Euclid

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

The English name 'Euclid' is the anglicized version of the Ancient Greek name Εὐκλείδης.[6][lower-alpha 1] It is derived from 'eu-' (εὖ; 'well') and 'klês' (-κλῆς; 'fame'), meaning "renowned, glorious".[8] The word 'Euclid' less commonly also means "a copy of the same",[7] and is sometimes synonymous with 'geometry'.[3]

Like many ancient Greek mathematicians, Euclid's life is mostly unknown.[9] 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.[10][lower-alpha 2] 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".[12] The traditional narrative mainly follows the 5th century AD 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.[6][lower-alpha 3] According to Proclus, Euclid lived after the philosopher Plato (d. 347 BC) and before the mathematician Archimedes (c. 287 – c. 212 BC); specifically, Proclus placed Euclid during the rule of Ptolemy I (r. 305/304–282 BC).[10][9][lower-alpha 4] In his Collection, Pappus indicates that Euclid was active in Alexandria, where he founded a mathematical tradition.[10][14] Thus, the traditional outline—described by the historian Michalis Sialaros as the "dominant view"—holds that Euclid lived around 300 BC in Alexandria while Ptolemy I reigned.[6]

Euclid's birthdate is unknown; some scholars estimate around 330[15][16] or 325 BC,[3][17] but other sources avoid speculating a date entirely.[18] It is presumed that he was of Greek descent,[15] but his birthplace is unknown.[12][lower-alpha 5] Proclus held that Euclid followed the Platonic tradition, but there is no definitive confirmation for this.[20] It is unlikely he was contemporary with Plato, so it is often presumed that he was educated by Plato's disciples at the Platonic Academy in Athens.[21] 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 on.[22][23] The accuracy of these assertions has been questioned by Sialaros,[24] who stated that Heath's theory "must be treated merely as a conjecture".[6] Regardless of his actual attendance at the Platonic academy, the contents of his later work certainly suggest he was familiar with the Platonic geometry tradition, though they also demonstrate no observable influence from Aristotle.[15]

Alexander the Great founded Alexandria in 331 BC, where Euclid would later be active sometime around 300 BC.[25] The rule of Ptolemy I from 306 BC onwards gave the city a stability which was relatively unique in the Mediterranean, amid the chaotic wars over dividing Alexander's empire.[26] Ptolemy began a process of hellenization and commissioned numerous constructions, building the massive Musaeum institution, which was a leading center of education.[12][lower-alpha 6] 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.[25] According to Pappus, the later mathematician Apollonius of Perga was taught there by pupils of Euclid.[22] Euclid's date of death is unknown; it has been estimated that he died c. 270 BC, presumably in Alexandria.[25]

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.[6][18] Historically, medieval scholars frequently confused the mathematician and philosopher, mistakenly referring to the former in Latin as 'Megarensis' (lit. 'of Megara').[28] As a result, biographical information on the mathematician Euclid was long conflated with the lives of both Euclid of Alexandria and Euclid of Megara.[6] The only scholar of antiquity known to have confused the mathematician and philosopher was Valerius Maximus.[29][lower-alpha 7] However, this mistaken identification was relayed by many anonymous Byzantine sources and the Renaissance scholars Campanus of Novara and Theodore Metochites, the latter of whom included it in a 1842 translation of the Elements printed by Erhard Ratdolt.[29] After the mathematician Bartolomeo Zamberti (1473–1539) affirmed this presumption in his 1505 translation, all subsequent publications passed on this identification.[29][lower-alpha 8] Later Renaissance scholars, particularly Peter Ramus, reevaluated this claim, proving it false via issues in chronology and contradiction in early sources.[29]

Arab sources written many centuries after his death give vast amounts of information concerning Euclid's life, but are completely unverifiable.[6] Most scholars consider them of dubious authenticity;[10] Heath in particular contends that the fictionalization was done to strengthen the connection between a revered mathematician and the Arab world.[20] There are also numerous anecdotal stories concerning to Euclid, all of uncertain historicity, which "picture him as a kindly and gentle old man".[31] The best known of these is Proclus' story about Ptolemy asking Euclid if there was a quicker path to learning geometry than reading his Elements, which Euclid replied with "there is no royal road to geometry".[31] This anecdote is questionable since a very similar interaction between Menaechmus and Alexander the Great is recorded from Stobaeus.[32] Both the accounts were written in the 5th century AD, neither indicate their source, and neither story appears in ancient Greek literature.[33]

The traditional narrative of Euclid's activity c. 300 is complicated by no mathematicians of the 4th century BC indicating his existence.[6] Mathematicians of the 3rd century such as Archimedes and Apollonius "assume a part of his work to be known";[6] however, Archimedes strangely uses an older theory of proportions, rather than that of Euclid.[10] The Elements is dated to have been at least partly in circulation by the 3rd century BC.[6] Some ancient Greek mathematician mention him by name, but he is usually referred to as "ὁ στοιχειώτης" ("the author of Elements").[34] In the Middle Ages, some scholars contended Euclid was not a not a historical personage and that his name arose from a corruption of Greek mathematical terms.[35]

Euclid is best known for his thirteen-book treatise, the Elements (Greek: Στοιχεῖα; 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]

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 AD.[5]

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.

• Catoptrics concerns the mathematical theory of mirrors, particularly the images formed in plane and spherical concave mirrors, though the attribution is sometimes questioned.[40]
• The Data (Greek: Δεδομένα), is a somewhat short text which deals with the nature and implications of "given" information in geometrical problems.[40]
• On Divisions (Greek: Περὶ Διαιρέσεων‎) survives only partially in Arabic translation, and concerns the division of geometrical figures into two or more equal parts or into parts in given ratios. It includes thirty-six propositions and is similar to Apollonius' Conics.[40]
• The Optics (Greek: Ὀπτικά‎) is the earliest surviving Greek treatise on perspective. It includes an introductory discussion of geometrical optics and basic rules of perspective.[40]
• The Phaenomena (Greek: Φαινόμενα) is a treatise on spherical astronomy, survives in Greek; it is similar to On the Moving Sphere by Autolycus of Pitane, who flourished around 310 BC.[40]

Four other works are credibly attributed to Euclid, but have been lost.[11]

• The Conics (Greek: Κωνικά‎) was a four-book survey on conic sections, which was later superseded by a Apollonius' more comprehensive treatment of the same name.[41][40] The work's existence is known primarily from Pappus, who asserts that the first four books of Apollonius' Conics are largely based on Euclid's earlier work.[42] Doubt has been cast on this assertion by the historian Alexander Jones, owing to sparse evidence and no other corroboration of Pappus' account.[42]
• The Pseudaria (Greek: Ψευδάρια‎; lit. 'Fallacies'), was—according to Proclus in (70.1–18)—a text in geometrical reasoning, written to advise beginners in avoiding common fallacies.[41][40] Very little is known of its specific contents aside from its scope and a few extant lines.[43]
• The Porisms (Greek: Πορίσματα; lit. 'Corollaries') was, based on accounts from Pappus and Proclus, probably a three-book treatise with approximately 200 propositions.[41][40] The term 'porism' in this context does not refer to a corollary, but to "a third type of proposition—an intermediate between a theorem and a problem—the aim of which is to discover a feature of an existing geometrical entity, for example, to find the centre of a circle".[40] The mathematician Michel Chasles speculated that these now-lost propositions included content related to the modern theories of transversals and projective geometry.[41][lower-alpha 9]
• The Surface Loci (Greek: Τόποι πρὸς ἐπιφανείᾳ) is of virtually unknown contents, aside from speculation based on the work's title.[41] Conjecture based on later accounts has suggested it discussed cones and cylinders, among other subjects.[40]

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