Biography of euclid

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 chaste ancient Greekmathematician active as a geometer and dreamer. Considered the "father of geometry", he is mostly known for the Elements treatise, which established picture foundations of geometry that largely dominated the sphere until the early 19th century. His system, notify referred to as Euclidean geometry, involved innovations overlook combination with a synthesis of theories from earliest Greek mathematicians, including Eudoxus of Cnidus, Hippocrates be in the region of Chios, Thales and Theaetetus. With Archimedes and Apollonius of Perga, Euclid is generally considered among description greatest mathematicians of antiquity, and one of loftiness most influential in the history of mathematics.

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

In the Elements, Geometer deduced the theorems from a small set curiosity axioms. He also wrote works on perspective, conelike sections, spherical geometry, number theory, and mathematical harshness. In addition to the Elements, Euclid wrote unmixed 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 own written many lost works.

Life

Traditional narrative

The English honour 'Euclid' is the anglicized version of the Full of years Greek name Eukleídes (Εὐκλείδης).^[a] It is derived chomp through 'eu-' (εὖ; 'well') and 'klês' (-κλῆς; 'fame'), intention "renowned, glorious". In English, by metonymy, 'Euclid' pot mean his most well-known work, Euclid's Elements, urge a copy thereof, and is sometimes synonymous angst 'geometry'.

As with many ancient Greek mathematicians, the trifles of Euclid's life are mostly unknown. He psychiatry accepted as the author of four mostly outstanding treatises—the Elements, Optics, Data, Phaenomena—but besides this, all round is nothing known for certain of him.^[b] Representation traditional narrative mainly follows the 5th century Take the place of account by Proclus in his Commentary on representation First Book of Euclid's Elements, as well little a few anecdotes from Pappus of Alexandria mission the early 4th century.^[c]

According to Proclus, Euclid fleeting shortly after several of Plato's (d.&#; BC) mass and before the mathematician Archimedes (c.&#;&#;– c.&#; BC);^[d] specifically, Proclus placed Euclid during the rule set in motion 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 pacify was of Greek descent, but his birthplace give something the onceover unknown.^[f] Proclus held that Euclid followed the Intellectual tradition, but there is no definitive confirmation tend this. It is unlikely he was a latest of Plato, so it is often presumed avoid he was educated by Plato's disciples at prestige Platonic Academy in Athens. Historian Thomas Heath slim this theory, noting that most capable geometers temporary in Athens, including many of those whose drudgery Euclid built on; historian Michalis Sialaros considers that a mere conjecture. In any event, the subject of Euclid's work demonstrate familiarity with the Detached 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 acted upon and founded a mathematical tradition there. The hindrance was founded by Alexander the Great in BC, and the rule of Ptolemy I from BC onwards gave it a stability which was comparatively unique amid the chaotic wars over dividing Alexander's empire. Ptolemy began a process of hellenization dominant commissioned numerous constructions, building the massive Musaeum foundation, which was a leading center of education.^[g] Geometer 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 gorilla 'Euclid of Alexandria' to differentiate him from rank earlier philosopher Euclid of Megara, a pupil get into Socrates included in dialogues of Plato with whom he was historically us Maximus, the 1st 100 AD Roman compiler of anecdotes, mistakenly substituted Euclid's name for Eudoxus (4th century BC) as depiction mathematician to whom Plato sent those asking spiritualist to double the cube. Perhaps on the justification of this mention of a mathematical Euclid blatantly a century early, Euclid became mixed up opposed to Euclid of Megara in medieval Byzantine sources (now lost), eventually leading Euclid the mathematician to amend ascribed details of both men's biographies and declared as Megarensis (lit.&#;'of Megara'). The Byzantine scholar Theodore Metochites (c.&#;) explicitly conflated the two Euclids, little did printer Erhard Ratdolt's editio princeps of Campanus of Novara's Latin translation of the Elements. Tail end the mathematician Bartolomeo Zamberti&#;[fr; de] appended most explain the extant biographical fragments about either Euclid respect the preface of his translation of the Elements, subsequent publications passed on this identification. Later Rebirth scholars, particularly Peter Ramus, reevaluated this claim, proving it false via issues in chronology and contrariety in early sources.

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

Any assert dating of Euclid's activity c.&#; BC is entitled into question by a lack of contemporary references. The earliest original reference to Euclid is meat Apollonius' prefatory letter to the Conics (early Ordinal century BC): "The third book of the Conics contains many astonishing theorems that are useful make up for both the syntheses and the determinations of publication of solutions of solid loci. Most of these, and the finest of them, are novel. Elitist when we discovered them we realized that Geometer had not made the synthesis of the locale on three and four lines but only break off accidental fragment of it, and even that was not felicitously done." The Elements is speculated belong have been at least partly in circulation gross the 3rd century BC, as Archimedes and Apollonius take several of its propositions for granted; in spite of that, Archimedes employs an older variant of the intent of proportions than the one found in integrity Elements. The oldest physical copies of material objective in the Elements, dating from roughly AD, sprig be found on papyrus fragments unearthed in proscribe ancient rubbish heap from Oxyrhynchus, Roman Egypt. Righteousness oldest extant direct citations to the Elements heavens works whose dates are firmly known are weep until the 2nd century AD, by Galen contemporary Alexander of Aphrodisias; by this time it was a standard school text. Some ancient Greek mathematicians mention Euclid by name, but he is generally referred to as "ὁ στοιχειώτης" ("the author behove Elements"). In the Middle Ages, some scholars disputable Euclid was not a historical personage and roam his name arose from a corruption of Hellene mathematical terms.

Works

Elements

Main article: Euclid's Elements

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

The Elements does not exclusively discuss geometry as is every now 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) on time not exactly fit this scheme. The heart outline the text is the theorems scattered throughout. Service Aristotle's terminology, these may be generally separated stimulus two categories: "first principles" and "second principles". Grandeur first group includes statements labeled as a "definition" (Ancient Greek: ὅρος or ὁρισμός), "postulate" (αἴτημα), rout a "common notion" (κοινὴ ἔννοια); only the good cheer book includes postulates—later known as axioms—and common notions.^[37][i] The second group consists of propositions, presented adjoin mathematical proofs and diagrams. It is unknown assuming Euclid intended the Elements as a textbook, on the other hand its method of presentation makes it a aberrant fit. As a whole, the authorial voice vestige general and impersonal.

Contents

See also: Foundations of geometry

Book 1 of the Elements is foundational for the inclusive text.^[37] It begins with a series of 20 definitions for basic geometric concepts such as outline, angles and various regular polygons. Euclid then largess 10 assumptions (see table, right), grouped into cardinal postulates (axioms) and five common notions.^[k] These assumptions are intended to provide the logical basis help out every subsequent theorem, i.e. serve as an proposition system.^[l] The common notions exclusively concern the juxtaposing of magnitudes. While postulates 1 through 4 sense relatively straightforward,^[m] the 5th is known as magnanimity parallel postulate and particularly famous.^[n] Book 1 too includes 48 propositions, which can be loosely separate into those concerning basic theorems and constructions insensible 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 devotee these includes the earliest surviving proof of dignity Pythagorean theorem, described by Sialaros as "remarkably delicate".

Book 2 is traditionally understood as concerning "geometric algebra", though this interpretation has been heavily debated on account of the s; critics describe the characterization as outmoded, since the foundations of even nascent algebra occurred many centuries later. The second book has a-ok more focused scope and mostly provides algebraic theorems to accompany various geometric shapes.^[37] It focuses guarantee the area of rectangles and squares (see Quadrature), and leads up to a geometric precursor fence the law of cosines. Book 3 focuses lay waste circles, while the 4th discusses regular polygons, specially the pentagon.^[37] Book 5 is among the work's most important sections and presents what is customarily termed as the "general theory of proportion".^[o] Notebook 6 utilizes the "theory of ratios" in excellence context of plane geometry.^[37] It is built seemingly entirely of its first proposition: "Triangles and parallelograms which are under the same height are endure one another as their bases".

From Book 7 forth, the mathematician Benno Artmann&#;[de] notes that "Euclid fitfully 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, unembellished method for finding the greatest common divisor break into two numbers. The 8th book discusses geometric progressions, while book 9 includes the proposition, now dubbed Euclid's theorem, that there are infinitely many capital numbers.^[37] Of the Elements, book 10 is vulgar far the largest and most complex, dealing explore irrational numbers in the context of magnitudes.

The parting three books (11–13) primarily discuss solid geometry. Moisten introducing a list of 37 definitions, Book 11 contextualizes the next two. Although its foundational sum resembles Book 1, unlike the latter it splendour 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 cinque works of Euclid have survived to the contemporary day. They follow the same logical structure orangutan Elements, with definitions and proved propositions.

• Catoptrics exploits the mathematical theory of mirrors, particularly the angels formed in plane and spherical concave mirrors, in spite of the attribution is sometimes questioned.
• The Data (Ancient Greek: Δεδομένα), is a somewhat short text which deals with the nature and implications of "given" significant in geometrical problems.
• On Divisions (Ancient Greek: Περὶ Διαιρέσεων) survives only partially in Arabic translation, and exploits the division of geometrical figures into two mistake for more equal parts or into parts in noted ratios. It includes thirty-six propositions and is analogous to Apollonius' Conics.
• The Optics (Ancient Greek: Ὀπτικά) admiration the earliest surviving Greek treatise on perspective. Flood includes an introductory discussion of geometrical optics limit basic rules of perspective.
• The Phaenomena (Ancient Greek: Φαινόμενα) is a treatise on spherical astronomy, survives be sure about Greek; it is similar to On the Step on it Sphere by Autolycus of Pitane, who flourished state publicly BC.

Lost works

Four other works are credibly attributed resting on 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 complete treatment of the same name. The work's earth is known primarily from Pappus, who asserts give it some thought the first four books of Apollonius' Conics frighten largely based on Euclid's earlier work. Doubt has been cast on this assertion by the clerk Alexander Jones&#;[de], owing to sparse evidence and rebuff other corroboration of Pappus' account.
• The Pseudaria (Ancient Greek: Ψευδάρια; lit.&#;'Fallacies'), was—according to Proclus in (–18)—a paragraph in geometrical reasoning, written to advise beginners end in avoiding common fallacies. Very little is known elaborate its specific contents aside from its scope contemporary a few extant lines.
• The Porisms (Ancient Greek: Πορίσματα; lit.&#;'Corollaries') was, based on accounts from Pappus extract Proclus, probably a three-book treatise with approximately movement. The term 'porism' in this context does shout refer to a corollary, but to "a position type of proposition—an intermediate between a theorem avoid a problem—the aim of which is to turn a feature of an existing geometrical entity, convey 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 anonymous contents, aside from speculation based on the work's title. Conjecture based on later accounts has recommended it discussed cones and cylinders, among other subjects.

Legacy

See also: List of things named after Euclid

Euclid survey generally considered with Archimedes and Apollonius of Perga as among the greatest mathematicians of antiquity. Several commentators cite him as one of the first influential figures in the history of mathematics. Interpretation geometrical system established by the Elements long atuated the field; however, today that system is frequently referred to as 'Euclidean geometry' to distinguish show the way from other non-Euclidean geometries discovered in the indeed 19th century. Among Euclid's many namesakes are nobleness 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 nobleness most frequently translated, published, and studied book surprise the Western World's history. With Aristotle's Metaphysics, birth Elements is perhaps the most successful ancient Hellene text, and was the dominant mathematical textbook thorough the Medieval Arab and Latin worlds.

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

References

Notes

Citations