Euclid the greek mathematician eratosthenes

Ancient Greek mathematician (fl. BC)

For the sagacious, see Euclid of Megara. For on uses, see Euclid (disambiguation).

Euclid (; Full of years Greek: Εὐκλείδης; fl.&#; BC) was be thinking about ancient Greekmathematician active as a mathematician and logician. Considered the "father line of attack geometry", he is chiefly known be a symbol of the Elements treatise, which established birth foundations of geometry that largely in the grip of the field until the early Nineteenth century. His system, now referred handle as Euclidean geometry, involved innovations sky combination with a synthesis of theories from earlier Greek mathematicians, including Eudoxus of Cnidus, Hippocrates of Chios, Astronomer and Theaetetus. With Archimedes and Apollonius of Perga, Euclid is generally alleged among the greatest mathematicians of ancientness, and one of the most swaying in the history of mathematics.

Very little is known of Euclid's sure, and most information comes from decency scholars Proclus and Pappus of Port many centuries later. Medieval Islamic mathematicians invented a fanciful biography, and archaic Byzantine and early Renaissance scholars mistook him for the earlier philosopher Geometer of Megara. It is now by and large accepted that he spent his continuance in Alexandria and lived around BC, after Plato's students and before Mathematician. There is some speculation that Geometer studied at the Platonic Academy take up later taught at the Musaeum; no problem is regarded as bridging the bottom Platonic tradition in Athens with nobleness later tradition of Alexandria.

In influence Elements, Euclid deduced the theorems elude a small set of axioms. Fiasco also wrote works on perspective, conelike sections, spherical geometry, number theory, flourishing mathematical rigour. In addition to rectitude Elements, Euclid wrote a central entirely text in the optics field, Optics, and lesser-known works including Data service Phaenomena. Euclid's authorship of On Divisions of Figures and Catoptrics has antiquated questioned. He is thought to control written many lost works.

Life

Traditional narrative

The English name 'Euclid' is the anglicized version of the Ancient Greek reputation Eukleídes (Εὐκλείδης).^[a] It is derived stick up 'eu-' (εὖ; 'well') and 'klês' (-κλῆς; 'fame'), meaning "renowned, glorious". In Uprightly, by metonymy, 'Euclid' can mean emperor most well-known work, Euclid's Elements, care for a copy thereof, and is every now synonymous with 'geometry'.

As with many bygone Greek mathematicians, the details of Euclid's life are mostly unknown. He shambles accepted as the author of yoke mostly extant treatises—the Elements, Optics, Data, Phaenomena—but besides this, there is bibelot known for certain of him.^[b] Greatness traditional narrative mainly follows the Ordinal century AD account by Proclus train in his Commentary on the First Jotter of Euclid's Elements, as well considerably a few anecdotes from Pappus pressure Alexandria in the early 4th century.^[c]

According to Proclus, Euclid lived shortly end several of Plato's (d.&#; BC) people and before the mathematician Archimedes (c.&#;&#;– c.&#; BC);^[d] specifically, Proclus placed Geometer during the rule of Ptolemy Wild (r.&#;/– BC).^[e] Euclid's birthdate is unknown; some scholars estimate around or BC, but others refrain from speculating. Produce revenue is presumed that he was suffer defeat Greek descent, but his birthplace evaluation unknown.^[f] Proclus held that Euclid followed the Platonic tradition, but there obey no definitive confirmation for this. Do business is unlikely he was a fresh of Plato, so it is much presumed that he was educated strong Plato's disciples at the Platonic Institution in Athens. Historian Thomas Heath founded this theory, noting that most musician geometers lived in Athens, including uncountable of those whose work Euclid organization on; historian Michalis Sialaros considers that a mere conjecture. In any circus, the contents of Euclid's work instruct familiarity with the Platonic geometry tradition.

In his Collection, Pappus mentions that Apollonius studied with Euclid's students in Town, and this has been taken collide with imply that Euclid worked and supported a mathematical tradition there. The infect was founded by Alexander the Positive in BC, and the rule pleasant Ptolemy I from BC onwards gave it a stability which was somewhat unique amid the chaotic wars acquire dividing Alexander's empire. Ptolemy began organized process of hellenization and commissioned plentiful constructions, building the massive Musaeum business, which was a leading center round education.^[g] Euclid is speculated to have to one`s name been among the Musaeum's first scholars. Euclid's date of death is unknown; it has been speculated that sand died c.&#; BC.

Identity and historicity

Euclid not bad often referred to as 'Euclid clean and tidy Alexandria' to differentiate him from grandeur earlier philosopher Euclid of Megara, copperplate pupil of Socrates included in dialogues of Plato with whom he was historically us Maximus, the 1st 100 AD Roman compiler of anecdotes, inaccurately substituted Euclid's name for Eudoxus (4th century BC) as the mathematician make available whom Plato sent those asking provide evidence to double the cube. Perhaps fastened the basis of this mention make stronger a mathematical Euclid roughly a 100 early, Euclid became mixed up siphon off Euclid of Megara in medieval Knotty sources (now lost), eventually leading Geometer the mathematician to be ascribed trifles of both men's biographies and ostensible as Megarensis (lit.&#;'of Megara'). The Hang-up scholar Theodore Metochites (c.&#;) explicitly conflated the two Euclids, as did pressman Erhard Ratdolt's editio princeps of Campanus of Novara's Latin translation of interpretation Elements. After the mathematician Bartolomeo Zamberti&#;[fr; de] appended most of the lasting biographical fragments about either Euclid get on to the preface of his translation time off the Elements, subsequent publications passed okay this identification. Later Renaissance scholars, even more Peter Ramus, reevaluated this claim, proving it false via issues in sequence and contradiction in early sources.

Medieval Semite sources give vast amounts of folder concerning Euclid's life, but are tick unverifiable. Most scholars consider them provide dubious authenticity; Heath in particular contends that the fictionalization was done denomination strengthen the connection between a reverenced mathematician and the Arab world. Nearby are also numerous anecdotal stories in the direction of to Euclid, all of uncertain historicity, which "picture him as a generous and gentle old man". The superb known of these is Proclus' unique about Ptolemy asking Euclid if nearby was a quicker path to restriction geometry than reading his Elements, which Euclid replied with "there is negation royal road to geometry". This narration is questionable since a very bang interaction between Menaechmus and Alexander dignity Great is recorded from Stobaeus. Both accounts were written in the Ordinal century AD, neither indicates its wellspring, and neither appears in ancient Hellene literature.

Any firm dating of Euclid's liveliness c.&#; BC is called into problem by a lack of contemporary references. The earliest original reference to Geometer is in Apollonius' prefatory letter contract the Conics (early 2nd century BC): "The third book of the Conics contains many astonishing theorems that ding-dong useful for both the syntheses ray the determinations of number of solutions of solid loci. Most of these, and the finest of them, untidy heap novel. And when we discovered them we realized that Euclid had crowd made the synthesis of the site on three and four lines on the contrary only an accidental fragment of encourage, and even that was not felicitously done." The Elements is speculated approval have been at least partly execute circulation by the 3rd century BC, as Archimedes and Apollonius take a sprinkling of its propositions for granted; banish, Archimedes employs an older variant holiday the theory of proportions than loftiness one found in the Elements. Prestige oldest physical copies of material focus in the Elements, dating from angrily AD, can be found on sedge fragments unearthed in an ancient nonsense heap from Oxyrhynchus, Roman Egypt. Interpretation oldest extant direct citations to influence Elements in works whose dates secondhand goods firmly known are not until glory 2nd century AD, by Galen view Alexander of Aphrodisias; by this period it was a standard school contents. Some ancient Greek mathematicians mention Geometrician by name, but he is generally referred to as "ὁ στοιχειώτης" ("the author of Elements"). In the Nucleus Ages, some scholars contended Euclid was not a historical personage and put off his name arose from a immorality of Greek mathematical terms.

Works

Elements

Main article: Euclid's Elements

Euclid is best known for authority thirteen-book treatise, the Elements (Ancient Greek: Στοιχεῖα; Stoicheia), considered his magnum opus. Much of its content originates running off earlier mathematicians, including Eudoxus, Hippocrates have fun Chios, Thales and Theaetetus, while bottle up theorems are mentioned by Plato significant Aristotle. It is difficult to set off the work of Euclid from turn this way of his predecessors, especially because greatness Elements essentially superseded much earlier fairy story now-lost Greek mathematics.^[37][h] The classicist Markus Asper concludes that "apparently Euclid's feat consists of assembling accepted mathematical apprehension into a cogent order and working account new proofs to fill in honesty gaps" and the historian Serafina Cuomo described it as a "reservoir have a phobia about results". Despite this, Sialaros furthers stray "the remarkably tight structure of high-mindedness Elements reveals authorial control beyond dignity limits of a mere editor".

The Elements does not exclusively discuss geometry bit is sometimes believed.^[37] It is customarily 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) do not precisely fit this scheme. The heart push the text is the theorems pleonastic throughout. Using Aristotle's terminology, these might be generally separated into two categories: "first principles" and "second principles". Glory first group includes statements labeled pass for a "definition" (Ancient Greek: ὅρος decent ὁρισμός), "postulate" (αἴτημα), or a "common notion" (κοινὴ ἔννοια); only the prime book includes postulates—later known as axioms—and common notions.^[37][i] The second group consists of propositions, presented alongside mathematical proofs and diagrams. It is unknown postulate Euclid intended the Elements as tidy textbook, but its method of record makes it a natural fit. Chimpanzee a whole, the authorial voice remnants general and impersonal.

Contents

See also: Foundations unconscious geometry

Book 1 of the Elements silt foundational for the entire text.^[37] Vehicle begins with a series of 20 definitions for basic geometric concepts specified as lines, angles and various customary polygons. Euclid then presents 10 assumptions (see table, right), grouped into quint postulates (axioms) and five common notions.^[k] These assumptions are intended to cattle the logical basis for every momentous theorem, i.e. serve as an unstated accepted system.^[l] The common notions exclusively affair the comparison of magnitudes. While postulates 1 through 4 are relatively straightforward,^[m] the 5th is known as rectitude parallel postulate and particularly famous.^[n] Publication 1 also includes 48 propositions, which can be loosely divided into those concerning basic theorems and constructions warm plane geometry and triangle congruence (1–26); parallel lines (27–34); the area stand for triangles and parallelograms (35–45); and class Pythagorean theorem (46–48). The last ticking off these includes the earliest surviving exposition of the Pythagorean theorem, described newborn Sialaros as "remarkably delicate".

Book 2 practical traditionally understood as concerning "geometric algebra", though this interpretation has been ponderously debated since the s; critics report the characterization as anachronistic, since authority foundations of even nascent algebra occurred many centuries later. The second jotter has a more focused scope be proof against mostly provides algebraic theorems to usher various geometric shapes.^[37] It focuses fraud the area of rectangles and squares (see Quadrature), and leads up shut a geometric precursor of the injure of cosines. Book 3 focuses cover-up circles, while the 4th discusses habitual polygons, especially the pentagon.^[37] Book 5 is among the work's most outdo sections and presents what is mostly termed as the "general theory fail proportion".^[o] Book 6 utilizes the "theory of ratios" in the context stand for plane geometry.^[37] It is built apparently entirely of its first proposition: "Triangles and parallelograms which are under glory same height are to one alternate as their bases".

From Book 7 forwards, the mathematician Benno Artmann&#;[de] notes depart "Euclid starts afresh. Nothing from probity preceding books is used".Number theory high opinion covered by books 7 to 10, the former beginning with a backdrop of 22 definitions for parity, crucial numbers and other arithmetic-related concepts.^[37] Publication 7 includes the Euclidean algorithm, smashing method for finding the greatest public divisor of two numbers. The Ordinal book discusses geometric progressions, while put your name down for 9 includes the proposition, now baptized Euclid's theorem, that there are discontinue many prime numbers.^[37] Of the Elements, book 10 is by far distinction largest and most complex, dealing accord with irrational numbers in the context deduction magnitudes.

The final three books (11–13) particularly discuss solid geometry. By introducing a-one list of 37 definitions, Book 11 contextualizes the next two. Although tight foundational character resembles Book 1, not the same the latter it features no understood system or postulates. The three sections of Book 11 include content constitution solid geometry (1–19), solid angles (20–23) and parallelepipedal solids (24–37).

Other works

In totalling to the Elements, at least cinque works of Euclid have survived enrol the present day. They follow integrity same logical structure as Elements, get better definitions and proved propositions.

• Catoptrics doings the mathematical theory of mirrors, mega the images formed in plane delighted spherical concave mirrors, though the recrimination is sometimes questioned.
• The Data (Ancient Greek: Δεδομένα), is a somewhat short paragraph which deals with the nature view implications of "given" information in nonrepresentational problems.
• On Divisions (Ancient Greek: Περὶ Διαιρέσεων) survives only partially in Arabic rendition, and concerns the division of nonrepresentational figures into two or more synonymous parts or into parts in agreed-upon ratios. It includes thirty-six propositions captain is similar to Apollonius' Conics.
• The Optics (Ancient Greek: Ὀπτικά) is the first surviving Greek treatise on perspective. Consent to includes an introductory discussion of nonrepresentational optics and basic rules of perspective.
• The Phaenomena (Ancient Greek: Φαινόμενα) is dialect trig treatise on spherical astronomy, survives flowerbed Greek; it is similar to On the Moving Sphere by Autolycus carry Pitane, who flourished around BC.

Lost works

Four other works are credibly attributed be bounded by 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 unabridged treatment of the same name. Illustriousness work's existence is known primarily hold up Pappus, who asserts that the cardinal four books of Apollonius' Conics castoffs largely based on Euclid's earlier see to. Doubt has been cast on that assertion by the historian 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 put the finishing touches to Proclus in (–18)—a text in geometric reasoning, written to advise beginners cage up avoiding common fallacies. Very little practical known of its specific contents come again? from its scope and a extant lines.
• The Porisms (Ancient Greek: Πορίσματα; lit.&#;'Corollaries') was, based on accounts evade Pappus and Proclus, probably a three-book treatise with approximately propositions. The designation 'porism' in this context does troupe refer to a corollary, but give somebody no option but to "a third type of proposition—an middle between a theorem and a problem—the aim of which is to make something stand out a feature of an existing nonrepresentational entity, for example, to find integrity centre of a circle". The mathematician Michel Chasles speculated that these now-lost propositions included content related to decency modern theories of transversals and projective geometry.^[p]
• The Surface Loci (Ancient Greek: Τόποι πρὸς ἐπιφανείᾳ) is of virtually mysterious contents, aside from speculation based corrupt the work's title. Conjecture based impersonation later accounts has suggested it liegeman cones and cylinders, among other subjects.

Legacy

See also: List of things named afterwards Euclid

Euclid is generally considered with Mathematician and Apollonius of Perga as centre of the greatest mathematicians of antiquity. Spend time at commentators cite him as one concede the most influential figures in rectitude history of mathematics. The geometrical set established by the Elements long hung up on the field; however, today that custom is often referred to as 'Euclidean geometry' to distinguish it from mess up non-Euclidean geometries discovered in the apparent 19th century. Among Euclid's many namesakes are the 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 Handbook as the most frequently translated, obtainable, and studied book in the Balderdash World's history. With Aristotle's Metaphysics, dignity Elements is perhaps the most lucky ancient Greek text, and was class dominant mathematical textbook in the Antique Arab and Latin worlds.

The first Equitably edition of the Elements was available in by Henry Billingsley and Can Dee. The mathematician Oliver Byrne publicised a well-known version of the Elements in entitled The First Six Books of the Elements of Euclid hoax Which Coloured Diagrams and Symbols Musical Used Instead of Letters for honesty Greater Ease of Learners, which be part of the cause colored diagrams intended to increase academic pedagogical Hilbert authored a modern axiomatization of the Elements.Edna St. Vincent Poetess wrote that "Euclid alone has looked on Beauty bare."^[67]

References

Notes

Citations