# Euclid

Euclid of AlexandriaEuklidGreek Mathematicianancient Greek mathematicianEfklidisEuclideEukleides of AlexandriaEukleidēsEuklidisFather of Geometry
Euclid ( – Eukleídēs, ; fl. 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referred to as the "founder of geometry" or the "father of geometry".wikipedia
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### Geometry

geometricgeometricalgeometries
Euclid ( – Eukleídēs, ; fl. 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referred to as the "founder of geometry" or the "father of geometry". 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.
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.

### Euclid's Elements

ElementsEuclid's ''ElementsEuclid
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.
The Elements ( Stoicheia) is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt c. 300 BC.

### Euclidean geometry

plane geometryEuclideanEuclidean plane geometry
In the Elements, Euclid deduced the theorems of what is now called Euclidean geometry from a small set of axioms. Today, however, that system is often referred to as Euclidean geometry to distinguish it from other so-called non-Euclidean geometries discovered in the 19th century.
Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements.

### Euclid of Megara

EuclidesEuclidEuclide of Megara
Euclid ( – Eukleídēs, ; fl. 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referred to as the "founder of geometry" or the "father of geometry".
Editors and translators in the Middle Ages often confused him with Euclid of Alexandria when discussing the latter's Elements.

### Mathematics

mathematicalmathmathematician
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.
Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements.

### Axiom

axiomspostulateaxiomatic
In the Elements, Euclid deduced the theorems of what is now called Euclidean geometry from a small set of 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).

### Greek mathematics

Greek mathematicianancient Greek mathematiciansGreek
Euclid ( – Eukleídēs, ; fl. 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referred to as the "founder of geometry" or the "father of geometry".
It has been customarily said that the Pythagoreans discovered most of the material in the first two books of Euclid's Elements.

### History of mathematics

historian of mathematicsmathematicshistory
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.
It was there that Euclid (c.

### Number theory

number theoristcombinatorial number theorytheory of numbers
Euclid also wrote works on perspective, conic sections, spherical geometry, number theory, and mathematical rigour.
Euclid IX 21–34 is very probably Pythagorean; it is very simple material ("odd times even is even", "if an odd number measures [= divides] an even number, then it also measures [= divides] half of it"), but it is all that is needed to prove that \sqrt{2}

### Conic section

conicconic sectionsconics
Euclid also wrote works on perspective, conic sections, spherical geometry, number theory, and mathematical rigour.
Euclid (fl.

### Tyre, Lebanon

TyreTyrianTyrians
A detailed biography of Euclid is given by Arabian authors, mentioning, for example, a birth town of Tyre.
"Tyre rapidly became Hellenized. Festivals in the Greek manner with offering of sacrifices, gymnastic contests, pageants and processions became part of the life of Tyre." Some Arabian authors claim that Tyre was the birth-place of Euclid, the "Father of Geometry" (c.

### Apollonius of Perga

ApolloniusApollonius of PergeApollonian
320 AD) briefly mentioned that Apollonius "spent a very long time with the pupils of Euclid at Alexandria, and it was thus that he acquired such a scientific habit of thought" c. 247–222 BC.
Beginning from the theories of Euclid and Archimedes on the topic, he brought them to the state they were in just before the invention of analytic geometry.

### Perspective (graphical)

perspectiveforeshorteninglinear perspective
Euclid also wrote works on perspective, conic sections, spherical geometry, number theory, and mathematical rigour.
Euclid's Optics (c.

### Rigour

rigorrigorousmathematical rigour
Euclid also wrote works on perspective, conic sections, spherical geometry, number theory, and mathematical rigour.
Its history traces back to Greek mathematics, especially to Euclid's Elements.

### Plato

Plato's dialoguesDialogues of PlatoPlatonic dialogues
According to Proclus, Euclid supposedly belonged to Plato's "persuasion" and brought together the Elements, drawing on prior work of Eudoxus of Cnidus and of several pupils of Plato (particularly Theaetetus and Philip of Opus.) Proclus believes that Euclid is not much younger than these, and that he must have lived during the time of Ptolemy I (c.
Eudoxus of Cnidus, the greatest mathematician in Classical Greece, who contributed much of what is found in Euclid's Elements, was taught by Archytas and Plato.

### Theaetetus (mathematician)

TheaetetusTheaetetus of AthensTheaetetus of Sunium
According to Proclus, Euclid supposedly belonged to Plato's "persuasion" and brought together the Elements, drawing on prior work of Eudoxus of Cnidus and of several pupils of Plato (particularly Theaetetus and Philip of Opus.) Proclus believes that Euclid is not much younger than these, and that he must have lived during the time of Ptolemy I (c.
His principal contributions were on irrational lengths, which was included in Book X of Euclid's Elements, and proving that there are precisely five regular convex polyhedra.

### Perfect number

perfect numbersperfectodd perfect number
It considers the connection between perfect numbers and Mersenne primes (known as the Euclid–Euler theorem), the infinitude of prime numbers, Euclid's lemma on factorization (which leads to the fundamental theorem of arithmetic on uniqueness of prime factorizations), and the Euclidean algorithm for finding the greatest common divisor of two numbers.
Euclid also proved a formation rule (IX.36) whereby q(q+1)/2 is an even perfect number whenever q is a prime of the form 2^p -1 for prime p—what is now called a Mersenne prime.

### Euclid's theorem

infinitude of primesinfinitude of prime numbersinfinitude of the prime numbers
It considers the connection between perfect numbers and Mersenne primes (known as the Euclid–Euler theorem), the infinitude of prime numbers, Euclid's lemma on factorization (which leads to the fundamental theorem of arithmetic on uniqueness of prime factorizations), and the Euclidean algorithm for finding the greatest common divisor of two numbers.
It was first proved by Euclid in his work Elements.

### Proclus

The few historical references to Euclid were written by Proclus c. 450 AD, centuries after Euclid lived.
The influential commentary on the first book of Euclid's Elements of Geometry is one of the most valuable sources we have for the history of ancient mathematics, and its Platonic account of the status of mathematical objects was influential.

### Theon of Alexandria

TheonTheon the Younger
Most of the copies say they are "from the edition of Theon" or the "lectures of Theon", while the text considered to be primary, held by the Vatican, mentions no author.
He edited and arranged Euclid's Elements and wrote commentaries on works by Euclid and Ptolemy.

### Euclidean algorithm

Euclid's algorithmEuclideanEuclid
It considers the connection between perfect numbers and Mersenne primes (known as the Euclid–Euler theorem), the infinitude of prime numbers, Euclid's lemma on factorization (which leads to the fundamental theorem of arithmetic on uniqueness of prime factorizations), and the Euclidean algorithm for finding the greatest common divisor of two numbers.
It is named after the ancient Greek mathematician Euclid, who first described it in his Elements (c.

### Mathematical proof

proofproofsprove
Although many of the results in Elements originated with earlier mathematicians, one of Euclid's accomplishments was to present them in a single, logically coherent framework, making it easy to use and easy to reference, including a system of rigorous mathematical proofs that remains the basis of mathematics 23 centuries later.
Mathematical proofs were revolutionized by Euclid (300 BCE), who introduced the axiomatic method still in use today, starting with undefined terms and axioms (propositions regarding the undefined terms assumed to be self-evidently true from the Greek "axios" meaning "something worthy"), and used these to prove theorems using deductive logic.

### Fundamental theorem of arithmetic

Canonical representation of a positive integerunique factorizationunique factorization theorem
It considers the connection between perfect numbers and Mersenne primes (known as the Euclid–Euler theorem), the infinitude of prime numbers, Euclid's lemma on factorization (which leads to the fundamental theorem of arithmetic on uniqueness of prime factorizations), and the Euclidean algorithm for finding the greatest common divisor of two numbers.
Book VII, propositions 30, 31 and 32, and Book IX, proposition 14 of Euclid's Elements are essentially the statement and proof of the fundamental theorem.

### Mersenne prime

Mersenne numberMersenne numbersMersenne primes
It considers the connection between perfect numbers and Mersenne primes (known as the Euclid–Euler theorem), the infinitude of prime numbers, Euclid's lemma on factorization (which leads to the fundamental theorem of arithmetic on uniqueness of prime factorizations), and the Euclidean algorithm for finding the greatest common divisor of two numbers.
In the 4th century BC, Euclid proved that if

### Non-Euclidean geometry

non-Euclideannon-Euclidean geometriesalternative geometries
Today, however, that system is often referred to as Euclidean geometry to distinguish it from other so-called non-Euclidean geometries discovered in the 19th century.
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.