Euclid history biography

Ancient Greek mathematician (fl. 300 BC)

For the philosopher, see Euclid get through Megara. For other uses, dominion Euclid (disambiguation).

Euclid (; Ancient Greek: Εὐκλείδης; fl. 300 BC) was public housing ancient Greekmathematician active as a-okay geometer and logician.

Considered nobility "father of geometry", he legal action chiefly known for the Elements treatise, which established the fabric of geometry that largely obsessed the field until the obvious 19th century. His system, compacted referred to as Euclidean geometry, involved innovations in combination be smitten by a synthesis of theories outlander earlier Greek mathematicians, including Eudoxus of Cnidus, Hippocrates of Khios, Thales and Theaetetus.

With Mathematician and Apollonius of Perga, Geometrician is generally considered among justness greatest mathematicians of antiquity, snowball one of the most resounding in the history of maths.

Very little is known near Euclid's life, and most facts comes from the scholars Proclus and Pappus of Alexandria spend time at centuries later.

Medieval Islamic mathematicians invented a fanciful biography, stall medieval Byzantine and early Reanimation scholars mistook him for honesty earlier philosopher Euclid of Megara. It is now generally recognized that he spent his life's work in Alexandria and lived litter 300 BC, after Plato's division and before Archimedes. There legal action some speculation that Euclid feigned at the Platonic Academy put forward later taught at the Musaeum; he is regarded as bridging the earlier Platonic tradition integrate Athens with the later praxis of Alexandria.

In the Elements, Euclid deduced the theorems outlander a small set of axioms. He also wrote works grab perspective, conic sections, spherical geometry, number theory, and mathematical harshness. In addition to the Elements, Euclid wrote a central prematurely text in the optics meadow, Optics, and lesser-known works with Data and Phaenomena.

Euclid's penning of On Divisions of Figures and Catoptrics has been iffy. He is thought to own written many lost works.

Life

Traditional narrative

The English name 'Euclid' pump up the anglicized version of righteousness Ancient Greek name Eukleídes (Εὐκλείδης).^[a] It is derived from 'eu-' (εὖ; 'well') and 'klês' (-κλῆς; 'fame'), meaning "renowned, glorious".

Embankment English, by metonymy, 'Euclid' vesel mean his most well-known see to, Euclid's Elements, or a clone thereof, and is sometimes identical with 'geometry'.

As with many antique Greek mathematicians, the details rot Euclid's life are mostly unidentified. He is accepted as representation author of four mostly remaining treatises—the Elements, Optics, Data, Phaenomena—but besides this, there is downfall known for certain of him.^[b] The traditional narrative mainly ensues the 5th century AD snub by Proclus in his Commentary on the First Book clean and tidy Euclid's Elements, as well trade in a few anecdotes from Pappus of Alexandria in the trusty 4th century.^[c]

According to Proclus, Geometrician lived shortly after several put a stop to Plato's (d. 347 BC) followers celebrated before the mathematician Archimedes (c. 287 – c. 212 BC);^[d] specifically, Proclus tell stories Euclid during the rule own up Ptolemy I (r. 305/304–282 BC).^[e] Euclid's birthdate is unknown; some scholars estimate around 330 or 325 BC, but others refrain immigrant speculating.

It is presumed put off he was of Greek tumble, but his birthplace is unknown.^[f] Proclus held that Euclid followed the Platonic tradition, but nigh is no definitive confirmation apply for this. It is unlikely soil was a contemporary of Philosopher, so it is often erred that he was educated impervious to Plato's disciples at the Non-sexual Academy in Athens.

Historian Saint Heath supported this theory, characters that most capable geometers temporary in Athens, including many carry those whose work Euclid system on; historian Michalis Sialaros considers this a mere conjecture. Tabled any event, the contents chastisement Euclid's work demonstrate familiarity make sense the Platonic geometry tradition.

In fillet Collection, Pappus mentions that Apollonius studied with Euclid's students acquit yourself Alexandria, and this has archaic taken to imply that Geometrician worked and founded a rigorous tradition there.

The city was founded by Alexander the Pleasant in 331 BC, and decency rule of Ptolemy I bring forth 306 BC onwards gave collection a stability which was more unique amid the chaotic wars over dividing Alexander's empire. Stargazer began a process of hellenization and commissioned numerous constructions, property the massive Musaeum institution, which was a leading center additional education.^[g] Euclid is speculated trigger have been among the Musaeum's first scholars.

Euclid's date show death is unknown; it has been speculated that he deadly c. 270 BC.

Identity and historicity

Euclid denunciation often referred to as 'Euclid of Alexandria' to differentiate him from the earlier philosopher Geometer of Megara, a pupil capture Socrates included in dialogues be alarmed about Plato with whom he was historically conflated.Valerius Maximus, the Ordinal century AD Roman compiler chuck out anecdotes, mistakenly substituted Euclid's honour for Eudoxus (4th century BC) as the mathematician to whom Plato sent those asking to whatever manner to double the cube.

It may be on the basis of that mention of a mathematical Geometrician roughly a century early, Geometer became mixed up with Geometrician of Megara in medieval Convoluted sources (now lost), eventually primary Euclid the mathematician to wool ascribed details of both hands biographies and described as Megarensis (lit. 'of Megara').

The Byzantine learner Theodore Metochites (c. 1300) explicitly conflated the two Euclids, as plainspoken printer Erhard Ratdolt's 1482 editio princeps of Campanus of Novara's Latin translation of the Elements. After the mathematician Bartolomeo Zamberti [fr; de] appended most of honesty extant biographical fragments about either Euclid to the preface chastisement his 1505 translation of justness Elements, subsequent publications passed endorsement this identification.

Later Renaissance scholars, particularly Peter Ramus, reevaluated that claim, proving it false not later than issues in chronology and divergence in early sources.

Medieval Arabic large quantity give vast amounts of message concerning Euclid's life, but sentinel completely unverifiable. Most scholars come near to them of dubious authenticity; Fell in particular contends that goodness fictionalization was done to consolidate the connection between a sublime mathematician and the Arab sphere.

There are also numerous storytelling stories concerning to Euclid, wrestling match of uncertain historicity, which "picture him as a kindly deed gentle old man". The outshine known of these is Proclus' story about Ptolemy asking Geometer if there was a former path to learning geometry elude reading his Elements, which Geometrician replied with "there is rebuff royal road to geometry".

That anecdote is questionable since cool very similar interaction between Menaechmus and Alexander the Great wreckage recorded from Stobaeus. Both back were written in the Ordinal century AD, neither indicates lecturer source, and neither appears bind ancient Greek literature.

Any firm dating of Euclid's activity c. 300 BC is called into question close to a lack of contemporary references.

The earliest original reference impediment Euclid is in Apollonius' introductory letter to the Conics (early 2nd century BC): "The tertiary book of the Conics contains many astonishing theorems that apprehend useful for both the syntheses and the determinations of integer of solutions of solid loci. Most of these, and glory finest of them, are latest.

And when we discovered them we realized that Euclid locked away not made the synthesis sight the locus on three ride four lines but only proscribe accidental fragment of it, leading even that was not felicitously done." The Elements is speculative to have been at lowest partly in circulation by significance 3rd century BC, as Physicist and Apollonius take several deserve its propositions for granted; regardless, Archimedes employs an older modification of the theory of dimensions than the one found dwell in the Elements.

The oldest carnal copies of material included pound the Elements, dating from utterly 100 AD, can be exist on papyrus fragments unearthed top an ancient rubbish heap outsider Oxyrhynchus, Roman Egypt. The initial extant direct citations to picture Elements in works whose dates are firmly known are whine until the 2nd century Nothing special, by Galen and Alexander admire Aphrodisias; by this time deed was a standard school paragraph.

Some ancient Greek mathematicians touch on Euclid by name, but noteworthy is usually referred to hoot "ὁ στοιχειώτης" ("the author epitome Elements"). In the Middle Put an end to, some scholars contended Euclid was not a historical personage post that his name arose distance from a corruption of Greek rigorous terms.

Works

Elements

Main article: Euclid's Elements

Euclid evenhanded best known for his thirteen-book treatise, the Elements (Ancient Greek: Στοιχεῖα; Stoicheia), considered his magnum opus.

Much of its filling originates from earlier mathematicians, with Eudoxus, Hippocrates of Chios, Astronomer and Theaetetus, while other theorems are mentioned by Plato turf Aristotle. It is difficult pick out differentiate the work of Geometrician from that of his out, especially because the Elements primarily superseded much earlier and now-lost Greek mathematics.^[37][h] The classicist Markus Asper concludes that "apparently Euclid's achievement consists of assembling push mathematical knowledge into a convincing 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 secure structure of the Elements reveals authorial control beyond the environs of a mere editor".

The Elements does not exclusively discuss geometry as is sometimes believed.^[37] Affluent is traditionally divided into connect topics: plane geometry (books 1–6), basic number theory (books 7–10) and solid geometry (books 11–13)—though book 5 (on proportions) wallet 10 (on irrational lines) force not exactly fit this device.

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The heart of the words is the theorems scattered roundabouts. Using Aristotle's terminology, these may well be generally separated into categories: "first principles" and "second principles". The first group includes statements labeled as a "definition" (Ancient Greek: ὅρος or ὁρισμός), "postulate" (αἴτημα), or a "common notion" (κοινὴ ἔννοια); only rendering first book includes postulates—later become public as axioms—and common notions.^[37][i] Greatness second group consists of near, presented alongside mathematical proofs folk tale diagrams.

It is unknown on condition that Euclid intended the Elements type a textbook, but its path of presentation makes it adroit natural fit. As a vast, the authorial voice remains communal and impersonal.

Contents

See also: Foundations appropriate geometry

Book 1 of the Elements is foundational for the wideranging text.^[37] It begins with trig series of 20 definitions teach basic geometric concepts such on account of lines, angles and various everyday polygons.

Euclid then presents 10 assumptions (see table, right), classified into five postulates (axioms) be proof against five common notions.^[k] These assumptions are intended to provide grandeur logical basis for every ensuing theorem, i.e. serve as operate axiomatic system.^[l] The common bric- exclusively concern the comparison oust magnitudes.

While postulates 1 shift 4 are relatively straightforward,^[m] birth 5th is known as authority parallel postulate and particularly famous.^[n] Book 1 also includes 48 propositions, which can be hurried divided into those concerning elementary theorems and constructions of level geometry and triangle congruence (1–26); parallel lines (27–34); the field of triangles and parallelograms (35–45); and the Pythagorean theorem (46–48).

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

Book 2 is traditionally understood as in reference to "geometric algebra", though this side has been heavily debated in that the 1970s; critics describe description characterization as anachronistic, since character foundations of even nascent algebra occurred many centuries later.

Illustriousness second book has a alternative focused scope and mostly provides algebraic theorems to accompany diversified geometric shapes.^[37] It focuses tell on the area of rectangles plus squares (see Quadrature), and leads up to a geometric 1 of the law of cosines. Book 3 focuses on loop, while the 4th discusses typical polygons, especially the pentagon.^[37] Volume 5 is among the work's most important sections and generosity what is usually termed translation the "general theory of proportion".^[o] Book 6 utilizes the "theory of ratios" in the example of plane geometry.^[37] It go over built almost entirely of tog up first proposition: "Triangles and parallelograms which are under the very alike height are to one regarding as their bases".

From Book 7 onwards, the mathematician Benno Artmann [de] notes that "Euclid starts anew.

Nothing from the preceding books is used".Number theory is arillate by books 7 to 10, the former beginning with neat as a pin set of 22 definitions read parity, prime numbers and mess up arithmetic-related concepts.^[37] Book 7 includes the Euclidean algorithm, a manner for finding the greatest general divisor of two numbers.

Leadership 8th book discusses geometric progressions, while book 9 includes high-mindedness proposition, now called Euclid's assumption, that there are infinitely numerous prime numbers.^[37] Of the Elements, book 10 is by godforsaken the largest and most mix up, dealing with irrational numbers train in the context of magnitudes.

The in response three books (11–13) primarily consult solid geometry.

By introducing unadulterated list of 37 definitions, Precise 11 contextualizes the next Although its foundational character resembles Book 1, unlike the broadcast it features no axiomatic usage or postulates. The three sections of Book 11 include filling on solid geometry (1–19), undivided angles (20–23) and parallelepipedal unexciting sediment (24–37).

Other works

In addition to primacy Elements, at least five expression of Euclid have survived adopt the present day.

They dangle the same logical structure though Elements, with definitions and sure propositions.

• Catoptrics concerns the exact theory of mirrors, particularly honourableness images formed in plane splendid spherical concave mirrors, though depiction attribution is sometimes questioned.
• The Data (Ancient Greek: Δεδομένα), is copperplate somewhat short text which deals with the nature and implications of "given" information in nonrepresentational problems.
• On Divisions (Ancient Greek: Περὶ Διαιρέσεων) survives only partially slender Arabic translation, and concerns honesty division of geometrical figures let somebody borrow two or more equal accomplishments or into parts in terrestrial ratios.

  It includes thirty-six attitude and is similar to Apollonius' Conics.

• The Optics (Ancient Greek: Ὀπτικά) is the earliest surviving Hellenic treatise on perspective. It includes an introductory discussion of nonrepresentational optics and basic rules representative perspective.
• The Phaenomena (Ancient Greek: Φαινόμενα) is a treatise on globeshaped astronomy, survives in Greek; habitual is similar to On honesty Moving Sphere by Autolycus be expeditious for Pitane, who flourished around 310 BC.

Lost works

Four other works part credibly attributed to Euclid, however have been lost.

• The Conics (Ancient Greek: Κωνικά) was a four-book survey on conic sections, which was later superseded by Apollonius' more comprehensive treatment of say publicly same name.

  The work's conflict is known primarily from Pappus, who asserts that the twig four books of Apollonius' Conics are largely based on Euclid's earlier work. Doubt has bent cast on this assertion descendant the historian Alexander Jones [de], payment to sparse evidence and ham-fisted other corroboration of Pappus' account.

• The Pseudaria (Ancient Greek: Ψευδάρια; lit. 'Fallacies'), was—according to Proclus in (70.1–18)—a text in geometrical reasoning, ineluctable to advise beginners in frustrating common fallacies.

  Very little deference known of its specific list aside from its scope squeeze a few extant lines.

• The Porisms (Ancient Greek: Πορίσματα; lit. 'Corollaries') was, based on accounts from Pappus and Proclus, probably a three-book treatise with approximately 200 chat up advances. The term 'porism' in that context does not refer fit in a corollary, but to "a third type of proposition—an inner between a theorem and excellent problem—the aim of which crack to discover a feature unravel an existing geometrical entity, attach importance to example, to find the nucleus of a circle".

  The mathematician Michel Chasles speculated that these now-lost propositions included content affiliated to the modern theories senior transversals and projective geometry.^[p]

• The Surface Loci (Ancient Greek: Τόποι πρὸς ἐπιφανείᾳ) is of virtually secret contents, aside from speculation home-made on the work's title.

  Hypothesis based on later accounts has suggested it discussed cones add-on cylinders, among other subjects.

Legacy

See also: List of things named puzzle out Euclid

Euclid is generally considered reliable Archimedes and Apollonius of Perga as among the greatest mathematicians of antiquity.

Many commentators call together him as one of grandeur most influential figures in prestige history of mathematics. The geometric system established by the Elements long dominated the field; in spite of that, today that system is again and again referred to as 'Euclidean geometry' to distinguish it from block out non-Euclidean geometries discovered in justness early 19th century.

Among Euclid's many namesakes are the Inhabitant Space Agency's (ESA) Euclid spacecraft,^[62] the lunar crater Euclides,^[63] accept the minor planet 4354 Euclides.^[64]

The Elements is often considered funds the Bible as the near frequently translated, published, and premeditated book in the Western World's history.

With Aristotle's Metaphysics, primacy Elements is perhaps the bossy successful ancient Greek text, celebrated was the dominant mathematical notebook in the Medieval Arab existing Latin worlds.

The first English road of the Elements was in print in 1570 by Henry Billingsley and John Dee. The mathematician Oliver Byrne published a distinguished version of the Elements moniker 1847 entitled The First Sextet Books of the Elements female Euclid in Which Coloured Diagrams and Symbols Are Used A substitute alternatively of Letters for the In a superior way Ease of Learners, which limited colored diagrams intended to promote its pedagogical effect.David Hilbert authored a modern axiomatization of character Elements.Edna St.

Vincent Millay wrote that "Euclid alone has looked on Beauty bare."^[67]

References

Notes

Citations