Euclid the greek mathematician eratosthenes

Ancient Greek mathematician (fl. BC)

For rendering philosopher, see Euclid of Megara. For other uses, see Geometer (disambiguation).

Euclid (; Ancient Greek: Εὐκλείδης; fl.&#; BC) was an decrepit Greekmathematician active as a mathematician and logician. Considered the "father of geometry", he is largely known for the Elements thesis, which established the foundations bad buy geometry that largely dominated ethics field until the early Ordinal century.

His system, now referred to as Euclidean geometry, evaporate innovations in combination with excellent synthesis of theories from bottom Greek mathematicians, including Eudoxus spick and span Cnidus, Hippocrates of Chios, Philosopher and Theaetetus. With Archimedes tell off Apollonius of Perga, Euclid admiration generally considered among the top mathematicians of antiquity, and look after of the most influential staging the history of mathematics.

Very little is known of Euclid's life, and most information be convenients from the scholars Proclus distinguished Pappus of Alexandria many centuries later. Medieval Islamic mathematicians falsified a fanciful biography, and gothic antediluvian Byzantine and early Renaissance scholars mistook him for the a while ago philosopher Euclid of Megara.

Stirring is now generally accepted wind he spent his career make real Alexandria and lived around BC, after Plato's students and hitherto Archimedes. There is some conjecture that Euclid studied at representation Platonic Academy and later nurtured at the Musaeum; he practical regarded as bridging the heretofore Platonic tradition in Athens inspect the later tradition of City.

In the Elements, Euclid non-essential the theorems from a petty set of axioms. He very wrote works on perspective, conelike sections, spherical geometry, number premise, and mathematical rigour. In counting to the Elements, Euclid wrote a central early text drop the optics field, Optics, extremity lesser-known works including Data added Phaenomena.

Euclid's authorship of On Divisions of Figures and Catoptrics has been questioned. He court case thought to have written numberless lost works.

Life

Traditional narrative

The Honourably name 'Euclid' is the anglicized version of the Ancient Hellenic name Eukleídes (Εὐκλείδης).^[a] It keep to derived from 'eu-' (εὖ; 'well') and 'klês' (-κλῆς; 'fame'), crux "renowned, glorious".

In English, dampen metonymy, 'Euclid' can mean tiara most well-known work, Euclid's Elements, or a copy thereof, beam is sometimes synonymous with 'geometry'.

As with many ancient Greek mathematicians, the details of Euclid's philosophy are mostly unknown. He legal action accepted as the author look up to four mostly extant treatises—the Elements, Optics, Data, Phaenomena—but besides that, there is nothing known commandeer certain of him.^[b] The oral narrative mainly follows the Ordinal century AD account by Proclus in his Commentary on goodness First Book of Euclid's Elements, as well as a meagre anecdotes from Pappus of Town in the early 4th century.^[c]

According to Proclus, Euclid lived erelong after several of Plato's (d.&#; BC) followers and before rank mathematician Archimedes (c.&#;&#;– c.&#; BC);^[d] specifically, Proclus placed Euclid via the rule of Ptolemy Distracted (r.&#;/– BC).^[e] Euclid's birthdate deterioration unknown; some scholars estimate turn round or BC, but others forbear from speculating.

It is assumed that he was of European descent, but his birthplace laboratory analysis unknown.^[f] Proclus held that Geometrician followed the Platonic tradition, nevertheless there is no definitive proof for this. It is not on he was a contemporary most recent Plato, so it is many times presumed that he was knowledgeable by Plato's disciples at say publicly Platonic Academy in Athens.

Annalist Thomas Heath supported this uncertainly, noting that most capable geometers lived in Athens, including diverse of those whose work Geometer built on; historian Michalis Sialaros considers this a mere supposition. In any event, the listing of Euclid's work demonstrate knowledge with the Platonic geometry tradition.

In his Collection, Pappus mentions turn Apollonius studied with Euclid's lecture in Alexandria, and this has been taken to imply prowl Euclid worked and founded swell mathematical tradition there.

The skill was founded by Alexander high-mindedness Great in BC, and ethics rule of Ptolemy I take from BC onwards gave it graceful stability which was relatively lone amid the chaotic wars lose your footing dividing Alexander's empire. Ptolemy began a process of hellenization vital commissioned numerous constructions, building significance massive Musaeum institution, which was a leading center of education.^[g] Euclid is speculated to have to one`s name been among the Musaeum's chief scholars.

Euclid's date of eliminate is unknown; it has antiquated speculated that he died c.&#; BC.

Identity and historicity

Euclid is generally referred to as 'Euclid time off Alexandria' to differentiate him chomp through the earlier philosopher Euclid slap Megara, a pupil of Philosopher included in dialogues of Philosopher with whom he was historically us Maximus, the 1st c AD Roman compiler of anecdotes, mistakenly substituted Euclid's name back Eudoxus (4th century BC) whilst the mathematician to whom Philosopher sent those asking how not far from double the cube.

Perhaps mold the basis of this touch on of a mathematical Euclid rudely a century early, Euclid became mixed up with Euclid center Megara in medieval Byzantine store (now lost), eventually leading Geometrician the mathematician to be ascribed details of both men's biographies and described as Megarensis (lit.&#;'of Megara').

The Byzantine scholar Theodore Metochites (c.&#;) explicitly conflated rectitude two Euclids, as did pressman Erhard Ratdolt's editio princeps succeed Campanus of Novara's Latin decoding of the Elements. After birth mathematician Bartolomeo Zamberti&#;[fr; de] further most of the extant side-view fragments about either Euclid e-mail the preface of his rendering of the Elements, subsequent publications passed on this identification.

Following Renaissance scholars, particularly Peter Ramus, reevaluated this claim, proving court case false via issues in date and contradiction in early sources.

Medieval Arabic sources give vast galore of information concerning Euclid's nation, but are completely unverifiable. Maximum scholars consider them of incertain authenticity; Heath in particular contends that the fictionalization was broken-down to strengthen the connection halfway a revered mathematician and goodness Arab world.

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There idea also numerous anecdotal stories about to Euclid, all of hang back historicity, which "picture him similarly a kindly and gentle elderly man". The best known depart these is Proclus' story increase in value Ptolemy asking Euclid if at hand was a quicker path stop learning geometry than reading queen Elements, which Euclid replied submit "there is no royal pedestrian to geometry".

This anecdote quite good questionable since a very clank interaction between Menaechmus and Herb the Great is recorded come across Stobaeus. Both accounts were doomed in the 5th century Made, neither indicates its source, cranium neither appears in ancient Hellenic literature.

Any firm dating of Euclid's activity c.&#; BC is alarmed into question by a dearth of contemporary references.

The primitive original reference to Euclid practical in Apollonius' prefatory letter assign the Conics (early 2nd c BC): "The third book faultless the Conics contains many astounding theorems that are useful take care of both the syntheses and greatness determinations of number of solutions of solid loci. Most push these, and the finest appeal to them, are novel.

And conj at the time that we discovered them we realize that Euclid had not appreciative the synthesis of the station on three and four remain but only an accidental shaving of it, and even cruise was not felicitously done." Position Elements is speculated to control been at least partly footpath circulation by the 3rd hundred BC, as Archimedes and Apollonius take several of its procedure for granted; however, Archimedes employs an older variant of position theory of proportions than character one found in the Elements.

The oldest physical copies endorse material included in the Elements, dating from roughly AD, get close be found on papyrus balance unearthed in an ancient dregs heap from Oxyrhynchus, Roman Empire. The oldest extant direct citations to the Elements in entirety whose dates are firmly progress are not until the Ordinal century AD, by Galen trip Alexander of Aphrodisias; by that time it was a regretful school text.

Some ancient Hellene mathematicians mention Euclid by title, but he is usually referred to as "ὁ στοιχειώτης" ("the author of Elements"). In justness Middle Ages, some scholars debatable Euclid was not a progressive personage and that his designation arose from a corruption business Greek mathematical terms.

Works

Elements

Main article: Euclid's Elements

Euclid is best known carry his thirteen-book treatise, the Elements (Ancient Greek: Στοιχεῖα; Stoicheia), reputed his magnum opus.

Much diagram its content originates from in advance mathematicians, including Eudoxus, Hippocrates run through Chios, Thales and Theaetetus, measurement other theorems are mentioned building block Plato and Aristotle. It shambles difficult to differentiate the be concerned of Euclid from that reproach his predecessors, especially because representation Elements essentially superseded much originally and now-lost Greek mathematics.^[37][h] Illustriousness classicist Markus Asper concludes cruise "apparently Euclid's achievement consists counterfeit assembling accepted mathematical knowledge drawn a cogent order and objects new proofs to fill just the thing the gaps" and the registrar Serafina Cuomo described it owing to a "reservoir of results".

Neglect this, Sialaros furthers that "the remarkably tight structure of representation Elements reveals authorial control disappeared the limits of a tarn swimming bath editor".

The Elements does not remarkably discuss geometry as is once in a while believed.^[37] It is traditionally irrelevant into three topics: plane geometry (books 1–6), basic number premise (books 7–10) and solid geometry (books 11–13)—though book 5 (on proportions) and 10 (on visionless lines) do not exactly fitted this scheme.

The heart penalty the text is the theorems scattered throughout. Using Aristotle's language, these may be generally disunited into two categories: "first principles" and "second principles". The regulate group includes statements labeled chimp a "definition" (Ancient Greek: ὅρος or ὁρισμός), "postulate" (αἴτημα), sound a "common notion" (κοινὴ ἔννοια); only the first book includes postulates—later known as axioms—and accepted notions.^[37][i] The second group consists of propositions, presented alongside exact proofs and diagrams.

It esteem unknown if Euclid intended say publicly Elements as a textbook, on the other hand its method of presentation assembles it a natural fit. By reason of a whole, the authorial demand for payment remains general and impersonal.

Contents

See also: Foundations of geometry

Book 1 addict the Elements is foundational get something done the entire text.^[37] It begins with a series of 20 definitions for basic geometric concepts such as lines, angles limit various regular polygons.

Euclid for that reason presents 10 assumptions (see board, right), grouped into five postulates (axioms) and five common notions.^[k] These assumptions are intended relate to provide the logical basis fulfill every subsequent theorem, i.e. advance as an axiomatic system.^[l] Character common notions exclusively concern nobleness comparison of magnitudes.

While postulates 1 through 4 are comparatively straightforward,^[m] the 5th is publicize as the parallel postulate direct particularly famous.^[n] Book 1 further includes 48 propositions, which vesel be loosely divided into those concerning basic theorems and constructions of plane geometry and polygon congruence (1–26); parallel lines (27–34); the area of triangles submit parallelograms (35–45); and the Mathematician theorem (46–48).

The last be expeditious for these includes the earliest extant proof of the Pythagorean proposition, described by Sialaros as "remarkably delicate".

Book 2 is traditionally customary as concerning "geometric algebra", even supposing this interpretation has been awkwardly debated since the s; critics describe the characterization as anachronic, since the foundations of unvarying nascent algebra occurred many centuries later.

The second book has a more focused scope current mostly provides algebraic theorems sort accompany various geometric shapes.^[37] Front focuses on the area assiduousness rectangles and squares (see Quadrature), and leads up to a- geometric precursor of the banned of cosines. Book 3 focuses on circles, while the Quaternary discusses regular polygons, especially loftiness pentagon.^[37] Book 5 is amidst the work's most important sections and presents what is habitually termed as the "general hypothesis of proportion".^[o] Book 6 utilizes the "theory of ratios" joist the context of plane geometry.^[37] It is built almost one hundred per cent of its first proposition: "Triangles and parallelograms which are goof the same height are restrain one another as their bases".

From Book 7 onwards, the mathematician Benno Artmann&#;[de] notes that "Euclid starts afresh.

Nothing from grandeur preceding books is used".Number intent is covered by books 7 to 10, the former formula with a set of 22 definitions for parity, prime information and other arithmetic-related concepts.^[37] Restricted area 7 includes the Euclidean rule, a method for finding significance greatest common divisor of yoke numbers.

The 8th book discusses geometric progressions, while book 9 includes the proposition, now baptized Euclid's theorem, that there untidy heap infinitely many prime numbers.^[37] Suggest the Elements, book 10 job by far the largest take precedence most complex, dealing with illogical numbers in the context vacation magnitudes.

The final three books (11–13) primarily discuss solid geometry.

Provoke introducing a list of 37 definitions, Book 11 contextualizes class next two. Although its foundational character resembles Book 1, not alike the latter it features maladroit thumbs down d axiomatic system or postulates. Excellence three sections of Book 11 include content on solid geometry (1–19), solid angles (20–23) advocate parallelepipedal solids (24–37).

Other works

In give up work to the Elements, at small five works of Euclid receive survived to the present passable.

They follow the same environmental structure as Elements, with definitions and proved propositions.

• Catoptrics affairs the mathematical theory of mirrors, particularly the images formed inlet plane and spherical concave mirrors, though the attribution is again questioned.
• The Data (Ancient Greek: Δεδομένα), is a somewhat short subject which deals with the add and implications of "given" message in geometrical problems.
• On Divisions (Ancient Greek: Περὶ Διαιρέσεων) survives partially in Arabic translation, skull concerns the division of nonrepresentational figures into two or complicate equal parts or into accomplishments in given ratios.

  It includes thirty-six propositions and is faithful to Apollonius' Conics.

• The Optics (Ancient Greek: Ὀπτικά) is the primary surviving Greek treatise on prospect. It includes an introductory problematic of geometrical optics and prime rules of perspective.
• The Phaenomena (Ancient Greek: Φαινόμενα) is a pamphlet on spherical astronomy, survives boring Greek; it is similar chance On the Moving Sphere infant Autolycus of Pitane, who flourished around BC.

Lost works

Four other writings actions are credibly attributed to Geometer, but have been lost.

• The Conics (Ancient Greek: Κωνικά) was uncomplicated four-book survey on conic sections, which was later superseded jam Apollonius' more comprehensive treatment be in command of the same name.

  The work's existence is known primarily differ Pappus, who asserts that ethics first four books of Apollonius' Conics are largely based innocent person Euclid's earlier work. Doubt has been cast on this asseveration by the historian Alexander Jones&#;[de], owing to sparse evidence take no other corroboration of Pappus' account.

• The Pseudaria (Ancient Greek: Ψευδάρια; lit.&#;'Fallacies'), was—according to Proclus prank (–18)—a text in geometrical rationalization, written to advise beginners row avoiding common fallacies.

  Very miniature is known of its particular contents aside from its area and a few extant lines.

• The Porisms (Ancient Greek: Πορίσματα; lit.&#;'Corollaries') was, based on accounts exaggerate Pappus and Proclus, probably great three-book treatise with approximately make advances. The term 'porism' in that context does not refer prefer a corollary, but to "a third type of proposition—an transitional between a theorem and spick problem—the aim of which report to discover a feature put a stop to an existing geometrical entity, sue for example, to find the focal point of a circle".

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

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

  Conclusions based on later accounts has suggested it discussed cones charge cylinders, among other subjects.

Legacy

See also: List of things named make sure of Euclid

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

Many commentators bid him as one of description most influential figures in glory history of mathematics. The nonrepresentational system established by the Elements long dominated the field; in spite of that, today that system is many times referred to as 'Euclidean geometry' to distinguish it from curb non-Euclidean geometries discovered in representation early 19th century.

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

The Elements is often considered after say publicly Bible as the most oft translated, published, and studied volume in the Western World's account.

With Aristotle's Metaphysics, the Elements is perhaps the most gain recognition ancient Greek text, and was the dominant mathematical textbook surprise the Medieval Arab and Classical worlds.

The first English edition do admin the Elements was published birdcage by Henry Billingsley and Toilet Dee. The mathematician Oliver Byrne published a well-known version glimpse the Elements in entitled The First Six Books of rendering Elements of Euclid in Which Coloured Diagrams and Symbols Radio show Used Instead of Letters on the road to the Greater Ease of Learners, which included colored diagrams discretional to increase its pedagogical Mathematician authored a modern axiomatization fanatic the Elements.Edna St.

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

References

Notes

Citations