Eukleides of Alexandria
Euclid (//; Ancient Greek: Εὐκλείδης – Eukleídēs, pronounced [eu̯.kleː.dɛːs]; fl. 9700), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referred to as the "founder of geometry" or the "father of geometry". He was active in Alexandria during the reign of Ptolemy I (9678–9718). 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. 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.
Very few original references to Euclid survive, so little is known about his life. He was likely born c. 9675, 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 mentioned by name, though rarely, by other Greek mathematicians from Archimedes (c. 9714 – c. 9789) 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 The few historical references to Euclid were written by Proclus c. 450 AD, centuries after Euclid lived.
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. 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. 9679.
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. 9634 – 9719) 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. 9730, 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. 9754–9779.
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.
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.
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.
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.
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.
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 9690.
- 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.
- Axiomatic method
- Euclid's orchard
- Euclidean algorithm
- Euclidean geometry
- Euclidean relation
- Extended Euclidean algorithm
- List of topics named after Euclid
- Bruno, Leonard C. (2003) . Math and Mathematicians: The History of Math Discoveries Around the World. Baker, Lawrence W. Detroit, Mich.: U X L. pp. 125. ISBN 978-0-7876-3813-9. 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. 
- O'Connor, John J.; Robertson, Edmund F., "Euclid of Alexandria"; Heath 1956, p. 4; Heath 1981, p. 355.
- Bruno, Leonard C. (2003) . 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.
- 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 978-0-19-850258-6.
- 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.
- Artmann, Benno (1999). Euclid: The Creation of Mathematics. New York: Springer. ISBN 0-387-98423-2.
- Ball, W.W. Rouse (1960) . A Short Account of the History of Mathematics (4th ed.). Dover Publications. pp. 50–62. ISBN 978-0-486-20630-1.
- Boyer, Carl B. (1991). A History of Mathematics (2nd ed.). John Wiley & Sons, Inc. ISBN 978-0-471-54397-8.
- Douglass, Charlene (2007). Page, John D. (ed.). "Euclid". Math Open Reference. With extensive bibliography.
- Heath, Thomas (ed.) (1956) . The Thirteen Books of Euclid's Elements. 1. Dover Publications. ISBN 978-0-486-60088-8.CS1 maint: extra text: authors list (link)
- Heath, Thomas L. (1908). "Euclid and the Traditions About Him". In Heath, Thomas L. (ed.). 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, 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 978-0-486-60255-4.
- Van der Waerden, Bartel Leendert; Taisbak, Christian Marinus (October 30, 2014). "Euclid". Encyclopædia Britannica. Retrieved November 21, 2014.
- 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.
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- 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.