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Euclid (; grc-gre,
Εὐκλείδης Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of g ...
; BC) was an ancient Greek
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...
active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
that largely dominated the field until the early 19th century. His system, now referred to as Euclidean geometry, involved new innovations in combination with a synthesis of theories from earlier Greek mathematicians, including
Eudoxus of Cnidus Eudoxus of Cnidus (; grc, Εὔδοξος ὁ Κνίδιος, ''Eúdoxos ho Knídios''; ) was an ancient Greek astronomer, mathematician, scholar, and student of Archytas and Plato. All of his original works are lost, though some fragments are ...
,
Hippocrates of Chios Hippocrates of Chios ( grc-gre, Ἱπποκράτης ὁ Χῖος; c. 470 – c. 410 BC) was an ancient Greek mathematician, geometer, and astronomer. He was born on the isle of Chios, where he was originally a merchant. After some misadve ...
,
Thales Thales of Miletus ( ; grc-gre, Θαλῆς; ) was a Greek mathematician, astronomer, statesman, and pre-Socratic philosopher from Miletus in Ionia, Asia Minor. He was one of the Seven Sages of Greece. Many, most notably Aristotle, regarded him ...
and
Theaetetus Theaetetus (Θεαίτητος) is a Greek name which could refer to: * Theaetetus (mathematician) (c. 417 BC – 369 BC), Greek geometer * ''Theaetetus'' (dialogue), a dialogue by Plato, named after the geometer * Theaetetus (crater), a lunar imp ...
. With
Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...
and
Apollonius of Perga Apollonius of Perga ( grc-gre, Ἀπολλώνιος ὁ Περγαῖος, Apollṓnios ho Pergaîos; la, Apollonius Pergaeus; ) was an Ancient Greek geometer and astronomer known for his work on conic sections. Beginning from the contribution ...
, Euclid is generally considered among the greatest mathematicians of antiquity, and one of the most influential in the history of mathematics. Very little is known of Euclid's life, and most information comes from the philosophers
Proclus Proclus Lycius (; 8 February 412 – 17 April 485), called Proclus the Successor ( grc-gre, Πρόκλος ὁ Διάδοχος, ''Próklos ho Diádokhos''), was a Greek Neoplatonist philosopher, one of the last major classical philosophers ...
and
Pappus of Alexandria Pappus of Alexandria (; grc-gre, Πάππος ὁ Ἀλεξανδρεύς; AD) was one of the last great Greek mathematicians of antiquity known for his ''Synagoge'' (Συναγωγή) or ''Collection'' (), and for Pappus's hexagon theorem i ...
many centuries later. Until the early
Renaissance The Renaissance ( , ) , from , with the same meanings. is a period in European history marking the transition from the Middle Ages to modernity and covering the 15th and 16th centuries, characterized by an effort to revive and surpass ideas ...
he was often mistaken for the earlier philosopher
Euclid of Megara Euclid of Megara (; grc-gre, Εὐκλείδης ; c. 435 – c. 365 BC) was a Greek Socratic philosopher who founded the Megarian school of philosophy. He was a pupil of Socrates in the late 5th century BC, and was present at his death. He he ...
, causing his biography to be substantially revised. It is generally agreed that he spent his career under
Ptolemy I Ptolemy I Soter (; gr, Πτολεμαῖος Σωτήρ, ''Ptolemaîos Sōtḗr'' "Ptolemy the Savior"; c. 367 BC – January 282 BC) was a Macedonian Greek general, historian and companion of Alexander the Great from the Kingdom of Macedon ...
in
Alexandria Alexandria ( or ; ar, ٱلْإِسْكَنْدَرِيَّةُ ; grc-gre, Αλεξάνδρεια, Alexándria) is the second largest city in Egypt, and the largest city on the Mediterranean coast. Founded in by Alexander the Great, Alexandria ...
and lived around 300 BC, after
Plato Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
and before Archimedes. There is some speculation that Euclid was a student of the
Platonic Academy The Academy (Ancient Greek: Ἀκαδημία) was founded by Plato in c. 387 BC in Classical Athens, Athens. Aristotle studied there for twenty years (367–347 BC) before founding his own school, the Lyceum (classical), Lyceum. The Academy ...
and later taught at the Musaeum. Euclid is often regarded as bridging the earlier Platonic tradition in
Athens Athens ( ; el, Αθήνα, Athína ; grc, Ἀθῆναι, Athênai (pl.) ) is both the capital and largest city of Greece. With a population close to four million, it is also the seventh largest city in the European Union. Athens dominates ...
with the later tradition of Alexandria. In the ''Elements'', Euclid deduced the theorems from a small set of
axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
s. He also wrote works on perspective,
conic section In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a specia ...
s,
spherical geometry 300px, A sphere with a spherical triangle on it. Spherical geometry is the geometry of the two-dimensional surface of a sphere. In this context the word "sphere" refers only to the 2-dimensional surface and other terms like "ball" or "solid sp ...
,
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777 ...
, and
mathematical rigour Rigour (British English) or rigor (American English; see spelling differences) describes a condition of stiffness or strictness. These constraints may be environmentally imposed, such as "the rigours of famine"; logically imposed, such as m ...
. In addition to the ''Elements'', Euclid wrote a central early text in the
optics Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviole ...
field, ''
Optics Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviole ...
'', and lesser-known works including ''
Data In the pursuit of knowledge, data (; ) is a collection of discrete values that convey information, describing quantity, quality, fact, statistics, other basic units of meaning, or simply sequences of symbols that may be further interpreted ...
'' and ''Phaenomena''. Euclid's authorship of two other texts—''On Divisions of Figures'', ''Catoptrics''—has been questioned. He is thought to have written many now
lost works A lost work is a document, literary work, or piece of multimedia produced some time in the past, of which no surviving copies are known to exist. It can only be known through reference. This term most commonly applies to works from the classical ...
.


Life


Traditional narrative

The English name 'Euclid' is the anglicized version of the
Ancient Greek Ancient Greek includes the forms of the Greek language used in ancient Greece and the ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Dark Ages (), the Archaic peri ...
name
Εὐκλείδης Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of g ...
. It is derived from '
eu- This is a list of common affixes used when scientifically naming species, particularly extinct species for whom only their scientific names are used, along with their derivations. *a-, an-: ''Pronunciation'': /ə/, /a/, /ən/, /an/. ''Origin' ...
' ( εὖ; 'well') and 'klês' ( -κλῆς; 'fame'), meaning "renowned, glorious". The word 'Euclid' less commonly also means "a copy of the same", and is sometimes synonymous with 'geometry'. Like many ancient Greek mathematicians, Euclid's life is mostly unknown. He is accepted as the author of four mostly extant treatises—the '' Elements'', ''
Optics Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviole ...
'', ''
Data In the pursuit of knowledge, data (; ) is a collection of discrete values that convey information, describing quantity, quality, fact, statistics, other basic units of meaning, or simply sequences of symbols that may be further interpreted ...
'', ''Phaenomena''—but besides this, there is nothing known for certain of him. The historian Carl Benjamin Boyer has noted irony in that "Considering the fame of the author and of his best seller
he ''Elements'' He or HE may refer to: Language * He (pronoun), an English pronoun * He (kana), the romanization of the Japanese kana へ * He (letter), the fifth letter of many Semitic alphabets * He (Cyrillic), a letter of the Cyrillic script called ''He'' ...
remarkably little is known of Euclid". The traditional narrative mainly follows the 5th century AD account by
Proclus Proclus Lycius (; 8 February 412 – 17 April 485), called Proclus the Successor ( grc-gre, Πρόκλος ὁ Διάδοχος, ''Próklos ho Diádokhos''), was a Greek Neoplatonist philosopher, one of the last major classical philosophers ...
in his ''Commentary on the First Book of Euclid's Elements'', as well as a few anecdotes from
Pappus of Alexandria Pappus of Alexandria (; grc-gre, Πάππος ὁ Ἀλεξανδρεύς; AD) was one of the last great Greek mathematicians of antiquity known for his ''Synagoge'' (Συναγωγή) or ''Collection'' (), and for Pappus's hexagon theorem i ...
in the early 4th century. According to Proclus, Euclid lived after the philosopher
Plato Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
( BC) and before the mathematician
Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...
( BC); specifically, Proclus placed Euclid during the rule of
Ptolemy I Ptolemy I Soter (; gr, Πτολεμαῖος Σωτήρ, ''Ptolemaîos Sōtḗr'' "Ptolemy the Savior"; c. 367 BC – January 282 BC) was a Macedonian Greek general, historian and companion of Alexander the Great from the Kingdom of Macedon ...
( BC). In his ''Collection'', Pappus indicates that Euclid was active in
Alexandria Alexandria ( or ; ar, ٱلْإِسْكَنْدَرِيَّةُ ; grc-gre, Αλεξάνδρεια, Alexándria) is the second largest city in Egypt, and the largest city on the Mediterranean coast. Founded in by Alexander the Great, Alexandria ...
, where he founded a mathematical tradition. Thus, the traditional outline—described by the historian Michalis Sialaros as the "dominant view"—holds that Euclid lived around 300 BC in Alexandria while Ptolemy I reigned. Euclid's birthdate is unknown; some scholars estimate around 330 or 325 BC, but other sources avoid speculating a date entirely. It is presumed that he was of Greek descent, but his birthplace is unknown. Proclus held that Euclid followed the
Platonic tradition Platonism is the philosophy of Plato and philosophical systems closely derived from it, though contemporary platonists do not necessarily accept all of the doctrines of Plato. Platonism had a profound effect on Western thought. Platonism at l ...
, but there is no definitive confirmation for this. It is unlikely he was contemporary with Plato, so it is often presumed that he was educated by Plato's disciples at the
Platonic Academy The Academy (Ancient Greek: Ἀκαδημία) was founded by Plato in c. 387 BC in Classical Athens, Athens. Aristotle studied there for twenty years (367–347 BC) before founding his own school, the Lyceum (classical), Lyceum. The Academy ...
in Athens. The historian Thomas Heath supported this theory by noting that most capable geometers lived in Athens, which included many of the mathematicians whose work Euclid later built on. The accuracy of these assertions has been questioned by Sialaros, who stated that Heath's theory "must be treated merely as a conjecture". Regardless of his actual attendance at the Platonic academy, the contents of his later work certainly suggest he was familiar with the Platonic geometry tradition, though they also demonstrate no observable influence from
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of phil ...
.
Alexander the Great Alexander III of Macedon ( grc, wikt:Ἀλέξανδρος, Ἀλέξανδρος, Alexandros; 20/21 July 356 BC – 10/11 June 323 BC), commonly known as Alexander the Great, was a king of the Ancient Greece, ancient Greek kingdom of Maced ...
founded Alexandria in 331 BC, where Euclid would later be active sometime around 300 BC. The rule of Ptolemy I from 306 BC onwards gave the city a stability which was relatively unique in the Mediterranean, amid the chaotic wars over dividing Alexander's empire. Ptolemy began a process of
hellenization Hellenization (other British spelling Hellenisation) or Hellenism is the adoption of Greek culture, religion, language and identity by non-Greeks. In the ancient period, colonization often led to the Hellenization of indigenous peoples; in the ...
and commissioned numerous constructions, building the massive Musaeum institution, which was a leading center of education. On the basis of later anecdotes, Euclid is thought to have been among the Musaeum's first scholars and to have founded the Alexandrian school of mathematics there. According to Pappus, the later mathematician
Apollonius of Perga Apollonius of Perga ( grc-gre, Ἀπολλώνιος ὁ Περγαῖος, Apollṓnios ho Pergaîos; la, Apollonius Pergaeus; ) was an Ancient Greek geometer and astronomer known for his work on conic sections. Beginning from the contribution ...
was taught there by pupils of Euclid. Euclid's date of death is unknown; it has been estimated that he died BC, presumably in Alexandria.


Identity and historicity

Euclid is often referred to as 'Euclid of Alexandria' to differentiate him from the earlier philosopher
Euclid of Megara Euclid of Megara (; grc-gre, Εὐκλείδης ; c. 435 – c. 365 BC) was a Greek Socratic philosopher who founded the Megarian school of philosophy. He was a pupil of Socrates in the late 5th century BC, and was present at his death. He he ...
, a pupil of
Socrates Socrates (; ; –399 BC) was a Greek philosopher from Athens who is credited as the founder of Western philosophy and among the first moral philosophers of the ethical tradition of thought. An enigmatic figure, Socrates authored no te ...
who was included in the dialogues of Plato. Historically, medieval scholars frequently confused the mathematician and philosopher, mistakenly referring to the former in Latin as 'Megarensis' (). As a result, biographical information on the mathematician Euclid was long conflated with the lives of both Euclid of Alexandria and Euclid of Megara. The only scholar of antiquity known to have confused the mathematician and philosopher was Valerius Maximus. However, this mistaken identification was relayed by many anonymous
Byzantine The Byzantine Empire, also referred to as the Eastern Roman Empire or Byzantium, was the continuation of the Roman Empire primarily in its eastern provinces during Late Antiquity and the Middle Ages, when its capital city was Constantinopl ...
sources and the Renaissance scholars
Campanus of Novara Campanus of Novara ( 1220 – 1296) was an Italian mathematician, astronomer, astrologer, and physician who is best known for his work on Euclid's ''Elements''. In his writings he refers to himself as Campanus Nouariensis; contemporary documen ...
and
Theodore Metochites Theodore Metochites ( el, Θεόδωρος Μετοχίτης; 1270–1332) was a Byzantine Greek statesman, author, gentleman philosopher, and patron of the arts. From c. 1305 to 1328 he held the position of personal adviser ('' mesazōn'') to e ...
, which was included in a of 1482 translation of the latter by Erhard Ratdolt. After the mathematician (1473–1539) affirmed this presumption in his 1505 translation, all subsequent publications passed on this identification. Later Renaissance scholars, particularly
Peter Ramus Petrus Ramus (french: Pierre de La Ramée; Anglicized as Peter Ramus ; 1515 – 26 August 1572) was a French humanist, logician, and educational reformer. A Protestant convert, he was a victim of the St. Bartholomew's Day massacre. Early life ...
, reevaluated this claim, proving it false via issues in chronology and contradiction in early sources. Arab sources written many centuries after his death give vast amounts of information concerning Euclid's life, but are completely unverifiable. Most scholars consider them of dubious authenticity; Heath in particular contends that the fictionalization was done to strengthen the connection between a revered mathematician and the Arab world. There are also numerous anecdotal stories concerning to Euclid, all of uncertain historicity, which "picture him as a kindly and gentle old man". The best known of these is Proclus' story about Ptolemy asking Euclid if there was a quicker path to learning geometry than reading his ''Elements'', which Euclid replied with "there is no royal road to geometry". This anecdote is questionable since a very similar interaction between
Menaechmus :''There is also a Menaechmus in Plautus' play, ''The Menaechmi''.'' Menaechmus ( el, Μέναιχμος, 380–320 BC) was an ancient Greek mathematician, geometer and philosopher born in Alopeconnesus or Prokonnesos in the Thracian Chersonese, wh ...
and Alexander the Great is recorded from Stobaeus. Both the accounts were written in the 5th century AD, neither indicate their source, and neither story appears in ancient Greek literature. The traditional narrative of Euclid's activity is complicated by no mathematicians of the 4th century BC indicating his existence. Mathematicians of the 3rd century such as Archimedes and Apollonius "assume a part of his work to be known"; however, Archimedes strangely uses an older theory of proportions, rather than that of Euclid. The ''Elements'' is dated to have been at least partly in circulation by the 3rd century BC. Some ancient Greek mathematician mention him by name, but he is usually referred to as "ὁ στοιχειώτης" ("the author of ''Elements''"). In the Middle Ages, some scholars contended Euclid was not a historical personage and that his name arose from a corruption of Greek mathematical terms.


Works


''Elements''

Euclid is best known for his thirteen-book treatise, the ''Elements'' ( grc-gre, Στοιχεῖα; ), considered his '' magnum opus''. Much of its content originates from earlier mathematicians, including Eudoxus,
Hippocrates of Chios Hippocrates of Chios ( grc-gre, Ἱπποκράτης ὁ Χῖος; c. 470 – c. 410 BC) was an ancient Greek mathematician, geometer, and astronomer. He was born on the isle of Chios, where he was originally a merchant. After some misadve ...
,
Thales Thales of Miletus ( ; grc-gre, Θαλῆς; ) was a Greek mathematician, astronomer, statesman, and pre-Socratic philosopher from Miletus in Ionia, Asia Minor. He was one of the Seven Sages of Greece. Many, most notably Aristotle, regarded him ...
and
Theaetetus Theaetetus (Θεαίτητος) is a Greek name which could refer to: * Theaetetus (mathematician) (c. 417 BC – 369 BC), Greek geometer * ''Theaetetus'' (dialogue), a dialogue by Plato, named after the geometer * Theaetetus (crater), a lunar imp ...
, while other theorems are mentioned by Plato and Aristotle. It is difficult to differentiate the work of Euclid from that of his predecessors, especially because the ''Elements'' essentially superseded much earlier and now-lost Greek mathematics. The classicist Markus Asper concludes that "apparently Euclid's achievement consists of assembling accepted mathematical knowledge into a cogent order and adding new proofs to fill in the gaps". Despite this, Sialaros furthers that "the remarkably tight structure of the ''Elements'' reveals authorial control beyond the limits of a mere editor". The mathematician Serafina Cuomo described it as a "reservoir of results". The ''Elements'' does not exclusively discuss geometry as is sometimes believed. It is traditionally divided into three topics: plane geometry (books 1–6), basic
arithmetic Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...
(books 7–10:) and solid geometry (books 11–13)—though book 5 (on proportions) and 10 (on irrational lines) do not exactly fit this scheme. The heart of the text is the
theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of th ...
s scattered throughout. Using Aristotle's terminology, these may be generally separated into two categories: "first principles" and "second principles". The first group includes statements labeled as a "definition" ( grc-gre, ὅρος or grc-gre, ὁρισμός), "postulate" ( grc-gre, αἴτημα‎), or a "common notion" ( grc-gre, κοινὴ ἔννοια); only the first book includes postulates—later known as axioms—and common notions. The second group consists of propositions, presented alongside
mathematical proof A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proo ...
s and diagrams. It is unknown if Euclid intended the ''Elements'' as a textbook, but its method of presentation makes it a natural fit. As a whole, the authorial voice remains general and impersonal.


Contents


=Books 1–6

= Book 1 of the ''Elements'' is foundational for the entire text. It begins with a series of 20 definitions for basic concepts geometric concepts such as
line Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Arts ...
s,
angle In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle. Angles formed by two ...
s and various regular polygons. Euclid then presents 10 assumptions (see table, right), grouped into five postulates (axioms) and five common notions. These assumptions are intended to provide the logical basis for every subsequent theorem, i.e. serve as an axiomatic system. The common notions exclusively concern the comparison of magnitudes. While postulates 1 through 4 are relatively straight forward, the 5th is known as the parallel postulate and particularly famous. Book 1 also includes 48 propositions, which can be loosely divided into those concerning basics theorems of plane geometry (1–26); theories on
parallel line In geometry, parallel lines are coplanar straight lines that do not intersect at any point. Parallel planes are planes in the same three-dimensional space that never meet. ''Parallel curves'' are curves that do not touch each other or inters ...
s (27–32); theories on
parallelogram In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equa ...
s (33–45); and the
Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
(46–48). The last of these includes the earliest surviving proof of the Pythagorean theorem, described by Sialaros as "remarkably delicate". Book 2 is traditionally understood as concerning
geometric algebra In mathematics, a geometric algebra (also known as a real Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the ge ...
, though this interpretation has been heavily debated since the 1970s; critics describe the characterization as anachronistic, since the foundations of even nascent algebra occurred many centuries later. The second book has a more focused scope and mostly provides algebraic theorems to accompany various geometric shapes. Book 3 focuses on circles, while the 4th discusses regular polygons, especially the
pentagon In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simpl ...
. Book 5 is among the work's most important sections and presents what is usually termed as the "general theory of proportion". Book 6 utilizes the "theory of
ratio In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...
s" in the context of plane geometry. It is built almost entirely of its first proposition: "Triangles and parallelograms which are under the same height are to one another as their bases".


=Books 7–10

= From Book 7 onwards, the mathematician notes that "Euclid starts afresh. Nothing from the preceding books is used".
Number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777 ...
is covered by books 7 to 10, the former beginning with a set of 22 definitions for
parity Parity may refer to: * Parity (computing) ** Parity bit in computing, sets the parity of data for the purpose of error detection ** Parity flag in computing, indicates if the number of set bits is odd or even in the binary representation of the r ...
,
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s and other arithmetic-related concepts. Book 7 includes the
Euclidean algorithm In mathematics, the Euclidean algorithm,Some widely used textbooks, such as I. N. Herstein's ''Topics in Algebra'' and Serge Lang's ''Algebra'', use the term "Euclidean algorithm" to refer to Euclidean division or Euclid's algorithm, is an effi ...
, a method for finding the greatest common divisor of two numbers. The 8th book discusses geometric progressions, while book 9 includes a proof that there are an infinite amount of prime numbers. Of the ''Elements'', book 10 is by far the largest and most complex, dealing with irrational numbers in the context of magnitudes.


=Books 11–13

= Books 11 through 13 primarily discuss solid geometry.


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. * ''Catoptrics'' concerns the mathematical theory of mirrors, particularly the images formed in plane and spherical concave mirrors, though the attribution is sometimes questioned. * The ''
Data In the pursuit of knowledge, data (; ) is a collection of discrete values that convey information, describing quantity, quality, fact, statistics, other basic units of meaning, or simply sequences of symbols that may be further interpreted ...
'' ( grc-gre, Δεδομένα), is a somewhat short text which deals with the nature and implications of "given" information in geometrical problems. * ''On Divisions'' ( grc-gre, Περὶ Διαιρέσεων‎) survives only partially in
Arabic Arabic (, ' ; , ' or ) is a Semitic languages, Semitic language spoken primarily across the Arab world.Semitic languages: an international handbook / edited by Stefan Weninger; in collaboration with Geoffrey Khan, Michael P. Streck, Janet C ...
translation, and concerns the division of geometrical figures into two or more equal parts or into parts in given
ratio In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...
s. It includes thirty-six propositions and is similar to Apollonius' ''Conics''. * The ''
Optics Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviole ...
'' ( grc-gre, Ὀπτικά‎) is the earliest surviving Greek treatise on perspective. It includes an introductory discussion of geometrical optics and basic rules of perspective. * The ''Phaenomena'' ( grc-gre, Φαινόμενα) is a treatise on
spherical astronomy Spherical astronomy, or positional astronomy, is a branch of observational astronomy used to locate astronomical objects on the celestial sphere, as seen at a particular date, time, and location on Earth. It relies on the mathematical methods of ...
, survives in Greek; it is similar to ''On the Moving Sphere'' by Autolycus of Pitane, who flourished around 310 BC.


Lost works

Four other works are credibly attributed to Euclid, but have been lost. * The ''Conics'' ( grc-gre, Κωνικά‎) was a four-book survey on
conic section In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a specia ...
s, which was later superseded by a Apollonius' more comprehensive treatment of the same name. The work's existence is known primarily from Pappus, who asserts that the first four books of Apollonius' ''Conics'' are largely based on Euclid's earlier work. Doubt has been cast on this assertion by the historian , owing to sparse evidence and no other corroboration of Pappus' account. * The ''Pseudaria'' ( grc-gre, Ψευδάρια‎; ), was—according to Proclus in (70.1–18)—a text in geometrical
reasoning Reason is the capacity of consciously applying logic by drawing conclusions from new or existing information, with the aim of seeking the truth. It is closely associated with such characteristically human activities as philosophy, science, lang ...
, written to advise beginners in avoiding common fallacies. Very little is known of its specific contents aside from its scope and a few extant lines. * The ''Porisms'' ( grc-gre, Πορίσματα; ) was, based on accounts from Pappus and Proclus, probably a three-book treatise with approximately 200 propositions. The term '
porism A porism is a mathematical proposition or corollary. It has been used to refer to a direct consequence of a Mathematical proof, proof, analogous to how a corollary refers to a direct consequence of a theorem. In modern usage, it is a relationship ...
' in this context does not refer to a
corollary In mathematics and logic, a corollary ( , ) is a theorem of less importance which can be readily deduced from a previous, more notable statement. A corollary could, for instance, be a proposition which is incidentally proved while proving another ...
, but to "a third type of proposition—an intermediate between a theorem and a problem—the aim of which is to discover a feature of an existing geometrical entity, for example, to find the centre of a circle". The mathematician Michel Chasles speculated that these now-lost propositions included content related to the modern theories of transversals and projective geometry. * The ''Surface Loci'' ( grc-gre, Τόποι πρὸς ἐπιφανείᾳ) is of virtually unknown contents, aside from speculation based on the work's title. Conjecture based on later accounts has suggested it discussed cones and cylinders, among other subjects.


Legacy

Euclid is generally considered with Archimedes and Apollonius of Perga as among the greatest mathematicians of antiquity. Many commentators cite him as one of the most influential figures in the history of mathematics. The geometrical system established by the ''Elements'' long dominated the field; however, today that system is often referred to as ' Euclidean geometry' to distinguish it from other
non-Euclidean geometries In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geo ...
discovered in the early 19th century. Among Euclid's many namesakes are the
European Space Agency , owners = , headquarters = Paris, Île-de-France, France , coordinates = , spaceport = Guiana Space Centre , seal = File:ESA emblem seal.png , seal_size = 130px , image = Views in the Main Control Room (1205 ...
's (ESA)
Euclid Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...
spacecraft, the lunar crater Euclides, and the minor planet
4354 Euclides 4354 Euclides , provisional designation , is a dark Dorian asteroid from the central regions of the asteroid belt, approximately in diameter. It was discovered on 24 September 1960, by Dutch astronomer couple Ingrid and Cornelis van Houten on pho ...
. The ''Elements'' is often considered after the
Bible The Bible (from Koine Greek , , 'the books') is a collection of religious texts or scriptures that are held to be sacred in Christianity, Judaism, Samaritanism, and many other religions. The Bible is an anthologya compilation of texts of a ...
as the most frequently translated, published, and studied book in the
Western World The Western world, also known as the West, primarily refers to the various nations and state (polity), states in the regions of Europe, North America, and Oceania.
's history. With Aristotle's ''
Metaphysics Metaphysics is the branch of philosophy that studies the fundamental nature of reality, the first principles of being, identity and change, space and time, causality, necessity, and possibility. It includes questions about the nature of conscio ...
'', the ''Elements'' is perhaps the most successful ancient Greek text, and was the dominant mathematical textbook in the Medieval Arab and Latin worlds. The first English edition of the ''Elements'' was published in 1570 by
Henry Billingsley Sir Henry Billingsley (died 22 November 1606) was an English merchant, Lord Mayor of London and the first translator of Euclid into English. Early life He was a son of Sir William Billingsley, haberdasher and assay master of London, and his wif ...
and
John Dee John Dee (13 July 1527 – 1608 or 1609) was an English mathematician, astronomer, astrologer, teacher, occultist, and alchemist. He was the court astronomer for, and advisor to, Elizabeth I, and spent much of his time on alchemy, divinatio ...
. The mathematician Oliver Byrne published a well-known version of the ''Elements'' in 1847 entitled ''The First Six Books of the Elements of Euclid in Which Coloured Diagrams and Symbols Are Used Instead of Letters for the Greater Ease of Learners'', which included colored diagrams intended to increase its pedagogical effect.
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...
authored a modern axiomatization of the ''Elements''.


References


Notes


Citations


Sources

;Books and chapters * * * * * * * * * * * * * * * * * * * ;Journal and encyclopedia articles * * * * * * ;Online * * *


External links

; Works * * *
Euclid Collection
at
University College London , mottoeng = Let all come who by merit deserve the most reward , established = , type = Public research university , endowment = £143 million (2020) , budget = ...
(c.500 editions of works by Euclid) ; The ''Elements''
PDF copy
with the original Greek and an English translation on facing pages,
University of Texas The University of Texas at Austin (UT Austin, UT, or Texas) is a public research university in Austin, Texas. It was founded in 1883 and is the oldest institution in the University of Texas System. With 40,916 undergraduate students, 11,075 ...
.
All thirteen books
in several languages as Spanish, Catalan, English, German, Portuguese, Arabic, Italian, Russian and Chinese. {{Authority control 4th-century BC births 4th-century BC Egyptian people 4th-century BC Greek people 4th-century BC writers 3rd-century BC deaths 3rd-century BC Egyptian people 3rd-century BC Greek people 3rd-century BC mathematicians 3rd-century BC writers Ancient Alexandrians Ancient Greek geometers Number theorists Philosophers of mathematics