Euclid Source: en.wikipedia.org/wiki/Euclid

Euclid
Εὐκλείδης
Euclid by Jusepe de Ribera, c. 1630–1635[1]
Known for	The Elements Optics Data Various concepts Euclidean geometry Euclidean algorithm Euclid's theorem Euclid's formula Numerous other namesakes
Scientific career
Fields	Mathematics
Influences	Eudoxus, Hippocrates of Chios, Thales and Theaetetus
Influenced	Virtually all of subsequent Western and Middle Eastern mathematics[2]

Euclid (/ˈjuːklɪd/; Greek: Εὐκλείδης; fl. 300 BCE) was an ancient Greek mathematician active as a geometer and logician.^[3] Considered the "father of geometry",^[4] he is chiefly known for the Elements treatise, which established the foundations of geometry that largely dominated the field until the early 19th century. His system, now referred to as Euclidean geometry, involved new innovations in combination with a synthesis of theories from earlier Greek mathematicians, including Eudoxus of Cnidus, Hippocrates of Chios, Thales and Theaetetus. With Archimedes and Apollonius of Perga, Euclid is generally considered among the greatest mathematicians of antiquity, and among the most influential in the history of mathematics.

Very little is known of Euclid's life, and most information comes from the philosophers Proclus and Pappus of Alexandria many centuries later. Until the early Renaissance he was often mistaken for the earlier philosopher Euclid of Megara, causing his biography to be substantially revised. It is generally agreed that he spent his career under Ptolemy I in Alexandria and lived around 300 BCE, after Plato and before Archimedes. There is some speculation that Euclid was a student of the Platonic Academy. Euclid is often regarded as bridging between the earlier Platonic tradition in Athens with the later tradition of Alexandria.

In the Elements, Euclid deduced the theorems from a small set of axioms. He also wrote works on perspective, conic sections, spherical geometry, number theory, and mathematical rigour. In addition to the Elements, Euclid wrote a central early text in the optics field, Optics, and lesser-known works including Data and Phaenomena.^[5] Euclid's authorship of two other texts—On Divisions of Figures, Catoptrics—has been questioned. He is thought to have written many now lost works.^[6]

Life[edit]

Traditional narrative[edit]

A papyrus fragment of Euclid's Elements dated to c. 75–125 CE CE. Found at Oxyrhynchus, the diagram accompanies Book II, Proposition 5.^[7]

The English name 'Euclid' is the anglicized version of the Ancient Greek name Εὐκλείδης.^[8][a] It is derived from 'eu-' (εὖ) and 'klês' (-κλῆς), meaning "renowned, glorious".^[10] The word 'Euclid' less commonly also means "a copy of the same",^[9] and is sometimes synonymous with 'geometry'.^[3]

Like many ancient Greek mathematicians, Euclid's life is mostly unknown.^[11] He is accepted as the author of four mostly extant treatises—the Elements, Optics, Data, Phaenomena—but besides this, there is nothing known for certain of him.^[5][b] The historian Carl Benjamin Boyer has noted irony in that "Considering the fame of the author and of his best seller [the Elements], remarkably little is known of Euclid".^[13] The traditional narrative mainly follows the 5th century CE account by Proclus in his Commentary on the First Book of Euclid's Elements, as well as a few anecdotes from Pappus of Alexandria in the early 4th century.^[8][c] According to Proclus, Euclid lived after the philosopher Plato (d. 347 BCE) and before the mathematician Archimedes (c. 287 – c. 212 BCE); specifically, Proclus placed Euclid during the rule of Ptolemy I (r. 305/304–282 BCE).^[5][11][d] In his Collection, Pappus indicates that Euclid was active in Alexandria, where he founded a mathematical tradition.^[5][15] Thus, the traditional outline—described by the historian Michalis Sialaros as the "dominant view"—holds that Euclid lived around 300 BCE in Alexandria while Ptolemy I reigned.^[8]

Euclid's birthdate is unknown; some scholars estimate around 330^[16][17] or 325 BCE,^[3][18] but other sources avoid speculating a date entirely.^[19] It is presumed that he was of Greek descent,^[16] but his birthplace is unknown.^[13][e] Proclus held that Euclid followed the Platonic tradition, but there is no definitive confirmation for this.^[21] It is unlikely he was contemporary with Plato, so it is often presumed that he was education by Plato's disciples at the Platonic Academy in Athens.^[22] The historian Thomas Heath supported this theory by noting that most capable geometers lived in Athens, which included many of the mathematicians whose work Euclid later built off.^[23][24] The accuracy of these assertions have been questioned by Sialaros,^[25] who stated that Heath's theory "must be treated merely as a conjecture".^[8] Regardless of his actual attendance at the Platonic academy, the contents of his later work certainly suggests he was familiar with the Platonic geometry tradition, though they also demonstrate no observable influence from Aristotle.^[16]

Alexander the Great founded Alexandria in 331 BCE, where Euclid would later be active sometime around 300 BCE.^[26] The rule of Ptolemy I from 306 BCE onwards gave the city a stability which was relatively unique in the Mediterranean, amid the chaotic wars over dividing of Alexander's empire.^[27] Ptolemy began a process of hellenization and commissioned numerous constructions, building the massive Musaeum institution, which was a leading center of education.^[13][f] On the basis of later anecdotes, Euclid is thought to have been among the Musaeum's first scholars and to have founded the Alexandrian school of mathematics there.^[26] According to Pappus, the later mathematician Pappus of Alexandria was taught there by pupils of Euclid.^[23] Euclid's date of death is unknown; it has been estimated that he died c. 270 BCE, presumably in Alexandria.^[26]

Identity and historicity[edit]

Euclid is often referred to as 'Euclid of Alexandria' to differentiate him from the earlier philosopher Euclid of Megara, a pupil of Socrates who was included in the dialogues of Plato.^[8][19] Historically, medieval scholars frequently confused the mathematician and philosopher, mistakenly referring to the former in Latin as 'Megarensis' (lit. 'of Megara').^[29] As a result, biographical information on the mathematician Euclid was long conflated with the lives of both Euclid of Alexandria and Euclid of Megara.^[8] The only scholar of antiquity known to have confused the mathematician and philosopher was Valerius Maximus.^[30][g] In addition to the many anonymous Byzantine sources, this mistaken identification was relayed by the scholars Campanus of Novara and Theodore Metochites, and put into a publication of the latter's translated printed by Erhard Ratdolt in 1482.^[30] After the mathematician Bartolomeo Zamberti [fr] (1473–1539) affirmed this presumption in his 1505 translation, all subsequent publications passed on this identification.^[30][h] Early Renaissance scholars, particularly Peter Ramus, reevaluated this claim, proving it false via issues in chronology and contradiction in early sources.^[30]

He is mentioned by name, though rarely, by other ancient Greek mathematicians from Archimedes onward, and is usually referred to as "ὁ στοιχειώτης" ("the author of Elements").^[32] Proclus later retells a story that, when Ptolemy I asked if there was a shorter path to learning geometry than Euclid's Elements, "Euclid replied there is no royal road to geometry."^[33] This anecdote is questionable since it is similar to a story told about Menaechmus and Alexander the Great.^[34]

Because the lack of biographical information is unusual for the period (extensive biographies being available for most significant Greek mathematicians several centuries before and after Euclid), some researchers have proposed that Euclid was not a historical personage, and that his works were written by a team of mathematicians who took the name Euclid from Euclid of Megara (à la Bourbaki). However, this hypothesis is not well accepted by scholars and there is little evidence in its favor.^[35]

Works[edit]

Elements[edit]

Euclid is best known for the thirteen-book treatise Elements (Greek: Εὐκλείδης; Stoicheia).^[4][37] Although many of the results in the Elements originated with earlier Greek mathematicians, one of Euclid's accomplishments was to present them in a single, logically coherent framework, making it easy to use and easy to reference, including a system of rigorous mathematical proofs that remains the basis of mathematics 23 centuries later.^[38] Among the mathematicians whose work is featured includes Eudoxus, Hippocrates of Chios, Thales and Theaetetus.^[39]

There is no mention of Euclid in the earliest remaining copies of the Elements. Most of the copies say they are "from the edition of Theon" or the "lectures of Theon",^[40] while the text considered to be primary, held by the Vatican, mentions no author. Proclus provides the only reference ascribing the Elements to Euclid.

Although best known for its geometric results, the Elements also includes number theory. It considers the connection between perfect numbers and Mersenne primes (known as the Euclid–Euler theorem), the infinitude of prime numbers, Euclid's lemma on factorization (which leads to the fundamental theorem of arithmetic on uniqueness of prime factorizations), and the Euclidean algorithm for finding the greatest common divisor of two numbers.

The Papyrus Oxyrhynchus 29 (P. Oxy. 29) is a fragment of the second book of the Elements of Euclid, unearthed by Grenfell and Hunt 1897 in Oxyrhynchus. More recent scholarship suggests a date of 75–125 CE.^[7]

Other works[edit]

Euclid's construction of a regular dodecahedron.

In addition to the Elements, at least five works of Euclid have survived to the present day. They follow the same logical structure as Elements, with definitions and proved propositions.

• Data deals with the nature and implications of "given" information in geometrical problems; the subject matter is closely related to the first four books of the Elements.
• On Divisions of Figures, which survives only partially in Arabic translation, concerns the division of geometrical figures into two or more equal parts or into parts in given ratios. It is similar to a first-century CE work by Heron of Alexandria.
• Catoptrics, which concerns the mathematical theory of mirrors, particularly the images formed in plane and spherical concave mirrors. The attribution is held to be anachronistic however by J J O'Connor and E F Robertson who name Theon of Alexandria as a more likely author.^[41]
• Phaenomena, a treatise on spherical astronomy, survives in Greek; it is quite similar to On the Moving Sphere by Autolycus of Pitane, who flourished around 310 BCE.
• Optics is the earliest surviving Greek treatise on perspective. In its definitions Euclid follows the Platonic tradition that vision is caused by discrete rays which emanate from the eye. One important definition is the fourth: "Things seen under a greater angle appear greater, and those under a lesser angle less, while those under equal angles appear equal." In the 36 propositions that follow, Euclid relates the apparent size of an object to its distance from the eye and investigates the apparent shapes of cylinders and cones when viewed from different angles. Proposition 45 is interesting, proving that for any two unequal magnitudes, there is a point from which the two appear equal. Pappus believed these results to be important in astronomy and included Euclid's Optics, along with his Phaenomena, in the Little Astronomy, a compendium of smaller works to be studied before the Syntaxis (Almagest) of Claudius Ptolemy.

Lost works[edit]

Other works are credibly attributed to Euclid, but have been lost.

• Conics was a work on conic sections that was later extended by Apollonius of Perga into his famous work on the subject. It is likely that the first four books of Apollonius's work come directly from Euclid. According to Pappus, "Apollonius, having completed Euclid's four books of conics and added four others, handed down eight volumes of conics." The Conics of Apollonius quickly supplanted the former work, and by the time of Pappus, Euclid's work was already lost.
• Porisms might have been an outgrowth of Euclid's work with conic sections, but the exact meaning of the title is controversial.
• Pseudaria, or Book of Fallacies, was an elementary text about errors in reasoning.
• Surface Loci concerned either loci (sets of points) on surfaces or loci which were themselves surfaces; under the latter interpretation, it has been hypothesized that the work might have dealt with quadric surfaces.
• Several works on mechanics are attributed to Euclid by Arabic sources. On the Heavy and the Light contains, in nine definitions and five propositions, Aristotelian notions of moving bodies and the concept of specific gravity. On the Balance treats the theory of the lever in a similarly Euclidean manner, containing one definition, two axioms, and four propositions. A third fragment, on the circles described by the ends of a moving lever, contains four propositions. These three works complement each other in such a way that it has been suggested that they are remnants of a single treatise on mechanics written by Euclid.

Legacy[edit]

The cover page of Oliver Byrne's 1847 colored edition of the Elements

Euclid is generally considered with Archimedes and Apollonius of Perga as among the greatest mathematicians of antiquity.^[16] Some commentators cite him as one of the most influential figures in the history of mathematics.^[3]

The geometrical system described in the Elements was long known simply as 'geometry', and was considered to be the only geometry possible. Today, however, that system is often referred to as 'Euclidean geometry' to distinguish it from 'Non-Euclidean geometry' discovered in the 19th century.

The first English edition of the Elements was published in 1570 by Henry Billingsley and John Dee.^[30] 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 pedological effect.^[42] David Hilbert authored a modern axiomatization of the Elements.^[43]

Among Euclid's many namesakes are the European Space Agency's (ESA) Euclid spacecraft,^[44] the lunar crater Euclides, and the minor planet 4354 Euclides are named after him.^[45]

References[edit]

Notes[edit]

Citations[edit]

Sources[edit]

Early[edit]

• Proclus (1873) [5th century]. Friedlein, Godofredi (ed.). Procli Diadochi in Primum Euclidis Elementorum Librum Commentarii [Commentary on the First Book of Euclid's Elements]. Leipzig: Bibliotheca Scriptorum Graecorum et Romanorum Teubneriana. ark:/13960/t5gb2dm74. Reprint, Hildesheim: Olms, 1967.
• Proclus, A commentary on the First Book of Euclid's Elements, translated by Glenn Raymond Morrow, Princeton University Press, 1992. ISBN 978-0-691-02090-7.

Modern[edit]

Further reading[edit]

• Artmann, Benno (2012) [1999]. Euclid: The Creation of Mathematics. New York: Springer Publishing. ISBN 978-1-4612-1412-0.

External links[edit]

Works

• Works by Euclid at Project Gutenberg
• Works by or about Euclid at Internet Archive
• Works by Euclid at LibriVox (public domain audiobooks)

The Elements

• PDF copy, with the original Greek and an English translation on facing pages, University of Texas.
• All thirteen books, in several languages as Spanish, Catalan, English, German, Portuguese, Arabic, Italian, Russian and Chinese.