# Euclid

(Redirected from Euclides)
For other uses, see Euclid (disambiguation).
Euclid
Euclid (central figure, bald-pated, marking a slate with a pair of compasses) as featured in Raphael's School of Athens.
Born unknown  (mid-4th century BC)
Died unknown  (mid-3rd century BC)
Residence Alexandria, Egypt
Fields Mathematics
Known for

Euclid (; Greek: Εὐκλείδης Eukleidēs; fl. 300 BC), also known as Euclid of Alexandria, 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 "Good Glory".[4]

## Life

Very 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, if ever, referred to by name by other Greek mathematicians from Archimedes onward, who instead call him "ό στοιχειώτης" ("the author of Elements").[5] The few historical references to Euclid were written centuries after he lived, by Proclus c. 450 AD and Pappus of Alexandria c. 320 AD.[6]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 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.[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]

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]

Partly due to the lack of biographical information that is somewhat atypical for the period (extensive biographies are available for most significant Greek mathematicians for several centuries before and after Euclid), it has been proposed that Euclid was not, in fact, a historical character, and his works were written by a team of mathematicians (not unlike Bourbaki) who took the name Euclid from the historical character Euclid of Megara. 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, 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

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.
• 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 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.

## 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. ^ etymonline.com Retrieved 2011-12-04
5. ^ Heath (1981), p. 357
6. ^ Joyce, David. Euclid. Clark University Department of Mathematics and Computer Science. [1]
7. ^ Proclus, p. XXX
8. ^ a b c Euclid of Alexandria
9. ^ The MacTutor History of Mathematics archive.
10. ^ Proclus, p. 57
11. ^ Boyer, p. 96.
12. ^ Heath (1956), p. 2.
13. ^
14. ^ Jean Itard (1962). Les livres arithmétiques d'Euclide.
15. ^ Bill Casselman. "One of the Oldest Extant Diagrams from Euclid". University of British Columbia. Retrieved 2008-09-26.
16. ^ Struik p. 51 ("their logical structure has influenced scientific thinking perhaps more than any other text in the world").
17. ^ Heath (1981), p. 360.
18. ^ Theon article at their institution

## References

• DeLacy, Estelle Allen (1963). Euclid and Geometry. New York: Franklin Watts.
• Knorr, Wilbur Richard (1975). The Evolution of the Euclidean Elements: A Study of the Theory of Incommensurable Magnitudes and Its Significance for Early Greek Geometry. Dordrecht, Holland: D. Reidel. ISBN 90-277-0509-7.
• Mueller, Ian (1981). Philosophy of Mathematics and Deductive Structure in Euclid's Elements. Cambridge, MA: MIT Press. ISBN 0-262-13163-3.
• Reid, Constance (1963). A Long Way from Euclid. New York: Crowell.
• Szabó, Árpád (1978). The Beginnings of Greek Mathematics. A.M. Ungar, trans. Dordrecht, Holland: D. Reidel. ISBN 90-277-0819-3.