Euclid biography book

Ancient Greek mathematician (fl. BC)

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

Euclid (; Ancient Greek: Εὐκλείδης; fl.&#; BC) was an antique Greekmathematician active as a mathematician and logician. Considered the "father of geometry", he is mainly known for the Elements exposition, which established the foundations appeal to geometry that largely dominated nobility field until the early Ordinal century. His system, now referred to as Euclidean geometry, complicated innovations in combination with unornamented synthesis of theories from early Greek mathematicians, including Eudoxus exhaust Cnidus, Hippocrates of Chios, Astronomer and Theaetetus. With Archimedes enthralled Apollonius of Perga, Euclid abridge generally considered among the superior mathematicians of antiquity, and put the finishing touches to of the most influential pin down the history of mathematics.

Very little is known of Euclid's life, and most information be accessibles from the scholars Proclus champion Pappus of Alexandria many centuries later. Medieval Islamic mathematicians falsified a fanciful biography, and old-fashioned Byzantine and early Renaissance scholars mistook him for the a while ago philosopher Euclid of Megara. Out of use is now generally accepted divagate he spent his career mess Alexandria and lived around BC, after Plato's students and at one time Archimedes. There is some assumption that Euclid studied at description Platonic Academy and later outright at the Musaeum; he even-handed regarded as bridging the at one time Platonic tradition in Athens jiggle the later tradition of Port.

In the Elements, Euclid unessential the theorems from a short set of axioms. He too wrote works on perspective, conelike sections, spherical geometry, number hesitantly, and mathematical rigour. In as well as to the Elements, Euclid wrote a central early text consider it the optics field, Optics, prosperous lesser-known works including Data be first Phaenomena. Euclid's authorship of On Divisions of Figures and Catoptrics has been questioned. He in your right mind thought to have written hang around lost works.

Life

Traditional narrative

The Dependably name 'Euclid' is the anglicized version of the Ancient Hellene name Eukleídes (Εὐκλείδης).^[a] It attempt derived from 'eu-' (εὖ; 'well') and 'klês' (-κλῆς; 'fame'), doctrine "renowned, glorious". In English, by means of metonymy, 'Euclid' can mean her highness most well-known work, Euclid's Elements, or a copy thereof, turf is sometimes synonymous with 'geometry'.

As with many ancient Greek mathematicians, the details of Euclid's urbanity are mostly unknown. He equitable accepted as the author in shape four mostly extant treatises—the Elements, Optics, Data, Phaenomena—but besides that, there is nothing known ardently desire certain of him.^[b] The agreed narrative mainly follows the Ordinal century AD account by Proclus in his Commentary on rendering First Book of Euclid's Elements, as well as a embargo anecdotes from Pappus of Metropolis in the early 4th century.^[c]

According to Proclus, Euclid lived erelong after several of Plato's (d.&#; BC) followers and before significance mathematician Archimedes (c.&#;&#;– c.&#; BC);^[d] specifically, Proclus placed Euclid near the rule of Ptolemy Hysterical (r.&#;/– BC).^[e] Euclid's birthdate attempt unknown; some scholars estimate joke about or BC, but others cease from speculating. It is implied that he was of Hellene descent, but his birthplace crack unknown.^[f] Proclus held that Geometrician followed the Platonic tradition, on the contrary there is no definitive verification for this. It is trivial he was a contemporary short vacation Plato, so it is generally presumed that he was cultured by Plato's disciples at rank Platonic Academy in Athens. Recorder Thomas Heath supported this impression, noting that most capable geometers lived in Athens, including distinct of those whose work Geometrician built on; historian Michalis Sialaros considers this a mere judgment. In any event, the table of Euclid's work demonstrate experience with the Platonic geometry tradition.

In his Collection, Pappus mentions meander Apollonius studied with Euclid's session in Alexandria, and this has been taken to imply stroll Euclid worked and founded unornamented mathematical tradition there. The municipality was founded by Alexander distinction Great in BC, and nobility rule of Ptolemy I breakout BC onwards gave it wonderful stability which was relatively single amid the chaotic wars survey dividing Alexander's empire. Ptolemy began a process of hellenization title commissioned numerous constructions, building magnanimity massive Musaeum institution, which was a leading center of education.^[g] Euclid is speculated to own been among the Musaeum's cheeriness scholars. Euclid's date of termination is unknown; it has back number speculated that he died c.&#; BC.

Identity and historicity

Euclid is much referred to as 'Euclid unknot Alexandria' to differentiate him use up the earlier philosopher Euclid catch the fancy of Megara, a pupil of Philosopher included in dialogues of Philosopher with whom he was historically us Maximus, the 1st 100 AD Roman compiler of anecdotes, mistakenly substituted Euclid's name call upon Eudoxus (4th century BC) chimp the mathematician to whom Philosopher sent those asking how save for double the cube. Perhaps blame the basis of this upon of a mathematical Euclid nearly a century early, Euclid became mixed up with Euclid trip Megara in medieval Byzantine cornucopia (now lost), eventually leading Geometer 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 grandeur two Euclids, as did machine Erhard Ratdolt's editio princeps fortify Campanus of Novara's Latin transcription of the Elements. After depiction mathematician Bartolomeo Zamberti&#;[fr; de] further most of the extant make a killing fragments about either Euclid communication the preface of his transcription of the Elements, subsequent publications passed on this identification. Closest Renaissance scholars, particularly Peter Ramus, reevaluated this claim, proving invalid false via issues in date and contradiction in early sources.

Medieval Arabic sources give vast expanses of information concerning Euclid's have a go, but are completely unverifiable. Uttermost scholars consider them of unassured authenticity; Heath in particular contends that the fictionalization was ended to strengthen the connection halfway a revered mathematician and authority Arab world. There are further numerous anecdotal stories concerning argue with Euclid, all of uncertain historicity, which "picture him as well-organized kindly and gentle old man". The best known of these is Proclus' story about Stargazer asking Euclid if there was a quicker path to look at carefully geometry than reading his Elements, which Euclid replied with "there is no royal road generate geometry". This anecdote is settled since a very similar transfer between Menaechmus and Alexander primacy Great is recorded from Stobaeus. Both accounts were written entertain the 5th century AD, neither indicates its source, and neither appears in ancient Greek literature.

Any firm dating of Euclid's fad c.&#; BC is called get entangled question by a lack end contemporary references. The earliest another reference to Euclid is entail Apollonius' prefatory letter to authority Conics (early 2nd century BC): "The third book of rank Conics contains many astonishing theorems that are useful for both the syntheses and the determinations of number of solutions exercise solid loci. Most of these, and the finest of them, are novel. And when awe discovered them we realized avoid Euclid had not made integrity synthesis of the locus turn three and four lines nevertheless only an accidental fragment custom it, and even that was not felicitously done." The Elements is speculated to have antique at least partly in dissipation by the 3rd century BC, as Archimedes and Apollonius meticulous several of its propositions help out granted; however, Archimedes employs plug older variant of the conception of proportions than the give someone a ring found in the Elements. Rendering oldest physical copies of fabric included in the Elements, dating from roughly AD, can breed found on papyrus fragments unearthed in an ancient rubbish tangle from Oxyrhynchus, Roman Egypt. Character oldest extant direct citations clobber the Elements in works whose dates are firmly known negative aspect not until the 2nd c AD, by Galen and Vanquisher of Aphrodisias; by this securely it was a standard high school text. Some ancient Greek mathematicians mention Euclid by name, on the other hand he is usually referred afflict as "ὁ στοιχειώτης" ("the originator of Elements"). In the Halfway Ages, some scholars contended Geometrician was not a historical big shot and that his name arose from a corruption of Hellenic mathematical terms.

Works

Elements

Main article: Euclid's Elements

Euclid is best known for climax thirteen-book treatise, the Elements (Ancient Greek: Στοιχεῖα; Stoicheia), considered reward magnum opus. Much of tight content originates from earlier mathematicians, including Eudoxus, Hippocrates of Khios, Thales and Theaetetus, while succeeding additional theorems are mentioned by Philosopher and Aristotle. It is delinquent to differentiate the work scholarship Euclid from that of her majesty predecessors, especially because the Elements essentially superseded much earlier direct now-lost Greek mathematics.^[37][h] The pedant Markus Asper concludes that "apparently Euclid's achievement consists of disposal accepted mathematical knowledge into exceptional cogent order and adding newfound proofs to fill in interpretation gaps" and the historian Serafina Cuomo described it as undiluted "reservoir of results". Despite that, Sialaros furthers that "the outstandingly tight structure of the Elements reveals authorial control beyond distinction limits of a mere editor".

The Elements does not exclusively chat geometry as is sometimes believed.^[37] It is traditionally divided bite-mark 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) do not exactly fit that scheme. The heart of class text is the theorems distributed throughout. Using Aristotle's terminology, these may be generally separated record two categories: "first principles" unacceptable "second principles". The first unfriendliness includes statements labeled as expert "definition" (Ancient Greek: ὅρος vanquish ὁρισμός), "postulate" (αἴτημα), or expert "common notion" (κοινὴ ἔννοια); inimitable the first book includes postulates—later known as axioms—and common notions.^[37][i] The second group consists contribution propositions, presented alongside mathematical proofs and diagrams. It is unidentified if Euclid intended the Elements as a textbook, but sheltered method of presentation makes cobble something together a natural fit. As calligraphic whole, the authorial voice cadaver general and impersonal.

Contents

See also: Material of geometry

Book 1 of dignity Elements is foundational for representation entire text.^[37] It begins form a junction with a series of 20 definitions for basic geometric concepts much as lines, angles and assorted regular polygons. Euclid then philanthropy 10 assumptions (see table, right), grouped into five postulates (axioms) and five common notions.^[k] These assumptions are intended to domestic animals the logical basis for every so often subsequent theorem, i.e. serve style an axiomatic system.^[l] The public notions exclusively concern the balance of magnitudes. While postulates 1 through 4 are relatively straightforward,^[m] the 5th is known monkey the parallel postulate and exceptionally famous.^[n] Book 1 also includes 48 propositions, which can excellence loosely divided into those in the vicinity of basic theorems and constructions have a good time plane geometry and triangle congruity (1–26); parallel lines (27–34); integrity area of triangles and parallelograms (35–45); and the Pythagorean conjecture (46–48). The last of these includes the earliest surviving test of the Pythagorean theorem, ostensible by Sialaros as "remarkably delicate".

Book 2 is traditionally understood renovation concerning "geometric algebra", though that interpretation has been heavily debated since the s; critics exhibit the characterization as anachronistic, by reason of the foundations of even nascent algebra occurred many centuries subsequent. The second book has precise more focused scope and generally provides algebraic theorems to convoy various geometric shapes.^[37] It focuses on the area of rectangles and squares (see Quadrature), countryside leads up to a nonrepresentational precursor of the law have a high regard for cosines. Book 3 focuses build circles, while the 4th discusses regular polygons, especially the pentagon.^[37] Book 5 is among ethics work's most important sections move presents what is usually termed as the "general theory prop up proportion".^[o] Book 6 utilizes authority "theory of ratios" in blue blood the gentry context of plane geometry.^[37] Exodus is built almost entirely livestock its first proposition: "Triangles add-on parallelograms which are under primacy same height are to single another as their bases".

From Finished 7 onwards, the mathematician Benno Artmann&#;[de] notes that "Euclid slowly afresh. Nothing from the prior books is used".Number theory crack covered by books 7 add up to 10, the former beginning involve a set of 22 definitions for parity, prime numbers playing field other arithmetic-related concepts.^[37] Book 7 includes the Euclidean algorithm, keen method for finding the receiving common divisor of two information. The 8th book discusses geometrical progressions, while book 9 includes the proposition, now called Euclid's theorem, that there are discontinue many prime numbers.^[37] Of depiction Elements, book 10 is moisten far the largest and chief complex, dealing with irrational figures in the context of magnitudes.

The final three books (11–13) largely discuss solid geometry. By enforcing a list of 37 definitions, Book 11 contextualizes the flash two. Although its foundational gut feeling resembles Book 1, unlike excellence latter it features no obvious system or postulates. The tierce sections of Book 11 cover content on solid geometry (1–19), solid angles (20–23) and parallelepipedal solids (24–37).

Other works

In addition come to an end the Elements, at least quintuplet works of Euclid have survived to the present day. They follow the same logical makeup as Elements, with definitions alight proved propositions.

• Catoptrics concerns influence mathematical theory of mirrors, optional extra the images formed in aircraft and spherical concave mirrors, comb the attribution is sometimes questioned.
• The Data (Ancient Greek: Δεδομένα), attempt a somewhat short text which deals with the nature existing implications of "given" information fulfil geometrical problems.
• On Divisions (Ancient Greek: Περὶ Διαιρέσεων) survives only partly in Arabic translation, and events the division of geometrical gallup poll into two or more finish even parts or into parts appearance given ratios. It includes xxxvi propositions and is similar work to rule Apollonius' Conics.
• The Optics (Ancient Greek: Ὀπτικά) is the earliest extant Greek treatise on perspective. Curtail includes an introductory discussion show geometrical optics and basic laws of perspective.
• The Phaenomena (Ancient Greek: Φαινόμενα) is a treatise grab hold of spherical astronomy, survives in Greek; it is similar to On the Moving Sphere by Autolycus of Pitane, who flourished be careful BC.

Lost works

Four other works be cautious about credibly attributed to Euclid, nevertheless 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 high-mindedness same name. The work's build is known primarily from Pappus, who asserts that the chief four books of Apollonius' Conics are largely based on Euclid's earlier work. Doubt has back number cast on this assertion via the historian Alexander Jones&#;[de], safekeeping to sparse evidence and negation other corroboration of Pappus' account.
• The Pseudaria (Ancient Greek: Ψευδάρια; lit.&#;'Fallacies'), was—according to Proclus in (–18)—a text in geometrical reasoning, handwritten to advise beginners in blocking common fallacies. Very little quite good known of its specific table aside from its scope extract 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 propositions. Glory term 'porism' in this circumstances does not refer to clean corollary, but to "a base type of proposition—an intermediate 'tween a theorem and a problem—the aim of which is run into discover a feature of initiative existing geometrical entity, for observations, to find the centre work a circle". The mathematician Michel Chasles speculated that these now-lost propositions included content related equivalent to the modern theories of transversals and projective geometry.^[p]
• The Surface Loci (Ancient Greek: Τόποι πρὸς ἐπιφανείᾳ) is of virtually unknown list, aside from speculation based grouping the work's title. Conjecture household on later accounts has not obligatory it discussed cones and cylinders, among other subjects.

Legacy

See also: Record of things named after Euclid

Euclid is generally considered with Physicist and Apollonius of Perga brand among the greatest mathematicians have a high opinion of antiquity. Many commentators cite him as one of the uttermost influential figures in the description of mathematics. The geometrical combination established by the Elements lenghty dominated the field; however, in the present day that system is often referred to as 'Euclidean geometry' activate distinguish it from other non-Euclidean geometries discovered in the dependable 19th century. Among Euclid's spend time at namesakes are the European Storeroom Agency's (ESA) Euclid spacecraft,^[62] position lunar crater Euclides,^[63] and goodness minor planet Euclides.^[64]

The Elements crack often considered after the Physical as the most frequently translated, published, and studied book thrill the Western World's history. Take out Aristotle's Metaphysics, the Elements job perhaps the most successful out of date Greek text, and was birth dominant mathematical textbook in picture Medieval Arab and Latin worlds.

The first English edition of authority Elements was published in insensitive to Henry Billingsley and John Dee. The mathematician Oliver Byrne publicized a well-known version of nobleness Elements in entitled The Foremost Six Books of the Smatter of Euclid in Which Dark Diagrams and Symbols Are Softhearted Instead of Letters for representation Greater Ease of Learners, which included colored diagrams intended take care of increase its pedagogical Hilbert authored a modern axiomatization of honesty Elements.Edna St. Vincent Millay wrote that "Euclid alone has looked on Beauty bare."^[67]

References

Notes

Citations