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Euclid's Elements

The Elements (Greek: Στοιχεῖα Stoikheîa) is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid c. 300 BC. It is a collection of definitions, postulates, propositions (theorems and constructions), and mathematical proofs of the propositions. The books cover plane and solid Euclidean geometry, elementary number theory, and incommensurable lines. Elements is the oldest extant large-scale deductive treatment of mathematics. It has proven instrumental in the development of logic and modern science, and its logical rigor was not surpassed until the 19th century .

Author

Mathematics

c. 300 BC

13 books

Euclid's Elements has been referred to as the most successful[a][b] and influential[c] textbook ever written. It was one of the very earliest mathematical works to be printed after the invention of the printing press and has been estimated to be second only to the Bible in the number of editions published since the first printing in 1482,[1] the number reaching well over one thousand.[d] For centuries, when the quadrivium 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.

Book 1 contains 5 postulates (including the infamous ) and 5 common notions, and covers important topics of plane geometry such as the Pythagorean theorem, equality of angles and areas, parallelism, the sum of the angles in a triangle, and the construction of various geometric figures.

parallel postulate

Book 2 contains a number of concerning the equality of rectangles and squares, sometimes referred to as "geometric algebra", and concludes with a construction of the golden ratio and a way of constructing a square equal in area to any rectilineal plane figure.

lemmas

Book 3 deals with circles and their properties: finding the center, angles, tangents, the power of a point, Thales' theorem.

inscribed

Book 4 constructs the and circumcircle of a triangle, as well as regular polygons with 4, 5, 6, and 15 sides.

incircle

Book 5, on proportions of , 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).

magnitudes

Book 6 applies proportions to plane geometry, especially the construction and recognition of figures.

similar

Book 7 deals with elementary number theory: , prime numbers and their relation to composite numbers, Euclid's algorithm for finding the greatest common divisor, finding the least common multiple.

divisibility

Book 8 deals with the construction and existence of of integers.

geometric sequences

Book 9 applies the results of the preceding two books and gives the and the construction of all even perfect numbers.

infinitude of prime numbers

Book 10 proves the irrationality of the square roots of non-square integers (e.g. ) and classifies the square roots of lines into thirteen disjoint categories. Euclid here introduces the term "irrational", which has a different meaning than the modern concept of irrational numbers. He also gives a formula to produce Pythagorean triples.[25]

incommensurable

Book 11 generalizes the results of book 6 to solid figures: perpendicularity, parallelism, volumes and similarity of .

parallelepipeds

Book 12 studies the volumes of , pyramids, and cylinders in detail by using the method of exhaustion, a precursor to integration, 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 is proportional to the cube of its radius (in modern language) by approximating its volume by a union of many pyramids.

cones

Book 13 constructs the five regular inscribed in a sphere and compares the ratios of their edges to the radius of the sphere.

Platonic solids

Criticism[edit]

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.[32]


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.[33] 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; 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.[34]


Mathematician and historian W. W. Rouse Ball put the criticisms in perspective, remarking that "the fact that for two thousand years [the Elements] was the usual text-book on the subject raises a strong presumption that it is not unsuitable for that purpose."[28]

Apocrypha[edit]

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.[35] The spurious Book XIV was probably written by Hypsicles on the basis of a treatise by 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


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.[f]

4th century, , 888 AD manuscript extant.

Theon of Alexandria

9th century, Pre-Theon Vat. gr. 190

Peyrard

Many medieval editions, pre 1482

1460s, (incomplete)

Regiomontanus

1482, (Venice), editio princeps (in Latin)[36][37]

Erhard Ratdolt

1533, of the Greek text by Simon Grynäus[38]

editio princeps

1557, by Jean Magnien and , reviewed by Stephanus Gracilis (only propositions, no full proofs, includes original Greek and the Latin translation)

Pierre de Montdoré

1572, Latin edition

Commandinus

1574,

Christoph Clavius

1883–1888,

Johan Ludvig Heiberg

Bride's Chair

by ratherthanpaer

Elements with highlights

Clark University Euclid's elements

Multilingual edition of Elementa in the Bibliotheca Polyglotta

Euclid (1997) [c. 300 BC]. David E. Joyce (ed.). . Retrieved 2006-08-30. In HTML with Java-based interactive figures.

"Elements"

(freely downloadable PDF, typeset in a two-column format with the original Greek beside a modern English translation; also available in print as ISBN 979-8589564587)

Richard Fitzpatrick's bilingual edition

Heath's English translation

vol. 1

(also hosted at archive.org)– an unusual version by Oliver Byrne who used color rather than labels such as ABC (scanned page images, public domain)

Oliver Byrne's 1847 edition

designed by Nicholas Rougeux

Web adapted version of Byrne’s Euclid

animated and explained by Sandy Bultena, contains books I-VII.

Video adaptation

by John Casey and Euclid scanned by Project Gutenberg.

The First Six Books of the Elements

– a course in how to read Euclid in the original Greek, with English translations and commentaries (HTML with figures)

Reading Euclid

's manuscript

Sir Thomas More

by Aethelhard of Bath

Latin translation

Greek HTML

Euclid Elements – The original Greek text

Historical Archive – The thirteen books of Euclid's Elements copied by Stephen the Clerk for Arethas of Patras, in Constantinople in 888 AD

Clay Mathematics Institute

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.

Kitāb Taḥrīr uṣūl li-Ūqlīdis

an open textbook based on the Elements

Euclid's Elements Redux

reprinted as part of the Complete Library of the Four Treasuries, or Siku Quanshu.

1607 Chinese translations