Английская Википедия:Euclid
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Euclid (Шаблон:IPAc-en; Шаблон:Lang-grc-gre; Шаблон:Fl. BC) was an ancient Greek mathematician active as a geometer and logician.Шаблон:Sfn Considered the "father of geometry",Шаблон:Sfn he is chiefly known for the Elements treatise, which established the foundations of geometry that largely dominated the field until the early 19th century. His system, now referred to as Euclidean geometry, involved new innovations in combination with a synthesis of theories from earlier Greek mathematicians, including Eudoxus of Cnidus, Hippocrates of Chios, and Theaetetus. With Archimedes and Apollonius of Perga, Euclid is generally considered among the greatest mathematicians of antiquity, and one of the most influential in the history of mathematics.
Very little is known of Euclid's life, and most information comes from the scholars Proclus and Pappus of Alexandria many centuries later. Medieval Islamic mathematicians invented a fanciful biography, and medieval Byzantine and early Renaissance scholars mistook him for the earlier philosopher Euclid of Megara. It is now generally accepted that he spent his career in Alexandria and lived around 300 BC, after Plato's students and before Archimedes. There is some speculation that Euclid studied at the Platonic Academy and later taught at the Musaeum; he is regarded as bridging the earlier Platonic tradition in Athens with the later tradition of Alexandria.
In the Elements, Euclid deduced the theorems from a small set of axioms. He also wrote works on perspective, conic sections, spherical geometry, number theory, and mathematical rigour. In addition to the Elements, Euclid wrote a central early text in the optics field, Optics, and lesser-known works including Data and Phaenomena. Euclid's authorship of two other texts—On Divisions of Figures, Catoptrics—has been questioned. He is thought to have written many now lost works. Шаблон:TOC limit
Life
Traditional narrative
The English name 'Euclid' is the anglicized version of the Ancient Greek name Шаблон:Transl (Шаблон:Lang).Шаблон:SfnШаблон:Efn It is derived from 'eu-' (εὖ; 'well') and 'klês' (-κλῆς; 'fame'), meaning "renowned, glorious".Шаблон:Sfn In English, by metonymy, 'Euclid' can mean his most well-known work, Euclid's Elements, or a copy thereof,Шаблон:Sfn and is sometimes synonymous with 'geometry'.Шаблон:Sfn
As with many ancient Greek mathematicians, the details of Euclid's life are mostly unknown.Шаблон:Sfn He is accepted as the author of four mostly extant treatises—the Elements, Optics, Data, Phaenomena—but besides this, there is nothing known for certain of him.Шаблон:SfnШаблон:Efn The traditional narrative mainly follows the 5th century AD account by Proclus in his Commentary on the First Book of Euclid's Elements, as well as a few anecdotes from Pappus of Alexandria in the early 4th century.Шаблон:SfnШаблон:Efn
According to Proclus, Euclid lived shortly after several of Plato's (Шаблон:Died in BC) followers and before the mathematician Archimedes (Шаблон:Circa BC);Шаблон:Efn specifically, Proclus placed Euclid during the rule of Ptolemy I (Шаблон:Reign BC).Шаблон:SfnШаблон:SfnШаблон:Efn Euclid's birthdate is unknown; some scholars estimate around 330Шаблон:SfnШаблон:Sfn or 325 BC,Шаблон:SfnШаблон:Sfn but others refrain from speculating.Шаблон:Sfn It is presumed that he was of Greek descent,Шаблон:Sfn but his birthplace is unknown.Шаблон:SfnШаблон:Efn
Proclus held that Euclid followed the Platonic tradition, but there is no definitive confirmation for this.Шаблон:Sfn It is unlikely he was contemporary with Plato, so it is often presumed that he was educated by Plato's disciples at the Platonic Academy in Athens.Шаблон:Sfn Historian Thomas Heath supported this theory, noting that most capable geometers lived in Athens, including many of those whose work Euclid built on;Шаблон:Sfn Sialaros considers this a mere conjecture.Шаблон:SfnШаблон:Sfn In any event, the contents of Euclid's work demonstrate familiarity with the Platonic geometry tradition.Шаблон:Sfn
In his Collection, Pappus mentions that Apollonius studied with Euclid's students in Alexandria, and this has been taken to imply that Euclid worked and founded a mathematical tradition there.Шаблон:SfnШаблон:SfnШаблон:Sfn The city was founded by Alexander the Great in 331 BC,Шаблон:Sfn and the rule of Ptolemy I from 306 BC onwards gave it a stability which was relatively unique amid the chaotic wars over dividing Alexander's empire.Шаблон:Sfn Ptolemy began a process of hellenization and commissioned numerous constructions, building the massive Musaeum institution, which was a leading center of education.Шаблон:SfnШаблон:Efn Euclid is speculated to have been among the Musaeum's first scholars.Шаблон:Sfn
Euclid's date of death is unknown; it has been speculated that he died Шаблон:Circa.Шаблон:Sfn
Identity and historicity
Euclid is often referred to as 'Euclid of Alexandria' to differentiate him from the earlier philosopher Euclid of Megara, a pupil of Socrates included in dialogues of Plato with whom he was historically conflated.Шаблон:SfnШаблон:Sfn Valerius Maximus, the 1st century AD Roman compiler of anecdotes, mistakenly substituted Euclid's name for Eudoxus (4th century BC) as the mathematician to whom Plato sent those asking how to double the cube.Шаблон:Sfn Perhaps on the basis of this mention of a mathematical Euclid roughly a century early, Euclid became mixed up with Euclid of Megara in medieval Byzantine sources (now lost),Шаблон:Sfn eventually leading Euclid the mathematician to be ascribed details of both men's biographies and described as Шаблон:Lang (Шаблон:Lit).Шаблон:SfnШаблон:Sfn The Byzantine scholar Theodore Metochites (Шаблон:C.) explicitly conflated the two Euclids, as did printer Erhard Ratdolt's 1482 Шаблон:Lang of Campanus of Novara's Latin translation of the Elements.Шаблон:Sfn After the mathematician Шаблон:Ill appended most of the extant biographical fragments about either Euclid to the preface of his 1505 translation of the Elements, subsequent publications passed on this identification.Шаблон:Sfn Later Renaissance scholars, particularly Peter Ramus, reevaluated this claim, proving it false via issues in chronology and contradiction in early sources.Шаблон:Sfn
Medieval Arabic sources give vast amounts of information concerning Euclid's life, but are completely unverifiable.Шаблон:Sfn Most scholars consider them of dubious authenticity;Шаблон:Sfn Heath in particular contends that the fictionalization was done to strengthen the connection between a revered mathematician and the Arab world.Шаблон:Sfn There are also numerous anecdotal stories concerning to Euclid, all of uncertain historicity, which "picture him as a kindly and gentle old man".Шаблон:Sfn The best known of these is Proclus' story about Ptolemy asking Euclid if there was a quicker path to learning geometry than reading his Elements, which Euclid replied with "there is no royal road to geometry".Шаблон:Sfn This anecdote is questionable since a very similar interaction between Menaechmus and Alexander the Great is recorded from Stobaeus.Шаблон:Sfn Both accounts were written in the 5th century AD, neither indicates its source, and neither appears in ancient Greek literature.Шаблон:Sfn
Any firm dating of Euclid's activity Шаблон:C. is called into question by a lack of contemporary references.Шаблон:Sfn The earliest original reference to Euclid is in Apollonius' prefatory letter to the Conics (early 2nd century BC): "The third book of the Conics contains many astonishing theorems that are useful for both the syntheses and the determinations of number of solutions of solid loci. Most of these, and the finest of them, are novel. And when we discovered them we realized that Euclid had not made the synthesis of the locus on three and four lines but only an accidental fragment of it, and even that was not felicitously done."Шаблон:Sfn The Elements is speculated to have been at least partly in circulation by the 3rd century BC, as Archimedes and Apollonius take several of its propositions for granted;Шаблон:Sfn however, Archimedes employs an older variant of the theory of proportions than the one found in the Elements.Шаблон:Sfn The oldest physical copies of material included in the Elements, dating from roughly 100 AD, can be found on papyrus fragments unearthed in an ancient rubbish heap from Oxyrhynchus, Roman Egypt. The oldest extant direct citations to the Elements in works whose dates are firmly known are not until the 2nd century AD, by Galen and Alexander of Aphrodisias; by this time it was a standard school text.Шаблон:Sfn Some ancient Greek mathematicians mention Euclid by name, but he is usually referred to as "ὁ στοιχειώτης" ("the author of Elements").Шаблон:Sfn In the Middle Ages, some scholars contended Euclid was not a historical personage and that his name arose from a corruption of Greek mathematical terms.Шаблон:Sfn
Works
Elements
Euclid is best known for his thirteen-book treatise, the Elements (Шаблон:Lang-grc-gre; Шаблон:Transliteration), considered his magnum opus.Шаблон:SfnШаблон:Sfn Much of its content originates from earlier mathematicians, including Eudoxus, Hippocrates of Chios, Thales and Theaetetus, while other theorems are mentioned by Plato and Aristotle.Шаблон:Sfn It is difficult to differentiate the work of Euclid from that of his predecessors, especially because the Elements essentially superseded much earlier and now-lost Greek mathematics.Шаблон:SfnШаблон:Efn The classicist Markus Asper concludes that "apparently Euclid's achievement consists of assembling accepted mathematical knowledge into a cogent order and adding new proofs to fill in the gaps" and the mathematician Serafina Cuomo described it as a "reservoir of results".Шаблон:SfnШаблон:Sfn Despite this, Sialaros furthers that "the remarkably tight structure of the Elements reveals authorial control beyond the limits of a mere editor".Шаблон:Sfn
The Elements does not exclusively discuss geometry as is sometimes believed.Шаблон:Sfn 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) do not exactly fit this scheme.Шаблон:SfnШаблон:Sfn The heart of the text is the theorems scattered throughout.Шаблон:Sfn Using Aristotle's terminology, these may be generally separated into two categories: "first principles" and "second principles".Шаблон:Sfn The first group includes statements labeled as a "definition" (Шаблон:Lang-grc-gre or Шаблон:Lang), "postulate" (Шаблон:Lang), or a "common notion" (Шаблон:Lang);Шаблон:SfnШаблон:Sfn only the first book includes postulates—later known as axioms—and common notions.Шаблон:SfnШаблон:Efn The second group consists of propositions, presented alongside mathematical proofs and diagrams.Шаблон:SfnШаблон:Efn It is unknown if Euclid intended the Elements as a textbook, but its method of presentation makes it a natural fit.Шаблон:Sfn As a whole, the authorial voice remains general and impersonal.Шаблон:Sfn
Contents
| Шаблон:Abbr | Postulates |
|---|---|
| Let the following be postulated: | |
| 1 | To draw a straight line from any point to any pointШаблон:Efn |
| 2 | To produce a finite straight line continuously in a straight line |
| 3 | To describe a circle with any centre and distance |
| 4 | That all right angles are equal to one another |
| 5 | That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles |
| Шаблон:Abbr | Common notions |
| 1 | Things which are equal to the same thing are also equal to one another |
| 2 | If equals be added to equals, the wholes are equal |
| 3 | If equals be subtracted from equals, the remainders are equal |
| 4 | Things which coincide with one another are equal to one another |
| 5 | The whole is greater than the part |
Book 1 of the Elements is foundational for the entire text.Шаблон:Sfn It begins with a series of 20 definitions for basic geometric concepts such as lines, angles and various regular polygons.Шаблон:Sfn Euclid then presents 10 assumptions (see table, right), grouped into five postulates (axioms) and five common notions.Шаблон:SfnШаблон:Efn These assumptions are intended to provide the logical basis for every subsequent theorem, i.e. serve as an axiomatic system.Шаблон:SfnШаблон:Efn The common notions exclusively concern the comparison of magnitudes.Шаблон:Sfn While postulates 1 through 4 are relatively straight forward,Шаблон:Efn the 5th is known as the parallel postulate and particularly famous.Шаблон:SfnШаблон:Efn
Book 1 also includes 48 propositions, which can be loosely divided into those concerning basic theorems and constructions of plane geometry and triangle congruence (1–26); parallel lines (27–34); the area of triangles and parallelograms (35–45); and the Pythagorean theorem (46–48).Шаблон:Sfn The last of these includes the earliest surviving proof of the Pythagorean theorem, described by Sialaros as "remarkably delicate".Шаблон:Sfn
Book 2 is traditionally understood as concerning "geometric algebra", though this interpretation has been heavily debated since the 1970s; critics describe the characterization as anachronistic, since the foundations of even nascent algebra occurred many centuries later.Шаблон:Sfn The second book has a more focused scope and mostly provides algebraic theorems to accompany various geometric shapes.Шаблон:SfnШаблон:Sfn It focuses on the area of rectangles and squares (see Quadrature), and leads up to a geometric precursor of the law of cosines.
Book 3 focuses on circles, while the 4th discusses regular polygons, especially the pentagon.Шаблон:SfnШаблон:Sfn Book 5 is among the work's most important sections and presents what is usually termed as the "general theory of proportion".Шаблон:SfnШаблон:Efn Book 6 utilizes the "theory of ratios" in the context of plane geometry.Шаблон:Sfn It is built almost entirely of its first proposition:Шаблон:Sfn "Triangles and parallelograms which are under the same height are to one another as their bases".Шаблон:Sfn
From Book 7 onwards, the mathematician Шаблон:Ill notes that "Euclid starts afresh. Nothing from the preceding books is used".Шаблон:Sfn 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.Шаблон:Sfn Book 7 includes the Euclidean algorithm, a method for finding the greatest common divisor of two numbers.Шаблон:Sfn The 8th book discusses geometric progressions, while book 9 includes the proposition, now called Euclid's theorem, that there are infinitely many prime numbers.Шаблон:Sfn
Of the Elements, book 10 is by far the largest and most complex, dealing with irrational numbers in the context of magnitudes.Шаблон:Sfn
The final three books (11–13) primarily discuss solid geometry.Шаблон:Sfn By introducing a list of 37 definitions, Book 11 contextualizes the next two.Шаблон:Sfn Although its foundational character resembles Book 1, unlike the latter it features no axiomatic system or postulates.Шаблон:Sfn The three sections of Book 11 include content on solid geometry (1–19), solid angles (20–23) and parallelepipedal solids (24–37).Шаблон:Sfn
Other works
In addition to the Elements, at least five works of Euclid have survived to the present day. They follow the same logical structure as Elements, with definitions and proved propositions.
- Catoptrics concerns the mathematical theory of mirrors, particularly the images formed in plane and spherical concave mirrors, though the attribution is sometimes questioned.Шаблон:Sfn
- The Data (Шаблон:Lang-grc-gre), is a somewhat short text which deals with the nature and implications of "given" information in geometrical problems.Шаблон:Sfn
- On Divisions (Шаблон:Lang-grc-gre) survives only partially in Arabic translation, and concerns the division of geometrical figures into two or more equal parts or into parts in given ratios. It includes thirty-six propositions and is similar to Apollonius' Conics.Шаблон:Sfn
- The Optics (Шаблон:Lang-grc-gre) is the earliest surviving Greek treatise on perspective. It includes an introductory discussion of geometrical optics and basic rules of perspective.Шаблон:Sfn
- The Phaenomena (Шаблон:Lang-grc-gre) is a treatise on spherical astronomy, survives in Greek; it is similar to On the Moving Sphere by Autolycus of Pitane, who flourished around 310 BC.Шаблон:Sfn
Lost works
Four other works are credibly attributed to Euclid, but have been lost.Шаблон:Sfn
- The Conics (Шаблон:Lang-grc-gre) was a four-book survey on conic sections, which was later superseded by Apollonius' more comprehensive treatment of the same name.Шаблон:SfnШаблон:Sfn The work's existence is known primarily from Pappus, who asserts that the first four books of Apollonius' Conics are largely based on Euclid's earlier work.Шаблон:Sfn Doubt has been cast on this assertion by the historian Шаблон:Ill, owing to sparse evidence and no other corroboration of Pappus' account.Шаблон:Sfn
- The Pseudaria (Шаблон:Lang-grc-gre; Шаблон:Lit), was—according to Proclus in (70.1–18)—a text in geometrical reasoning, written to advise beginners in avoiding common fallacies.Шаблон:SfnШаблон:Sfn Very little is known of its specific contents aside from its scope and a few extant lines.Шаблон:Sfn
- The Porisms (Шаблон:Lang-grc-gre; Шаблон:Lit) was, based on accounts from Pappus and Proclus, probably a three-book treatise with approximately 200 propositions.Шаблон:SfnШаблон:Sfn The term 'porism' in this context does not refer to a corollary, but to "a third type of proposition—an intermediate between a theorem and a problem—the aim of which is to discover a feature of an existing geometrical entity, for example, to find the centre of a circle".Шаблон:Sfn The mathematician Michel Chasles speculated that these now-lost propositions included content related to the modern theories of transversals and projective geometry.Шаблон:SfnШаблон:Efn
- The Surface Loci (Шаблон:Lang-grc-gre) is of virtually unknown contents, aside from speculation based on the work's title.Шаблон:Sfn Conjecture based on later accounts has suggested it discussed cones and cylinders, among other subjects.Шаблон:Sfn
Collections
Euclid's works can be found in cultural institutions across the world. Many of these editions are digitised and available for public consultation.
- University College London holds c.500 works of Euclid, including 63 editions published before 1580. The majority of the collection is from the library of John Thomas Graves, who bequeathed his books to the University in 1870. The collection has been digitised through the Stavros Niarchos Foundation Library.[1] [2]
Legacy
Euclid is generally considered with Archimedes and Apollonius of Perga as among the greatest mathematicians of antiquity.Шаблон:Sfn Many commentators cite him as one of the most influential figures in the history of mathematics.Шаблон:Sfn The geometrical system established by the Elements long dominated the field; however, today that system is often referred to as 'Euclidean geometry' to distinguish it from other non-Euclidean geometries discovered in the early 19th century.Шаблон:Sfn Among Euclid's many namesakes are the European Space Agency's (ESA) Euclid spacecraft,[3] the lunar crater Euclides,[4] and the minor planet 4354 Euclides.[5]
The Elements is often considered after the Bible as the most frequently translated, published, and studied book in the Western World's history.Шаблон:Sfn With Aristotle's Metaphysics, the Elements is perhaps the most successful ancient Greek text, and was the dominant mathematical textbook in the Medieval Arab and Latin worlds.Шаблон:Sfn
The first English edition of the Elements was published in 1570 by Henry Billingsley and John Dee.Шаблон:Sfn The mathematician Oliver Byrne published a well-known version of the Elements in 1847 entitled The First Six Books of the Elements of Euclid in Which Coloured Diagrams and Symbols Are Used Instead of Letters for the Greater Ease of Learners, which included colored diagrams intended to increase its pedagogical effect.[6] David Hilbert authored a modern axiomatization of the Elements.Шаблон:Sfn
References
Notes
Citations
Sources
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External links
- Works
- Шаблон:Gutenberg author
- Шаблон:Internet Archive author
- Шаблон:Librivox author
- Euclid Collection at University College London (c.500 editions of works by Euclid), available online through the Stavros Niarchos Foundation Digital Library.
- Scans of Johan Heiberg's edition of Euclid at wilbourhall.org
- The Elements
- PDF copy, with the original Greek and an English translation on facing pages, University of Texas.
- All thirteen books, in several languages as Spanish, Catalan, English, German, Portuguese, Arabic, Italian, Russian and Chinese.
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