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Euclid

Artist's depiction of Euclid
Born fl. 300 BC
Died unknown
Residence Alexandria, Egypt
Ethnicity Greek
Fields Mathematics
Known for Euclidean geometry
Euclid's Elements

Euclid (Ancient GreekΕὐκλείδης Eukleidēs), fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as 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.[1][2][3] In it, the principles of what is now called Euclidean geometry were deduced 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 — Eukleídēs, meaning "Good Glory".

Contents

Life

Little is known about Euclid's life, as there are only a handful of references to him. In fact, the key references to Euclid were written centuries after he lived, by Proclus and Pappus of Alexandria.[4] Proclus introduces Euclid only briefly in his Commentary on the Elements, written in the fifth century, where he writes that Euclid was the author of the Elements, that he was mentioned by Archimedes, and that when Ptolemy the First asked Euclid if there was no shorter road to geometry than the Elements, he replied, "there is no royal road to geometry." Although the purported 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 those of Archimedes.[5][6] In addition, the "royal road" anecdote is questionable since it is similar to a story told about Menaechmus and Alexander the Great.[7] In the only other key reference to Euclid, Pappus briefly mentioned in the fourth century 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."[8] It is further believed that Euclid may have studied at Plato's Academy in Greece.

The date and place of Euclid's birth and the date and circumstances of his death are unknown, and only roughly estimated in proximity to contemporary figures mentioned in references. No likeness or description of Euclid's physical appearance made during his lifetime survived antiquity. Therefore, Euclid's depiction in works of art is the product of the artist's imagination.

Elements

One of the oldest surviving fragments of Euclid's Elements, found at Oxyrhynchus and dated to circa AD 100. The diagram accompanies Book II, Proposition 5.[9]

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.[10]

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",[11] 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, 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.

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 third 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 to Euclid is doubtful. Its author may have been Theon of Alexandria.
  • 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.

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.

See also

Notes

  1. ^ Ball, pp. 50–62.
  2. ^ Boyer, pp. 100–19.
  3. ^ Macardle, et al. (2008). Scientists: Extraordinary People Who Altered the Course of History. New York: Metro Books. g. 12.
  4. ^ Joyce, David. Euclid. Clark University Department of Mathematics and Computer Science. [1]
  5. ^ Morrow, Glen. A Commentary on the first book of Euclid's Elements
  6. ^ Euclid of Alexandria. The MacTutor History of Mathematics archive.
  7. ^ Boyer, p. 1.
  8. ^ Heath (1956), p. 2.
  9. ^ Bill Casselman. "One of the Oldest Extant Diagrams from Euclid". University of British Columbia. http://www.math.ubc.ca/~cass/Euclid/papyrus/papyrus.html. Retrieved 2008-09-26. 
  10. ^ Struik p. 51 ("their logical structure has influenced scientific thinking perhaps more than any other text in the world").
  11. ^ Heath (1981), p. 360.

References

External links

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Quotes

Up to date as of January 14, 2010

From Wikiquote

There is no royal road to Geometry.

Euclid (Greek: Εὐκλείδης), also known as Euclid of Alexandria, was a Greek mathematician who flourished in Alexandria, Egypt, almost certainly during the reign of Ptolemy I (323–283 BC). Neither the year nor place of his birth have been established, nor the circumstances of his death. He is famous for writing one of the earliest comprehensive mathematics textbooks, the Elements.

Attributed

  • There is no royal road to geometry.
    • Proclus (412–485 AD) in Commentary on the First Book of Euclid's Elements as translated by Glenn R. Morrow (1970)
  • Give him a coin, since he must profit by what he learns.
    • Attributed by Proclus as a remark about a reluctant student
  • The laws of nature are but the mathematical thoughts of God.
    • Unsourced

External links

Wikipedia
Wikipedia has an article about:

1911 encyclopedia

Up to date as of January 14, 2010

From LoveToKnow 1911

EUCLID, Greek mathematician of the 3rd century B.C.; we are ignorant not only of the dates of his birth and death, but also of his parentage, his teachers, and the residence of his early years. In some of the editions of his works he is called Megarensis, as if he had been born at Megara in Greece, a mistake which arose from confounding him with another Euclid, a disciple of Socrates. Proclus (A.D. 412-485), the authority for most of our information regarding Euclid, states in his commentary on the first book of the Elements that Euclid lived in the time of Ptolemy I., king of Egypt, who reigned from 323 to 285 B.C., that he was younger than the associates of Plato, but older than Eratosthenes (276196 B.C.) and Archimedes (287-212 B.C.). Euclid is said to have founded the mathematical school of Alexandria, which was at that time becoming a centre, not only of commerce, but of learning and research, and for this service to the cause of exact science he would have deserved commemoration, even if his writings had not secured him a worthier title to fame. Proclus preserves a reply made by Euclid to King Ptolemy, who asked whether he could not learn geometry more easily than by studying the Elements - " There is no royal road to geometry." Pappus of Alexandria, in his Mathematical Collection, says that Euclid was a man of mild and inoffensive temperament, unpretending, and kind to all genuine students of mathematics. This being all that is known of the life and character of Euclid, it only remains therefore to speak of his works.

Among those which have come down to us the most remarkable is the Elements (lrotxeia) (see Geometry). They consist of thirteen books; two more are frequently added, but there is reason to believe that they are the work of a later mathematician, Hypsicles of Alexandria.

The question has often been mooted, to what extent Euclid, in his Elements, is a discoverer or a compiler. To this question no entirely satisfactory answer can be given, for scarcely any of the writings of earlier geometers have come down to our times. We are mainly dependent on Pappus and Proclus for the scanty notices we have of Euclid's predecessors, and of the problems which engaged their attention; for the solution of problems, and not the discovery of theorems, would seem to have been their principal object. From these authors we learn that the property of the right-angled triangle had been found out, the principles of geometrical analysis laid down, the restriction of constructions in plane geometry to the straight line and the circle agreed upon, the doctrine of proportion, for both commensurables and incommensurables, as well as loci, plane and solid, and some of the properties of the conic sections investigated, the five regular solids (often called the Platonic bodies) and the relation between the volume of a cone or pyramid and that of its circumscribed cylinder or prism discovered. Elementary works had been written, and the famous problem of the duplication of the cube reduced to the determination of two mean proportionals between two given straight lines. Notwithstanding this amount of discovery, and all that it implied, Euclid must have made a great advance beyond his predecessors (we are told that " he arranged the discoveries of Eudoxus, perfected those of Theaetetus, and reduced to invincible demonstration many things that had previously been more loosely proved "), for his Elements supplanted all similar treatises, and, as Apollonius received the title of " the great geometer," so Euclid has come down to later ages as " the elementator." For the past twenty centuries parts of the Elements, notably the first six books, have been used as an introduction to geometry. Though they are now to some extent superseded in most countries, their long retention is a proof that they were, at any rate, not unsuitable for such a purpose. They are, speaking generally, not too difficult for novices in the science; the demonstrations are rigorous, ingenious and often elegant; the mixture of problems and theorems gives perhaps some variety, and makes their study less monotonous; and, if regard be had merely to the metrical properties of space as distinguished from the graphical, hardly any cardinal geometrical truths are omitted. With these excellences are combined a good many defects, some of them inevitable to a system based on a very few axioms and postulates. Thus the arrangement of the propositions seems arbitrary; associated theorems and problems are not grouped together; the classification, in short, is imperfect. Other objections, not to mention minor blemishes, are the prolixity of the style, arising partly from a defective nomenclature, the treatment of parallels depending on an axiom which is not axiomatic, and the sparing use of superposition as a method of proof.

Of the thirty-three ancient books subservient to geometrical analysis, Pappus enumerates first the Data (De60 thva) of Euclid. He says it contained 90 propositions, the scope of which he describes; it now consists of 95. It is not easy to explain this discrepancy, unless we suppose that some of the propositions, as they existed in the time of Pappus, have since been split into two, or that what were once scholia have since been erected into propositions. The object of the Data is to show that when certain things - lines, angles, spaces, ratios, &c. - are given by hypothesis, certain other things are given, that is, are determinable. The book, as we are expressly told, and as we may gather from its contents, was intended for the investigation of problems; and it has been conjectured that Euclid must have extended the method of the Data to the investigation of theorems. What prompts this conjecture is the similarity between the analysis of a theorem and the method, common enough in the Elements, of reductio ad absurdum - the one setting out from the supposition that the theorem is true, the other from the supposition that it is false, thence in both cases deducing a chain of consequences which ends in a conclusion previously known to be true or false. The Introduction to Harmony (Eioa v yw r yrt a pµovuci i), and the Section of the Scale (KaTaro,ui rcavovos), treat of music. There is good reason for believing that one at any rate, and probably both, of these books are not by Euclid. No mention is made of them by any writer previous to Ptolemy (A.D. 140), or by Ptolemy himself, and in no ancient codex are they ascribed to Euclid.

The Phaenomena (c Dacv6 €va) contains an exposition of the appearances produced by the motion attributed to the celestial sphere. Pappus, in the few remarks prefatory to his sixth book, complains of the faults, both of omission and commission, of writers on astronomy, and cites as an example of the former the second theorem of Euclid's Phaenomena, whence, and from the interpolation of other proofs, David Gregory infers that this treatise is corrupt.

The Optics and Catoptrics (' Orr uca, KaTOIrTpu(a) are ascribed to Euclid by Proclus, and by Marinus in his preface to the Data, but no mention is made of them by Pappus. This latter circumstance, taken in connexion with the fact that two of the propositions in the sixth book of the Mathematical Collection prove the same things as three in the Optics, is one of the reasons given by Gregory for deeming that work spurious. Several other reasons will be found in Gregory's preface to his edition of Euclid's works.

In some editions of Euclid's works there is given a book on the Divisions of Superficies, which consists of a few propositions, showing how a straight line may be drawn to divide in a given ratio triangles, quadrilaterals and pentagons. This was supposed by John Dee of London, who transcribed or translated it, and entrusted it for publication to his friend Federico Commandino of Urbino, to be the treatise of Euclid referred to by Proclus as TO 7repL &cupEaEwv 13e(3Xiov. Dee mentions that, in the copy from which he wrote, the book was ascribed to Machomet of Bagdad, and adduces two or three reasons for thinking it to be Euclid's. This opinion, however, he does not seem to have held very strongly, nor does it appear that it was adopted by Commandino. The book does not exist in Greek.

The fragment, in Latin, De levi et ponderoso, which is of no value, and was printed at the end of Gregory's edition only in order that nothing might be left out, is mentioned neither by Pappus nor Proclus, and occurs first in Bartholomew Zamberti's edition of There is no reason for supposing it to be genuine. The following works attributed to Euclid are not now extant: 1. Three books on Porisms (llepi ropcorµaTwv) are mentioned both by Pappus and Proclus, and the former gives an abstract of them, with the lemmas assumed. (See PoxlsM.) 2. Two books are mentioned, named Throw lrpos &4 aveia, which is rendered Locorum ad supericiem by Commandino and subsequent geometers. These books were subservient to tile analysis of loci, but the four lemmas which refer to them and which occur at the end of the seventh book of the Mathematical Collection, throw very little light'on their contents. R. Simson's opinion was that they treated of curves of double curvature, and he intended at one time to write a treatise on the subject. (See Trail's Life of Dr Simson). 3. Pappus says that Euclid wrote four books on the Conic Sections (3e3Xia TEaaapa which Apollonius amplified, and to which he added four more. It is known that, in the time of Euclid, the parabola was considered as the section of a rightangled cone, the ellipse that of an acute-angled cone, the hyperbola that of an obtuse-angled cone, and that Apollonius was the first who showed that the three sections could be obtained from any cone. There is good ground therefore for supposing that the first four books of Apollonius's Conics, which are still extant, resemble Euclid's Conics even less than Euclid's Elements do those of Eudoxus and Theaetetus.

4. A book on Fallacies (IIEpi l/iev6apiwv) is mentioned by Proclus, who says that Euclid wrote it for the purpose of exercising beginners in the detection of errors in reasoning.

This notice of Euclid would be incomplete without some account of the earliest and the most important editions of his works. Passing over the commentators of the Alexandrian school, the first European translator of any part of Euclid is Boetius (500), author of the De consolatione philosophiae. H:s Euclidis Megarensis geometriae libri duo contain nearly all the definitions of the first three books of the Elements, the postulates, and most of the axioms. The enunciations, with diagrams but no proofs, are given of most of the propositions in the first, second and fourth books, and a few from the third. Some centuries afterwards, Euclid was translated into Arabic, but the only printed version in that language is the one made of the thirteen books of the Elements by Nasir Al-Din AlTusi (13th century), which appeared at Rome in 1594.

The first printed edition of Euclid was a translation of the fifteen books of the Elements from the Arabic, made, it is supposed, by Adelard of Bath (12th century), with the comments of Campanus of Novara. It appeared at Venice in 1482, printed by Erhardus Ratdolt, and dedicated to the doge Giovanni Mocenigo. This edition represents Euclid very inadequately; the comments are often foolish, propositions are sometimes omitted, sometimes joined together, useless cases are interpolated, and now and then Euclid's order changed.

The first printed translation from the Greek is that of Bartholomew Zamberti, which appeared at Venice in 1505. Its contents will be seen from the title: Euclidis megaresis philosophi platonici Mathematicarua disciplinarie Janitoris: Habent in hoc volumine ad mathematica substantia aspirat: elemetorum libros xiii cu expositione Theonis insignis mathematici. Quibus. adjuncta. Deputatum scilicet Euclidi volume xiiii eft expositioe Hypsi. Alex. Itideqa Phaeno. Specu. Perspe. cum expositione Theonis ac mirandus ille liber Datorum cum expositioe Pappi Mechanici una ca Marini dialectici protheoria. Bar. Zaber. Vene. Interpte. The first printed Greek text was published at Basel, in with the title EbK%Eisov EIwv /3/ A. eEwvc auvovo &,v. It was edited by Simon Grynaeus from two MSS. sent to him, the one from Venice by Lazarus Bayfius, and the other from Paris by John Ruellius. The four books of Proclus's commentary are given at the end from an Oxford MS. supplied by John Claymundus.

The English edition, the only one which contains all the extant works attributed to Euclid, is that of Dr David Gregory, published at Oxford in 1703, with the title, Ta Euclidis quae supersunt omnia. The text is that of the Basel edition, corrected from the MSS. bequeathed by Sir Henry Savile, and from Savile's annotations on his own copy. The Latin translation, which accompanies the Greek on the same page, is for the most part that of Commandino. The French edition has the title, Les CEuvres d'Euclide, traduites en Latin et en Francais, d'apres un manuscrit tres-ancien qui etait reste inconnu jusqu'd nos jours. Par F. Peyrard, Traducteur des oeuvres d'Archimede. It was published at Paris in three volumes, the first of which appeared in 1814, the second in 1816 and the third in 1818. It contains the Elements and the Data, which are, says the editor, certainly the only works which remain to us of this ever-celebrated geometer. The texts of the Basel and Oxford editions were collated with 23 MSS., one of which belonged to the library of the Vatican, but had been sent to Paris by the comte de Peluse (Monge). The Vatican MS. was supposed to date from the 9th century; and to its readings Peyrard gave the greatest weight. What may be called the German edition has the title Ein AEieov Firoex€%a. Euclidis Elementa ex optimis libris in usum Tironum Graece editor ab Ernesto Ferdinando August. It was published at Berlin in two parts, the first of which appeared in 1826 and the second in 1829. The above mentioned texts were collated with three other MSS. Modern standard editions are by Dr Heiberg of Copenhagen, Euclidis Elementa, edidit et Latine interpretatus est J. L. Heiberg. vols. i.-v. (Lipsiae, 1883-1888), and by T. L. Heath, The Thirteen Books of Euclid's Elements, vols. i.-iii. (Cambridge, 1908).

Of translations of the Elements into modern languages the number is very large. The first English translation, published at London in 1570, has the title, The Elements of Geometric of the most auncient Philosopher Euclide of Megara. Faithfully (now first) translated into the Englishe toung, by H. Billingsley, Citizen of London. Whereunto are annexed certaine Scholies, Annotations and Inventions, of the best Mathematiciens, both of time past and in this our age. The first French translation of the whole of the Elements has the title, Les Quinze Livres des Elements d'Euclide. Traduicts de Latin en Francois. Par D. Henrion, Mathematicien. The first edition of it was published at Paris in 1615, and a second, corrected and augmented, in 1623. Pierre Forcadel de Bezies had published at Paris in 1564 a translation of the first six books of the Elements, and in 1565 of the seventh, eighth and ninth books. An Italian translation, with the title, Euclide Megarense acutissimo philosopho solo introduttore delle Scientie Mathematice. Diligentemente rassettato, et all y integrity ridotto, per it degno professore di tal Scientie Nicola Tartalea Brisciano, was published at Venice in 1569, and Federico Commandino's translation appeared at Urbino in 1575; a Spanish version, Los Seis Libros primeros de la geometria de Euclides. Traduzidos en legua Espanola por Rodrigo Camorano, Astrologo y Mathematico, at Seville in 1576; and a Turkish one, translated from the edition of J. Bonnycastle by Husain R;fki, at Bulak in 1825. Dr Robert Simson's editions of the first six and the eleventh and twelfth books of the Elements, and of the Data. - The authors and editions above referred to; Fabricius, Bibliotheca Graeca, vol. iv.; Murhard's Litteratur der mathematischen Wissenschaften; Heilbronner's Historia matheseos universae; De Morgan's article ` L Eucleides " in Smith's Dictionary of Biography and Mythology; Moritz Cantor's Geschichte der Mathematik, vol. i. (J. S. M.)


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Wiktionary

Up to date as of January 15, 2010

Definition from Wiktionary, a free dictionary

Contents

English

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Wikipedia

Proper noun

Singular
Euclid

Plural
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Euclid

  1. Greek mathematician

Related terms

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Simple English

[[File:|thumb|160px|Statue of Euclid, Oxford Museum of Natural History.]] [[File:|thumb|160px|Illustration of the most famous theorem in Euclid, Pythagoras' theorem: the square on the hypothenuse of a right-angled triangle is equal to the sum of the squares on the other two sides.]]

File:Euclid Elements Book 3 Proposition 35
Euclid's Elements Book III, Proposition 35: "If in a circle two straight lines cut one another, then the rectangle contained by the segments of the one equals the rectangle contained by the segments of the other".

Euclid of Alexandria (Greek: Εὐκλείδης) (about 325 BC–265 BC) was a Greek mathematician who lived in Alexandria, Egypt and worked at the Library of Alexandria. Little is known about this person, but people think he lived there when Ptolemy I was Pharaoh. It is not known where or when he was born.

Contents

The Elements

Euclid collected together all that was known of geometry, which is part of mathematics. His Elements is the main source of ancient geometry. Textbooks based on Euclid have been used up to the present day. In the book, he starts out from a small set of axioms (that is, a group of things that everyone thinks are true). Euclid then shows the properties of geometric objects and of whole numbers, based on those axioms.

The Elements also includes works on perspective, conic sections, spherical geometry, and possibly quadric surfaces. Apart from geometry, the work also includes number theory. Euclid came up with the idea of greatest common divisors. They were in his Elements. The greatest common divisor of two numbers is the greatest number that can fit evenly in both of the 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 that system is referred to as Euclidean geometry to distinguish it from other so-called non-Euclidean geometries that mathematicians discovered in the 19th century.

Other works

File:Oxyrhynchus papyrus with Euclid's
One of the oldest surviving fragments of Euclid's Elements, dated to circa AD 100. The diagram accompanies Book II, Proposition 5. [1]

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 third 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 to Euclid is doubtful. Its author may have been Theon of Alexandria.
  • 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.

Tributes to Euclid

References

  1. Bill Casselman. "One of the oldest extant diagrams from Euclid". University of British Columbia. http://www.math.ubc.ca/~cass/Euclid/papyrus/papyrus.html. Retrieved 2008-09-26. 


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