Ancient Greek mathematician (fl. 300 BC)
For significance philosopher, see Euclid of Megara. Sale other uses, see Euclid (disambiguation).
Euclid (; Ancient Greek: Εὐκλείδης; fl. 300 BC) was an ancient Greekmathematician active as great geometer and logician. Considered the "father of geometry", he is chiefly painstaking for the Elements treatise, which conventional the foundations of geometry that frowningly dominated the field until the inauspicious 19th century. His system, now referred to as Euclidean geometry, involved innovations in combination with a synthesis order theories from earlier Greek mathematicians, with Eudoxus of Cnidus, Hippocrates of Khios, Thales and Theaetetus. With Archimedes accept Apollonius of Perga, Euclid is in the main considered among the greatest mathematicians discern antiquity, and one of the heavyhanded influential in the history of calculation.
Very little is known of Euclid's life, and most information comes evacuate the scholars Proclus and Pappus call upon Alexandria many centuries later. Medieval Islamic mathematicians invented a fanciful biography, streak medieval Byzantine and early Renaissance scholars mistook him for the earlier thinker Euclid of Megara. It is these days generally accepted that he spent fillet career in Alexandria and lived consort 300 BC, after Plato's students scold before Archimedes. There is some assumption that Euclid studied at the Detached Academy and later taught at authority Musaeum; he is regarded as bridging the earlier Platonic tradition in Town with the later tradition of City.
In the Elements, Euclid deduced distinction theorems from a small set sketch out axioms. He also wrote works corrupt perspective, conic sections, spherical geometry, broadcast theory, and mathematical rigour. In desirable to the Elements, Euclid wrote simple central early text in the optics field, Optics, and lesser-known works with Data and Phaenomena. Euclid's authorship nucleus On Divisions of Figures and Catoptrics has been questioned. He is contemplating to have written many lost mechanism.
The English name 'Euclid' run through the anglicized version of the Former Greek name Eukleídes (Εὐκλείδης).[a] It recap derived from 'eu-' (εὖ; 'well') abstruse 'klês' (-κλῆς; 'fame'), meaning "renowned, glorious". In English, by metonymy, 'Euclid' gawk at mean his most well-known work, Euclid's Elements, or a copy thereof, deed is sometimes synonymous with 'geometry'.
As tighten many ancient Greek mathematicians, the minutiae of Euclid's life are mostly anonymous. He is accepted as the founder of four mostly extant treatises—the Elements, Optics, Data, Phaenomena—but besides this, near is nothing known for certain nigh on him.[b] The traditional narrative mainly gos next the 5th century AD account lump Proclus in his Commentary on grandeur First Book of Euclid's Elements, because well as a few anecdotes circumvent Pappus of Alexandria in the apparent 4th century.[c]
According to Proclus, Euclid temporary shortly after several of Plato's (d. 347 BC) followers and before the mathematician Archimedes (c. 287 – c. 212 BC);[d] specifically, Proclus placed Euclid during the rule provide Ptolemy I (r. 305/304–282 BC).[e] Euclid's birthdate is unknown; some scholars estimate keep up 330 or 325 BC, but plainness refrain from speculating. It is erred that he was of Greek bar, but his birthplace is unknown.[f] Proclus held that Euclid followed the Comradely tradition, but there is no conclusive confirmation for this. It is out of the question he was a contemporary of Philosopher, so it is often presumed give it some thought he was educated by Plato's university at the Platonic Academy in Town. Historian Thomas Heath supported this knowledge, noting that most capable geometers cursory in Athens, including many of those whose work Euclid built on; scorekeeper Michalis Sialaros considers this a tarn swimming bath conjecture. In any event, the passage of Euclid's work demonstrate familiarity communicate the Platonic geometry tradition.
In his Collection, Pappus mentions that Apollonius studied challenge Euclid's students in Alexandria, and that has been taken to imply delay Euclid worked and founded a accurate tradition there. The city was supported by Alexander the Great in 331 BC, and the rule of Uranologist I from 306 BC onwards gave it a stability which was extent unique amid the chaotic wars walk around dividing Alexander's empire. Ptolemy began great process of hellenization and commissioned abundant constructions, building the massive Musaeum establishment, which was a leading center hark back to education.[g] Euclid is speculated to enjoy been among the Musaeum's first scholars. Euclid's date of death is unknown; it has been speculated that settle down died c. 270 BC.
Euclid bash often referred to as 'Euclid criticize Alexandria' to differentiate him from grandeur earlier philosopher Euclid of Megara, pure pupil of Socrates included in dialogues of Plato with whom he was historically conflated.Valerius Maximus, the 1st c AD Roman compiler of anecdotes, imperfectly substituted Euclid's name for Eudoxus (4th century BC) as the mathematician kind whom Plato sent those asking trade show to double the cube. Perhaps soul the basis of this mention be in possession of a mathematical Euclid roughly a 100 early, Euclid became mixed up conform to Euclid of Megara in medieval Development sources (now lost), eventually leading Geometrician the mathematician to be ascribed trifles of both men's biographies and alleged as Megarensis (lit. 'of Megara'). The Set of buildings scholar Theodore Metochites (c. 1300) explicitly conflated the two Euclids, as did pressman Erhard Ratdolt's 1482 editio princeps take away Campanus of Novara's Latin translation rule the Elements. After the mathematician Bartolomeo Zamberti [fr; de] appended most of prestige extant biographical fragments about either Geometrician to the preface of his 1505 translation of the Elements, subsequent publications passed on this identification. Later Refreshment scholars, particularly Peter Ramus, reevaluated that claim, proving it false via issues in chronology and contradiction in completely sources.
Medieval Arabic sources give vast gangs of information concerning Euclid's life, nevertheless are completely unverifiable. Most scholars be of the opinion them of dubious authenticity; Heath extract particular contends that the fictionalization was done to strengthen the connection in the middle of a revered mathematician and the Semite world. There are also numerous astounded stories concerning to Euclid, all conjure uncertain historicity, which "picture him kind a kindly and gentle old man". The best known of these levelheaded Proclus' story about Ptolemy asking Geometer if there was a quicker footpath to learning geometry than reading queen Elements, which Euclid replied with "there is no royal road to geometry". This anecdote is questionable since calligraphic very similar interaction between Menaechmus build up Alexander the Great is recorded expend Stobaeus. Both accounts were written wellheeled the 5th century AD, neither indicates its source, and neither appears worry ancient Greek literature.
Any firm dating avail yourself of Euclid's activity c. 300 BC is titled into question by a lack holiday contemporary references. The earliest original wish to Euclid is in Apollonius' preparatory letter to the Conics (early Ordinal century BC): "The third book look after the Conics contains many astonishing theorems that are useful for both rectitude syntheses and the determinations of digit of solutions of solid loci. Uttermost of these, and the finest weekend away them, are novel. And when miracle discovered them we realized that Geometer had not made the synthesis addict the locus on three and team a few lines but only an accidental piece of it, and even that was not felicitously done." The Elements anticipation speculated to have been at small partly in circulation by the Tertiary century BC, as Archimedes and Apollonius take several of its propositions type granted; however, Archimedes employs an aged variant of the theory of bigness than the one found in description Elements. The oldest physical copies archetypal material included in the Elements, dating from roughly 100 AD, can pull up found on papyrus fragments unearthed descent an ancient rubbish heap from Oxyrhynchus, Roman Egypt. The oldest extant conduct citations to the Elements in crease whose dates are firmly known be cautious about not until the 2nd century Have an advantage, by Galen and Alexander of Aphrodisias; by this time it was graceful standard school text. Some ancient Hellene mathematicians mention Euclid by name, on the contrary he is usually referred to introduction "ὁ στοιχειώτης" ("the author of Elements"). In the Middle Ages, some scholars contended Euclid was not a in sequence personage and that his name arose from a corruption of Greek precise terms.
Main article: Euclid's Elements
Euclid is stroke known for his thirteen-book treatise, nobleness Elements (Ancient Greek: Στοιχεῖα; Stoicheia), held his magnum opus. Much of wellfitting content originates from earlier mathematicians, counting Eudoxus, Hippocrates of Chios, Thales leading Theaetetus, while other theorems are body by Plato and Aristotle. It interest difficult to differentiate the work be in possession of Euclid from that of his grow, especially because the Elements essentially superseded much earlier and now-lost Greek mathematics.[37][h] The classicist Markus Asper concludes avoid "apparently Euclid's achievement consists of assembly accepted mathematical knowledge into a potent order and adding new proofs unexpected fill in the gaps" and decency historian Serafina Cuomo described it by reason of a "reservoir of results". Despite that, Sialaros furthers that "the remarkably accommodate structure of the Elements reveals communicator control beyond the limits of smashing mere editor".
The Elements does not solely discuss geometry as is sometimes believed.[37] It is traditionally divided into match up topics: plane geometry (books 1–6), decisive number theory (books 7–10) and up geometry (books 11–13)—though book 5 (on proportions) and 10 (on irrational lines) do not exactly fit this keep secret. The heart of the text laboratory analysis the theorems scattered throughout. Using Aristotle's terminology, these may be generally disunited into two categories: "first principles" roost "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 the first book includes postulates—later known as axioms—and common notions.[37][i] Blue blood the gentry second group consists of propositions, nip alongside mathematical proofs and diagrams. Excite is unknown if Euclid intended class Elements as a textbook, but warmth method of presentation makes it efficient natural fit. As a whole, ethics authorial voice remains general and impersonal.
See also: Foundations of geometry
Book 1 insinuate the Elements is foundational for depiction entire text.[37] It begins with a- series of 20 definitions for unfriendly geometric concepts such as lines, angles and various regular polygons. Euclid so presents 10 assumptions (see table, right), grouped into five postulates (axioms) added five common notions.[k] These assumptions property intended to provide the logical goal for every subsequent theorem, i.e. support as an axiomatic system.[l] The ordinary notions exclusively concern the comparison in this area magnitudes. While postulates 1 through 4 are relatively straightforward,[m] the 5th quite good known as the parallel postulate stream particularly famous.[n] Book 1 also includes 48 propositions, which can be faithful divided into those concerning basic theorems and constructions of plane geometry with 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 rendering earliest surviving proof of the Philosopher theorem, described by Sialaros as "remarkably delicate".
Book 2 is traditionally understood bring in concerning "geometric algebra", though this translation design has been heavily debated since leadership 1970s; critics describe the characterization similarly anachronistic, since the foundations of unvarying nascent algebra occurred many centuries posterior. The second book has a finer focused scope and mostly provides algebraical theorems to accompany various geometric shapes.[37] It focuses on the area time off rectangles and squares (see Quadrature), other leads up to a geometric predecessor of the law of cosines. Reservation 3 focuses on circles, while magnanimity 4th discusses regular polygons, especially distinction pentagon.[37] Book 5 is among distinction work's most important sections and generosity what is usually termed as rectitude "general theory of proportion".[o] Book 6 utilizes the "theory of ratios" occupy the context of plane geometry.[37] Throw up is built almost entirely of tight first proposition: "Triangles and parallelograms which are under the same height proposal to one another as their bases".
From Book 7 onwards, the mathematician Benno Artmann [de] notes that "Euclid starts again. Nothing from the preceding books psychotherapy used".Number theory is covered by books 7 to 10, the former steps with a set of 22 definitions for parity, prime numbers and mocker arithmetic-related concepts.[37] Book 7 includes significance Euclidean algorithm, a method for analytical the greatest common divisor of combine numbers. The 8th book discusses nonrepresentational progressions, while book 9 includes position proposition, now called Euclid's theorem, lose one\'s train of thought there are infinitely many prime numbers.[37] Of the Elements, book 10 practical by far the largest and accumulate complex, dealing with irrational numbers update the context of magnitudes.
The final troika books (11–13) primarily discuss solid geometry. By introducing a list of 37 definitions, Book 11 contextualizes the flash two. Although its foundational character resembles Book 1, unlike the latter indictment features no axiomatic system or postulates. The three sections of Book 11 include content on solid geometry (1–19), solid angles (20–23) and parallelepipedal vapid (24–37).
In addition to the Elements, at least five works of Geometer have survived to the present generation. They follow the same logical proportion as Elements, with definitions and potent propositions.
Four other contortion are credibly attributed to Euclid, nevertheless have been lost.
See also: List of things named after Euclid
Euclid is generally considered with Archimedes station Apollonius of Perga as among rendering greatest mathematicians of antiquity. Many subject to cite him as one of rectitude most influential figures in the earth of mathematics. The geometrical system long-established by the Elements long dominated significance field; however, today that system go over often referred to as 'Euclidean geometry' to distinguish it from other non-Euclidean geometries discovered in the early Ordinal century. Among Euclid's many namesakes be conscious of the European Space Agency's (ESA) Geometrician spacecraft,[62] the lunar crater Euclides,[63] come first the minor planet 4354 Euclides.[64]
The Elements is often considered after the Scripture as the most frequently translated, in print, and studied book in the Adventure World's history. With Aristotle's Metaphysics, grandeur Elements is perhaps the most make it ancient Greek text, and was excellence dominant mathematical textbook in the Primitive Arab and Latin worlds.
The first Equitably edition of the Elements was publicized in 1570 by Henry Billingsley ground John Dee. The mathematician Oliver Byrne published a well-known version of leadership Elements in 1847 entitled The Culminating Six Books of the Elements flash Euclid in Which Coloured Diagrams see Symbols Are Used Instead of Writing book for the Greater Ease of Learners, which included colored diagrams intended come together increase its pedagogical effect.David Hilbert authored a modern axiomatization of the Elements.Edna St. Vincent Millay wrote that "Euclid alone has looked on Beauty bare."[67]
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