# Euclid

Euclid | |
---|---|

Eukleides of Alexandria | |

Born | Mid-4th century BC |

Died | Mid-3rd century BC |

Residence | Alexandria, Hellenistic Egypt |

Known for | |

Scientific career | |

Fields | Mathematics |

**Euclid** (/ˈjuːklɪd/; Ancient Greek: Εὐκλείδης – *Eukleídēs*, pronounced [eu̯.klěː.dɛːs]; fl. 300 BC), sometimes given the name **Euclid of Alexandria**^{[1]} to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the "founder of geometry"^{[1]} or 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.^{[2]}^{[3]}^{[4]} In the *Elements*, Euclid deduced the theorems 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, which means "renowned, glorious".^{[5]}

## Contents

## Life

Very few original references to Euclid survive, so little is known about his life. He was likely born c. 325 BC, although the place and circumstances of both his birth and death are unknown and may only be estimated roughly relative to other people mentioned with him. He is rarely mentioned by name by other Greek mathematicians from Archimedes (c. 287 BC – c. 212 BC) onward, and is usually referred to as "ὁ στοιχειώτης" ("the author of Elements").^{[6]} The few historical references to Euclid were written centuries after he lived, namely by Proclus c. 450 AD.^{[7]}

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 fictitious.^{[8]} If he came from Alexandria, he would have known the Serapeum of Alexandria, and the Library of Alexandria, and may have worked there during his time. Euclid's arrival in Alexandria came about ten years after its founding by Alexander the Great, which means he arrived c. 322 BC.^{[9]}

Proclus introduces Euclid only briefly in his *Commentary on the Elements*. According to Proclus, Euclid supposedly belonged to Plato's "persuasion" and brought together the *Elements*, drawing on prior work of Eudoxus of Cnidus and of several pupils of Plato (particularly 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 (c. 367 BC – 282 BC) because he was mentioned by Archimedes. 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 Archimedes wrote his.^{[10]} 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."^{[11]} This anecdote is questionable since it is similar to a story told about Menaechmus and Alexander the Great.^{[12]}

Euclid died c. 270 BC, presumably in Alexandria.^{[9]} In the only other key reference to Euclid, Pappus of Alexandria (c. 320 AD) briefly mentioned 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" c. 247–222 BC.^{[13]}^{[14]}

Because the lack of biographical information is unusual for the period (extensive biographies being available for most significant Greek mathematicians several centuries before and after Euclid), some researchers have proposed that Euclid was not a historical personage, and that his works were written by a team of mathematicians who took the name Euclid from Euclid of Megara (à la Bourbaki). However, this hypothesis is not well accepted by scholars and there is little evidence in its favor.^{[15]}

*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.^{[17]}

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",^{[18]} 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 referred to as *Euclidean geometry* to distinguish it from other so-called *non-Euclidean geometries* that mathematicians discovered in the 19th century.

### Fragments

The Papyrus Oxyrhynchus 29 (P. Oxy. 29) is a fragment of the second book of the *Elements* of Euclid, unearthed by Grenfell and Hunt 1897 in Oxyrhynchus. More recent scholarship suggests a date of 75–125 AD.^{[19]}

The classic translation of T. L. Heath, reads:^{[20]}

If a straight line be cut into equal and unequal segments, the rectangle contained by the unequal segments of the whole together with the square on the straight line between the points of section is equal to the square on the half.

## 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 first-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.^{[21]}*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.

### Lost works

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.

## Legacy

The European Space Agency's (ESA) Euclid spacecraft was named in his honor.^{[22]}

## See also

- Axiomatic method
- Euclid's orchard
- Euclidean algorithm
- Euclidean geometry
- Euclidean relation
- Extended Euclidean algorithm
- List of topics named after Euclid

## References

- ^
^{a}^{b}Bruno, Leonard C. (2003) [1999].*Math and mathematicians : the history of math discoveries around the world*. Baker, Lawrence W. Detroit, Mich.: U X L. p. 125. ISBN 0787638137. OCLC 41497065. **^**Ball, pp. 50–62.**^**Boyer, pp. 100–19.**^**Macardle, et al. (2008).*Scientists: Extraordinary People Who Altered the Course of History.*New York: Metro Books. g. 12.**^**Harper, Douglas. "Euclidean (adj.)".*Online Etymology Dictionary*. Retrieved March 18, 2015.**^**Heath (1981), p. 357**^**Joyce, David.*Euclid*. Clark University Department of Mathematics and Computer Science. [1]**^**O'Connor, John J.; Robertson, Edmund F., "Euclid of Alexandria"; Heath 1956, p. 4; Heath 1981, p. 355.- ^
^{a}^{b}Bruno, Leonard C. (2003) [1999].*Math and mathematicians : the history of math discoveries around the world*. Baker, Lawrence W. Detroit, Mich.: U X L. p. 126. ISBN 0787638137. OCLC 41497065. **^**Proclus, p. XXX; O'Connor, John J.; Robertson, Edmund F., "Euclid of Alexandria"**^**Proclus, p. 57**^**Boyer, p. 96.**^**Heath (1956), p. 2.**^**"Conic Sections in Ancient Greece".**^**O'Connor, John J.; Robertson, Edmund F., "Euclid of Alexandria"; Jean Itard (1962).*Les livres arithmétiques d'Euclide*.**^**Bill Casselman. "One of the Oldest Extant Diagrams from Euclid". University of British Columbia. Retrieved 2008-09-26.**^**Struik p. 51 ("their logical structure has influenced scientific thinking perhaps more than any other text in the world").**^**Heath (1981), p. 360.**^**Fowler, David (1999).*The Mathematics of Plato's Academy*(Second ed.). Oxford: Clarendon Press. ISBN 0-19-850258-3.**^**Bill Casselman, One of the oldest extant diagrams from Euclid**^**O'Connor, John J.; Robertson, Edmund F., "Theon of Alexandria"**^**"NASA Delivers Detectors for ESA's Euclid Spacecraft".*NASA*. 2017.

### Works cited

- Artmann, Benno (1999).
*Euclid: The Creation of Mathematics*. New York: Springer. ISBN 0-387-98423-2. - Ball, W.W. Rouse (1960) [1908].
*A Short Account of the History of Mathematics*(4th ed.). Dover Publications. pp. 50–62. ISBN 0-486-20630-0. - Boyer, Carl B. (1991).
*A History of Mathematics*(2nd ed.). John Wiley & Sons, Inc. ISBN 0-471-54397-7. - Douglass, Charlene (2007). Page, John D., ed. "Euclid".
*Math Open Reference*. With extensive bibliography. - Heath, Thomas (ed.) (1956) [1908].
*The Thirteen Books of Euclid's Elements*.**1**. Dover Publications. ISBN 0-486-60088-2. - Heath, Thomas L. (1908). "Euclid and the Traditions About Him". In Heath, Thomas L.
*Euclid,**Elements*.**1**. pp. 1–6. As reproduced in the Perseus Digital Library. - 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. - O'Connor, John J.; Robertson, Edmund F., "Euclid of Alexandria",
*MacTutor History of Mathematics archive*, University of St Andrews. - O'Connor, John J.; Robertson, Edmund F., "Theon of Alexandria",
*MacTutor History of Mathematics archive*, University of St Andrews. - Proclus,
*A commentary on the First Book of Euclid's Elements*, translated by Glenn Raymond Morrow, Princeton University Press, 1992. ISBN 978-0-691-02090-7. - Struik, Dirk J. (1967).
*A Concise History of Mathematics*. Dover Publications. ISBN 0-486-60255-9. - Van der Waerden, Bartel Leendert; Taisbak, Christian Marinus (October 30, 2014). "Euclid".
*Encyclopædia Britannica*. Retrieved November 21, 2014.

## Further reading

- 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.

## External links

Wikiquote has quotations related to: Euclid |

Wikisource has original works written by or about:Euclid |

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Wikisource has the text of the 1911 Encyclopædia Britannica article .Euclid |

- Works by Euclid at Project Gutenberg
- Works by or about Euclid at Internet Archive
- Works by Euclid at LibriVox (public domain audiobooks)
- 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, books I–VI, in English pdf, in a Project Gutenberg Victorian textbook edition with diagrams.
- 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.- 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. - The Elements of Geometrie of the Most Auncient Philosopher Euclide of Megara (1570) from the English Printing Collection in the Rare Book and Special Collection Division at the Library of Congress