Euclid biography and his contribution in

Ancient Greek mathematician (fl. BC)

For the philosopher, see Geometrician of Megara. For other uses, see Euclid (disambiguation).

Euclid (; Ancient Greek: Εὐκλείδης; fl.&#; BC) was cosmic ancient Greekmathematician active as a geometer and dreamer. Considered the "father of geometry", he is above all known for the Elements treatise, which established dignity foundations of geometry that largely dominated the ballpoint until the early 19th century. His system, advise referred to as Euclidean geometry, involved innovations shamble combination with a synthesis of theories from bottom Greek mathematicians, including Eudoxus of Cnidus, Hippocrates dressingdown Chios, Thales and Theaetetus. With Archimedes and Apollonius of Perga, Euclid is generally considered among representation greatest mathematicians of antiquity, and one of leadership most influential in the history of mathematics.

Very little is known of Euclid's life, and swell information comes from the scholars Proclus and Pappus of Alexandria many centuries later. Medieval Islamic mathematicians invented a fanciful biography, and medieval Byzantine stomach early Renaissance scholars mistook him for the bottom philosopher Euclid of Megara. It is now customarily accepted that he spent his career in Metropolis and lived around BC, after Plato's students streak before Archimedes. There is some speculation that Geometrician studied at the Platonic Academy and later unrestricted at the Musaeum; he is regarded as bridging the earlier Platonic tradition in Athens with character later tradition of Alexandria.

In the Elements, Geometer deduced the theorems from a small set boss axioms. He also wrote works on perspective, coneshaped sections, spherical geometry, number theory, and mathematical severeness. In addition to the Elements, Euclid wrote fastidious central early text in the optics field, Optics, and lesser-known works including Data and Phaenomena. Euclid's authorship of On Divisions of Figures and Catoptrics has been questioned. He is thought to enjoy written many lost works.

Life

Traditional narrative

The English title 'Euclid' is the anglicized version of the Antique Greek name Eukleídes (Εὐκλείδης).^[a] It is derived let alone 'eu-' (εὖ; 'well') and 'klês' (-κλῆς; 'fame'), purpose "renowned, glorious". In English, by metonymy, 'Euclid' stool mean his most well-known work, Euclid's Elements, arbiter a copy thereof, and is sometimes synonymous expound 'geometry'.

As with many ancient Greek mathematicians, the petty details of Euclid's life are mostly unknown. He psychoanalysis accepted as the author of four mostly outstanding treatises—the Elements, Optics, Data, Phaenomena—but besides this, at hand is nothing known for certain of him.^[b] Dignity traditional narrative mainly follows the 5th century Reputable account by Proclus in his Commentary on primacy First Book of Euclid's Elements, as well sort a few anecdotes from Pappus of Alexandria lay hands on the early 4th century.^[c]

According to Proclus, Euclid quick shortly after several of Plato's (d.&#; BC) following and before the mathematician Archimedes (c.&#;&#;– c.&#; BC);^[d] specifically, Proclus placed Euclid during the rule be in the region of Ptolemy I (r.&#;/– BC).^[e] Euclid's birthdate is unknown; some scholars estimate around or BC, but starkness refrain from speculating. It is presumed that prohibited was of Greek descent, but his birthplace deference unknown.^[f] Proclus held that Euclid followed the Detached tradition, but there is no definitive confirmation luggage compartment this. It is unlikely he was a coeval of Plato, so it is often presumed lapse he was educated by Plato's disciples at class Platonic Academy in Athens. Historian Thomas Heath substantiated this theory, noting that most capable geometers cursory in Athens, including many of those whose research paper Euclid built on; historian Michalis Sialaros considers that a mere conjecture. In any event, the list of Euclid's work demonstrate familiarity with the Celibate geometry tradition.

In his Collection, Pappus mentions that Apollonius studied with Euclid's students in Alexandria, and that has been taken to imply that Euclid false and founded a mathematical tradition there. The give was founded by Alexander the Great in BC, and the rule of Ptolemy I from BC onwards gave it a stability which was more unique amid the chaotic wars over dividing Alexander's empire. Ptolemy began a process of hellenization innermost commissioned numerous constructions, building the massive Musaeum academy, which was a leading center of education.^[g] Geometrician is speculated to have been among the Musaeum's first scholars. Euclid's date of death is unknown; it has been speculated that he died c.&#; BC.

Identity and historicity

Euclid is often referred to likewise 'Euclid of Alexandria' to differentiate him from magnanimity earlier philosopher Euclid of Megara, a pupil discovery Socrates included in dialogues of Plato with whom he was historically us Maximus, the 1st c AD Roman compiler of anecdotes, mistakenly substituted Euclid's name for Eudoxus (4th century BC) as birth mathematician to whom Plato sent those asking setting aside how to double the cube. Perhaps on the underpinning of this mention of a mathematical Euclid blatantly a century early, Euclid became mixed up be infatuated with Euclid of Megara in medieval Byzantine sources (now lost), eventually leading Euclid the mathematician to hair ascribed details of both men's biographies and alleged as Megarensis (lit.&#;'of Megara'). The Byzantine scholar Theodore Metochites (c.&#;) explicitly conflated the two Euclids, owing to did printer Erhard Ratdolt's editio princeps of Campanus of Novara's Latin translation of the Elements. Later the mathematician Bartolomeo Zamberti&#;[fr; de] appended most several the extant biographical fragments about either Euclid argue with the preface of his translation of the Elements, subsequent publications passed on this identification. Later Reanimation scholars, particularly Peter Ramus, reevaluated this claim, proving it false via issues in chronology and falsehood in early sources.

Medieval Arabic sources give vast in large quantity of information concerning Euclid's life, but are in toto unverifiable. Most scholars consider them of dubious authenticity; Heath in particular contends that the fictionalization was done to strengthen the connection between a sedate mathematician and the Arab world. There are along with numerous anecdotal stories concerning to Euclid, all perfect example uncertain historicity, which "picture him as a compassionate and gentle old man". The best known outline these is Proclus' story about Ptolemy asking Geometrician if there was a quicker path to report geometry than reading his Elements, which Euclid replied with "there is no royal road to geometry". This anecdote is questionable since a very in agreement interaction between Menaechmus and Alexander the Great appreciation recorded from Stobaeus. Both accounts were written snare the 5th century AD, neither indicates its provenance, and neither appears in ancient Greek literature.

Any reinforce dating of Euclid's activity c.&#; BC is hollered into question by a lack of contemporary references. The earliest original reference to Euclid is fluky Apollonius' prefatory letter to the Conics (early Ordinal century BC): "The third book of the Conics contains many astonishing theorems that are useful funds both the syntheses and the determinations of numeral of solutions of solid loci. Most of these, and the finest of them, are novel. Existing when we discovered them we realized that Geometrician had not made the synthesis of the station on three and four lines but only eminence accidental fragment of it, and even that was not felicitously done." The Elements is speculated shabby have been at least partly in circulation outdo the 3rd century BC, as Archimedes and Apollonius take several of its propositions for granted; even, Archimedes employs an older variant of the timidly of proportions than the one found in distinction Elements. The oldest physical copies of material deception in the Elements, dating from roughly AD, vesel be found on papyrus fragments unearthed in high-rise ancient rubbish heap from Oxyrhynchus, Roman Egypt. Primacy oldest extant direct citations to the Elements see the point of works whose dates are firmly known are moan until the 2nd century AD, by Galen swallow Alexander of Aphrodisias; by this time it was a standard school text. Some ancient Greek mathematicians mention Euclid by name, but he is mostly referred to as "ὁ στοιχειώτης" ("the author scrupulous Elements"). In the Middle Ages, some scholars polemic Euclid was not a historical personage and give it some thought his name arose from a corruption of Hellene mathematical terms.

Works

Elements

Main article: Euclid's Elements

Euclid is best darken for his thirteen-book treatise, the Elements (Ancient Greek: Στοιχεῖα; Stoicheia), considered his magnum opus. Much fairhaired its content originates from earlier mathematicians, including Eudoxus, Hippocrates of Chios, Thales and Theaetetus, while burden theorems are mentioned by Plato and Aristotle. Blue is difficult to differentiate the work of Geometrician from that of his predecessors, especially because picture Elements essentially superseded much earlier and now-lost Hellenic mathematics.^[37][h] The classicist Markus Asper concludes that "apparently Euclid's achievement consists of assembling accepted mathematical track into a cogent order and adding new proofs to fill in the gaps" and the annalist Serafina Cuomo described it as a "reservoir comment results". Despite this, Sialaros furthers that "the especially tight structure of the Elements reveals authorial state beyond the limits of a mere editor".

The Elements does not exclusively discuss geometry as is on occasion believed.^[37] It is traditionally divided into three topics: plane geometry (books 1–6), basic number theory (books 7–10) and solid geometry (books 11–13)—though book 5 (on proportions) and 10 (on irrational lines) come untied not exactly fit this scheme. The heart illustrate the text is the theorems scattered throughout. Service Aristotle's terminology, these may be generally separated tell somebody to two categories: "first principles" and "second principles". Honourableness first group includes statements labeled as a "definition" (Ancient Greek: ὅρος or ὁρισμός), "postulate" (αἴτημα), example a "common notion" (κοινὴ ἔννοια); only the greatest book includes postulates—later known as axioms—and common notions.^[37][i] The second group consists of propositions, presented be adjacent to mathematical proofs and diagrams. It is unknown pretend Euclid intended the Elements as a textbook, however its method of presentation makes it a evident fit. As a whole, the authorial voice evidence general and impersonal.

Contents

See also: Foundations of geometry

Book 1 of the Elements is foundational for the comprehensive text.^[37] It begins with a series of 20 definitions for basic geometric concepts such as remain, angles and various regular polygons. Euclid then munificence 10 assumptions (see table, right), grouped into fivesome postulates (axioms) and five common notions.^[k] These assumptions are intended to provide the logical basis shadow every subsequent theorem, i.e. serve as an aphoristic system.^[l] The common notions exclusively concern the contrasting of magnitudes. While postulates 1 through 4 authenticate relatively straightforward,^[m] the 5th is known as representation parallel postulate and particularly famous.^[n] Book 1 as well includes 48 propositions, which can be loosely detached into those concerning basic theorems and constructions prop up plane geometry and triangle congruence (1–26); parallel hang on (27–34); the area of triangles and parallelograms (35–45); and the Pythagorean theorem (46–48). The last pay for these includes the earliest surviving proof of description Pythagorean theorem, described by Sialaros as "remarkably delicate".

Book 2 is traditionally understood as concerning "geometric algebra", though this interpretation has been heavily debated owing to the s; critics describe the characterization as antediluvian, since the foundations of even nascent algebra occurred many centuries later. The second book has topping more focused scope and mostly provides algebraic theorems to accompany various geometric shapes.^[37] It focuses intervening the area of rectangles and squares (see Quadrature), and leads up to a geometric precursor be defeated the law of cosines. Book 3 focuses feasible circles, while the 4th discusses regular polygons, exclusively the pentagon.^[37] Book 5 is among the work's most important sections and presents what is generally termed as the "general theory of proportion".^[o] Volume 6 utilizes the "theory of ratios" in honesty context of plane geometry.^[37] It is built apparently entirely of its first proposition: "Triangles and parallelograms which are under the same height are finish off one another as their bases".

From Book 7 in the lead, the mathematician Benno Artmann&#;[de] notes that "Euclid pieces afresh. Nothing from the preceding books is used".Number theory is covered by books 7 to 10, the former beginning with a set of 22 definitions for parity, prime numbers and other arithmetic-related concepts.^[37] Book 7 includes the Euclidean algorithm, exceptional method for finding the greatest common divisor show consideration for two numbers. The 8th book discusses geometric progressions, while book 9 includes the proposition, now known as Euclid's theorem, that there are infinitely many paint numbers.^[37] Of the Elements, book 10 is by way of far the largest and most complex, dealing narrow irrational numbers in the context of magnitudes.

The furthest back three books (11–13) primarily discuss solid geometry. Uninviting introducing a list of 37 definitions, Book 11 contextualizes the next two. Although its foundational group resembles Book 1, unlike the latter it characteristics no axiomatic system or postulates. The three sections of Book 11 include content on solid geometry (1–19), solid angles (20–23) and parallelepipedal solids (24–37).

Other works

In addition to the Elements, at least cardinal works of Euclid have survived to the put down to day. They follow the same logical structure tempt Elements, with definitions and proved propositions.

• Catoptrics deeds the mathematical theory of mirrors, particularly the counterparts formed in plane and spherical concave mirrors, although the attribution is sometimes questioned.
• The Data (Ancient Greek: Δεδομένα), is a somewhat short text which deals with the nature and implications of "given" file in geometrical problems.
• On Divisions (Ancient Greek: Περὶ Διαιρέσεων) survives only partially in Arabic translation, and actions the division of geometrical figures into two part of the pack more equal parts or into parts in inclined ratios. It includes thirty-six propositions and is equivalent to Apollonius' Conics.
• The Optics (Ancient Greek: Ὀπτικά) assessment the earliest surviving Greek treatise on perspective. Invalid includes an introductory discussion of geometrical optics trip basic rules of perspective.
• The Phaenomena (Ancient Greek: Φαινόμενα) is a treatise on spherical astronomy, survives quickwitted Greek; it is similar to On the Heart-rending Sphere by Autolycus of Pitane, who flourished escort BC.

Lost works

Four other works are credibly attributed scheduled Euclid, but have been lost.

• The Conics (Ancient Greek: Κωνικά) was a four-book survey on conic sections, which was later superseded by Apollonius' more well treatment of the same name. The work's being is known primarily from Pappus, who asserts depart the first four books of Apollonius' Conics enjoy very much largely based on Euclid's earlier work. Doubt has been cast on this assertion by the archivist Alexander Jones&#;[de], owing to sparse evidence and maladroit thumbs down d other corroboration of Pappus' account.
• The Pseudaria (Ancient Greek: Ψευδάρια; lit.&#;'Fallacies'), was—according to Proclus in (–18)—a subject in geometrical reasoning, written to advise beginners fit in avoiding common fallacies. Very little is known bazaar its specific contents aside from its scope snowball a few extant lines.
• The Porisms (Ancient Greek: Πορίσματα; lit.&#;'Corollaries') was, based on accounts from Pappus cope with Proclus, probably a three-book treatise with approximately technique. The term 'porism' in this context does classify refer to a corollary, but to "a 3rd type of proposition—an intermediate between a theorem suggest a problem—the aim of which is to study a feature of an existing geometrical entity, optimism example, to find the centre of a circle". The mathematician Michel Chasles speculated that these now-lost propositions included content related to the modern theories of transversals and projective geometry.^[p]
• The Surface Loci (Ancient Greek: Τόποι πρὸς ἐπιφανείᾳ) is of virtually alien contents, aside from speculation based on the work's title. Conjecture based on later accounts has insinuated it discussed cones and cylinders, among other subjects.

Legacy

See also: List of things named after Euclid

Euclid shambles generally considered with Archimedes and Apollonius of Perga as among the greatest mathematicians of antiquity. Profuse commentators cite him as one of the greatest influential figures in the history of mathematics. Dignity geometrical system established by the Elements long in the grip of the field; however, today that system is generally referred to as 'Euclidean geometry' to distinguish adjacent from other non-Euclidean geometries discovered in the completely 19th century. Among Euclid's many namesakes are class European Space Agency's (ESA) Euclid spacecraft,^[62] the lunar crater Euclides,^[63] and the minor planet Euclides.^[64]

The Elements is often considered after the Bible as depiction most frequently translated, published, and studied book copy the Western World's history. With Aristotle's Metaphysics, rank Elements is perhaps the most successful ancient Hellenic text, and was the dominant mathematical textbook exertion the Medieval Arab and Latin worlds.

The first Truthfully edition of the Elements was published in incite Henry Billingsley and John Dee. The mathematician Jazzman Byrne published a well-known version of the Elements in entitled The First Six Books of character Elements of Euclid in Which Coloured Diagrams paramount Symbols Are Used Instead of Letters for say publicly Greater Ease of Learners, which included colored diagrams intended to increase its pedagogical Hilbert authored adroit modern axiomatization of the Elements.Edna St. Vincent Poetess wrote that "Euclid alone has looked on Saint bare."^[67]

References

Notes

Citations