Euclid

• Born: ca. 325 B.C.
• Birthplace:
• Died: ca. 265 B.C.
• Best Known As: Alexandrian mathematician and father of geometry

Euclid was a mathematician whose third century B.C. textbook Elements served as the western world's unchallenged standard for two millennia. Nothing is known about Euclid's life or physical appearance, and what little is known about his career comes from inferences in later sources. It is generally agreed that he taught geometry in Hellenistic Egypt, at Alexandria during the reign of Ptolemy I, between 305 and 285 B.C. He is credited with the thirteen volumes of Elements, a work that accumulated mathematical knowledge and codified it into a deductive system of proofs. Euclidean geometry was the geometry until the 19th century, when mathematicians began to challenge Euclid's assumptions about parallel lines when considering measurements over very large distances of, say, billions of light years.

Greek mathematician (c. 330 bc–260 bc)

Euclid is one of the best known and most influential of classical Greek mathematicians but almost nothing is known about his life. He was a founder and member of the academy in Alexandria, and may have been a pupil of Plato in Athens. Despite his great fame Euclid was not one of the greatest of Greek mathematicians and not of the same caliber as Archimedes.

Euclid's most celebrated work is the Elements, which is primarily a treatise on geometry contained in 13 books. The influence of this work not only on the future development of geometry, mathematics, and science, but on the whole of Western thought is hard to exaggerate. Some idea of the importance that has been attached to the Elements is gained from the fact that there have probably been more commentaries written on it than on the Bible. The Elements systematized and organized the work of many previous Greek geometers, such as Theaetetus and Eudoxus, as well as containing many new discoveries that Euclid had made himself. Although mainly concerned with geometry it also deals with such topics as number theory and the theory of irrational quantities. One of the most celebrated number theoretic results is Euclid's proof that there are an infinite number of primes. The Elements is in many ways a synthesis and culmination of Greek mathematics. Euclid and Apollonius of Perga were the last Greek mathematicians of any distinction, and after their time Greek civilization as a whole soon became decadent and sterile.

Euclid's Elements owed its enormously high status to a number of reasons. The most influential single feature was Euclid's use of the axiomatic method whereby all the theorems were laid out as deductions from certain self-evident basic propositions or axioms in such a way that in each successive proof only propositions already proved or axioms were used. This became accepted as the paradigmatically rigorous way of setting out any body of knowledge, and attempts were made to apply it not just to mathematics, but to natural science, theology, and even philosophy and ethics.

However, despite being revered as an almost perfect example of rigorous thinking for almost 2000 years there are considerable defects in Euclid's reasoning. A number of his proofs were found to contain mistakes, the status of the initial axioms themselves was increasingly considered to be problematic, and the definitions of such basic terms as ‘line’ and ‘point’ were found to be unsatisfactory. The most celebrated case is that of the parallel axiom, which states that there is only one straight line passing through a given point and parallel to a given straight line. The status of this axiom was long recognized as problematic, and many unsuccessful attempts were made to deduce it from the remaining axioms. The question was only settled in the 19th century when Janos Bolyai and Nicolai Lobachevski showed that it was perfectly possible to construct a consistent geometry in which Euclid's other axioms were true but in which the parallel axiom was false. This epoch-making discovery displaced Euclidean geometry from the privileged position it had occupied. The question of the relation of Euclid's geometry to the properties of physical space had to wait until the early 20th century for a full answer. Until then it was believed that Euclid's geometry gave a fully accurate description of physical space. No less a thinker than Immanuel Kant had thought that it was logically impossible for space to obey any other geometry. However when Albert Einstein developed his theory of relativity he found that the appropriate geometry for space was not Euclid's but that developed by Georg Riemann. It was subsequently experimentally verified that the geometry of space is indeed non-Euclidean.

In mathematical terms too, the discovery of non-Euclidean geometries was of great importance, since it led to a broadening of the conception of geometry and the development by such mathematicians as Felix Klein of many new geometries very different from Euclid's. It also made mathematicians scrutinize the logical structure of Euclid's geometry far more closely and in 1899 David Hilbert at last gave a definitively rigorous axiomatic treatment of geometry and made an exhaustive investigation of the relations of dependence and independence between the axioms, and of the consistency of the various possible geometries so produced.

Euclid wrote a number of other works besides the Elements, although many of them are now lost and known only through references to them by other classical authors. Those that do survive include Data, containing 94 propositions, On Divisions, and the Optics. One of his sayings has come down to us. When asked by Ptolemy I Soter, the reigning king of Egypt, if there was any quicker way to master geometry than by studying the Elements Euclid replied “There is no royal road to geometry.”

The Greek mathematician Euclid (active 300 B.C.) wrote the "Elements", a collection of geometrical theorems. The oldest extant major mathematical work in the Western world, it set a standard for logical exposition for over 2,000 years.

Virtually nothing is known of Euclid personally. It is not even known for certain whether he was a creative mathematician himself or was simply good at compiling the work of others. Most of the information about Euclid comes from Proclus, a 5th-century-A.D. Greek scholar. Since Archimedes refers to Euclid and Archimedes lived immediately after the time of Ptolemy I, King of Egypt (ca. 306-283 B.C.), Proclus concludes they were contemporaries. Euclid's mathematical education may well have been obtained from Plato's pupils in Athens, since it was there that most of the earlier mathematicians upon whose work the Elements is based had studied and taught.

No earlier writings comparable to the Elements of Euclid have survived. One reason is that Euclid's Elements superseded all previous writings of this type, making it unnecessary to preserve them. This makes it difficult for the historian to investigate those earlier mathematicians whose works were probably more important in the development of Greek mathematics than Euclid's. About 600 B.C. the Greek mathematician Thales is said to have discovered a number of theorems that appear in the Elements. It might be noted too that Eudoxus is also given credit for the discovery of the method of exhaustion, whereby the area of a circle and volume of a sphere and other figures can be calculated. Book XII of the Elements makes use of this method. Although mathematics may have been initiated by concrete problems, such as determining areas and volumes, by the time of Euclid mathematics had developed into an abstract construction, an intellectual occupation for philosophers rather than scientists.

The Elements

The Elements consists of 13 books. Within each book is a sequence of propositions or theorems, varying from about 10 to 100, preceded by definitions. In Book I, 23 definitions are followed by five postulates. After the postulates, five common notions or axioms are listed. The first is, "Things which are equal to the same thing are also equal to each other." Next are 48 propositions which relate some of the objects that were defined and which lead up to Pythagoras's theorem: in right-angled triangles the square on the side subtending the right angle is equal to the sum of the squares on the sides containing the right angle. The usual elementary course in Euclidean geometry is based on Book I.

The remaining books, although not so well known, are more advanced mathematically. Book II is a continuation of Book I, proving geometrically what today would be called algebraic identities, such as (a + b)² = a² + b² + 2ab, and generalizing some propositions of Book I. Book III is on circles, intersections of circles, and properties of tangents to circles. Book IV continues with circles, emphasizing inscribed and circumscribed rectilinear figures.

Book V of the Elements is one of the finest works in Greek mathematics. The theory of proportions discovered by Eudoxus is here expounded masterfully by Euclid. The theory of proportions is concerned with the ratios of magnitudes (rational or irrational numbers) and their integral multiples. Book VI applies the propositions of Book V to the figures of plane geometry. A basic proposition in this book is that a line parallel to one side of a triangle will divide the other two sides in the same ratio.

As in Book V, Books VII, VIII, and IX are concerned with properties of (positive integral) numbers. In Book VII a prime number is defined as that which is measured by a unit alone (a prime number is divisible only by itself and 1). In Book IX proposition 20 asserts that there are infinitely many prime numbers, and Euclid's proof is essentially the one usually given in modern algebra textbooks. Book X is an impressively well-finished treatment of irrational numbers or, more precisely, straight lines whose lengths cannot be measured exactly by a given line assumed as rational.

Books XI-XIII are principally concerned with three-dimensional figures. In Book XII the method of exhaustion is used extensively. The final book shows how to construct and circumscribe by a sphere the five Platonic, or regular, solids: the regular pyramid or tetrahedron, octahedron, cube, icosahedron, and dodecahedron.

Manuscript translations of the Elements were made in Latin and Arabic, but it was not until the first printed edition, published in Venice in 1482, that geometry, which meant in effect the Elements, became important in European education. The first complete English translation was printed in 1570. It was during the most active mathematical period in England, about 1700, that Greek mathematics was studied most intensively. Euclid was admired, mastered, and utilized by all major mathematicians, including Isaac Newton.

The growing predominance of the sciences and mathematics in the 18th and 19th centuries helped to keep Euclid in a prominent place in the curriculum of schools and universities throughout the Western world. But also the Elements was considered educational as a primer in logic.

Euclid's Other Works

Some of Euclid's other works are known only through references by other writers. The Data is on plane geometry. The word "data" means "things given." The treatise contains 94 propositions concerned with the kind of problem where certain data are given about a figure and from which other data can be deduced, for example: if a triangle has one angle given, the rectangle contained by the sides including the angle has to the area of the triangle a given ratio.

On Division (of figures), also on plane geometry, is known only in the Arabic, from which English translations were made. Proclus refers to it when speaking of dividing a figure into other figures different in kind, for example, dividing a triangle into a triangle and a quadrilateral. On Division is concerned with more general problems of division. As an example, one problem is to draw in a given circle two parallel chords cutting off between them a given fraction of the area of the circle.

The Conics appears to have been lost by the time of the Greek astronomer Pappus (late 3d century A.D.). It is frequently referred to by Archimedes. As the name suggests, it dealt with the conic sections: the ellipse, parabola, and hyperbola, to use the names given them later by Apollonius of Perga.

A work which has survived is Phaenomena. This is what today would be called applied mathematics; it is about the geometry of spheres applicable to astronomy. Another applied work which has survived is the Optics. It was maintained by some that the sun and other heavenly bodies are actually the size they appear to be to the eye. This work refuted such a view by analyzing the relationship between what the eye sees of an object and what the object actually is. For example, the eye always sees less than half of a sphere, and as the observer moves closer to the sphere the part of it seen is decreased although it appears larger.

Another lost work is the Porisms, known only through Pappus. A porism is intermediate between a theorem and a problem; that is, rather than something to be proved or something to be constructed, a porism is concerned with bringing out another aspect of something that is already there. To find the center of a circle or to find the greatest common divisor of two numbers are examples of porisms. This work appears to have been more advanced than the Elements and perhaps if known would give Euclid a higher place in the history of mathematics.

Further Reading

The standard English translation of the Elements is Thomas L. Heath, The Thirteen Books of Euclid's Elements (3 vols., 1908; 2d ed. 1926); the introduction and commentary contain much information on Euclid. A full-length study of Euclid is Thomas Smith, Euclid: His Life and System (1902). General studies with good discussions of Euclid include Thomas L. Heath, A Manual of Greek Mathematics (1931); Carl B. Boyer, The History of the Calculus and Its Conceptual Development (1939; rev. ed. 1949); Bartel L. van der Waerden, Science Awakening (1950; trans. 1954); Morris Kline, Mathematics in Western Culture (1953); Joseph Frederick Scott, A History of Mathematics: From Antiquity to the Beginning of the Nineteenth Century (1958); and Howard Eves, Fundamentals of Geometry (1969).

(fl. c. 300 BC) Greek geometer. One of the greatest mathematicians, Euclid lived under Ptolemy I and taught at Alexandria. His Elements contain thirteen books: six on plane geometry, three on the theory of numbers, one on irrationals, and three on solid geometry. Euclidean geometry is the greatest example of the pure axiomatic method, and as such had incalculable philosophical influence as a paradigm of rational certainty. It had no competitor until the 19th century when it was realized that the fifth axiom of his system (parallel lines never meet) could be denied without inconsistency, leading to Riemannian spherical geometry. The fifth chapter of Euclid's Elements is attributed to the mathematician Eudoxus, and contains a precise development of the real numbers, work which remained unappreciated until rediscovered in the 19th century. See also geometry.

An ancient Greek mathematician; the founder of the study of geometry. Euclid's Elements is the basis for modern school textbooks in geometry. One of the basic statements, or postulates, of Euclid's geometry is that if a line and a point separate from it are given, only one line parallel to the first line can pass through the point.

For other uses, see Euclid (disambiguation).

Euclid
Artist's depiction of Euclid
Born	fl. 300 BC
Residence	Alexandria, Egypt
Ethnicity	Greek
Fields	Mathematics
Known for	Euclidean geometry Euclid's Elements

Euclid (Greek: Εὐκλείδης — Eukleídēs), fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician and is often referred to as the "Father of Geometry." He was active in Hellenistic Alexandria during the reign of Ptolemy I (323–283 BC). His Elements is the most successful textbook and one of the most influential works in the history of mathematics, serving as the main textbook for teaching mathematics (especially geometry) from the time of its publication until the late 19th or early 20th century.^[1][2][3] In it, the principles of what is now called Euclidean geometry were deduced from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, number theory and rigor.

"Euclid" is the anglicized version of the Greek name Εὐκλείδης — Eukleídēs, meaning "Good Glory".

Contents 1 Life 2 Elements 3 Other works 4 See also 5 Notes 6 References 7 External links

Life

Little is known about Euclid's life, as there are only a handful of references to him. In fact, the key references to Euclid were written centuries after he lived, by Proclus and Pappus of Alexandria.^[4] Proclus introduces Euclid only briefly in his Commentary on the Elements, written in the fifth century, where he writes that Euclid was the author of the Elements, that he was mentioned by Archimedes, and that when Ptolemy the First asked Euclid if there was no shorter road to geometry than the Elements, he replied, "there is no royal road to geometry." Although the purported citation of Euclid by Archimedes has been judged to be an interpolation by later editors of his works, it is still believed that Euclid wrote his works before those of Archimedes.^[5][6] In addition, the "royal road" anecdote is questionable since it is similar to a story told about Menaechmus and Alexander the Great.^[7] In the only other key reference to Euclid, Pappus briefly mentioned in the fourth century that Apollonius "spent a very long time with the pupils of Euclid at Alexandria, and it was thus that he acquired such a scientific habit of thought."^[8] It is further believed that Euclid may have studied at Plato's Academy in Greece.

The date and place of Euclid's birth and the date and circumstances of his death are unknown, and only roughly estimated in proximity to contemporary figures mentioned in references. No likeness or description of Euclid's physical appearance made during his lifetime survived antiquity. Therefore, Euclid's depiction in works of art is the product of the artist's imagination.

Elements

Main article: Euclid's Elements

One of the oldest surviving fragments of Euclid's Elements, found at Oxyrhynchus and dated to circa AD 100. The diagram accompanies Book II, Proposition 5.^[9]

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, making it easy to use and easy to reference, including a system of rigorous mathematical proofs that remains the basis of mathematics 23 centuries later.^[10]

There is no mention of Euclid in the earliest remaining copies of the Elements, and most of the copies say they are "from the edition of Theon" or the "lectures of Theon",^[11] while the text considered to be primary, held by the Vatican, mentions no author. The only reference that historians rely on of Euclid having written the Elements was from Proclus, who briefly in his Commentary on the Elements ascribes Euclid as its author.

Although best-known for its geometric results, the Elements also includes number theory. It considers the connection between perfect numbers and Mersenne primes, the infinitude of prime numbers, Euclid's lemma on factorization (which leads to the fundamental theorem of arithmetic on uniqueness of prime factorizations), and the Euclidean algorithm for finding the greatest common divisor of two numbers.

The geometrical system described in the Elements was long known simply as geometry, and was considered to be the only geometry possible. Today, however, that system is often referred to as Euclidean geometry to distinguish it from other so-called non-Euclidean geometries that mathematicians discovered in the 19th century.

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.

• 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.
• Catoptrics, which concerns the mathematical theory of mirrors, particularly the images formed in plane and spherical concave mirrors. The attribution to Euclid is doubtful. Its author may have been Theon of Alexandria.
• Phaenomena, a treatise on spherical astronomy, survives in Greek; it is quite similar to On the Moving Sphere by Autolycus of Pitane, who flourished around 310 BC.
• Optics is the earliest surviving Greek treatise on perspective. In its definitions Euclid follows the Platonic tradition that vision is caused by discrete rays which emanate from the eye. One important definition is the fourth: "Things seen under a greater angle appear greater, and those under a lesser angle less, while those under equal angles appear equal." In the 36
  

  

  Statue of Euclid in the Oxford University Museum of Natural History
  propositions that follow, Euclid relates the apparent size of an object to its distance from the eye and investigates the apparent shapes of cylinders and cones when viewed from different angles. Proposition 45 is interesting, proving that for any two unequal magnitudes, there is a point from which the two appear equal. Pappus believed these results to be important in astronomy and included Euclid's Optics, along with his Phaenomena, in the Little Astronomy, a compendium of smaller works to be studied before the Syntaxis (Almagest) of Claudius Ptolemy.

Other works are credibly attributed to Euclid, but 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. It is likely that the first four books of Apollonius's work come directly from Euclid. According to Pappus, "Apollonius, having completed Euclid's four books of conics and added four others, handed down eight volumes of conics." The Conics of Apollonius quickly supplanted the former work, and by the time of Pappus, Euclid's work was already lost.
• 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.
• Several works on mechanics are attributed to Euclid by Arabic sources. On the Heavy and the Light contains, in nine definitions and five propositions, Aristotelian notions of moving bodies and the concept of specific gravity. On the Balance treats the theory of the lever in a similarly Euclidean manner, containing one definition, two axioms, and four propositions. A third fragment, on the circles described by the ends of a moving lever, contains four propositions. These three works complement each other in such a way that it has been suggested that they are remnants of a single treatise on mechanics written by Euclid.

See also

• Axiomatic method
• Euclid's orchard
• Euclidean algorithm
• Euclidean relation

Notes

1. ^ Ball, pp. 50–62.
2. ^ Boyer, pp. 100–19.
3. ^ Macardle, et al. (2008). Scientists: Extraordinary People Who Altered the Course of History. New York: Metro Books. g. 12.
4. ^ Joyce, David. Euclid. Clark University Department of Mathematics and Computer Science. [1]
5. ^ Morrow, Glen. A Commentary on the first book of Euclid's Elements
6. ^ Euclid of Alexandria. The MacTutor History of Mathematics archive.
7. ^ Boyer, p. 1.
8. ^ Heath (1956), p. 2.
9. ^ Bill Casselman. "One of the Oldest Extant Diagrams from Euclid". University of British Columbia. http://www.math.ubc.ca/~cass/Euclid/papyrus/papyrus.html. Retrieved 2008-09-26. 
10. ^ Struik p. 51 ("their logical structure has influenced scientific thinking perhaps more than any other text in the world").
11. ^ Heath (1981), p. 360.

References

• "Euclid (Greek mathematician)". Encyclopædia Britannica, Inc. 2008. http://www.britannica.com/EBchecked/topic/194880/Euclid. Retrieved 2008-04-18. 
• Artmann, Benno (1999). Euclid: The Creation of Mathematics. New York: Springer. ISBN 0387984232.
• Ball, W.W. Rouse (1960) [1908]. A Short Account of the History of Mathematics (4th ed.). Dover Publications. pp. 50–62. ISBN 0486206300. 
• Boyer, Carl B. (1991). A History of Mathematics (2nd ed.). John Wiley & Sons, Inc.. ISBN 0471543977. 
• Heath, Thomas (1956) [1908]. The Thirteen Books of Euclid's Elements. vol.1. Dover Publications. ISBN 0486600882. 
• Heath, Thomas L. (1981). A History of Greek Mathematics, 2 Vols. New York: Dover Publications. ISBN 0486240738 / ISBN 0486240746.
• Kline, Morris (1980). Mathematics: The Loss of Certainty. Oxford: Oxford University Press. ISBN 019502754X.
• O'Connor, John J.; Robertson, Edmund F., "Euclid of Alexandria", MacTutor History of Mathematics archive, http://www-history.mcs.st-andrews.ac.uk/Biographies/Euclid.html .
• Struik, Dirk J. (1967). A Concise History of Mathematics. Dover Publications. ISBN 486-60255-9. 

External links

Wikiquote has a collection of quotations related to: Euclid

Wikimedia Commons has media related to: Euclid

• Euclid's elements, All thirteen books, with interactive diagrams using Java. Clark University
• Euclid's elements, with the original Greek and an English translation on facing pages (includes PDF version for printing). University of Texas.
• Euclid's elements, All thirteen books, in several languages as Spanish, Catalan, English, German, Portuguese, Arabic, Italian, Russian and Chinese .
• Elementa Geometriae 1482, Venice. From Rare Book Room.
• Elementa 888 AD, Byzantine. From Rare Book Room.
• Euclid biography by Charlene Douglass With extensive bibliography.
• Texts on Ancient Mathematics and Mathematical Astronomy PDF scans (Note: many are very large files). Includes editions and translations of Euclid's Elements, Data, and Optica, Proclus's Commentary on Euclid, and other historical sources.

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Persondata
NAME	Euclid
ALTERNATIVE NAMES	Euclid of Alexandria; Εὐκλείδης
SHORT DESCRIPTION	Greek mathematician
DATE OF BIRTH	325 BCE
PLACE OF BIRTH
DATE OF DEATH	265 BCE
PLACE OF DEATH

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