Euclid  

Eukleides of Alexandria  
Born  Mid4th century BC 
Died  Mid3rd century BC 
Residence  Alexandria, Hellenistic Egypt 
Known for  
Scientific career  
Fields  Mathematics 
Euclid (/ˈjuːklɪd/; Ancient Greek: Εὐκλείδης – Eukleídēs, pronounced [eu̯.kleː.dɛːs]; fl. 300 BC), sometimes called Euclid of Alexandria^{[1]} to distinguish him from Euclid 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 mathematical rigour.
The English name Euclid is the anglicized version of the Greek name Εὐκλείδης, which means "renowned, glorious".^{[5]}
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Transcription
"The laws of nature are but the mathematical thoughts of God." And this is a quote by Euclid of Alexandria, who was a Greek mathematician and philosopher who lived about 300 years before Christ. And the reason why I include this quote is because Euclid is considered to be the father of geometry. And it is a neat quote. Regardless of your views of God, whether or not God exists or the nature of God, it says something very fundamental about nature. The laws of nature are but the mathematical thoughts of God. That math underpins all of the laws of nature. And the word geometry itself has Greek roots. Geo comes from Greek for Earth. Metry comes from Greek for measurement. You're probably used to something like the metric system. And Euclid is considered to be the father of geometry not because he was the first person who studied geometry. You could imagine the very first humans might have studied geometry. They might have looked at two twigs on the ground that looked something like that and they might have looked at another pair of twigs that looked like that and said, this is a bigger opening. What is the relationship here? Or they might have looked at a tree that had a branch that came off it like that. And they said, oh there's something similar about this opening here and this opening here. Or they might have asked themselves, what is the ratio? Or what is the relationship between the distance around a circle and the distance across it? And is that the same for all circles? And is there a way for us to feel really good that that is definitely true? And then once you got to the early Greeks, they started to get even more thoughtful essentially about geometric things when you talk about Greek mathematicians like Pythagoras, who came before Euclid. But the reason why Euclid is considered to be the father of geometry, and why we often talk about Euclidean geometry, is around 300 BC and this right over here is a picture of Euclid painted by Raphael. And no one really knows what Euclid looked like, even when he was born or when he died. So this is just Raphael's impression of what Euclid might have looked like when he was teaching in Alexandria. But what made Euclid the father of geometry is really his writing of Euclid's Elements. And what the Elements were were essentially a 13 volume textbook. And arguably the most famous textbook of all time. And what he did in those 13 volumes is he essentially did a rigorous, thoughtful, logical march through geometry and number theory, and then also solid geometry. So geometry in three dimensions. And this right over here is the frontispiece piece for the English version, or the first translation of the English version of Euclid's Elements. And this was done in 1570. But it was obviously first written in Greek. And then during much of the Middle Ages, that knowledge was shepherded by the Arabs and it was translated into Arabic. And then eventually in the late Middle Ages, translated into Latin, and then obviously eventually English. And when I say that he did a rigorous march, what Euclid did is he didn't just say, oh well, I think if you take the length of one side of a right triangle and the length of the other side of the right triangle, it's going to be the same as the square of the hypotenuse, all of these other things. And we'll go into depth about what all of these things mean. He says, I don't want to just feel good that it's probably true. I want to prove to myself that it is true. And so what he did in Elements, especially the six books that are concerned with planar geometry, in fact, he did all of them, but from a geometrical point of view, he started with basic assumptions. So he started with basic assumptions and those basic assumptions in geometric speak are called axioms or postulates. And from them, he proved, he deduced other statements or propositions. Or these are sometimes called theorems. And then he says, now I know if this is true and this is true, this must be true. And he could also prove that other things cannot be true. So then he could prove that this is not going to be the truth. He didn't just say, well, every circle I've said has this property. He says, I've now proven that this is true. And then from there, we can go and deduce other propositions or theorems, and we can use some of our original axioms to do that. And what's special about that is no one had really done that before, rigorously proven beyond a shadow of a doubt across a whole broad sweep of knowledge. So not just one proof here or there. He did it for an entire set of knowledge that we're talking about. A rigorous march through a subject so that he could build this scaffold of axioms and postulates and theorems and propositions. And theorems and propositions are essentially the same thing. And essentially for about 2,000 years after Euclid so this is unbelievable shelf life for a textbook people didn't view you as educated if you did not read and understand Euclid's Elements. And Euclid's Elements, the book itself, was the second most printed book in the Western world after the Bible. This is a math textbook. It was second only to the Bible. When the first printing presses came out, they said OK, let's print the Bible. What do we print next? Let's print Euclid's Elements. And to show that this is relevant into the fairly recent past although whether or not you argue that about 150, 160 years ago is the recent past this right here is a direct quote from Abraham Lincoln, obviously one of the great American presidents. I like this picture of Abraham Lincoln. This is actually a photograph of Lincoln in his late 30s. But he was a huge fan of Euclid's Elements. He would actually use it to fine tune his mind. While he was riding his horse, he would read Euclid's Elements. While was in the White House, he would read Euclid's Elements. But this is a direct quote from Lincoln. "In the course of my law reading, I constantly came upon the word demonstrate. I thought at first that I understood its meaning, but soon became satisfied that I did not. I said to myself, what do I do when I demonstrate more than when I reason or prove? How does demonstration differ from any other proof?" So Lincoln's saying, there's this word demonstration that means something more. Proving beyond doubt. Something more rigorous. More than just simple feeling good about something or reasoning through it. "I consulted Webster's Dictionary." So Webster's Dictionary was around even when Lincoln was around. "They told of certain proof. Proof beyond the possibility of doubt. But I could form no idea of what sort of proof that was. "I thought a great many things were proved beyond the possibility of doubt, without recourse to any such extraordinary process of reasoning as I understood demonstration to be. I consulted all the dictionaries and books of reference I could find, but with no better results. You might as well have defined blue to a blind man. "At last I said, Lincoln" he's talking to himself. "At last I said, Lincoln, you never can make a lawyer if you do not understand what demonstrate means. And I left my situation in Springfield, went home to my father's house, and stayed there till I could give any proposition in the six books of Euclid at sight." So the six books concerned with planar geometry. "I then found out what demonstrate means and went back to my law studies." So one of the greatest American presidents of all time felt that in order to be a great lawyer, he had to understand, be able to prove any proposition in the six books of Euclid's Elements at sight. And also once he was in the White House, he continued to do this to make him, in his mind, to fine tune his mind to become a great president. And so what we're going to be doing in the geometry play list is essentially that. What we're going to study is we're going to think about how do we really tightly, rigorously prove things? We're essentially going to be, in a slightly more modern form, studying what Euclid studied 2,300 years ago. Really tighten our reasoning of different statements and being able to make sure that when we say something, we can really prove what we're saying. And this is really some of the most fundamental, real mathematics that you will do. Arithmetic was really just computation. Now in geometry and what we're going to be doing is really Euclidean geometry this is really what math is about. Making some assumptions and then deducing other things from those assumptions.
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 by Proclus c. 450 AD, centuries after Euclid lived.^{[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 socalled nonEuclidean 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 firstcentury 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 9780787638139. 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 9780787638139. 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 20080926.
 ^ 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 9780198502586.
 ^ 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 0387984232.
 Ball, W.W. Rouse (1960) [1908]. A Short Account of the History of Mathematics (4th ed.). Dover Publications. pp. 50–62. ISBN 9780486206301.
 Boyer, Carl B. (1991). A History of Mathematics (2nd ed.). John Wiley & Sons, Inc. ISBN 9780471543978.
 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 9780486600888.CS1 maint: Extra text: authors list (link)
 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 0486240738, 0486240746.
 Kline, Morris (1980). Mathematics: The Loss of Certainty. Oxford: Oxford University Press. ISBN 019502754X.
 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 9780691020907.
 Struik, Dirk J. (1967). A Concise History of Mathematics. Dover Publications. ISBN 9780486602554.
 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 9789027705099.
 Mueller, Ian (1981). Philosophy of Mathematics and Deductive Structure in Euclid's Elements. Cambridge, MA: MIT Press. ISBN 9780262131636.
 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 9789027708199.
External links
Wikiquote has quotations related to: Euclid 
Wikisource has original works written by or about: Euclid 
Wikimedia Commons has media related to Euclid. 
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