Euclid's Elements

The ''Elements'' ( ) is a mathematical treatise written 300 BC by the Ancient Greek mathematician Euclid.

''Elements'' is the oldest extant large-scale deductive treatment of mathematics. Drawing on the works of earlier mathematicians such as Hippocrates of Chios, Eudoxus of Cnidus and Theaetetus, the ''Elements'' is a collection in 13 books of definitions, postulates, propositions and mathematical proofs that covers plane and solid Euclidean geometry, elementary number theory, and incommensurable lines. These include Pythagorean theorem, Thales' theorem, the Euclidean algorithm for greatest common divisors, Euclid's theorem that there are infinitely many prime numbers, and the construction of regular polygons and polyhedra.

Often referred to as the most successful textbook ever written, the ''Elements'' has continued to be used for introductory geometry from the time it was written up through the present day. It was translated into Arabic and Latin in the medieval period, where it exerted a great deal of influence on mathematics in the medieval Islamic world and in Western Europe, and has proven instrumental in the development of logic and modern science, where its logical rigor was not surpassed until the 19th century.

Background

Proclus (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". Scholars believe that the ''Elements'' is largely a compilation of propositions based on books by earlier Greek mathematicians, including Eudoxus, Hippocrates of Chios, Thales and Theaetetus, while other theorems are mentioned by Plato and Aristotle. It is difficult to differentiate the work of Euclid from that of his predecessors, especially because the ''Elements'' essentially superseded much earlier and now-lost Greek mathematics. The ''Elements'' version available today also includes "post-Euclidean" mathematics, probably added later by later editors such as the mathematician Theon of Alexandria in the 4th century. The classicist Markus Asper concludes that "apparently Euclid's achievement consists of assembling accepted mathematical knowledge into a cogent order and adding new proofs to fill in the gaps" and the historian Serafina Cuomo described it as a "reservoir of results". Despite this, Sialaros furthers that "the remarkably tight structure of the ''Elements'' reveals authorial control beyond the limits of a mere editor".

Pythagoras ( 570–495 BC) was probably the source for most of books I and II, Hippocrates of Chios ( 470–410 BC, not the better known Hippocrates of Kos) for book III, and Eudoxus of Cnidus ( 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.

Other similar works are also reported to have been written by Theudius of Magnesia, Leon, and Hermotimus of Colophon.

Contents

The ''Elements'' does not exclusively discuss geometry as is sometimes believed. It is traditionally divided into three topics: plane geometry (books I–VI), basic number theory (books VII–X) and solid geometry (books XI–XIII)—though book V (on proportions) and X (on irrational lines) do not exactly fit this scheme. The heart of the text is the theorems scattered throughout. Using Aristotle's terminology, these may be generally separated into two categories: "first principles" and "second principles". The first group includes statements labeled as a "definition" ( or ), "postulate" (), or a "common notion" (); only the first book includes postulates—later known as axioms—and common notions following a distinction made by Proclus. The second group consists of propositions, presented alongside mathematical proofs and diagrams. It is unknown if Euclid intended the ''Elements'' as a textbook, but its method of presentation makes it a natural fit. As a whole, the authorial voice remains general and impersonal.

Books I to VI: Plane geometry

Book I

Book I of the ''Elements'' is foundational for the entire text. It begins with a series of 20 definitions for basic geometric concepts such as lines, angles and various regular polygons. Euclid then presents 10 assumptions (see table, right), grouped into five postulates (axioms) and five common notions. These assumptions are intended to provide the logical basis for every subsequent theorem, i.e. serve as an axiomatic system. The common notions exclusively concern the comparison of magnitudes. While postulates 1 through 4 are relatively straightforward, the 5th is known as the parallel postulate and particularly famous.

Book I also includes 48 propositions, which can be loosely divided into: basic theorems and constructions of plane geometry and triangle congruence (1–26), parallel lines (27-34), the area of triangles and parallelograms (35–45), and the Pythagorean theorem (46–48).

The last of these includes the earliest surviving proof of the Pythagorean theorem, described by Sialaros as "remarkably delicate".

Book II

The second book has a more focused scope and mostly provides algebraic theorems to accompany various geometric shapes. It focuses on the area of rectangles and squares (see Quadrature), and leads up to a geometric precursor of the law of cosines. Book II is traditionally understood as concerning "geometric algebra", though this interpretation has been heavily debated since the 1970s; critics describe the characterization as anachronistic, since the foundations of even nascent algebra occurred many centuries later.

Book III

Book III begins with a list of 11 definitions, and follows with 37 propositions that deal with circles and their properties: finding the center (1), chords, intersecting and tangent circles (2-15), tangent lines to circles (16-19), inscribed angles (20-22), chords, arcs, and angles (23-30), angles in circles, including Thales' theorem (31-34), and intersecting chords and tangents, including the Intersecting secants theorem and the Tangent-secant theorem (35-39).

Book IV

Book IV treats four problems systematically for different polygons: Inscribing a polygon within a circle, Circumscribing a polygon about a circle, inscribing a circle within a polygon, circumscribing a circle about a polygon. These problems are solved in sequence for triangles, as well as regular polygons with 4, 5, 6, and 15 sides.

Book V

Book V, which is independent of the previous four books, concerns proportions of magnitudes.

Much of Book V was probably ascertained from earlier mathematicians, perhaps Eudoxus, although certain propositions, such as V.16, dealing with "alternation" (if ''a'' : ''b'' :: ''c'' : ''d'', then ''a'' : ''c'' :: ''b'' : ''d'') likely predate him.

Book VI

Book VI utilizes the theory of proportions from Book V in the context of plane geometry, especially the construction and recognition of similar figures. It is built almost entirely of its first proposition: "Triangles and parallelograms which are under the same height are to one another as their bases".

Books VII to X: Number theory

Number theory is covered by books VII to X, the former beginning with a set of 22 definitions for parity, prime numbers and other arithmetic-related concepts.

Book VII

Book VII deals with elementary number theory, and includes 39 propositions, which can be loosely divided into: Euclidean algorithm, a method for finding the greatest common divisor (1-4), fractions (5-10), theory of proportions for numbers (11-19), prime and relatively prime numbers (20-32), and Least common multiples (33-39).

Book VIII

Book VIII deals with the construction and existence of geometric sequences of integers. Propositions 1 to 10 deal with geometric progressions in general, while 11 to 27 deal with square and cube numbers.

Book IX

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

Book X

Of the ''Elements'', book X is by far the largest and most complex, dealing with irrational numbers in the context of magnitudes. Book X proves the irrationality of the square roots of non-square integers (e.g. ) 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 numbers. He also gives a formula to produce Pythagorean triples.

Books XI to XIII: Solid geometry

The final three books primarily discuss solid geometry. By introducing a list of 37 definitions, Book XI contextualizes the next two. Although its foundational character resembles Book I, unlike the latter it features no axiomatic system or postulates.

Book XI

Book XI generalizes the results of book VI to solid figures: perpendicularity, parallelism, volumes and similarity of parallelepipeds. The three sections of Book XI include content on: solid geometry (1-19), solid angles (20-23), and parallelepipeds (24-37).

Book XII

Book XII studies the volumes of cones, 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.

Book XIII

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

Apocryphal books

Two additional books, that were not written by Euclid, Books XIV and XV, have been transmitted in the manuscripts of the Elements:
* Book XIV, probably written by Hypsicles on the basis of a treatise by Apollonius, 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
$\sqrt = \sqrt.$
* Book XV, probably written, at least in part, by Isidore of Miletus, 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.

It was not uncommon in ancient times to attribute works to celebrated authors that were not written by them. It is by these means that the apocryphal books of the ''Elements'' were sometimes included in the collection.

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. 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.

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'' 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.

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

Reception

Euclid's ''Elements'' has been referred to as the most successful textbook ever written. The ''Elements'' is often considered after the Bible as the most frequently translated, published, and studied book in history. With Aristotle's ''Metaphysics'', the ''Elements'' is perhaps the most successful ancient Greek text, and was the dominant mathematical textbook in the Medieval Islamic world and Western Europe. In historical context, it has proven enormously influential in many areas of science. 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, the number reaching well over one thousand. Scientists Nicolaus Copernicus, Johannes Kepler, Galileo Galilei, Albert Einstein and Sir Isaac Newton were all influenced by the ''Elements'', and applied their knowledge of it to their work.

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. Mathematicians and philosophers, such as Thomas Hobbes, Baruch Spinoza, Alfred North Whitehead, and Bertrand Russell, have attempted to create their own foundational "Elements" for their respective disciplines, by adopting the axiomatized deductive structures that Euclid's work introduced.

In Classical antiquity

The oldest extant evidence for Euclid's Elements are a set of six ostraca found among the Elephantine papyri and ostraca, from the 3rd century BC that deal with propositions XIII.10 and XIII.16, on the conctruction of a dodecahedron. A papyrus recovered from Herculaneum contains an essay by the Epicurean philosopher Demetrius Lacon on Euclid's Elements. The earliest extant papyrus containing the actual text of the Elements is Papyrus Oxyrhynchus 29, a fragment containing the text of Book II, Proposition 5 and an accompanying diagram, dated to .

These 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 reconstructing the history of the text. Such analyses are conducted by J. L. Heiberg and Sir Thomas Little Heath in their editions of the text.

In the 4th century AD, Theon of Alexandria 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 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 is a tiny fragment of an even older manuscript, but only contains the statement of one proposition.

Medieval era

Although Euclid was known to Cicero, for instance, no record exists 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 around 760; this version was translated into Arabic under Harun al-Rashid ( 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 translated it into Latin from an Arabic translation. A relatively recent discovery was made of a Greek-to-Latin translation from the 12th century at Palermo, Sicily. The name of the translator is not known other than he was an anonymous medical student from Salerno who was visiting Palermo in order to translate the Almagest to Latin. The Euclid manuscript is extant and quite complete.

After the translation by Adelard of Bath (known as Adelard I), there was a flurry of translations from Arabic. Notable translators in this period include Herman of Carinthia who wrote an edition around 1140, Robert of Chester (his manuscripts are referred to collectively as Adelard II, written on or before 1251), Johannes de Tinemue, possibly also known as John of Tynemouth (his manuscripts are referred to collectively as Adelard III), late 12th century, and Gerard of Cremona (sometime after 1120 but before 1187). The exact details concerning these translations is still an active area of research. Campanus of Novara relied heavily on these Arabic translations to create his edition (sometime before 1260) which ultimately came to dominate Latin editions until the availability of Greek manuscripts in the 16th century. There are more than 100 pre-1482 Campanus manuscripts still available today.

Copies of the Greek text still exist, some of which can be found in the Vatican Library and the Bodleian 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. 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 further study.

Renaissance and early modern period

The first printed edition appeared in 1482 (based on Campanus's translation), and since then it has been translated into many languages and published in about a thousand different editions. Theon's Greek edition was recovered and published in 1533 based on Paris gr. 2343 and Venetus Marcianus 301. In 1570, John Dee provided a widely respected "Mathematical Preface", along with copious notes and supplementary material, to the first English edition by Henry Billingsley.

The first English edition of the ''Elements'' was published in 1570 by Henry Billingsley and John Dee.

In 1607, The Italian Jesuit Matteo Ricci and the Chinese mathematician Xu Guangqi published the first Chinese edition of Euclid's Elements.

In modern mathematics

The ''Elements'' is still considered a masterpiece in the application of logic to mathematics. The mathematician Oliver Byrne published a well-known version of the ''Elements'' in 1847 entitled ''The First Six Books of the Elements of Euclid in Which Coloured Diagrams and Symbols Are Used Instead of Letters for the Greater Ease of Learners'', which included colored diagrams intended to increase its pedagogical effect. David Hilbert authored a modern axiomatization of the ''Elements''.

Non-Euclidean geometry

The geometrical system established by the ''Elements'' long dominated the field; however, today that system is often referred to as 'Euclidean geometry' to distinguish it from other non-Euclidean geometries discovered in the early 19th century.

One of the most notable influences of Euclid on modern mathematics is the discussion of the parallel postulate. In Book I, Euclid lists five postulates, the fifth of which stipulates

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 published a description of acute geometry (or hyperbolic geometry), 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). If one takes the fifth postulate as a given, the result is Euclidean geometry.

Criticisms

Some of the foundational theorems are proved using axioms that Euclid did not state explicitly, such as Pasch's axiom. Early attempts to construct a more complete set of axioms include Hilbert's geometry axioms and Tarski's. Later editors have added Euclid's implicit axiomatic assumptions in their list of formal axioms. In 2017, Michael Beeson et al. used computer proof assistants to create a new set of axioms similar to Euclid's and generate proofs that were valid with those axioms.

A few proofs also rely on assumptions that are intuitive but not explicitly proven. 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. Mathematician and historian W. W. Rouse Ball 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."

Notes

Editions and translations

*1460s, Regiomontanus (incomplete)
*1482, Erhard Ratdolt (Venice), ''editio princeps'' (in Latin)
*1533, ''editio princeps'' of the Greek text 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, Commandinus Latin edition
*1574, Christoph Clavius
*
*1883–1888, Johan Ludvig Heiberg
*''Euclid's Elements – All thirteen books complete in one volume'', Based on Heath's translation, edited by Dana Densmore, et al. Green Lion Press .
*
*
*''The Elements: Books I–XIII – Complete and Unabridged'' (2006), Translated by Sir Thomas Heath, Barnes & Noble .
*''Plane Geometry (Euclid's elements Redux) Books I–VI'', based on John Casey's translation, edited by Daniel Callahan,
* ''The first six books of the Elements of Euclid'', edited by Werner Oechslin, Taschen, 2010, , a facsimile of Byrne (1847).
*''Oliver Byrne's Elements of Euclid'', Art Meets Science, 2022, , a facsimile of Byrne (1847).

''Euclid’s Elements: Completing Oliver Byrne's work''
Kronecker Wallis, 2019, a modern redrawing extended to the rest of the ''Elements'', originally launched on Kickstarter.

References

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External links

''Elements'' with highlights
by ratherthanpaper
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 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
Video adaptation
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.

Reading Euclid
– a course in how to read Euclid in the original Greek, with English translations and commentaries (HTML with figures)
* Sir Thomas More'
manuscript
by Aethelhard of Bath
* !-- http://www.physics.ntua.gr/Faculty/mourmouras/euclid/index.html -->http://www.physics.ntua.gr/~mourmouras/euclid/index.html Euclid ''Elements'' – The original Greek textGreek HTML
* Clay Mathematics Institute 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 b
Islamic Heritage Project

Euclid's ''Elements'' Redux
an open textbook based on the ''Elements''
1607 Chinese translations
reprinted as part of the '' Complete Library of the Four Treasuries'', or ''Siku Quanshu''.

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