Euclid

Not to be confused with Euclid of Megara. For other uses, see Euclid (disambiguation).

Euclid
Εὐκλείδης
Detail from Raphael's The School of Athens (1509–1511) presumed to represent Donato Bramante as Euclid
Known for	The Elements Various concepts Optics Euclidean geometry Euclidean algorithm Euclid's theorem Euclid's formula Numerous other namesakes
Scientific career
Fields	Mathematics

Euclid of Alexandria (/ˈjuːklɪd/; Greek: Εὐκλείδης; fl. 300 BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the Elements treatise, which established the foundations of geometry which were largely undisputed until the early 19th century. Much of his work involved synthesizing the theories of earlier Greek mathematicians, including Eudoxus, Hippocrates, Thales and Theaetetus. Euclid is regarded as bridging between the earlier Platonic tradition in Athens and the later tradition of Alexandria, where he spent his career under Ptolemy I Soter.

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 mathematical rigour. With Archimedes and Apollonius of Perga, Euclid is generally considered among the greatest mathematicians of antiquity, and among the most influential in the history of mathematics.

Life

Euclidis quae supersunt omnia (1704)

The English name 'Euclid' is the anglicized version of the Ancient Greek name Εὐκλείδης. It is derived from 'eu-' (εὖ) and 'klês' (-κλῆς), meaning "renowned, glorious".

Euclid is often referred to as 'Euclid of Alexandria' to differentiate him from the earlier philosopher Euclid of Megara, a pupil of Socrates who was included in the dialogues of Plato. Historically, medieval scholars frequently confused the mathematician and philosopher, mistakenly referring to the former in Latin as 'Megarensis' (lit. 'of Megara'). As a result, biographical information on the mathematician Euclid was long conflated with the lives of both Euclid of Alexandria and Euclid of Megara. The only scholar of antiquity known to have confused the mathematician and philosopher was Valerius Maximus. In addition to the many anonymous Byzantine sources, this mistaken identification was relayed by the scholars Campanus of Novara and Theodore Metochites, and put into a publication of the latter's translated printed by Erhard Ratdolt in 1482. After the mathematician Bartolomeo Zamberti (1473–1539) affirmed this presumption in his 1505 translation, all subsequent publications passed on this identification. Early Renaissance scholars, particularly Peter Ramus, reevaluated this claim, proving it false via issues in chronology and contradiction in early sources.

Besides his authorship of certain works, extremely little is known of Euclid, and most is inferential or speculative. He was likely born around 325 BC, although the place and circumstances of both his birth and death are unknown and may only be estimated relative to other people mentioned with him. He is mentioned by name, though rarely, 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"). The few historical references to Euclid were written by Proclus c. 450 AD, eight centuries after Euclid lived.

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

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. 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." This anecdote is questionable since it is similar to a story told about Menaechmus and Alexander the Great.

Euclid died c. 270 BC, presumably in Alexandria. 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.

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.

Works

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.

Main article: Euclid's Elements

Although many of the results in the 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.

There is no mention of Euclid in the earliest remaining copies of the Elements. Most of the copies say they are "from the edition of Theon" or the "lectures of Theon", while the text considered to be primary, held by the Vatican, mentions no author. Proclus provides the only reference ascribing the Elements to Euclid.

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 discovered in the 19th century. There have been many editions, translations, and adaptations of The Elements, including a pictorial version by Oliver Byrne and a modern axiomatization by David Hilbert.

Fragments

Further information: Oxyrhynchus Papyri

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.

The fragment contains the statement of the 5th proposition of Book 2, which in the translation of T. L. Heath reads:

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

Euclid's construction of a regular dodecahedron.

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.
• 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

19th-century statue of Euclid by Joseph Durham in the Oxford University Museum of Natural History

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

See also: List of things named after Euclid

The first English edition of the Elements was published in 1570 by Henry Billingsley and John Dee.

Among Euclid's many namesakes are the European Space Agency's (ESA) Euclid spacecraft, the lunar crater Euclides, and the minor planet 4354 Euclides are named after him.

See also

• Euclid's orchard
• Euclidean relation
• Extended Euclidean algorithm
• Oliver Byrne (mathematician)

References

Sources

• Asper, Markus (2010). "Euclid". In Gagarin, Michael (ed.). The Oxford Encyclopedia of Ancient Greece and Rome. Oxford: Oxford University Press. ISBN 978-0-19-517072-6.
• Ball, W.W. Rouse (1960) [1908]. A Short Account of the History of Mathematics (4th ed.). Dover Publications. pp. 50–62. ISBN 978-0-486-20630-1.
• Bruno, Leonard C. (2003) [1999]. Math and Mathematicians: The History of Math Discoveries Around the World. Baker, Lawrence W. Detroit: U X L. ISBN 978-0-7876-3813-9. OCLC 41497065.
• Boyer, Carl B. (1991). A History of Mathematics (2nd ed.). John Wiley & Sons, Inc. ISBN 978-0-471-54397-8.
• Fowler, David (1999). The Mathematics of Plato's Academy (2nd ed.). Oxford: Clarendon Press. ISBN 978-0-19-850258-6.
• Heath, Thomas, ed. (1956) [1908]. The Thirteen Books of Euclid's Elements. Vol. 1. Dover Publications. ISBN 978-0-486-60088-8.
• Heath, Thomas L. (1981). A History of Greek Mathematics. Vol. 2 Vols. New York: Dover Publications. ISBN 0-486-24073-8, 0-486-24074-6
• 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.
• Sialaros, Michalis (2018). "How Much Does a Theorem Cost?". In Sialaros, Michalis (ed.). Revolutions and Continuity in Greek Mathematics. Berlin: De Gruyter. pp. 89–106. ISBN 978-3-11-056595-9.
• Sialaros, Michalis (2021) [2015]. "Euclid". Oxford Classical Dictionary. Oxford: Oxford University Press. doi:10.1093/acrefore/9780199381135.013.2521.
• Smorynski, Craig (2008). History of Mathematics: A Supplement. New York: Springer Publishing. ISBN 978-0-387-75480-2.
• Struik, Dirk J. (1967). A Concise History of Mathematics. Dover Publications. ISBN 978-0-486-60255-4.
• Thoenes, Christof (2005). Raphael. Cologne: Taschen. ISBN 978-3-8228-2203-6.
• Taisbak, Christian Marinus; Waerden, Bartel Leendert van der (5 January 2021). "Euclid". Encyclopædia Britannica. Chicago: Encyclopædia Britannica, Inc.

Further reading

• Artmann, Benno (2012) [1999]. Euclid: The Creation of Mathematics. New York: Springer Publishing. ISBN 978-1-4612-1412-0.
• DeLacy, Estelle Allen (1963). Euclid and Geometry. New York: Franklin Watts.

External links

Works

• Works by Euclid at Project Gutenberg
• Works by or about Euclid at Internet Archive
• Works by Euclid at LibriVox (public domain audiobooks)

The Elements

• PDF copy, with the original Greek and an English translation on facing pages, University of Texas.
• All thirteen books, in several languages as Spanish, Catalan, English, German, Portuguese, Arabic, Italian, Russian and Chinese.