Euclid fl.Floruit , abbreviated fl. , is a Latin verb meaning "flourished", denoting the period of time during which something was active...
300 BC, also known as
Euclid of Alexandria, was a
Greek mathematicianGreek mathematics, as that term is used in this article, is the mathematics written in Greek, developed from the 7th century BC to the 4th century AD around the Eastern shores of the Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean, from Italy to...
, often referred to as the "Father of Geometry". He was active in
AlexandriaAlexandria is the second-largest city of Egypt, with a population of 4.1 million, extending about along the coast of the Mediterranean Sea in the north central part of the country; it is also the largest city lying directly on the Mediterranean coast. It is Egypt's largest seaport, serving...
during the reign of Ptolemy I (323–283 BC). His
ElementsEuclid's Elements is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria c. 300 BC. It is a collection of definitions, postulates , propositions , and mathematical proofs of the propositions...
is one of the most influential works in the
history of mathematicsThe area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past....
, serving as the main
textbookA textbook or coursebook is a manual of instruction in any branch of study. Textbooks are produced according to the demands of educational institutions...
for teaching
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
(especially
geometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
) from the time of its publication until the late 19th or early 20th century. In the
Elements, Euclid deduced the principles of what is now called
Euclidean geometryEuclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
from a small set of
axiomIn traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...
s. Euclid also wrote works on
perspectivePerspective, in context of vision and visual perception, is the way in which objects appear to the eye based on their spatial attributes; or their dimensions and the position of the eye relative to the objects...
,
conic sectionIn mathematics, a conic section is a curve obtained by intersecting a cone with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2...
s,
spherical geometrySpherical geometry is the geometry of the two-dimensional surface of a sphere. It is an example of a geometry which is not Euclidean. Two practical applications of the principles of spherical geometry are to navigation and astronomy....
,
number theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
and rigor.
"Euclid" is the anglicized version of the
GreekGreek is an independent branch of the Indo-European family of languages. Native to the southern Balkans, it has the longest documented history of any Indo-European language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history;...
name , meaning "Good Glory".
Life
Little is known about Euclid's life, as there are only a handful of references to him. The date and place of Euclid's birth and the date and circumstances of his death are unknown, and only roughly estimated in proximity to contemporary figures mentioned in references. No likeness or description of Euclid's physical appearance made during his lifetime survived antiquity. Therefore, Euclid's depiction in works of art is the product of the artist's imagination.
The few historical references to Euclid were written centuries after he lived, by
ProclusProclus Lycaeus , called "The Successor" or "Diadochos" , was a Greek Neoplatonist philosopher, one of the last major Classical philosophers . He set forth one of the most elaborate and fully developed systems of Neoplatonism...
and
Pappus of AlexandriaPappus of Alexandria was one of the last great Greek mathematicians of Antiquity, known for his Synagoge or Collection , and for Pappus's Theorem in projective geometry...
. Proclus introduces Euclid only briefly in his fifth-century
Commentary on the Elements, as the author of
Elements, that he was mentioned by
ArchimedesArchimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...
, and that when
King PtolemyPtolemy I Soter I , also known as Ptolemy Lagides, c. 367 BC – c. 283 BC, was a Macedonian general under Alexander the Great, who became ruler of Egypt and founder of both the Ptolemaic Kingdom and the Ptolemaic Dynasty...
asked if there was a shorter path to learning geometry than Euclid's
Elements, "Euclid replied there is no royal road to geometry." Although the purported citation of Euclid by Archimedes has been judged to be an interpolation by later editors of his works, it is still believed that Euclid wrote his works before those of Archimedes. In addition, the "royal road" anecdote is questionable since it is similar to a story told about
MenaechmusMenaechmus was an ancient Greek mathematician and geometer born in Alopeconnesus in the Thracian Chersonese, who was known for his friendship with the renowned philosopher Plato and for his apparent discovery of conic sections and his solution to the then-long-standing problem of doubling the cube...
and
Alexander the Great. In the only other key reference to Euclid, Pappus briefly mentioned in the fourth century 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." It is further believed that Euclid may have studied at
Plato's AcademyThe Academy was founded by Plato in ca. 387 BC in Athens. Aristotle studied there for twenty years before founding his own school, the Lyceum. The Academy persisted throughout the Hellenistic period as a skeptical school, until coming to an end after the death of Philo of Larissa in 83 BC...
in
AthensAthens , is the capital and largest city of Greece. Athens dominates the Attica region and is one of the world's oldest cities, as its recorded history spans around 3,400 years. Classical Athens was a powerful city-state...
.
Elements
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 proofIn mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...
s that remains the basis of mathematics 23 centuries later.
There is no mention of Euclid in the earliest remaining copies of the
Elements, and most of the copies say they are "from the edition of
TheonTheon was a Greek scholar and mathematician who lived in Alexandria, Egypt. He edited and arranged Euclid's Elements and Ptolemy's Handy Tables, as well as writing various commentaries...
" or the "lectures of Theon", while the text considered to be primary, held by the Vatican, mentions no author. The only reference that historians rely on of Euclid having written the
Elements was from Proclus, who briefly in his
Commentary on the Elements ascribes Euclid as its author.
Although best known for its geometric results, the
Elements also includes
number theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
. It considers the connection between perfect numbers and Mersenne primes, the infinitude of
prime numberA prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
s,
Euclid's lemmaIn mathematics, Euclid's lemma is an important lemma regarding divisibility and prime numbers. In its simplest form, the lemma states that a prime number that divides a product of two integers must divide one of the two integers...
on factorization (which leads to the
fundamental theorem of arithmeticIn number theory, the fundamental theorem of arithmetic states that any integer greater than 1 can be written as a unique product of prime numbers...
on uniqueness of
prime factorizationsIn number theory, integer factorization or prime factorization is the decomposition of a composite number into smaller non-trivial divisors, which when multiplied together equal the original integer....
), and the
Euclidean algorithmIn mathematics, the Euclidean algorithm is an efficient method for computing the greatest common divisor of two integers, also known as the greatest common factor or highest common factor...
for finding the
greatest common divisorIn mathematics, the greatest common divisor , also known as the greatest common factor , or highest common factor , of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder.For example, the GCD of 8 and 12 is 4.This notion can be extended to...
of two numbers.
The geometrical system described in the
Elements was long known simply as
geometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, and was considered to be the only geometry possible. Today, however, that system is often referred to as
Euclidean geometryEuclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
to distinguish it from other so-called
non-Euclidean geometriesNon-Euclidean geometry is the term used to refer to two specific geometries which are, loosely speaking, obtained by negating the Euclidean parallel postulate, namely hyperbolic and elliptic geometry. This is one term which, for historical reasons, has a meaning in mathematics which is much...
that mathematicians discovered in the 19th century.
Other works
In addition to the
Elements, at least five works of Euclid have survived to the present day. They follow the same logical structure as
Elements, with definitions and proved propositions.
- Data
Data is a work by Euclid. It deals with the nature and implications of "given" information in geometrical problems. The subject matter is closely related to the first four books of Euclid's Elements....
deals with the nature and implications of "given" information in geometrical problems; the subject matter is closely related to the first four books of the Elements.
- On Divisions of Figures, which survives only partially in Arabic
Arabic is a name applied to the descendants of the Classical Arabic language of the 6th century AD, used most prominently in the Quran, the Islamic Holy Book...
translation, concerns the division of geometrical figures into two or more equal parts or into parts in given ratioIn mathematics, a ratio is a relationship between two numbers of the same kind , usually expressed as "a to b" or a:b, sometimes expressed arithmetically as a dimensionless quotient of the two which explicitly indicates how many times the first number contains the second In mathematics, a ratio is...
s. It is similar to a third century AD work by Heron of Alexandria.
- Catoptrics
Catoptrics deals with the phenomena of reflected light and image-forming optical systems using mirrors. From the Greek κατοπτρικός ....
, which concerns the mathematical theory of mirrors, particularly the images formed in plane and spherical concave mirrors. The attribution to Euclid is doubtful. Its author may have been Theon of AlexandriaTheon was a Greek scholar and mathematician who lived in Alexandria, Egypt. He edited and arranged Euclid's Elements and Ptolemy's Handy Tables, as well as writing various commentaries...
.
- Phaenomena, a treatise on spherical astronomy
Spherical astronomy or positional astronomy is the branch of astronomy that is used to determine the location of objects on the celestial sphere, as seen at a particular date, time, and location on the Earth. It relies on the mathematical methods of spherical geometry and the measurements of...
, survives in Greek; it is quite similar to On the Moving Sphere by Autolycus of PitaneAutolycus of Pitane was a Greek astronomer, mathematician, and geographer. The lunar crater Autolycus was named in his honour.- Life and work :Autolycus was born in Pitane, a town of Aeolis within Western Anatolia...
, who flourished around 310 BC.
- Optics
Euclid's Optics, is a work on the geometry of vision written by the Greek mathematician Euclid around 300 BC. The earliest surviving manuscript of Optics is in Greek and dates from the 10th century AD....
is the earliest surviving Greek treatise on perspective. In its definitions Euclid follows the Platonic tradition that vision is caused by discrete rays which emanate from the eyeEmission theory or extramission theory is the proposal that visual perception is accomplished by rays of light emitted by the eyes. This theory has been replaced by intromission theory, which states that visual perception comes from something representative of the object entering the eyes...
. One important definition is the fourth: "Things seen under a greater angle appear greater, and those under a lesser angle less, while those under equal angles appear equal." In the 36 propositions that follow, Euclid relates the apparent size of an object to its distance from the eye and investigates the apparent shapes of cylinders and cones when viewed from different angles. Proposition 45 is interesting, proving that for any two unequal magnitudes, there is a point from which the two appear equal. PappusPappus of Alexandria was one of the last great Greek mathematicians of Antiquity, known for his Synagoge or Collection , and for Pappus's Theorem in projective geometry...
believed these results to be important in astronomy and included Euclid's Optics, along with his Phaenomena, in the Little Astronomy, a compendium of smaller works to be studied before the Syntaxis (Almagest) of Claudius Ptolemy.
Other works are credibly attributed to Euclid, but have been lost.
- Conics was a work on conic section
In mathematics, a conic section is a curve obtained by intersecting a cone with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2...
s that was later extended by Apollonius of PergaApollonius of Perga [Pergaeus] was a Greek geometer and astronomer noted for his writings on conic sections. His innovative methodology and terminology, especially in the field of conics, influenced many later scholars including Ptolemy, Francesco Maurolico, Isaac Newton, and René Descartes...
into his famous work on the subject. It is likely that the first four books of Apollonius's work come directly from Euclid. According to Pappus, "Apollonius, having completed Euclid's four books of conics and added four others, handed down eight volumes of conics." The Conics of Apollonius quickly supplanted the former work, and by the time of Pappus, Euclid's work was already lost.
- Porism
A porism is a mathematical proposition or corollary. In particular, the term porism has been used to refer to a direct result of a proof, analogous to how a corollary refers to a direct result of a theorem.-Beginnings:...
s might have been an outgrowth of Euclid's work with conic sections, but the exact meaning of the title is controversial.
- Pseudaria, or Book of Fallacies, was an elementary text about errors in reasoning.
- Surface Loci concerned either loci
In geometry, a locus is a collection of points which share a property. For example a circle may be defined as the locus of points in a plane at a fixed distance from a given point....
(sets of points) on surfaces or loci which were themselves surfaces; under the latter interpretation, it has been hypothesized that the work might have dealt with quadric surfacesIn mathematics, a quadric, or quadric surface, is any D-dimensional hypersurface in -dimensional space defined as the locus of zeros of a quadratic polynomial...
.
- Several works on mechanics
Mechanics is the branch of physics concerned with the behavior of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment....
are attributed to Euclid by Arabic sources. On the Heavy and the Light contains, in nine definitions and five propositions, Aristotelian notions of moving bodies and the concept of specific gravity. On the Balance treats the theory of the lever in a similarly Euclidean manner, containing one definition, two axioms, and four propositions. A third fragment, on the circles described by the ends of a moving lever, contains four propositions. These three works complement each other in such a way that it has been suggested that they are remnants of a single treatise on mechanics written by Euclid.
See also
- Axiomatic method
- Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
- Euclid's orchard
In mathematics Euclid's orchard is an array of one-dimensional trees of unit height planted at the lattice points in one quadrant of a square lattice...
- Euclidean relation
In mathematics, Euclidean relations are a class of binary relations that satisfy a weakened form of transitivity that formalizes Euclid's "Common Notion 1" in The Elements: things which equal the same thing also equal one another.-Definition:...
- Euclidean algorithm
In mathematics, the Euclidean algorithm is an efficient method for computing the greatest common divisor of two integers, also known as the greatest common factor or highest common factor...
- Extended Euclidean algorithm
The extended Euclidean algorithm is an extension to the Euclidean algorithm. Besides finding the greatest common divisor of integers a and b, as the Euclidean algorithm does, it also finds integers x and y that satisfy Bézout's identityThe extended Euclidean algorithm is particularly useful when a...
- List of topics named after Euclid
- Papyrus Oxyrhynchus 29
Papyrus Oxyrhynchus 29 is a fragment of the second book of the Elements of Euclid in Greek. It was discovered by Grenfell and Hunt in 1897 in Oxyrhynchus. The fragment was originally dated to the end third century or the beginning of the fourth century, although more recent scholarship suggests a...
External links
- Euclid's Elements, All thirteen books, with interactive diagrams using Java. Clark University
Clark University is a private research university and liberal arts college in Worcester, Massachusetts.Founded in 1887, it is the oldest educational institution founded as an all-graduate university. Clark now also educates undergraduates...
- Euclid's Elements, with the original Greek and an English translation on facing pages (includes PDF version for printing). University of Texas.
- Euclid's Elements, books I-VI, in English pdf, in a Project Gutenberg Victorian textbook edition with diagrams.
- Euclid's Elements, All thirteen books, in several languages as Spanish, Catalan, English, German, Portuguese, Arabic, Italian, Russian and Chinese.
- Elementa Geometriae 1482, Venice. From Rare Book Room
Rare Book Room is an educational website for the repository of digitally scanned rare books made freely available to the public.Starting around 1996 the California based company Octavo began scanning rare and important books from libraries around the world. These scans were done at extremely high...
.
- Elementa 888 AD, Byzantine. From Rare Book Room
Rare Book Room is an educational website for the repository of digitally scanned rare books made freely available to the public.Starting around 1996 the California based company Octavo began scanning rare and important books from libraries around the world. These scans were done at extremely high...
.
- Euclid biography by Charlene Douglass With extensive bibliography.
- Texts on Ancient Mathematics and Mathematical Astronomy PDF scans (Note: many are very large files). Includes editions and translations of Euclid's Elements, Data, and Optica, Proclus's Commentary on Euclid, and other historical sources.