Euclid

Euclid (/ˈjuːklɪd/; Ancient Greek: ΕὐκλείδηςEukleídēs, pronounced [eu̯.kleː.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]

Euclid
Eukleides of Alexandria
BornMid-4th century BC
DiedMid-3rd century BC
ResidenceAlexandria, Hellenistic Egypt
Known for
Scientific career
FieldsMathematics
Bramante as Euclid or Archimedes in the School of Athens.

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(c. 287 BC – c. 212 BC) onward, and is usually referred to as "ὁ στοιχειώτης" ("the author of Elements").[6] The few historical references to Euclid were written centuries after he lived, namely by Proclus c. 450 AD.[7]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]

Euclidis quae supersunt omnia (1704)

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

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.[16]

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

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 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.
19th-century statue of Euclid by Joseph Durham 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.

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]

References

1. ^ 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 978-0-7876-3813-9. OCLC 41497065.
2. ^ Ball, pp. 50–62.
3. ^ Boyer, pp. 100–19.
4. ^ Macardle, et al. (2008). Scientists: Extraordinary People Who Altered the Course of History. New York: Metro Books. g. 12.
5. ^ Harper, Douglas. "Euclidean (adj.)". Online Etymology Dictionary. Retrieved March 18, 2015.
6. ^ Heath (1981), p. 357
7. ^ Joyce, David. Euclid. Clark University Department of Mathematics and Computer Science. [1]
8. ^ O'Connor, John J.; Robertson, Edmund F., "Euclid of Alexandria"; Heath 1956, p. 4; Heath 1981, p. 355.
9. ^ 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 978-0-7876-3813-9. OCLC 41497065.
10. ^ Proclus, p. XXX; O'Connor, John J.; Robertson, Edmund F., "Euclid of Alexandria"
11. ^ Proclus, p. 57
12. ^ Boyer, p. 96.
13. ^ Heath (1956), p. 2.
14. ^
15. ^ O'Connor, John J.; Robertson, Edmund F., "Euclid of Alexandria"; Jean Itard (1962). Les livres arithmétiques d'Euclide.
16. ^ Bill Casselman. "One of the Oldest Extant Diagrams from Euclid". University of British Columbia. Retrieved 2008-09-26.
17. ^ Struik p. 51 ("their logical structure has influenced scientific thinking perhaps more than any other text in the world").
18. ^ Heath (1981), p. 360.
19. ^ Fowler, David (1999). The Mathematics of Plato's Academy (Second ed.). Oxford: Clarendon Press. ISBN 978-0-19-850258-6.
20. ^
21. ^ O'Connor, John J.; Robertson, Edmund F., "Theon of Alexandria"
22. ^

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 978-90-277-0509-9.
• Mueller, Ian (1981). Philosophy of Mathematics and Deductive Structure in Euclid's Elements. Cambridge, MA: MIT Press. ISBN 978-0-262-13163-6.
• 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 978-90-277-0819-9.