a cube.
19th-century statue of Euclid by Joseph Durham in the Oxford University Museum of Natural History

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.

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 classical approach is well-illustrated by Euclid's Elements, where a list of postulates is given (common-sensical geometric facts drawn from our experience), followed by a list of "common notions" (very basic, self-evident assertions).

The Pythagorean theorem has at least 370 known proofs

Theorem

The Pythagorean theorem has at least 370 known proofs
A planar map with five colors such that no two regions with the same color meet. It can actually be colored in this way with only four colors. The four color theorem states that such colorings are possible for any planar map, but every known proof involves a computational search that is too long to check by hand.
The Collatz conjecture: one way to illustrate its complexity is to extend the iteration from the natural numbers to the complex numbers. The result is a fractal, which (in accordance with universality) resembles the Mandelbrot set.
This diagram shows the syntactic entities that can be constructed from formal languages. The symbols and strings of symbols may be broadly divided into nonsense and well-formed formulas. A formal language can be thought of as identical to the set of its well-formed formulas. The set of well-formed formulas may be broadly divided into theorems and non-theorems.

In mathematics, a theorem is a statement that has been proved, or can be proved.

A proposition is a theorem of lesser importance, or one that is considered so elementary or immediately obvious, that it may be stated without proof. This should not be confused with "proposition" as used in propositional logic. In classical geometry the term "proposition" was used differently: in Euclid's Elements (c. 300 BCE), all theorems and geometric constructions were called "propositions" regardless of their importance.

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.

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

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

4) Ἡ καθόλου πραγματεία, a work on the general principles of mathematics that perhaps included Apollonius's criticisms and suggestions for the improvement of Euclid's Elements

The combined area of these three shapes is approximately 15.57 squares.

Area

Quantity that expresses the extent of a region on the plane or on a curved surface.

Quantity that expresses the extent of a region on the plane or on a curved surface.

The combined area of these three shapes is approximately 15.57 squares.
This square and this disk both have the same area (see: squaring the circle).
A square metre quadrat made of PVC pipe.
Although there are 10 mm in 1 cm, there are 100 mm2 in 1 cm2.
The area of this rectangle is lw.
A diagram showing how a parallelogram can be re-arranged into the shape of a rectangle.
A parallelogram split into two equal triangles.
A circle can be divided into sectors which rearrange to form an approximate parallelogram.
Archimedes showed that the surface area of a sphere is exactly four times the area of a flat disk of the same radius, and the volume enclosed by the sphere is exactly 2/3 of the volume of a cylinder of the same height and radius.
Integration can be thought of as measuring the area under a curve, defined by f(x), between two points (here a and b).
The area between two graphs can be evaluated by calculating the difference between the integrals of the two functions

In mathematics, the unit square is defined to have area one, and the area of any other shape or surface is a dimensionless real number.

Subsequently, Book I of Euclid's Elements dealt with equality of areas between two-dimensional figures.

Pythagorean theorem The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c).

Pythagorean theorem

The theorem can be generalized in various ways: to higher-dimensional spaces, to spaces that are not Euclidean, to objects that are not right triangles, and to objects that are not triangles at all but n-dimensional solids.

The theorem can be generalized in various ways: to higher-dimensional spaces, to spaces that are not Euclidean, to objects that are not right triangles, and to objects that are not triangles at all but n-dimensional solids.

Pythagorean theorem The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c).
The rearrangement proof (click to view animation)
Proof using similar triangles
Proof in Euclid's Elements
Illustration including the new lines
Showing the two congruent triangles of half the area of rectangle BDLK and square BAGF
Diagram of the two algebraic proofs
Right triangle on the hypotenuse dissected into two similar right triangles on the legs, according to Einstein's proof
Diagram for differential proof
The spiral of Theodorus: A construction for line segments with lengths whose ratios are the square root of a positive integer
The absolute value of a complex number z is the distance r from z to the origin
Similar right triangles showing sine and cosine of angle θ
The area of a parallelogram as a cross product; vectors a and b identify a plane and a × b is normal to this plane.
The separation s of two points (r1, θ1) and (r2, θ2) in polar coordinates is given by the law of cosines. Interior angle Δθ = θ1−θ2.
Generalization of Pythagoras' theorem by Tâbit ibn Qorra. Lower panel: reflection of triangle CAD (top) to form triangle DAC, similar to triangle ABC (top).
Generalization for arbitrary triangles, green area
Construction for proof of parallelogram generalization
Pythagoras' theorem in three dimensions relates the diagonal AD to the three sides.
A tetrahedron with outward facing right-angle corner
Vectors involved in the parallelogram law
Spherical triangle
Hyperbolic triangle
Distance between infinitesimally separated points in Cartesian coordinates (top) and polar coordinates (bottom), as given by Pythagoras' theorem
The Plimpton 322 tablet records Pythagorean triples from Babylonian times.
Geometric proof of the Pythagorean theorem from the Zhoubi Suanjing.
Rearrangement proof of the Pythagorean theorem.
Visual proof of the Pythagorean theorem by area-preserving shearing.

English mathematician Sir Thomas Heath gives this proof in his commentary on Proposition I.47 in Euclid's Elements, and mentions the proposals of German mathematicians Carl Anton Bretschneider and Hermann Hankel that Pythagoras may have known this proof.

Around 300 BC, in Euclid's Elements, the oldest extant axiomatic proof of the theorem is presented.