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Euclid (/ ̍ juː k l ɪ d / ; Greek: Εὐκλείδης Eukleidēs; fl. 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy   I (323–283 BC). His Elements is 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 the Elements, Euclid deduced the principles of what is now called Euclidean geometry 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, meaning "renowned, celebrated".[4]

Contents 1 Life 2 Elements 3 Other works 4 See also 5 Notes 6 References 7 Further reading 8 External links

LifeVery few original references to Euclid survive, so little is known about his life. The date, place and circumstances of both his birth and death are unknown and may only be estimated roughly relative to other figures mentioned alongside him. He is rarely mentioned by name by other Greek mathematicians from Archimedes onward, who usually call him "ὁ στοιχειώτης" ("the author of Elements").[5] The few historical references to Euclid were written centuries after he lived, by Proclus c. 450 AD and Pappus of Alexandria c. 320 AD.[6]

Proclus introduces Euclid only briefly in his Commentary on the Elements. According to Proclus, Euclid belonged to Plato's "persuasion" and brought together the Elements, drawing on prior work by several pupils of Plato (particularly Eudoxus of Cnidus, Theaetetus and Philip of Opus.) Proclus believes that Euclid is not much younger than these, and that he must have lived during the time of Ptolemy I because he was mentioned by Archimedes (287-212 BC). Although the apparent 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.[7][8][9]

Proclus later retells a story that, when Ptolemy I asked if there was a shorter path to learning geometry than Euclid's Elements, "Euclid replied there is no royal road to geometry."[10] This anecdote is questionable since it is similar to a story told about Menaechmus and Alexander the Great.[11]

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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" circa 247-222 BC.[12][13]

A detailed biography of Euclid is given by Arabian authors, mentioning, for example, a birth town of Tyre. This biography is generally believed to be completely fictitious.[8]

Because the lack of biographical information is unusual for the period (extensive biographies are available for most significant Greek mathematicians for several centuries before and after Euclid), some researchers have proposed that Euclid was not, in fact, a historical character and that his works were written by a team of mathematicians who took the name Euclid from the historical character Euclid of Megara (compare Bourbaki). However, this hypothesis is not well accepted by scholars and there is little evidence in its favor.[8][14]

Elements

One of the oldest surviving fragments of Euclid's Elements, found at Oxyrhynchus and dated to circa AD 100 (P. Oxy. 29). The diagram accompanies Book II, Proposition 5.[15]

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, 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.[16]

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",[17] 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 (known as the Euclid–Euler theorem), 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

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referred to as Euclidean geometry to distinguish it from other so-called non-Euclidean geometries that mathematicians discovered in the 19th century.

Other works

Euclid's construction of a regular dodecahedron.

Construction of a dodecahedron by placing faces on the edges of a cube.

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 is held to be anachronistic however by J J O'Connor and E F Robertson who name Theon of Alexandria as a more likely author.[18]

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.

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Statue in honor of Euclid in the Oxford University Museum of Natural History

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 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

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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.

Euclid

Greek mathematician

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Alternate title: EukleidesTable of Contents

Introduction Life Sources and contents of the Elements Renditions of the Elements Other writings Assessment Major Works

Euclid, Greek Eukleides   (born c. 300 bce, Alexandria, Egypt), the most prominent mathematician of Greco-Roman antiquity, best known for his treatise on geometry, the Elements.

LifeOf Euclid’s life nothing is known except what the Greek philosopher Proclus (c. 410–485 ce) reports in his “summary” of famous Greek mathematicians. According to him, Euclid taught at Alexandria in the time of Ptolemy I Soter, who reigned over Egypt from 323 to 285 bce.

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Medieval translators and editors often confused him with the philosopher Eukleides of Megara, a contemporary of Plato about a century before, and therefore called him Megarensis. Proclus supported his date for Euclid by writing “Ptolemy once asked Euclid if there was not a shorter road to geometry than through the Elements, and Euclid replied that there was no royal road to geometry.” Today few historians challenge the consensus that Euclid was older than Archimedes (c. 290/280–212/211 bce).

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Sources and contents of the ElementsEuclid compiled his Elements from a number of works of earlier men. Among these are Hippocrates of Chios (flourished c. 460 bce), not to be confused with the physician Hippocrates of Cos (c. 460–377 bce). The latest compiler before Euclid was Theudius, whose textbook was used in the Academy and was probably the one used by Aristotle (384–322 bce). The older elements were at once superseded by Euclid’s and then forgotten. For his subject matter Euclid doubtless drew upon all his predecessors, but it is clear that the whole design of his work was his own, culminating in the construction of the five regular solids, now known as the Platonic solids.

A brief survey of the Elements belies a common belief that it concerns only geometry. This misconception may be caused by reading no further than Books I through IV, which cover elementary plane geometry. Euclid understood that building a logical and rigorous geometry (and mathematics) depends on the foundation—a foundation that Euclid began in Book I with 23 definitions (such as “a point is that which has no part” and “a line is a length without breadth”), five unproved assumptions that Euclid called postulates (now known as axioms), and five further unproved assumptions that he called common notions. Book I then proves elementary theorems about triangles and parallelograms and ends with the Pythagorean theorem. (For Euclid’s proof of the theorem, see Sidebar: Euclid’s Windmill Proof.)

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Euclid’s axioms Euclid’s common notions

01 Given two points there is one straight line that joins them.

02 A straight line segment can be prolonged indefinitely.

03 A circle can be constructed when a point for its centre and a distance for its radius are given.

04 All right angles are equal.

05 If a straight line falling on two straight lines makes 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 the angles are less than the two right angles.

06 Things equal to the same thing are equal.

07 If equals are added to equals, the wholes are equal.

08 If equals are subtracted from equals, the remainders are equal.

09 Things that coincide with one another are equal.

10 The whole is greater than a part.

The subject of Book II has been called geometric algebra because it states algebraic identities as theorems about equivalent geometric figures. Book II contains a construction of “the section,” the division of a line into two parts such that the ratio of the larger to the smaller segment is equal to the ratio of the original line to the larger segment. (This division was renamed the golden section in the Renaissance after artists and architects rediscovered its pleasing proportions.) Book II also generalizes the Pythagorean theorem to arbitrary triangles, a result that is equivalent to the law of cosines (see plane trigonometry). Book III deals with properties of circles and Book IV with the construction of regular polygons, in particular the pentagon.

Book V shifts from plane geometry to expound a general theory of ratios and proportions that is attributed by Proclus (along with Book XII) to Eudoxus of Cnidus (c. 390–350 bce). While Book V can be read independently of the rest of the Elements, its solution to the problem of incommensurables (irrational numbers) is essential to later books. In addition, it formed the foundation for a geometric theory of numbers until an analytic theory developed in the late 19th century. Book VI applies this theory of ratios to plane geometry, mainly triangles and parallelograms, culminating in the “application of areas,” a procedure for solving quadratic problems by geometric means.

Books VII–IX contain elements of number theory, where number (arithmos) means positive integers greater than 1. Beginning with 22 new definitions—such as unity, even, odd, and

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prime—these books develop various properties of the positive integers. For instance, Book VII describes a method, antanaresis (now known as the Euclidean algorithm), for finding the greatest common divisor of two or more numbers; Book VIII examines numbers in continued proportions, now known as geometric sequences (such as ax, ax2, ax3, ax4…); and Book IX proves that there are an infinite number of primes.

According to Proclus, Books X and XIII incorporate the work of the Pythagorean Theaetetus (c. 417–369 bce). Book X, which comprises roughly one-fourth of the Elements, seems disproportionate to the importance of its classification of incommensurable lines and areas (although study of this book would inspire Johannes Kepler [1571–1630] in his search for a cosmological model).

Books XI–XIII examine three-dimensional figures, in Greek stereometria. Book XI concerns the intersections of planes, lines, and parallelepipeds (solids with parallel parallelograms as opposite faces). Book XII applies Eudoxus’s method of exhaustion to prove that the areas of circles are to one another as the squares of their diameters and that the volumes of spheres are to one another as the cubes of their diameters. Book XIII culminates with the construction of the five regular Platonic solids (pyramid, cube, octahedron, dodecahedron, icosahedron) in a given sphere, as displayed in the animation.

The unevenness of the several books and the varied mathematical levels may give the impression that Euclid was but an editor of treatises written by other mathematicians. To some extent this is certainly true, although it is probably impossible to figure out which parts are his own and which were adaptations from his predecessors. Euclid’s contemporaries considered his work final and authoritative; if more was to be said, it had to be as commentaries to the Elements.