Euclid

Euclid of Alexandria (Greek: Εὐκλείδης) (ca. 325 BC–265 BC) was a Greek mathematician who taught at Alexandria in Egypt almost certainly during the reign (323 BC–283 BC) of Ptolemy I. Now known as "the father of geometry," his most famous work is Elements, widely considered to be history's most successful textbook. Within it, the properties of geometrical objects and integers are deduced from a small set of axioms, thereby anticipating (and partly inspiring) the axiomatic method of modern mathematics. Centuries: 5th century BC - 4th century BC - 3rd century BC Decades: 370s BC 360s BC 350s BC 340s BC 330s BC - 320s BC - 310s BC 300s BC 290s BC 280s BC 270s BC 330 BC 329 BC 328 BC 327 BC 326 BC - 325 BC - 324 BC 323 BC 322... Centuries: 4th century BC - 3rd century BC - 2nd century BC Decades: 310s BC 300s BC 290s BC 280s BC 270s BC - 260s BC - 250s BC 240s BC 230s BC 220s BC 210s BC Years: 270 BC 269 BC 268 BC 267 BC 266 BC - 265 BC - 264 BC 263 BC... A mathematician is a person whose area of study and research is mathematics. ... Antiquity and modernity stand cheek-by-jowl in Egypts chief Mediterranean seaport Located on the Mediterranean Sea coast, Alexandria (in Arabic, الإسكندرية, transliterated al-ʼIskandariyyah) is the chief seaport in Egypt, and that countrys second largest city, and the capital of the Al Iskandariyah governate. ... Centuries: 5th century BC - 4th century BC - 3rd century BC Decades: 370s BC 360s BC 350s BC 340s BC 330s BC - 320s BC - 310s BC 300s BC 290s BC 280s BC 270s BC 328 BC 327 BC 326 BC 325 BC 324 BC - 323 BC - 322 BC 321 BC 320... Centuries: 4th century BC - 3rd century BC - 2nd century BC Decades: 330s BC 320s BC 310s BC 300s BC 290s BC - 280s BC - 270s BC 260s BC 250s BC 240s BC 230s BC 288 BC 287 BC 286 BC 285 BC 284 BC 283 BC 282 BC 281 BC 280... For the unrelated astronomer, see Ptolemy Ptolemy I Soter (367 BC–283 BC), ruler of Egypt (reigned 323 BC - 283 BC) and founder of the Ptolemaic dynasty. ... Geometry (Greek Γεωμετρια, geo = earth, metria = measure ) arose as the field of knowledge dealing with spatial relationships. ... Euclids Elements (Greek Στοιχεία) is a mathematical and geometric treatise, consisting of 13 books, written by the Greek mathematician Euclid around 300 BC. It comprises a collection of definitions, postulates (axioms), propositions (theorems) and proofs thereof. ... Textbooks are defined as a manual of instruction, a standard book in any branch of study. They are further defined by both the age of the person who is to study the text and the classification of the subject matter itself. ... Geometry (Greek Γεωμετρια, geo = earth, metria = measure ) arose as the field of knowledge dealing with spatial relationships. ... The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up. ... In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. ... Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ...

Euclid also wrote works on perspective, conic sections, spherical geometry, and possibly quadric surfaces. Neither the year nor place of his birth have been established, nor the circumstances of his death. Perspective when used in the context of vision and visual perception refers to the way in which objects appear to the eye based on their spatial attributes or dimension and the position of the eye relative to the objects. ... In mathematics, a conic section (or just conic) is a curved locus of points, fby intersecting a cone with a plane. ... Spherical geometry is the geometry of the two-dimensional surface of a sphere. ... Ellipsoid Elliptic Paraboloid Hyperbolic Paraboloid Hyperboloid of One Sheet Hyperboloid of Two Sheets Cone Elliptic Cylinder Hyperbolic Cylinder Parabolic Cylinder In mathematics a quadric, or quadric surface, is any D-dimensional (hyper-)surface represented by a second-order equation in spatial variables (coordinates). ...

The Elements

Main article: Euclid's Elements

Although many of the results in Elements originated with earlier mathematicians, one of Euclid's accomplishments was to present them in a single, logically coherent framework. In addition to providing some missing proofs, Euclid's text also includes sections on number theory and three-dimensional geometry. Euclids Elements (Greek Στοιχεία) is a mathematical and geometric treatise, consisting of 13 books, written by the Greek mathematician Euclid around 300 BC. It comprises a collection of definitions, postulates (axioms), propositions (theorems) and proofs thereof. ... In mathematics, a proof is a demonstration that, given certain axioms, some statement of interest is necessarily true. ... Traditionally, number theory is that branch of pure mathematics concerned with the properties of integers. ... In mathematics, solid geometry was the traditional name for the geometry of three-dimensional Euclidean space — for practical purposes the kind of space we live in. ...

The geometrical system described in Elements was long known simply as "the" geometry. Today, however, it is often referred to as Euclidean geometry to distinguish it from other so-called non-Euclidean geometries which were discovered in the 19th century. These new geometries grew out of more than two millennia of investigation into Euclid's fifth postulate, one of the most-studied axioms in all of mathematics. Most of these investigations involved attempts to prove the relatively complex and presumably non-intuitive fifth postulate using the other four (a feat which, if successful, would have shown the postulate to be in fact a theorem). In mathematics, Euclidean geometry is the familiar kind of geometry on the plane or in three dimensions. ... Behavior of lines with a common perpendicular in each of the three types of geometry The term non-Euclidean geometry (also spelled: non-Euclidian geometry) describes both hyperbolic and elliptic geometry, which are contrasted with Euclidean geometry. ... Alternative meaning: Nineteenth Century (periodical) (18th century — 19th century — 20th century — more centuries) As a means of recording the passage of time, the 19th century was that century which lasted from 1801-1900 in the sense of the Gregorian calendar. ... A millennium is a period of time, literally equal to one thousand years. ... a and b are parallel, the transversal t produces congruent angles. ... In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up. ... A theorem is a proposition that has been or is to be proved on the basis of explicit assumptions. ...

While the Elements was used well into the 20th century as a geometry textbook and has been considered a fine example of the formally precise axiomatic method, Euclid's treatment does not hold up to modern standards of rigor; some logically necessary axioms are missing, and the definitions of primitive terms appeal to spatial intuition. The first correct axiomatic treatment of geometry by modern standards was provided by David Hilbert in 1899, in his Grundlagen der Geometrie. (19th century - 20th century - 21st century - more centuries) Decades: 1900s 1910s 1920s 1930s 1940s 1950s 1960s 1970s 1980s 1990s As a means of recording the passage of time, the 20th century was that century which lasted from 1901–2000 in the sense of the Gregorian calendar (1900–1999 in the... David Hilbert David Hilbert (January 23, 1862 – February 14, 1943) was a German mathematician born in Wehlau, near Königsberg, Prussia (now Znamensk, near Kaliningrad, Russia) who is recognized as one of the most influential mathematicians of the 19th and early 20th centuries. ... 1899 was a common year starting on Sunday (see link for calendar). ...

Other works

In addition to the Elements, four works of Euclid have survived to the present day.

• Data deals with the nature and implications of "given" information in geometrical problems; the subject matter is closely related to the first four books of the Elements.
• On Divisions of Figures, which survives only partially in Arabic translation, concerns the division of geometrical figures into two or more equal parts or into parts in given ratios. It is similar to a third century (AD) work by Heron of Alexandria, except Euclid's work characteristically lacks any numerical calculations.
• Phaenomena concerns the application of spherical geometry to problems of astronomy.
• Optics, the earliest surviving Greek treatise on perspective, contains propositions on the apparent sizes and shapes of objects viewed from different distances and angles.

All of these works follow the basic logical structure of the Elements, containing definitions and proved propositions. Data is the plural of datum. ... Arabic (العربية al-arabiyyah, or less formally arabi) is the largest member of the Semitic branch of the Afro-Asiatic language family (classification: South Central Semitic) and is closely related to Hebrew and Aramaic. ... In algebra, a ratio is the relationship between two quantities. ... (2nd century - 3rd century - 4th century - other centuries) Events The Sassanid dynasty of Persia launches a war to reconquer lost lands in the Roman east. ... Heros aeolipile Hero (or Heron) of Alexandria (c. ... Spherical geometry is the geometry of the two-dimensional surface of a sphere. ... Astrometry: the study of the position of objects in the sky and their changes of position. ... Perspective is the choice of a single point of view from which to sense, categorize, measure or codify experience, typically for comparing with another. ...

There are four works credibly attributed to Euclid which have been lost

• Conics was a work on conic sections that was later extended by Apollonius of Perga into his famous work on the subject.
• Porisms might have been an outgrowth of Euclid's work with conic sections, but the exact meaning of the title is controversial.
• Pseudaria, or Book of Fallacies, was an elementary text about errors in reasoning.
• Surface Loci concerned either loci (sets of points) on surfaces or loci which were themselves surfaces; under the latter interpretation, it has been hypothesized that the work might have dealt with quadric surfaces.

In mathematics, a conic section (or just conic) is a curved locus of points, fby intersecting a cone with a plane. ... Apollonius of Perga or Perge (ca. ... The subject of porisms is perplexed by the multitude of different views which have been held by geometers as to what a porism really was and is. ... Reasoning is the act of using reason to derive a conclusion from certain premises. ... In mathematics, a locus (plural loci) is a collection of points which share a common property. ... Ellipsoid Elliptic Paraboloid Hyperbolic Paraboloid Hyperboloid of One Sheet Hyperboloid of Two Sheets Cone Elliptic Cylinder Hyperbolic Cylinder Parabolic Cylinder In mathematics a quadric, or quadric surface, is any D-dimensional (hyper-)surface represented by a second-order equation in spatial variables (coordinates). ...

Biographical sources

Almost nothing is known about Euclid outside of what is presented in Elements and his few other surviving books. What little biographical information we do have comes largely from commentaries by Proclus and Pappus of Alexandria: he was active at the great library in Alexandria and may have studied at Plato's Academe in Greece, but his exact lifespan and place of birth are unknown. Proclus Lycaeus (February 8, 412 – April 17, 487), surnamed The Successor (Greek Πρόκλος ὁ Διάδοχος Próklos ho Diádokhos), was a Greek Neoplatonist philosopher. ... Pappus of Alexandria is one of the most important mathematicians of ancient Greek time, known for his work Synagoge or Collection (c. ... The Royal Library of Alexandria was once the largest in the world. ... Statue of a philosopher, presumably Plato, in Delphi. ... Plato is credited with the inception of academia: the body of knowledge, its development and transmission across generations. ...

In the Middle Ages, writers sometimes referred to him as Euclid of Megara, confusing him with a Greek Socratic philosopher who lived approximately one century earlier. The Middle Ages formed the middle period in a traditional schematic division of European history into three ages: the classical civilization of Antiquity, the Middle Ages, and modern times, beginning with the Renaissance. ... Euclid of Megara, a Greek Socratic philosopher who lived around 400 BC, was the follower of Socrates. ... Socrates This article is about the ancient Greek philosopher, for all other uses see: Socrates (disambiguation) Socrates (June 4, ca. ... A philosopher is a person devoted to studying and producing results in philosophy. ...

References

• Bulmer-Thomas, Ivor (1971). "Euclid." Dictionary of Scientific Biography.
• Heath, Thomas L. (1956). The Thirteen Books of Euclid's Elements, Vol. 1 (2nd ed.). New York: Dover Publications. ISBN 0-486-60088-2.
• Heath, Thomas L. (1981). A History of Greek Mathematics, 2 Vols. New York: Dover Publications. ISBN 0-486-24073-8 / ISBN 0-486-24074-6.
• Kline, Morris (1980). Mathematics: The Loss of Certainty. Oxford: Oxford University Press. ISBN 0-19-502754-X.

External links

• Euclid entry at the MacTutor History of Mathematics archive
• Library search at WorldCat for The Medieval Latin translation of the Data of Euclid by Shuntaro Ito