TheInfoList

The ''Elements'' ( grc, Στοιχεῖα ''Stoikheîa'') is a
mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
treatise A treatise is a formal Formal, formality, informal or informality imply the complying with, or not complying with, some set theory, set of requirements (substantial form, forms, in Ancient Greek). They may refer to: Dress code and events * For ...
consisting of 13 books attributed to the ancient Greek mathematician
Euclid Euclid (; grc-gre, Εὐκλείδης Euclid (; grc, Εὐκλείδης – ''Eukleídēs'', ; fl. 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referre ...

in
Alexandria Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic language, Coptic: Rakodī; el, Αλεξάνδρεια ''Alexandria'') is the List of cities and towns in Egypt, third-largest city in Egypt after Cairo and Giza, ...

,
Ptolemaic Egypt The Ptolemaic Kingdom (; grc-koi, Πτολεμαϊκὴ βασιλεία, Ptolemaïkḕ basileía) was an Ancient Greek Ancient Greek includes the forms of the Greek language Greek ( el, label=Modern Greek Modern Greek (, , o ...
300 BC. It is a collection of definitions, postulates,
propositions In logic and linguistics, a proposition is the meaning of a declarative sentence (linguistics), sentence. In philosophy, "Meaning (philosophy), meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same mea ...
(
theorem In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
s and
constructions Construction is the process of producing buildings and other infrastructure. Construction also may refer to: * Additional physical/mechanical senses: ** Offshore construction, the installation of structures in marine environments * Primarily abstr ...
), and
mathematical proof A mathematical proof is an inferential argument In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek ...
s of the propositions. The books cover plane and solid
Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandria Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic Coptic may refer to: Afro-Asia * Copts, an ethnoreligious group mainly in the area of modern ...
, elementary
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

, and incommensurable lines. ''Elements'' is the oldest extant large-scale deductive treatment of
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
. It has proven instrumental in the development of
logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents statements and ar ...

and modern
science Science () is a systematic enterprise that builds and organizes knowledge Knowledge is a familiarity or awareness, of someone or something, such as facts A fact is something that is truth, true. The usual test for a statement of ...

, and its logical rigor was not surpassed until the 19th century. Euclid's ''Elements'' has been referred to as the most successful and influential
textbook A textbook is a book A book is a medium for recording information Information is processed, organised and structured data. It provides context for data and enables decision making process. For example, a single customer’s sale at ...
ever written. It was one of the very earliest mathematical works to be printed after the
invention of the printing press Movable Type is a blog software, weblog publishing system developed by the company Six Apart. It was publicly announced on September 3, 2001; version 1.0 was publicly released on October 8, 2001. The current version is 7.0. Movable Type is proprie ...
and has been estimated to be second only to the
Bible The Bible (from Koine Greek Koine Greek (, , Greek approximately ;. , , , lit. "Common Greek"), also known as Alexandrian dialect, common Attic, Hellenistic or Biblical Greek, was the koiné language, common supra-regional form of Gree ...

in the number of editions published since the first printing in 1482, the number reaching well over one thousand. For centuries, when the
quadrivium In liberal arts education, the ''quadrivium'' (plural: quadrivia) consists of the four subjects or arts (arithmetic, geometry, music, and astronomy) taught after the trivium (education), ''trivium''. The word is Latin, meaning 'four ways', and its ...
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 other school textbooks, did it cease to be considered something all educated people had read.
Geometry emerged as an indispensable part of the standard education of the English gentleman in the eighteenth century; by the
Victorian period In the history of the United Kingdom, the Victorian era was the period of Queen Victoria Victoria (Alexandrina Victoria; 24 May 1819 – 22 January 1901) was Queen of the United Kingdom of Great Britain and Ireland from 20 June 18 ...
it was also becoming an important part of the education of artisans, children at Board Schools, colonial subjects and, to a rather lesser degree, women. The standard textbook for this purpose was none other than Euclid's ''The Elements''.

# History

## Basis in earlier work

Scholars believe that the ''Elements'' is largely a compilation of propositions based on books by earlier Greek mathematicians.
Proclus Proclus Lycius (; 410/411/ 7 Feb. or 8 Feb. 412 –17 April 485 AD), called Proclus the Successor, Proclus the Platonic Successor, or Proclus of Athens (Greek: Προκλου Διαδοχου ''Próklos Diádochos'', ''"''in some Manuscript ...
(412–485 AD), a Greek mathematician who lived around seven centuries after Euclid, wrote in his commentary on the ''Elements'': "Euclid, who put together the ''Elements'', collecting many of Eudoxus' theorems, perfecting many of Theaetetus', and also bringing to irrefragable demonstration the things which were only somewhat loosely proved by his predecessors".
Pythagoras Pythagoras of Samos, or simply ; in Ionian Greek () was an ancient Ionians, Ionian Ancient Greek philosophy, Greek philosopher and the eponymous founder of Pythagoreanism. His political and religious teachings were well known in Magna Graec ...

( 570–495 BC) was probably the source for most of books I and II,
Hippocrates of Chios Hippocrates of Kos (; grc-gre, Ἱπποκράτης ὁ Κῷος, Hippokrátēs ho Kôios; ), also known as Hippocrates II, was a Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), ...
( 470–410 BC, not the better known
Hippocrates of Kos Hippocrates of Kos (; grc-gre, Ἱπποκράτης ὁ Κῷος, Hippokrátēs ho Kṓos; ), also known as Hippocrates (physicians), Hippocrates II, was a Greeks, Greek physician of the Age of Pericles (Classical Greece), who is considered on ...

) for book III, and
Eudoxus of Cnidus Eudoxus of Cnidus (; grc, Εὔδοξος ὁ Κνίδιος, ''Eúdoxos ho Knídios''; ) was an ancient Greek Ancient Greek includes the forms of the Greek language used in ancient Greece and the classical antiquity, ancient world from ...
( 408–355 BC) for book V, while books IV, VI, XI, and XII probably came from other Pythagorean or Athenian mathematicians. 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

Theon of Alexandria Theon of Alexandria (; grc, Θέων ὁ Ἀλεξανδρεύς;  335 – c. 405) was a Greeks, Greek scholar and mathematician who lived in Alexandria, Egypt. He edited and arranged Euclid's ''Euclid's Elements, Elements'' and wrot ...
produced an edition of Euclid which was so widely used that it became the only surviving source until
François Peyrard François Peyrard (1760–1822) was a French mathematician, educator and librarian. During the French Revolution The French Revolution ( ) refers to the period that began with the Estates General of 1789 and ended in coup of 18 Brumaire, ...
's 1808 discovery at the
Vatican Vatican City Vatican City (), officially the Vatican City State ( it, Stato della Città del Vaticano; la, Status Civitatis Vaticanae),—' * german: Vatikanstadt, cf. '—' (in Austria: ') * pl, Miasto Watykańskie, cf. '—' * pt, Ci ...
of a manuscript not derived from Theon's. This manuscript, the Heiberg manuscript, is from a Byzantine workshop around 900 and is the basis of modern editions.
Papyrus Oxyrhynchus 29 Papyrus Oxyrhynchus 29 (P. Oxy. 29) is a fragment of the second book of the ''Euclid's Elements, Elements'' of Euclid in Ancient Greek, Greek. It was discovered by Bernard Grenfell, Grenfell and Arthur Surridge Hunt, Hunt in 1897 in Oxyrhynchus. T ...
is a tiny fragment of an even older manuscript, but only contains the statement of one proposition. Although Euclid was known to
Cicero Marcus Tullius Cicero ( ; ; 3 January 106 BC – 7 December 43 BC) was a Ancient Rome, Roman statesman, lawyer, scholar, philosopher and Academic skepticism, Academic Skeptic, who tried to uphold optimate principles during crisis of ...

, for instance, no record exists of the text having been translated into Latin prior to
Boethius Anicius Manlius Severinus Boëthius, commonly called Boethius (; also Boetius ; 477 – 524 AD), was a Roman Roman Senate, senator, Roman consul, consul, ''magister officiorum'', and philosopher of the early 6th century. He was born about a ye ...

in the fifth or sixth century. The Arabs received the ''Elements'' from the Byzantines around 760; this version was translated into
Arabic Arabic (, ' or , ' or ) is a Semitic language The Semitic languages are a branch of the Afroasiatic language family originating in the Middle East The Middle East is a list of transcontinental countries, transcontinental region ...

under
Harun al Rashid Harun al-Rashid (; ar, هَارُون الرَشِيد ''Hārūn Ar-Rašīd'', "Aaron the Just" or "Aaron the Rightly-Guided"; 17 March 763 or February 766 – 24 March 809 CE / 148–193 AH) was the fifth Abbasid The Abbasid Caliph ...
( 800). The Byzantine scholar Arethas 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 about 1120, when the English monk
Adelard of Bath Adelard of Bath ( la, Adelardus Bathensis; 1080? 1142-1152?) was a 12th-century English natural philosopher. He is known both for his original works and for translating many important Arabic Arabic (, ' or , ' or ) is a Semitic languag ...
translated it into Latin from an Arabic translation. The first printed edition appeared in 1482 (based on
Campanus of Novara Campanus of Novara ( 1220 – 1296) was an Italian mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theo ...

'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 (13 July 1527 – 1608 or 1609) was an English mathematician, astronomer An astronomer is a in the field of who focuses their studies on a specific question or field outside the scope of . They observe s such as s, s, , s and ...
provided a widely respected "Mathematical Preface", along with copious notes and supplementary material, to the first English edition by
Henry Billingsley Sir Henry Billingsley (died 22 November 1606) was an English merchant, Lord Mayor of London The Lord Mayor of London is the Mayors in England, mayor of the City of London and the Leader of the council, leader of the City of London Corporation. ...
. Copies of the Greek text still exist, some of which can be found in the
Vatican Library The Vatican Apostolic Library ( la, Bibliotheca Apostolica Vaticana, it, Biblioteca Apostolica Vaticana), more commonly known as the Vatican Library or informally as the Vat, is the library A library is a collection of materials, books o ...
and the
Bodleian Library The Bodleian Library () is the main research library A research library is a library A library is a collection of materials, books or media that are easily accessible for use and not just for display purposes. It is responsible for housi ...

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 and Sir
Thomas Little Heath Sir Thomas Little Heath (; 5 October 1861 – 16 March 1940) was a British civil servant, mathematician, classical Classical may refer to: European antiquity *Classical antiquity, a period of history from roughly the 7th or 8th century B.C.E. ...

in their editions of the text. Also of importance are the
scholia Scholia (singular scholium or scholion, from grc, σχόλιον, "comment, interpretation") are grammatical In linguistics Linguistics is the science, scientific study of language. It encompasses the analysis of every aspect of langu ...
, 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 further study.

# Influence

The ''Elements'' is still considered a masterpiece in the application of
logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents statements and ar ...

to
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
. In historical context, it has proven enormously influential in many areas of
science Science () is a systematic enterprise that builds and organizes knowledge Knowledge is a familiarity or awareness, of someone or something, such as facts A fact is something that is truth, true. The usual test for a statement of ...

. Scientists
Nicolaus Copernicus Nicolaus Copernicus (; pl, Mikołaj Kopernik; gml, link=no, Niclas Koppernigk, modern: ''Nikolaus Kopernikus''; 19 February 1473 – 24 May 1543) was a Renaissance polymath, active as a mathematician, astronomer, and Catholic Church, C ...

,
Johannes Kepler Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer An astronomer is a in the field of who focuses their studies on a specific question or field outside the scope of . They observe s such as s, s, , s and ...

,
Galileo Galilei Galileo di Vincenzo Bonaiuti de' Galilei ( , ; 15 February 1564 – 8 January 1642), commonly referred to as Galileo, was an astronomer An astronomer is a scientist in the field of astronomy who focuses their studies on a specific q ...

,
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest physicists of all time. Einstein is known for developing the theory of relativity The theo ...

and Sir
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics a ...

were all influenced by the ''Elements'', and applied their knowledge of it to their work. Mathematicians and philosophers, such as
Thomas Hobbes Thomas Hobbes ( ; sometimes known as Thomas Hobbes of Malmesbury; 5 April 1588 – 4 December 1679) was an English English usually refers to: * English language English is a West Germanic languages, West Germanic language first sp ...
,
Baruch Spinoza Baruch (de) Spinoza (; ; ; born Baruch Espinosa; later as an author and a correspondent Benedictus de Spinoza, anglicized to Benedict de Spinoza; 24 November 1632 – 21 February 1677) was a Dutch philosopher of Spanish and Portuguese Jews, Por ...

,
Alfred North Whitehead Alfred North Whitehead (15 February 1861 – 30 December 1947) was an English mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics ...
, and
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell (18 May 1872 – 2 February 1970) was a British polymath A polymath ( el, πολυμαθής, , "having learned much"; la, homo universalis, "universal human") is an individual whose know ...
, 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 Abraham Lincoln (; February 12, 1809 – April 15, 1865) was an American statesman and lawyer who served as the 16th president of the United States The president of the United States (POTUS) is the head of state and head of governme ...

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 Springfield may refer to: * Springfield (toponym) Springfield is a famously common place-name in the English-speaking world, especially in the United States. According to the USGS, U.S. Geological Survey there are currently 34 populated places na ...
, 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 (February 22, 1892 – October 19, 1950) was an American lyric poetry, lyrical poet and playwright. Encouraged to read the classics at home, she was too rebellious to make a success of formal education, but she won poet ...

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!". Albert 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. Furthermore, its logical, axiomatic approach and rigorous proofs remain the cornerstone of mathematics.

## In modern mathematics

One of the most notable influences of Euclid on modern mathematics is the discussion of the
parallel postulate In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...

. In Book I, Euclid lists five postulates, the fifth of which stipulates
''If a
line segment In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

intersects two straight
lines Long interspersed nuclear elements (LINEs) (also known as long interspersed nucleotide elements or long interspersed elements) are a group of non-LTR (long terminal repeat A long terminal repeat (LTR) is a pair of identical sequences of DNA ...
forming two interior angles on the same side that sum to less than two
right angle In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

s, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.''
This postulate plagued mathematicians for centuries due to its apparent complexity compared with the other four postulates. Many attempts were made to prove the fifth postulate based on the other four, but they never succeeded. Eventually in 1829, mathematician
Nikolai Lobachevsky Nikolai Ivanovich Lobachevsky ( rus, Никола́й Ива́нович Лобаче́вский, p=nʲikɐˈlaj ɪˈvanəvʲɪtɕ ləbɐˈtɕɛfskʲɪj, a=Ru-Nikolai_Ivanovich_Lobachevsky.ogg; – ) was a Russian mathematician and geometer, kno ...
published a description of acute geometry (or
hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For an ...
), a geometry which assumed a different form of the parallel postulate. It is in fact possible to create a valid geometry without the fifth postulate entirely, or with different versions of the fifth postulate (
elliptic geometry Elliptic geometry is an example of a geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is conc ...
). If one takes the fifth postulate as a given, the result is
Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandria Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic Coptic may refer to: Afro-Asia * Copts, an ethnoreligious group mainly in the area of modern ...
.

# Contents

* Book 1 contains 5 postulates (including the famous
parallel postulate In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...

) and 5 common notions, and covers important topics of plane geometry such as the
Pythagorean theorem In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

, equality of angles and
area Area is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value in ...

s, parallelism, the sum of the angles in a triangle, and the construction of various geometric figures. * Book 2 contains a number of
lemmas Lemma may refer to: Language and linguistics * Lemma (morphology), the canonical, dictionary or citation form of a word * Lemma (psycholinguistics), a mental abstraction of a word about to be uttered * Headword, under which a set of related dict ...
concerning the equality of rectangles and squares, sometimes referred to as "
geometric algebra In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
", and concludes with a construction of the
golden ratio In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

and a way of constructing a square equal in area to any rectilineal plane figure. * Book 3 deals with circles and their properties: finding the center,
inscribe{{unreferenced, date=August 2012 frame, Inscribed circles of various polygons An inscribed triangle of a circle In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurem ...
d angles,
tangent In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

s, the power of a point,
Thales' theorem In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

. * Book 4 constructs the
incircle In geometry, the incircle or inscribed circle of a triangle is the largest circle contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. An excircle ...

and
circumcircle In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...

of a triangle, as well as
regular polygon In , a regular polygon is a that is (all angles are equal in measure) and (all sides have the same length). Regular polygons may be either or . In the , a sequence of regular polygons with an increasing number of sides approximates a , if the ...
s with 4, 5, 6, and 15 sides. * Book 5, on proportions of magnitudes, gives the highly sophisticated theory of proportion probably developed by Eudoxus, and proves properties such as "alternation" (if ''a'' : ''b'' :: ''c'' : ''d'', then ''a'' : ''c'' :: ''b'' : ''d''). * Book 6 applies proportions to plane geometry, especially the construction and recognition of similar figures. * Book 7 deals with elementary number theory:
divisibility In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
,
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), 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 ...
s and their relation to
composite number A composite number is a positive integer In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calcul ...
s,
Euclid's algorithm In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
for finding the
greatest common divisor In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is ...

, finding the
least common multiple In arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', ...

. * Book 8 deals with the construction and existence of
geometric sequences In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero 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 exa ...

of integers. * Book 9 applies the results of the preceding two books and gives the infinitude of prime numbers and the construction of all even
perfect number In number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number ...
s. * Book 10 proves the irrationality of the square roots of non-square integers (e.g. $\sqrt$) and classifies the square roots of incommensurable lines into thirteen disjoint categories. Euclid here introduces the term "irrational", which has a different meaning than the modern concept of
irrational number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
s. He also gives a
formula In science Science () is a systematic enterprise that Scientific method, builds and organizes knowledge in the form of Testability, testable explanations and predictions about the universe."... modern science is a discovery as well a ...
to produce
Pythagorean triples A Pythagorean triple consists of three positive integer An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around ...
. * Book 11 generalizes the results of book 6 to solid figures: perpendicularity, parallelism, volumes and similarity of
parallelepiped In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of f ...

s. * Book 12 studies the volumes of
cones A cone is a three-dimensional space, three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the Apex (geometry), apex or vertex (geometry), vertex. A cone is fo ...

,
pyramids A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilater ...

, and
cylinders A cylinder (from ) has traditionally been a Solid geometry, three-dimensional solid, one of the most basic of curvilinear geometric shapes. Geometrically, it can be considered as a Prism (geometry), prism with a circle as its base. This traditi ...

in detail by using the
method of exhaustion The method of exhaustion (; ) is a method of finding the area Area is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of ...

, a precursor to , and shows, for example, that the volume of a cone is a third of the volume of the corresponding cylinder. It concludes by showing that the volume of a
sphere A sphere (from Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appr ...

is proportional to the cube of its radius (in modern language) by approximating its volume by a union of many pyramids. * Book 13 constructs the five regular
Platonic solid In three-dimensional space, a Platonic solid is a Regular polyhedron, regular, Convex set, convex polyhedron. It is constructed by Congruence (geometry), congruent (identical in shape and size), regular polygon, regular (all angles equal and all sid ...
s inscribed in a sphere and compares the ratios of their edges to the radius of the sphere.

# Euclid's method and style of presentation

Euclid's axiomatic approach and constructive methods were widely influential. Many of Euclid's propositions were constructive, demonstrating the existence of some figure by detailing the steps he used to construct the object using a
compass and straightedge Straightedge and compass construction, also known as ruler-and-compass construction or classical construction, is the construction of lengths, angle In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexan ...
. His constructive approach appears even in his geometry's postulates, as the first and third postulates stating the existence of a line and circle are constructive. Instead of stating that lines and circles exist per his prior definitions, he states that it is possible to 'construct' a line and circle. It also appears that, for him to use a figure in one of his proofs, he needs to construct it in an earlier proposition. For example, he proves the Pythagorean theorem by first inscribing a square on the sides of a right triangle, but only after constructing a square on a given line one proposition earlier. As was common in ancient mathematical texts, when a proposition needed
proof Proof may refer to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Formal sciences * Formal proof, a construct in proof theory * Mathematical proof, a co ...
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 often interpolated their own proofs of these cases. 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. The presentation of each result is given in a stylized form, which, although not invented by Euclid, is recognized as typically classical. It has six different parts: First is the 'enunciation', which states the result in general terms (i.e., the statement of the proposition). Then comes the 'setting-out', which gives the figure and denotes particular geometrical objects by letters. Next comes the 'definition' or 'specification', which restates the enunciation in terms of the particular figure. Then the 'construction' or 'machinery' follows. Here, the original figure is extended to forward the proof. Then, the 'proof' itself follows. Finally, the 'conclusion' connects the proof to the enunciation by stating the specific conclusions drawn in the proof, in the general terms of the enunciation. No indication is given of the method of reasoning that led to the result, although the ''
Data Data (; ) are individual facts A fact is something that is truth, true. The usual test for a statement of fact is verifiability—that is whether it can be demonstrated to correspond to experience. Standard reference works are often used ...
'' 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.

# Criticism

Euclid's list of axioms in the ''Elements'' 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. For example, in the first construction of Book 1, Euclid used a premise that was neither postulated nor proved: that two circles with centers at the distance of their radius will intersect in two points. Later, in the fourth construction, he used superposition (moving the triangles on top of each other) to prove that if two sides and their angles are equal, then they are
congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In modu ...
; during these considerations he uses some properties of superposition, but these properties are not described explicitly in the treatise. If superposition is to be considered a valid method of geometric proof, all of geometry would be full of such proofs. For example, propositions I.2 and I.3 can be proved trivially by using superposition. Mathematician and historian
W. W. Rouse Ball Walter William Rouse Ball (14 August 1850 – 4 April 1925), known as W. W. Rouse Ball, was a British mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) ...
put the criticisms in perspective, remarking that "the fact that for two thousand years he ''Elements''was the usual text-book on the subject raises a strong presumption that it is not unsuitable for that purpose."

# Apocrypha

It was not uncommon in ancient times 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 on the basis of a treatise by Apollonius of Perga, Apollonius. 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 inscribed in the same sphere is the same as the ratio of their volumes, the ratio being $\sqrt = \sqrt.$ The spurious Book XV was probably written, at least in part, by Isidore of Miletus. 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 (incomplete) *1482, Erhard Ratdolt (Venice), first printed edition *1533, ''editio princeps'' by Simon Grynäus *1557, by Jean Magnien and , reviewed by Stephanus Gracilis (only propositions, no full proofs, includes original Greek and the Latin translation) *1572, Federico Commandino, Commandinus Latin edition *1574, Christoph Clavius

## Translations

*1505, (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, de Béziers (French) *1570,
Henry Billingsley Sir Henry Billingsley (died 22 November 1606) was an English merchant, Lord Mayor of London The Lord Mayor of London is the Mayors in England, mayor of the City of London and the Leader of the council, leader of the City of London Corporation. ...
(English) *1572, Federico Commandino, Commandinus (Latin) *1575, Commandinus (Italian) *1576, Rodrigo de Zamorano (Spanish) *1594, Typographia Medicea (edition of the Arabic translation of ''The Recension of Euclid's "Elements"'' *1604, de Bar-le-Duc (French) *1606, Jan Pieterszoon Dou (Dutch) *1607, Matteo Ricci, Xu Guangqi (Chinese) *1613, Pietro Cataldi (Italian) *1615, Denis Henrion (French) *1617, Frans van Schooten (Dutch) *1637, L. Carduchi (Spanish) *1639, Pierre Hérigone (French) *1651, Heinrich Hoffmann (German) *1651, Thomas Rudd (English) *1660, Isaac Barrow (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, Claes Jansz Vooght (Dutch) *1697, Samuel Reyher (German) *1702, Hendrik Coets (Dutch) *1705, Charles Scarborough (English) *1708, John Keill (English) *1714, Chr. Schessler (German) *1714, W. Whiston (English) *1720s, Jagannatha Samrat (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 (English) *1763, Pibo Steenstra (Dutch) *1768, Angelo Brunelli (Portuguese) *1773, 1781, J. F. Lorenz (German) *1780, Baruch Schick of Shklov (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,
François Peyrard François Peyrard (1760–1822) was a French mathematician, educator and librarian. During the French Revolution The French Revolution ( ) refers to the period that began with the Estates General of 1789 and ended in coup of 18 Brumaire, ...
(French). Peyrard discovered in 1808 the ''Vaticanus Graecus 190'', which enables him to provide a first definitive version in 1814–1818 *1807, Józef Czech (Polish based on Greek, Latin and English editions) *1807, J. K. F. Hauff (German) *1818, Vincenzo Flauti (Italian) *1820, Benjamin of Lesbos (Modern Greek) *1826, George Phillips (English) *1828, Joh. Josh and Ign. Hoffmann (German) *1828, Dionysius Lardner (English) *1833, E. S. Unger (German) *1833, Thomas Perronet Thompson (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 (English) *1865, Sámuel Brassai (Hungarian) *1873, Masakuni Yamada (Japanese) *1880, Mykhailo Vaschenko-Zakharchenko, Vachtchenko-Zakhartchenko (Russian) *1897, Thyra Eibe (Danish) *1901, Max Simon (German) *1907, František Servít (Czech) *1908,
Thomas Little Heath Sir Thomas Little Heath (; 5 October 1861 – 16 March 1940) was a British civil servant, mathematician, classical Classical may refer to: European antiquity *Classical antiquity, a period of history from roughly the 7th or 8th century B.C.E. ...

(English) *1939, R. Catesby Taliaferro (English) *1999, Maja Hudoletnjak Grgić (Book I-VI) (Croatian) *2009, Irineu Bicudo (Brazilian Portuguese) *2019, Ali Sinan Sertöz (Turkish)

## Currently in print

*''Euclid's Elements – All thirteen books complete in one volume'', Based on Heath's translation, Green Lion Press . *''The Elements: Books I–XIII – Complete and Unabridged,'' (2006) Translated by Sir Thomas Heath, Barnes & Noble . *''The Thirteen Books of Euclid's Elements'', translation and commentaries by Heath, Thomas L. (1956) in three volumes. Dover Publications. (vol. 1), (vol. 2), (vol. 3)

## Free versions

*''Euclid's Elements Redux, Volume 1'', contains books I–III, based on John Casey's translation. *''Euclid's Elements Redux, Volume 2'', contains books IV–VIII, based on John Casey's translation.

# References

## Sources

* *Benno Artmann, Artmann, Benno: ''Euclid – The Creation of Mathematics.'' New York, Berlin, Heidelberg: Springer 1999, * * * * * * * * * * Heath's authoritative translation plus extensive historical research and detailed commentary throughout the text. * * * * * * * * * * * *

Multilingual edition of ''Elementa'' in the Bibliotheca Polyglotta
* In HTML with Java-based interactive figures.
Richard Fitzpatrick's bilingual edition
(freely downloadable PDF, typeset in a two-column format with the original Greek beside a modern English translation; also available in print as )
Heath's English translation
(HTML, without the figures, public domain) (accessed February 4, 2010) ** Heath's English translation and commentary, with the figures (Google Books)
vol. 1vol. 2vol. 3vol. 3 c. 2

(also hosted a
archive.org
– an unusual version by Oliver Byrne (mathematician), Oliver Byrne who used color rather than labels such as ABC (scanned page images, public domain)
Web adapted version of Byrne’s Euclid
designed by Nicholas Rougeux
animated and explained by Sandy Bultena, contains books I-VII.
The First Six Books of the ''Elements''
by John Casey and Euclid scanned by Project Gutenberg.