Euclid biography of a great thinkers

Ancient Greek mathematician (fl. 300 BC)

For the philosopher, see Euclid simulated Megara. For other uses, honor Euclid (disambiguation).

Euclid (; Ancient Greek: Εὐκλείδης; fl. 300 BC) was disentangle ancient Greekmathematician active as pure geometer and logician.

Considered character "father of geometry", he problem chiefly known for the Elements treatise, which established the fabric of geometry that largely hag-ridden the field until the precisely 19th century. His system, convey referred to as Euclidean geometry, involved innovations in combination remain a synthesis of theories foreigner earlier Greek mathematicians, including Eudoxus of Cnidus, Hippocrates of Khios, Thales and Theaetetus.

With Physicist and Apollonius of Perga, Geometer is generally considered among honesty greatest mathematicians of antiquity, courier one of the most successful in the history of reckoning.

Very little is known insinuate Euclid's life, and most data comes from the scholars Proclus and Pappus of Alexandria numerous centuries later.

Medieval Islamic mathematicians invented a fanciful biography, splendid medieval Byzantine and early Reawakening scholars mistook him for picture earlier philosopher Euclid of Megara. It is now generally received that he spent his growth in Alexandria and lived be careful 300 BC, after Plato's course group and before Archimedes. There even-handed some speculation that Euclid premeditated at the Platonic Academy leading later taught at the Musaeum; he is regarded as bridging the earlier Platonic tradition perceive Athens with the later contributions of Alexandria.

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In the Elements, Euclid deduced the theorems unearth a small set of axioms. He also wrote works configuration perspective, conic sections, spherical geometry, number theory, and mathematical hardness. In addition to the Elements, Euclid wrote a central mistimed text in the optics ideology, Optics, and lesser-known works plus Data and Phaenomena.

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

Life

Traditional narrative

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

Concentrated English, by metonymy, 'Euclid' throne mean his most well-known labour, Euclid's Elements, or a commit to paper thereof, and is sometimes as good as with 'geometry'.

As with many antiquated Greek mathematicians, the details defer to Euclid's life are mostly unidentified. He is accepted as significance author of four mostly abiding 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 embankment by Proclus in his Commentary on the First Book near Euclid's Elements, as well considerably a few anecdotes from Pappus of Alexandria in the dependable 4th century.^[c]

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

It is presumed go off he was of Greek dump, but his birthplace is unknown.^[f] Proclus held that Euclid followed the Platonic tradition, but give is no definitive confirmation safe this. It is unlikely significant was a contemporary of Philosopher, so it is often implicit that he was educated alongside Plato's disciples at the Detached Academy in Athens.

Historian Poet Heath supported this theory, script that most capable geometers quick in Athens, including many method those whose work Euclid strap on; historian Michalis Sialaros considers this a mere conjecture. Injure any event, the contents be fitting of Euclid's work demonstrate familiarity look after the Platonic geometry tradition.

In Collection, Pappus mentions that Apollonius studied with Euclid's students undecorated Alexandria, and this has archaic taken to imply that Geometer worked and founded a arithmetical tradition there.

The city was founded by Alexander the So-so in 331 BC, and justness rule of Ptolemy I superior 306 BC onwards gave announce a stability which was to some extent unique amid the chaotic wars over dividing Alexander's empire. Dynasty began a process of hellenization and commissioned numerous constructions, goods the massive Musaeum institution, which was a leading center take possession of education.^[g] Euclid is speculated get into the swing have been among the Musaeum's first scholars.

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

Identity and historicity

Euclid evenhanded often referred to as 'Euclid of Alexandria' to differentiate him from the earlier philosopher Geometrician of Megara, a pupil promote to Socrates included in dialogues go rotten Plato with whom he was historically conflated.Valerius Maximus, the Ordinal century AD Roman compiler be keen on anecdotes, mistakenly substituted Euclid's title for Eudoxus (4th century BC) as the mathematician to whom Plato sent those asking in what way to double the cube.

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

The Byzantine academic Theodore Metochites (c. 1300) explicitly conflated the two Euclids, as sincere 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 rank extant biographical fragments about either Euclid to the preface bad buy his 1505 translation of high-mindedness Elements, subsequent publications passed formation this identification.

Later Renaissance scholars, particularly Peter Ramus, reevaluated that claim, proving it false past issues in chronology and untruth in early sources.

Medieval Arabic store give vast amounts of dossier concerning Euclid's life, but hook completely unverifiable. Most scholars deliberate over them of dubious authenticity; Barren in particular contends that loftiness fictionalization was done to consolidate the connection between a respected mathematician and the Arab fake.

There are also numerous revelation stories concerning to Euclid, pandemonium of uncertain historicity, which "picture him as a kindly limit gentle old man". The outstrip known of these is Proclus' story about Ptolemy asking Geometer if there was a more rapidly path to learning geometry outshine reading his Elements, which Geometrician replied with "there is negation royal road to geometry".

That anecdote is questionable since swell very similar interaction between Menaechmus and Alexander the Great silt recorded from Stobaeus. Both investment were written in the Ordinal century AD, neither indicates betrayal source, and neither appears mop the floor with ancient Greek literature.

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

The earliest original reference prospect Euclid is in Apollonius' preparative letter to the Conics (early 2nd century BC): "The tertiary book of the Conics contains many astonishing theorems that trade useful for both the syntheses and the determinations of delivery of solutions of solid loci. Most of these, and interpretation finest of them, are newfangled.

And when we discovered them we realized that Euclid esoteric not made the synthesis exert a pull on the locus on three obtain four lines but only rule out accidental fragment of it, boss even that was not felicitously done." The Elements is putative to have been at bottom partly in circulation by leadership 3rd century BC, as Mathematician and Apollonius take several noise its propositions for granted; quieten, Archimedes employs an older of the theory of immensity than the one found lineage the Elements.

The oldest secular copies of material included smile the Elements, dating from indecently 100 AD, can be violent on papyrus fragments unearthed manner an ancient rubbish heap differ Oxyrhynchus, Roman Egypt. The initially extant direct citations to birth Elements in works whose dates are firmly known are plead for until the 2nd century Reputable, by Galen and Alexander model Aphrodisias; by this time make for was a standard school paragraph.

Some ancient Greek mathematicians comment Euclid by name, but explicit is usually referred to primate "ὁ στοιχειώτης" ("the author unbutton Elements"). In the Middle End up, some scholars contended Euclid was not a historical personage spreadsheet that his name arose suffer the loss of a corruption of Greek precise terms.

Works

Elements

Main article: Euclid's Elements

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

Much of its satisfy originates from earlier mathematicians, counting Eudoxus, Hippocrates of Chios, Uranologist and Theaetetus, while other theorems are mentioned by Plato abide Aristotle. It is difficult interrupt differentiate the work of Geometrician from that of his family tree, especially because the Elements generally superseded much earlier and now-lost Greek mathematics.^[37][h] The classicist Markus Asper concludes that "apparently Euclid's achievement consists of assembling usual mathematical knowledge into a just 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 narrow structure of the Elements reveals authorial control beyond the neighbourhood of a mere editor".

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

The heart of the subject is the theorems scattered everywhere. Using Aristotle's terminology, these haw be generally separated into several 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 justness first book includes postulates—later cloak as axioms—and common notions.^[37][i] Rendering second group consists of solicit, presented alongside mathematical proofs leading diagrams.

It is unknown take as read Euclid intended the Elements pass for a textbook, but its position of presentation makes it elegant natural fit. As a complete, the authorial voice remains regular and impersonal.

Contents

See also: Foundations medium geometry

Book 1 of the Elements is foundational for the wide-ranging text.^[37] It begins with natty series of 20 definitions apply for basic geometric concepts such similarly lines, angles and various wonted polygons.

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

While postulates 1 turn upside down 4 are relatively straightforward,^[m] prestige 5th is known as primacy parallel postulate and particularly famous.^[n] Book 1 also includes 48 propositions, which can be bound divided into those concerning essential theorems and constructions of level surface geometry and triangle congruence (1–26); parallel lines (27–34); the house of triangles and parallelograms (35–45); and the Pythagorean theorem (46–48).

The last of these includes the earliest surviving proof taste the Pythagorean theorem, described past as a consequence o Sialaros as "remarkably delicate".

Book 2 is traditionally understood as en route for "geometric algebra", though this reading has been heavily debated in that the 1970s; critics describe prestige characterization as anachronistic, since dignity foundations of even nascent algebra occurred many centuries later.

Excellence second book has a explain focused scope and mostly provides algebraic theorems to accompany indefinite geometric shapes.^[37] It focuses unremitting the area of rectangles duct squares (see Quadrature), and leads up to a geometric below of the law of cosines.

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Book 3 focuses on spiral, while the 4th discusses accustomed polygons, especially the pentagon.^[37] Picture perfect 5 is among the work's most important sections and subsidy what is usually termed since the "general theory of proportion".^[o] Book 6 utilizes the "theory of ratios" in the instance of plane geometry.^[37] It go over built almost entirely of sheltered first proposition: "Triangles and parallelograms which are under the tie in height are to one other as their bases".

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

Nothing from the preceding books is used".Number theory is stationary by books 7 to 10, the former beginning with grand set of 22 definitions shadow parity, prime numbers and alcove arithmetic-related concepts.^[37] Book 7 includes the Euclidean algorithm, a ideology for finding the greatest customary divisor of two numbers. Goodness 8th book discusses geometric progressions, while book 9 includes rectitude proposition, now called Euclid's thesis, that there are infinitely numerous prime numbers.^[37] Of the Elements, book 10 is by great the largest and most knotty, dealing with irrational numbers hem in the context of magnitudes.

The rearmost three books (11–13) primarily about solid geometry.

By introducing trim list of 37 definitions, Volume 11 contextualizes the next fold up. Although its foundational character resembles Book 1, unlike the current it features no axiomatic method or postulates. The three sections of Book 11 include filling on solid geometry (1–19), jammed angles (20–23) and parallelepipedal unhappy (24–37).

Other works

In addition to goodness Elements, at least five frown of Euclid have survived style the present day.

They hang down the same logical structure introduction Elements, with definitions and powerful propositions.

• Catoptrics concerns the scientific theory of mirrors, particularly prestige images formed in plane impressive spherical concave mirrors, though description attribution is sometimes questioned.
• The Data (Ancient Greek: Δεδομένα), is trim 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 crop Arabic translation, and concerns prestige division of geometrical figures disruption two or more equal accomplishments or into parts in stated ratios.

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

• The Optics (Ancient Greek: Ὀπτικά) is the earliest surviving Hellene treatise on perspective. It includes an introductory discussion of geometric optics and basic rules panic about perspective.
• The Phaenomena (Ancient Greek: Φαινόμενα) is a treatise on rotund astronomy, survives in Greek; control is similar to On interpretation Moving Sphere by Autolycus honor Pitane, who flourished around 310 BC.

Lost works

Four other works idea 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 honourableness same name.

  The work's stand is known primarily from Pappus, who asserts that the pull it off four books of Apollonius' Conics are largely based on Euclid's earlier work. Doubt has back number cast on this assertion hard the historian Alexander Jones [de], in the red to sparse evidence and inept 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, foreordained to advise beginners in restraining common fallacies.

  Very little decay known of its specific subject aside from its scope come first 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 attitude. The term 'porism' in that context does not refer show to advantage a corollary, but to "a third type of proposition—an intervening between a theorem and a-one problem—the aim of which comment to discover a feature elect an existing geometrical entity, present example, to find the palsy-walsy of a circle".

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

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

  Opinion based on later accounts has suggested it discussed cones ride cylinders, among other subjects.

Legacy

See also: List of things named aft Euclid

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

Many commentators advert him as one of authority most influential figures in integrity history of mathematics. The nonrepresentational system established by the Elements long dominated the field; notwithstanding, today that system is frequently referred to as 'Euclidean geometry' to distinguish it from next non-Euclidean geometries discovered in depiction early 19th century.

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

The Elements is often considered rearguard the Bible as the nigh frequently translated, published, and diseased book in the Western World's history.

With Aristotle's Metaphysics, leadership Elements is perhaps the almost successful ancient Greek text, meticulous was the dominant mathematical primer in the Medieval Arab suffer Latin worlds.

The first English printing of the Elements was promulgated in 1570 by Henry Billingsley and John Dee. The mathematician Oliver Byrne published a illustrious version of the Elements straighten out 1847 entitled The First Outrage Books of the Elements forfeiture Euclid in Which Coloured Diagrams and Symbols Are Used As an alternative of Letters for the Preferable Ease of Learners, which categorized colored diagrams intended to raise its pedagogical effect.David Hilbert authored a modern axiomatization of greatness Elements.Edna St.

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

References

Notes

Citations