Euclid's Elements

Euclid's Elements

Overview

Euclid's Elements (Greek
Greek language
Greek 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;...

: Stoicheia) is a mathematical
Mathematics
Mathematics 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...

 and geometric
Geometry
Geometry 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 ....

 treatise
Treatise
A treatise is a formal and systematic written discourse on some subject, generally longer and treating it in greater depth than an essay, and more concerned with investigating or exposing the principles of the subject.-Noteworthy treatises:...

 consisting of 13 books written by the Greek mathematician
Greek mathematics
Greek 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...

 Euclid
Euclid
Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...

 in Alexandria
Alexandria
Alexandria 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...

 c. 300 BC. It is a collection of definitions, postulates (axiom
Axiom
In 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), propositions (theorem
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...

s and constructions), and mathematical proof
Mathematical proof
In 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 of the propositions. The thirteen books cover Euclidean geometry
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...

 and the ancient Greek version of elementary number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

. The work also includes an algebraic system that has become known as geometric algebra, which is powerful enough to solve many algebraic problems, including the problem of finding the square root
Square root
In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x...

.
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Euclid's Elements (Greek
Greek language
Greek 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;...

: Stoicheia) is a mathematical
Mathematics
Mathematics 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...

 and geometric
Geometry
Geometry 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 ....

 treatise
Treatise
A treatise is a formal and systematic written discourse on some subject, generally longer and treating it in greater depth than an essay, and more concerned with investigating or exposing the principles of the subject.-Noteworthy treatises:...

 consisting of 13 books written by the Greek mathematician
Greek mathematics
Greek 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...

 Euclid
Euclid
Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...

 in Alexandria
Alexandria
Alexandria 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...

 c. 300 BC. It is a collection of definitions, postulates (axiom
Axiom
In 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), propositions (theorem
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...

s and constructions), and mathematical proof
Mathematical proof
In 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 of the propositions. The thirteen books cover Euclidean geometry
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...

 and the ancient Greek version of elementary number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

. The work also includes an algebraic system that has become known as geometric algebra, which is powerful enough to solve many algebraic problems, including the problem of finding the square root
Square root
In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x...

. With the exception of Autolycus'
Autolycus of Pitane
Autolycus 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...

 On the Moving Sphere, the Elements is one of the oldest extant Greek mathematical treatises and it is the oldest extant axiomatic deductive treatment of mathematics
Mathematics
Mathematics 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...

. It has proven instrumental in the development of logic
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...

 and modern science
Science
Science is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe...

.
The name Elements comes from the plural of 'element'. According to Proclus
Proclus
Proclus 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...

 the term was used to describe a theorem that is all-pervading and helps furnishing proofs of many other theorems. The word 'element' is in the Greek language the same as 'letter'. This suggests that theorems in the Elements should be seen as standing in the same relation to geometry as letters to language. Later commentators give a slightly different meaning to the term 'element', emphasizing on how the propositions progress in small steps, and continue to build on previous propositions in a well-defined order.

Euclid's Elements has been referred to as the most successful and influential textbook ever written. Being first set in type in Venice
Venice
Venice is a city in northern Italy which is renowned for the beauty of its setting, its architecture and its artworks. It is the capital of the Veneto region...

 in 1482, it is one of the very earliest mathematical works to be printed after the invention of the printing press
Printing press
A printing press is a device for applying pressure to an inked surface resting upon a print medium , thereby transferring the ink...

 and was estimated by Carl Benjamin Boyer
Carl Benjamin Boyer
Carl Benjamin Boyer was a historian of sciences, and especially mathematics. David Foster Wallace called him the "Gibbon of math history"....

 to be second only to the Bible
Bible
The Bible refers to any one of the collections of the primary religious texts of Judaism and Christianity. There is no common version of the Bible, as the individual books , their contents and their order vary among denominations...

 in the number of editions published, with the number reaching well over one thousand. For centuries, when the quadrivium
Quadrivium
The quadrivium comprised the four subjects, or arts, taught in medieval universities, after teaching the trivium. The word is Latin, meaning "the four ways" , and its use for the 4 subjects has been attributed to Boethius or Cassiodorus in the 6th century...

 was included in the curriculum of all university students, knowledge of at least part of Euclid's Elements was required of all students. Not until the 20th century, by which time its content was universally taught through school books, did it cease to be considered something all educated people had read.

Basis in earlier work



Scholars believe that the Elements is largely a collection of theorems proved by other mathematicians supplemented by some original work. Proclus
Proclus
Proclus 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...

, a Greek mathematician who lived several centuries after Euclid, wrote in his commentary of the Elements: "Euclid, who put together the Elements, collecting many of Eudoxus
Eudoxus of Cnidus
Eudoxus of Cnidus was a Greek astronomer, mathematician, scholar and student of Plato. Since all his own works are lost, our knowledge of him is obtained from secondary sources, such as Aratus's poem on astronomy...

' theorems, perfecting many of Theaetetus
Theaetetus (mathematician)
Theaetetus, Theaitētos, of Athens, possibly son of Euphronius, of the Athenian deme Sunium, was a classical Greek mathematician...

', and also bringing to irrefragable demonstration the things which were only somewhat loosely proved by his predecessors". Pythagoras
Pythagoras
Pythagoras of Samos was an Ionian Greek philosopher, mathematician, and founder of the religious movement called Pythagoreanism. Most of the information about Pythagoras was written down centuries after he lived, so very little reliable information is known about him...

 was probably the source of most of books I and II, Hippocrates of Chios
Hippocrates of Chios
Hippocrates of Chios was an ancient Greek mathematician, , and astronomer, who lived c. 470 – c. 410 BCE.He was born on the isle of Chios, where he originally was a merchant. After some misadventures he went to Athens, possibly for litigation...

 (not the better known Hippocrates of Kos
Hippocrates
Hippocrates of Cos or Hippokrates of Kos was an ancient Greek physician of the Age of Pericles , and is considered one of the most outstanding figures in the history of medicine...

) of book III, and Eudoxus book V, while books IV, VI, XI, and XII probably came from other Pythagorean or Athenian mathematicians. Euclid often replaced fallacious proofs with his own, more rigorous versions. The use of definitions, postulates, and axioms dated back to Plato
Plato
Plato , was a Classical Greek philosopher, mathematician, student of Socrates, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the Western world. Along with his mentor, Socrates, and his student, Aristotle, Plato helped to lay the...

. The Elements may have been based on an earlier textbook by Hippocrates of Chios, who also may have originated the use of letters to refer to figures.

Transmission of the text


In the fourth century AD Theon of Alexandria
Theon of Alexandria
Theon 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...

 produced an edition of Euclid which was so widely used that it became the only surviving source until François Peyrard's 1808 discovery at the Vatican
Vatican Library
The Vatican Library is the library of the Holy See, currently located in Vatican City. It is one of the oldest libraries in the world and contains one of the most significant collections of historical texts. Formally established in 1475, though in fact much older, it has 75,000 codices from...

 of a manuscript not derived from Theon's. This manuscript, the Heiberg
Johan Ludvig Heiberg (historian)
Johan Ludvig Heiberg was a Danish philologist and historian. He is best known for his discovery of previously unknown texts in the Archimedes Palimpsest, and for his edition of Euclid's Elements that T. L. Heath translated into English...

 manuscript, is from a Byzantine
Byzantine
Byzantine usually refers to the Roman Empire during the Middle Ages.Byzantine may also refer to:* A citizen of the Byzantine Empire, or native Greek during the Middle Ages...

 workshop c. 900 and is the basis of modern editions. Papyrus Oxyrhynchus 29
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...

 is a tiny fragment of an even older manuscript, but only contains the statement of one proposition.

Although known to, for instance, Cicero
Cicero
Marcus Tullius Cicero , was a Roman philosopher, statesman, lawyer, political theorist, and Roman constitutionalist. He came from a wealthy municipal family of the equestrian order, and is widely considered one of Rome's greatest orators and prose stylists.He introduced the Romans to the chief...

, there is no extant record of the text having been translated into Latin prior to Boethius in the fifth or sixth century. The Arabs received the Elements from the Byzantines in approximately 760; this version, by a pupil of Euclid called Proclo
Proclo
Proclo was a later pupil of the Greek geometer Euclid whose version of Euclid's Elements was translated into Arabic....

, was translated into Arabic under Harun al Rashid c. 800. The Byzantine scholar Arethas
Arethas of Caesarea
Arethas of Caesarea became Archbishop of Caesarea early in the 10th century, and is reckoned one of the most scholarly theologians of the Greek Orthodox Church.-Life:He was born at Patrae . He was a disciple of Photius...

 commissioned the copying of one of the extant Greek manuscripts of Euclid in the late ninth century. Although known in Byzantium, the Elements was lost to Western Europe until c. 1120, when the English monk Adelard of Bath
Adelard of Bath
Adelard of Bath was a 12th century English scholar. He is known both for his original works and for translating many important Greek and Arabic scientific works of astrology, astronomy, philosophy and mathematics into Latin from Arabic versions, which were then introduced to Western Europe...

 translated it into Latin from an Arabic translation.

The first printed edition appeared in 1482 (based on Campanus of Novara's 1260 edition), and since then it has been translated into many languages and published in about a thousand different editions. Theon's Greek edition was recovered in 1533. In 1570, John Dee
John Dee (mathematician)
John Dee was an English mathematician, astronomer, astrologer, occultist, navigator, imperialist and consultant to Queen Elizabeth I. He devoted much of his life to the study of alchemy, divination and Hermetic philosophy....

 provided a widely respected "Mathematical Preface", along with copious notes and supplementary material, to the first English edition by Henry Billingsley
Henry Billingsley
Sir Henry Billingsley was Lord Mayor of London and the first translator of Euclid into English.-Early Life:He was a son of William Billingsley, haberdasher and assaymaster of London, and his wife, Elizabeth Harlowe. He entered St...

.

Copies of the Greek text still exist, some of which can be found in the Vatican Library
Vatican Library
The Vatican Library is the library of the Holy See, currently located in Vatican City. It is one of the oldest libraries in the world and contains one of the most significant collections of historical texts. Formally established in 1475, though in fact much older, it has 75,000 codices from...

 and the Bodleian Library
Bodleian Library
The Bodleian Library , the main research library of the University of Oxford, is one of the oldest libraries in Europe, and in Britain is second in size only to the British Library...

 in Oxford. The manuscripts available are of variable quality, and invariably incomplete. By careful analysis of the translations and originals, hypotheses have been made about the contents of the original text (copies of which are no longer available).

Ancient texts which refer to the Elements itself and to other mathematical theories that were current at the time it was written are also important in this process. Such analyses are conducted by J. L. Heiberg
Johan Ludvig Heiberg (historian)
Johan Ludvig Heiberg was a Danish philologist and historian. He is best known for his discovery of previously unknown texts in the Archimedes Palimpsest, and for his edition of Euclid's Elements that T. L. Heath translated into English...

 and Sir Thomas Little Heath in their editions of the text.

Also of importance are the scholia, or annotations to the text. These additions, which often distinguished themselves from the main text (depending on the manuscript), gradually accumulated over time as opinions varied upon what was worthy of explanation or elucidation.

Influence


The Elements is still considered a masterpiece in the application of logic
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...

 to mathematics
Mathematics
Mathematics 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...

. In historical context, it has proven enormously influential in many areas of science
Science
Science is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe...

. Scientists Nicolaus Copernicus
Nicolaus Copernicus
Nicolaus Copernicus was a Renaissance astronomer and the first person to formulate a comprehensive heliocentric cosmology which displaced the Earth from the center of the universe....

, Johannes Kepler
Johannes Kepler
Johannes Kepler was a German mathematician, astronomer and astrologer. A key figure in the 17th century scientific revolution, he is best known for his eponymous laws of planetary motion, codified by later astronomers, based on his works Astronomia nova, Harmonices Mundi, and Epitome of Copernican...

, Galileo Galilei
Galileo Galilei
Galileo Galilei , was an Italian physicist, mathematician, astronomer, and philosopher who played a major role in the Scientific Revolution. His achievements include improvements to the telescope and consequent astronomical observations and support for Copernicanism...

, and Sir Isaac Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

 were all influenced by the Elements, and applied their knowledge of it to their work. Mathematicians and philosophers, such as Bertrand Russell
Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...

, Alfred North Whitehead
Alfred North Whitehead
Alfred North Whitehead, OM FRS was an English mathematician who became a philosopher. He wrote on algebra, logic, foundations of mathematics, philosophy of science, physics, metaphysics, and education...

, and Baruch Spinoza
Baruch Spinoza
Baruch de Spinoza and later Benedict de Spinoza was a Dutch Jewish philosopher. Revealing considerable scientific aptitude, the breadth and importance of Spinoza's work was not fully realized until years after his death...

, have attempted to create their own foundational "Elements" for their respective disciplines, by adopting the axiomatized deductive structures that Euclid's work introduced.

The austere beauty of Euclidean geometry has been seen by many in western culture as a glimpse of an otherworldly system of perfection and certainty. Abraham Lincoln kept a copy of Euclid in his saddlebag, and studied it late at night by lamplight; he related that he said to himself, "You never can make a lawyer if you do not understand what demonstrate means; and I left my situation in Springfield, went home to my father's house, and stayed there till I could give any proposition in the six books of Euclid at sight". Edna St. Vincent Millay
Edna St. Vincent Millay
Edna St. Vincent Millay was an American lyrical poet, playwright and feminist. She received the Pulitzer Prize for Poetry, and was known for her activism and her many love affairs. She used the pseudonym Nancy Boyd for her prose work...

 wrote in her sonnet Euclid Alone Has Looked on Beauty Bare, "O blinding hour, O holy, terrible day, When first the shaft into his vision shone Of light anatomized!". Einstein recalled a copy of the Elements and a magnetic compass as two gifts that had a great influence on him as a boy, referring to the Euclid as the "holy little geometry book".

The success of the Elements is due primarily to its logical presentation of most of the mathematical knowledge available to Euclid. Much of the material is not original to him, although many of the proofs are his. However, Euclid's systematic development of his subject, from a small set of axioms to deep results, and the consistency of his approach throughout the Elements, encouraged its use as a textbook for about 2,000 years. The Elements still influences modern geometry books. Further, its logical axiomatic approach and rigorous proofs remain the cornerstone of mathematics.

Outline of Elements





Contents of the books


Books 1 through 4 deal with plane geometry:
  • Book 1 contains Euclid's 10 axioms (5 named postulates—including the parallel postulate
    Parallel postulate
    In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in Euclidean geometry...

    —and 5 named axioms) and the basic propositions of geometry: the pons asinorum
    Pons asinorum
    Pons asinorum is the name given to Euclid's fifth proposition in Book 1 of his Elements of geometry, also known as the theorem on isosceles triangles. It states that the angles opposite the equal sides of an isosceles triangle are equal...

     (proposition 5), the Pythagorean theorem
    Pythagorean theorem
    In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...

     (Proposition 47), equality of angles and area
    Area
    Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...

    s, parallelism, the sum of the angles in a triangle, and the three cases in which triangles are "equal" (have the same area).
  • Book 2 is commonly called the "book of geometric algebra" because most of the propositions can be seen as geometric interpretations of algebraic identities, such as a(b + c + ...) = ab + ac + ... or (2a + b)2 + b2 = 2(a2 + (a + b)2). It also contains a method of finding the square root
    Square root
    In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x...

     of a given number.
  • Book 3 deals with circles and their properties: inscribe
    Inscribe
    right|thumb|An inscribed triangle of a circleIn geometry, an inscribed planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. To say that "Figure F is inscribed in figure G" means precisely the same thing as "figure G is circumscribed about...

    d angles, tangent
    Tangent
    In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where f...

    s, the power of a point, Thales' theorem
    Thales' theorem
    In geometry, Thales' theorem states that if A, B and C are points on a circle where the line AC is a diameter of the circle, then the angle ABC is a right angle. Thales' theorem is a special case of the inscribed angle theorem...

    .
  • Book 4 constructs the incircle and circumcircle of a triangle, and constructs regular polygon
    Regular polygon
    A regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.-General properties:...

    s with 4, 5, 6, and 15 sides.


Books 5 through 10 introduce ratio
Ratio
In 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 and proportions
Proportionality (mathematics)
In mathematics, two variable quantities are proportional if one of them is always the product of the other and a constant quantity, called the coefficient of proportionality or proportionality constant. In other words, are proportional if the ratio \tfrac yx is constant. We also say that one...

:
  • Book 5 is a treatise on proportions of magnitudes
    Magnitude (mathematics)
    The magnitude of an object in mathematics is its size: a property by which it can be compared as larger or smaller than other objects of the same kind; in technical terms, an ordering of the class of objects to which it belongs....

    . Proposition 25 has as a special case the inequality of arithmetic and geometric means
    Inequality of arithmetic and geometric means
    In mathematics, the inequality of arithmetic and geometric means, or more briefly the AM–GM inequality, states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list; and further, that the two means are equal if and only if...

    .
  • Book 6 applies proportions to geometry: Similar figures.
  • Book 7 deals strictly with elementary number theory: divisibility, prime number
    Prime number
    A 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 algorithm for finding the greatest common divisor
    Greatest common divisor
    In 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...

    , least common multiple
    Least common multiple
    In arithmetic and number theory, the least common multiple of two integers a and b, usually denoted by LCM, is the smallest positive integer that is a multiple of both a and b...

    . Propositions 30 and 32 together are essentially equivalent to the fundamental theorem of arithmetic
    Fundamental theorem of arithmetic
    In number theory, the fundamental theorem of arithmetic states that any integer greater than 1 can be written as a unique product of prime numbers...

     stating that every positive integer can be written as a product of primes in an essentially unique way, though Euclid would have had trouble stating it in this modern form as he did not use the product of more than 3 numbers.
  • Book 8 deals with proportions in number theory and geometric sequences
    Geometric progression
    In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio. For example, the sequence 2, 6, 18, 54, ... is a geometric progression...

    .
  • Book 9 applies the results of the preceding two books and gives the infinitude of prime numbers (proposition 20), the sum of a geometric series (proposition 35), and the construction of even perfect number
    Perfect number
    In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself . Equivalently, a perfect number is a number that is half the sum of all of its positive divisors i.e...

    s (proposition 36).
  • Book 10 attempts to classify incommensurable
    Commensurability (mathematics)
    In mathematics, two non-zero real numbers a and b are said to be commensurable if a/b is a rational number.-History of the concept:...

     (in modern language, irrational
    Irrational number
    In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....

    ) magnitudes by using the method of exhaustion
    Method of exhaustion
    The method of exhaustion is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in area between the n-th polygon and the containing shape will...

    , a precursor to integration
    Integral
    Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

    .


Books 11 through to 13 deal with spatial geometry:
  • Book 11 generalizes the results of Books 1–6 to space: perpendicularity, parallelism, volumes of parallelepiped
    Parallelepiped
    In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms. By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclidean geometry, its definition encompasses all four concepts...

    s.
  • Book 12 studies volumes of cones
    Cone (geometry)
    A cone is an n-dimensional geometric shape that tapers smoothly from a base to a point called the apex or vertex. Formally, it is the solid figure formed by the locus of all straight line segments that join the apex to the base...

    , pyramids
    Pyramid (geometry)
    In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle. It is a conic solid with polygonal base....

    , and cylinders
    Cylinder (geometry)
    A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder...

     in detail, and shows for example that the volume of a cone is a third of the volume of the corresponding cylinder. It concludes by showing the volume of a sphere
    Sphere
    A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

     is proportional to the cube of its radius by approximating it by a union of many pyramids.
  • Book 13 constructs the five regular Platonic solid
    Platonic solid
    In geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex; thus, all its edges are congruent, as are its vertices and...

    s inscribed in a sphere, calculates the ratio of their edges to the radius of the sphere, and proves that there are no further regular solids.

Euclid's method and style of presentation



Euclid's axiomatic approach and constructive methods were widely influential.

As was common in ancient mathematical texts, when a proposition needed proof
Mathematical proof
In 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...

 in several different cases, Euclid often proved only one of them (often the most difficult), leaving the others to the reader. Later editors such as Theon
Theon of Alexandria
Theon 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...

 often interpolated their own proofs of these cases.

Euclid's list of axioms was not exhaustive, but represented the principles that were the most important. His proofs often invoke axiomatic notions which were not originally presented in his list of axioms. Later editors have interpolated Euclid's implicit axiomatic assumptions in the list of formal axioms.

Euclid's presentation was limited by the mathematical ideas and notations in common currency in his era, and this causes the treatment to seem awkward to the modern reader in some places. For example, there was no notion of an angle greater than two right angles, the number 1 was sometimes treated separately from other positive integers, and as multiplication was treated geometrically he did not use the product of more than 3 different numbers. The geometrical treatment of number theory may have been because the alternative would have been the extremely awkward Alexandrian system of numerals
Greek numerals
Greek numerals are a system of representing numbers using letters of the Greek alphabet. They are also known by the names Ionian numerals, Milesian numerals , Alexandrian numerals, or alphabetic numerals...

.

The presentation of each result is given in a stylized form, which originated with Euclid: enunciation, statement, construction, proof, and conclusion. No indication is given of the method of reasoning that led to the result, although the Data
Data (Euclid)
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....

 does provide instruction about how to approach the types of problems encountered in the first four books of the Elements. Some scholars have tried to find fault in Euclid's use of figures in his proofs, accusing him of writing proofs that depended on the specific figures drawn rather than the general underlying logic, especially concerning Proposition II of Book I. However, Euclid's original proof of this proposition is general, valid, and does not depend on the figure used as an example to illustrate one given configuration.

Apocrypha


It was not uncommon in ancient time to attribute to celebrated authors works that were not written by them. It is by these means that the apocryphal books XIV and XV of the Elements were sometimes included in the collection. The spurious Book XIV was probably written by Hypsicles
Hypsicles
This article is about Hypsicles of Alexandria. For the historian, see Hyspicrates .Hypsicles was an ancient Greek mathematician and astronomer known for authoring On Ascensions and the spurious Book XIV of Euclid's Elements.- Life and work :Although little is known about the life of Hypsicles,...

 on the basis of a treatise by Apollonius
Apollonius of Perga
Apollonius 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...

. The book continues Euclid's comparison of regular solids inscribed in spheres, with the chief result being that the ratio of the surfaces of the dodecahedron and icosahedron
Icosahedron
In geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids....

 inscribed in the same sphere is the same as the ratio of their volumes, the ratio being

The spurious Book XV was probably written, at least in part, by Isidore of Miletus
Isidore of Miletus
Isidore of Miletus was one of the two main Byzantine architects that Emperor Justinian I commissioned to design the church of Hagia Sophia in Constantinople from 532-537A.D.-Summary:...

. This book covers topics such as counting the number of edges and solid angles in the regular solids, and finding the measure of dihedral angles of faces that meet at an edge.

Editions


  • 1460s, Regiomontanus
    Regiomontanus
    Johannes Müller von Königsberg , today best known by his Latin toponym Regiomontanus, was a German mathematician, astronomer, astrologer, translator and instrument maker....

     (incomplete)
  • 1533, editio princeps
    Editio princeps
    In classical scholarship, editio princeps is a term of art. It means, roughly, the first printed edition of a work that previously had existed only in manuscripts, which could be circulated only after being copied by hand....

     by Simon Grynäus
  • 1557, by Jean Magnien and Pierre de Montdoré, reviewed by Stephanus Gracilis (only propositions, no full proofs, includes original Greek and the Latin translation)
  • 1572, Commandinus Latin edition
  • 1574, Christoph Clavius

Translations

  • 1505, Bartolomeo Zamberti (Latin)
  • 1543, Niccolò Tartaglia (Italian)
  • 1557, Jean Magnien and Pierre de Montdoré, reviewed by Stephanus Gracilis (Greek to Latin)
  • 1558, Johann Scheubel (German)
  • 1562, Jacob Kündig (German)
  • 1562, Wilhelm Holtzmann (German)
  • 1564-1566, Pierre Forcadel de Béziers (French)
  • 1570, Henry Billingsley
    Henry Billingsley
    Sir Henry Billingsley was Lord Mayor of London and the first translator of Euclid into English.-Early Life:He was a son of William Billingsley, haberdasher and assaymaster of London, and his wife, Elizabeth Harlowe. He entered St...

     (English)
  • 1575, Commandinus (Italian)
  • 1576, Rodrigo de Zamorano (Spanish)
  • 1594, Typografia Medicea (edition of the Arabic translation of Nasir al-Din al-Tusi
    Nasir al-Din al-Tusi
    Khawaja Muḥammad ibn Muḥammad ibn Ḥasan Ṭūsī , better known as Naṣīr al-Dīn al-Ṭūsī , was a Persian polymath and prolific writer: an astronomer, biologist, chemist, mathematician, philosopher, physician, physicist, scientist, theologian and Marja Taqleed...

    )
  • 1604, Jean Errard de Bar-le-Duc (French)
  • 1606, Jan Pieterszoon Dou (Dutch)
  • 1607, Matteo Ricci
    Matteo Ricci
    Matteo Ricci, SJ was an Italian Jesuit priest, and one of the founding figures of the Jesuit China Mission, as it existed in the 17th-18th centuries. His current title is Servant of God....

    , Xu Guangqi
    Xu Guangqi
    Xu Guangqi , was a Chinese scholar-bureaucrat, agricultural scientist, astronomer, and mathematician in the Ming Dynasty. Xu was a colleague and collaborator of the Italian Jesuits Matteo Ricci and Sabatino de Ursis and they translated several classic Western texts into Chinese, including part of...

     (Chinese)
  • 1613, Pietro Cataldi
    Pietro Cataldi
    Pietro Antonio Cataldi was an Italian mathematician. A citizen of Bologna, he taught mathematics and astronomy and also worked on military problems. His work included the development of continued fractions and a method for their representation. He was one of many mathematicians who attempted to...

     (Italian)
  • 1615, Denis Henrion
    Denis Henrion
    Denis Henrion, was a French mathematician born at the end of the 16th century, who died in Paris around 1640. He co-edited the works of Viète.-See also:* Alexander Anderson* Marin Getaldić* Pierre Hérigone...

     (French)
  • 1617, Frans van Schooten (Dutch)
  • 1637, L. Carduchi (Spanish)
  • 1639, Pierre Hérigone
    Pierre Hérigone
    Pierre Hérigone was a French mathematician and astronomer.Of Basque origin, Hérigone taught in Paris for most of his life.-Works:...

     (French)
  • 1651, Heinrich Hoffmann (German)
  • 1651, Thomas Rudd
    Thomas Rudd
    -Life:The eldest son of Thomas Rudd of Higham Ferrars, Northamptonshire, he was born in 1583 or 1584. He served during his earlier years as a military engineer in the Low Countries. On 10 July 1627 Charles I appointed him ‘chief engineer of all castles, forts, and fortifications within Wales,’ at a...

     (English)
  • 1660, Isaac Barrow
    Isaac Barrow
    Isaac Barrow was an English Christian theologian, and mathematician who is generally given credit for his early role in the development of infinitesimal calculus; in particular, for the discovery of the fundamental theorem of calculus. His work centered on the properties of the tangent; Barrow was...

     (English)
  • 1661, John Leeke and Geo. Serle (English)
  • 1663, Domenico Magni (Italian from Latin)
  • 1672, Claude François Milliet Dechales (French)
  • 1680, Vitale Giordano (Italian)
  • 1685, William Halifax (English)
  • 1689, Jacob Knesa (Spanish)
  • 1690, Vincenzo Viviani (Italian)
  • 1694, Ant. Ernst Burkh v. Pirckenstein (German)
  • 1695, C. J. Vooght (Dutch)
  • 1697, Samuel Reyher (German)
  • 1702, Hendrik Coets (Dutch)
  • 1705, Edmund Scarburgh (English)
  • 1708, John Keill (English)
  • 1714, Chr. Schessler (German)
  • 1714, W. Whiston (English)
  • 1720s Jagannatha Samrat
    Jagannatha Samrat
    Pandita Jagannatha Samrat was an Indian astronomer and mathematician in the court of Jai Singh II of Amber. He learned Arabic and Persian in order to study Islamic astronomy...

     (Sanskrit, based on the Arabic translation of Nasir al-Din al-Tusi)
  • 1731, Guido Grandi (abbreviation to Italian)
  • 1738, Ivan Satarov (Russian from French)
  • 1744, Mårten Strömer (Swedish)
  • 1749, Dechales (Italian)
  • 1745, Ernest Gottlieb Ziegenbalg (Danish)
  • 1752, Leonardo Ximenes (Italian)
  • 1756, Robert Simson
    Robert Simson
    Robert Simson was a Scottish mathematician and professor of mathematics at the University of Glasgow. The pedal line of a triangle is sometimes called the "Simson line" after him.-Life:...

     (English)
  • 1763, Pubo Steenstra (Dutch)
  • 1773, 1781, J. F. Lorenz (German)
  • 1780, Baruch Ben-Yaakov Mshkelab (Hebrew)
  • 1781, 1788 James Williamson (English)
  • 1781, William Austin (English)
  • 1789, Pr. Suvoroff nad Yos. Nikitin (Russian from Greek)
  • 1795, John PLayfair (English)
  • 1803, H.C. Linderup (Danish)
  • 1804, F. Peyrard (French)
  • 1807, Józef Czech (Polish based on Greek, Latin and English editions)
  • 1807, J. K. F. Hauff (German)
  • 1817, Jo. Czencha (Polish)
  • 1818, Vincenzo Flauti (Italian)
  • 1820, Benjamin of Lesbos (Modern Greek)
  • 1826, George Phillips (English)
  • 1828, Joh. Josh and Ign. Hoffmann (German)
  • 1828, Dionysius Lardner
    Dionysius Lardner
    Dionysius Lardner , was an Irish scientific writer who popularised science and technology, and edited the 133-volume Cabinet Cyclopedia.-Early life in Dublin:...

     (English)
  • 1833, E. S. Unger (German)
  • 1833, Thomas Perronet Thompson
    Thomas Perronet Thompson
    Thomas Perronet Thompson was a British Parliamentarian, a Governor of Sierra Leone and a radical reformer.Thompson was born in Kingston upon Hull in 1783. He was son of Thomas Thompson, a merchant of Hull and his wife, Philothea Perronet Briggs...

     (English)
  • 1836, H. Falk (Swedish)
  • 1844, 1845, 1859 P. R. Bråkenhjelm (Swedish)
  • 1850, F. A. A. Lundgren (Swedish)
  • 1850, H. A. Witt and M. E. Areskong (Swedish)
  • 1862, Isaac Todhunter
    Isaac Todhunter
    Isaac Todhunter FRS , was an English mathematician who is best known today for the books he wrote on mathematics and its history.- Life and work :...

     (English)
  • 1880, Vachtchenko-Zakhartchenk (Russian)
  • 1901, Max Simon (German)
  • 1908, Thomas Little Heath (English)

Currently in print

  • Euclid's Elements – All thirteen books in one volume, Based on Heath's translation, Green Lion Press ISBN 1-888009-18-7.
  • The Elements: Books I-XIII-Complete and Unabridged, (2006) Translated by Sir Thomas Heath, Barnes & Noble ISBN 0-7607-6312-7.
  • The Thirteen Books of Euclid's Elements, translation and commentaries by Heath, Thomas L. (1956) in three volumes. Dover Publications. ISBN 0-486-60088-2 (vol. 1), ISBN 0-486-60089-0 (vol. 2), ISBN 0-486-60090-4 (vol. 3)

External links



In HTML with Java-based interactive figures.
  • Richard Fitzpatrick a bilingual edition (typset in PDF format, with the original Greek and an English translation on facing pages; free in PDF form, available in print) ISBN 978-0615179841
  • Heath's English translation (HTML, without the figures, public domain) (accessed February 4, 2010)
  • Euclid's Elements in ancient Greek (typeset in PDF format, public domain. available in print--free download)
  • Oliver Byrne's 1847 edition – an unusual version by Oliver Byrne (mathematician)
    Oliver Byrne (mathematician)
    Oliver Byrne was a civil engineer and prolific author of works on subjects including mathematics, geometry, and engineering. He is best known for his 'coloured' book of Euclid's Elements. He was a large contributor to Spon's Dictionary of Engineering...

     who used color rather than labels such as ABC (scanned page images, public domain)
  • The First Six Books of the Elements by John Casey and Euclid scanned by Project Gutenberg
    Project Gutenberg
    Project Gutenberg is a volunteer effort to digitize and archive cultural works, to "encourage the creation and distribution of eBooks". Founded in 1971 by Michael S. Hart, it is the oldest digital library. Most of the items in its collection are the full texts of public domain books...

    .
  • Reading Euclid – a course in how to read Euclid in the original Greek, with English translations and commentaries (HTML with figures)
  • Sir Thomas More
    Thomas More
    Sir Thomas More , also known by Catholics as Saint Thomas More, was an English lawyer, social philosopher, author, statesman and noted Renaissance humanist. He was an important councillor to Henry VIII of England and, for three years toward the end of his life, Lord Chancellor...

    's manuscript
  • Latin translation by Aethelhard of Bath
  • Euclid Elements – The original Greek text Greek HTML
  • Clay Mathematics Institute
    Clay Mathematics Institute
    The Clay Mathematics Institute is a private, non-profit foundation, based in Cambridge, Massachusetts. The Institute is dedicated to increasing and disseminating mathematical knowledge. It gives out various awards and sponsorships to promising mathematicians. The institute was founded in 1998...

    Historical Archive – The thirteen books of Euclid's Elements copied by Stephen the Clerk for Arethas of Patras, in Constantinople in 888 AD
  • Kitāb Taḥrīr uṣūl li-Ūqlīdis Arabic translation of the thirteen books of Euclid's Elements by Nasīr al-Dīn al-Ṭūsī. Published by Medici Oriental Press(also, Typographia Medicea). Facsimile hosted by Islamic Heritage Project.