|
|
| Line 1: |
Line 1: |
| {{Infobox book
| | Nothing to say about me at all.<br>Nice to be here and a part of this community.<br>I really hope Im useful in one way .<br><br>Also visit my homepage ... [http://safedietplans.com/bmr-calculator bmr calculator] |
| | italic title = Elements
| |
| | name = Elements
| |
| | image = [[File:Title page of Sir Henry Billingsley's first English version of Euclid's Elements, 1570 (560x900).jpg|frameless|alt=Cover]]
| |
| | caption = The [[Book frontispiece|frontispiece]] of Sir Henry Billingsley's first English version of Euclid's ''Elements'', 1570
| |
| | author = Euclid, and translators
| |
| | title_orig =
| |
| | translator =
| |
| | illustrator =
| |
| | cover_artist =
| |
| | country =
| |
| | language = [[Ancient Greek]], translations
| |
| | series =
| |
| | subject = [[Euclidean geometry]], elementary [[number theory]]
| |
| | genre = Mathematics
| |
| | publisher =
| |
| | publisher2 =
| |
| | pub_date = c. 300 BC
| |
| | english_pub_date =
| |
| | media_type =
| |
| | pages = 13 books, or more in translation with [[scholia]]
| |
| | awards =
| |
| | isbn =
| |
| | oclc =
| |
| | dewey =
| |
| | congress =
| |
| | preceded_by =
| |
| | followed_by =
| |
| }}
| |
| {{DISPLAYTITLE:Euclid's ''Elements''}}
| |
| '''Euclid's ''Elements''''' ({{lang-grc|Στοιχεῖα}} ''Stoicheia'') is a [[mathematics|mathematical]] and [[geometry|geometric]] [[treatise]] consisting of 13 books written by the ancient [[Greek mathematics|Greek mathematician]] [[Euclid]] in [[Alexandria]] c. 300 BC. It is a collection of definitions, postulates ([[axiom]]s), propositions ([[theorem]]s and [[Compass and straightedge constructions|constructions]]), and [[mathematical proof]]s of the propositions. The thirteen books cover [[Euclidean geometry]] and the ancient Greek version of elementary [[number theory]]. The work also includes an algebraic system that has become known as [[Greek geometric algebra|geometric algebra]], which is powerful enough to solve many algebraic problems,<ref>Heath (1956) (vol. 1), p. 372</ref> including the problem of finding the [[square root]] of a number.<ref>Heath (1956) (vol. 1), p. 409</ref> With the exception of [[Autolycus of Pitane|Autolycus']] ''On the Moving Sphere'', the ''Elements'' is one of the oldest extant Greek mathematical treatises,<ref>{{cite book|last=Boyer|authorlink=Carl Benjamin Boyer|title=|year=1991|chapter=Euclid of Alexandria|page=101|quote=With the exception of the ''Sphere'' of Autolycus, surviving work by Euclid are the oldest Greek mathematical treatises extant; yet of what Euclid wrote more than half has been lost,}}</ref> and it is the oldest extant axiomatic deductive treatment of [[mathematics]]. It has proven instrumental in the development of [[logic]] and modern [[science]].
| |
| The name 'Elements' comes from the plural of 'element'. According to [[Proclus]] the term was used to describe a theorem that is all-pervading and helps furnishing proofs of many other theorems. The word 'element' is in the Greek language the same as 'letter'. This suggests that theorems in the ''Elements'' should be seen as standing in the same relation to geometry as letters to language. Later commentators give a slightly different meaning to the term 'element', emphasizing how the propositions have progressed in small steps, and continued to build on previous propositions in a well-defined order.<ref>Heath (1956) (vol. 1), p. 114</ref>
| |
| | |
| Euclid's ''Elements'' has been referred to as the most successful<ref>Encyclopedia of Ancient Greece (2006) by Nigel Guy Wilson, page 278. Published by Routledge Taylor and Francis Group. Quote:"Euclid's Elements subsequently became the basis of all mathematical education, not only in the Romand and Byzantine periods, but right down to the mid-20th century, and it could be argued that it is the most successful textbook ever written."</ref><ref name="Boyer Author of the Elements">{{cite book|last=Boyer|authorlink=Carl Benjamin Boyer|title=|year=1991|chapter=Euclid of Alexandria|page=100|quote=As teachers at the school he called a band of leading scholars, among whom was the author of the most fabulously successful mathematics textbook ever written – the ''Elements'' (''Stoichia'') of Euclid.}}</ref> and influential<ref name="Boyer Influence of the Elements">{{cite book|last=Boyer|authorlink=Carl Benjamin Boyer|title=|year=1991|chapter=Euclid of Alexandria|page=119|quote=The ''Elements'' of Euclid not only was the earliest major Greek mathematical work to come down to us, but also the most influential textbook of all times. [...]The first printed versions of the ''Elements'' appeared at Venice in 1482, one of the very earliest of mathematical books to be set in type; it has been estimated that since then at least a thousand editions have been published. Perhaps no book other than the Bible can boast so many editions, and certainly no mathematical work has had an influence comparable with that of Euclid's ''Elements''.}}</ref> textbook ever written. Being first set in type in [[Venice]] in 1482, it is one of the very earliest mathematical works to be printed after the invention of the [[printing press]] and was estimated by [[Carl Benjamin Boyer]] to be second only to the [[Bible]] in the number of editions published,<ref name="Boyer Influence of the Elements"/> with the number reaching well over one thousand.<ref>The Historical Roots of Elementary Mathematics by Lucas Nicolaas Hendrik Bunt, Phillip S. Jones, Jack D. Bedient (1988), page 142. Dover publications. Quote:"the ''Elements'' became known to Western Europe via the Arabs and the Moors. There the ''Elements'' became the foundation of mathematical education. More than 1000 editions of the ''Elements'' are known. In all probability it is, next to the ''Bible'', the most widely spread book in the civilization of the Western world."</ref> For centuries, when the [[quadrivium]] was included in the curriculum of all university students, knowledge of at least part of Euclid's ''Elements'' was required of all students. Not until the 20th century, by which time its content was universally taught through other school textbooks, did it cease to be considered something all educated people had read. <ref>From the introduction by Amit Hagar to ''Euclid and His Modern Rivals'' by Lewis Carroll (2009, Barnes & Noble) pg. xxviii: <blockquote>Geometry emerged as an indispensable part of the standard education of the English gentleman in the eighteenth century; by the Victorian period it was also becoming an important part of the education of artisans, children at Board Schools, colonial subjects and, to a rather lesser degree, women. ... The standard textbook for this purpose was none other than Euclid's ''The Elements''. </blockquote></ref>
| |
| | |
| == History ==
| |
| | |
| ===Basis in earlier work===
| |
| [[Image:Woman teaching geometry.jpg|thumb|The frontispiece of an [[Adelard of Bath]] Latin translation of Euclid's ''Elements'', c. 1309–1316; the oldest surviving Latin translation of the ''Elements'' is a 12th-century work by Adelard, which translates to Latin from the Arabic.<ref name="Russell" />]]
| |
| Scholars believe that the ''Elements'' is largely a collection of theorems proven by other mathematicians, supplemented by some original work. [[Proclus]], a Greek mathematician who lived several centuries after Euclid, wrote in his commentary of the ''Elements'': "Euclid, who put together the Elements, collecting many of [[Eudoxus of Cnidus|Eudoxus]]' theorems, perfecting many of [[Theaetetus (mathematician)|Theaetetus]]', and also bringing to irrefragable demonstration the things which were only somewhat loosely proved by his predecessors". However, Proclus' papyrus manuscript has been copied to a paper which became available in Europe after 800 years. Therefore, the originality of Proclus' work might have been altered since it was copied by hand.{{Citation needed|date=June 2013}} [[Pythagoras]] was probably the source for most of books I and II, [[Hippocrates of Chios]] (not the better known [[Hippocrates|Hippocrates of Kos]]) for book III, and Eudoxus for book V, while books IV, VI, XI, and XII probably came from other Pythagorean or Athenian mathematicians.<ref>W.W. Rouse Ball, A Short Account of the History of Mathematics, 4th ed., 1908, p. 54</ref> Euclid often replaced fallacious proofs with his own, more rigorous versions.<ref>{{cite book|author=Daniel Shanks|title=Solved and Unsolved Problems in Number Theory|year=2002|publisher=American Mathematical Society}}</ref> The use of definitions, and postulates or axioms dated back to [[Plato]], almost a century earlier.<ref>Ball, p. 43</ref> The ''Elements'' may have been based on an earlier textbook by Hippocrates of Chios, who also may have originated the use of letters to refer to figures.<ref>Ball, p. 38</ref>
| |
| | |
| ===Transmission of the text===
| |
| In the fourth century AD, [[Theon of Alexandria]] produced an edition of Euclid which was so widely used that it became the only surviving source until [[:fr:François Peyrard|François Peyrard]]'s 1808 discovery at the [[Vatican Library|Vatican]] of a manuscript not derived from Theon's. This manuscript, the [[Johan Ludvig Heiberg (historian)|Heiberg]] manuscript, is from a [[Byzantine]] workshop c. 900 and is the basis of modern editions.<ref>[<!-- http://www.historyofscience.com/G2I/timeline/index.php?id=2749 -->http://historyofinformation.com/expanded.php?id=2749 The Earliest Surviving Manuscript Closest to Euclid's Original Text (Circa 850)]; an [http://www.ibiblio.org/expo/vatican.exhibit/exhibit/d-mathematics/Greek_math.html image] of one page</ref> [[Papyrus Oxyrhynchus 29]] is a tiny fragment of an even older manuscript, but only contains the statement of one proposition.
| |
| | |
| Although known to, for instance, [[Cicero]], there is no extant record of the text having been translated into Latin prior to [[Boethius]] in the fifth or sixth century.<ref name="Russell">Russell, Bertrand. ''A History of Western Philosophy''. p. 212.</ref> The Arabs received the ''Elements'' from the Byzantines in approximately 760; this version was translated into [[Arabic]] under [[Harun al Rashid]] c. 800.<ref name="Russell" /> The Byzantine scholar [[Arethas of Caesarea|Arethas]] commissioned the copying of one of the extant Greek manuscripts of Euclid in the late ninth century.<ref>L.D. Reynolds and Nigel G. Wilson, ''Scribes and Scholars'' 2nd. ed. (Oxford, 1974) p. 57</ref> Although known in Byzantium, the ''Elements'' was lost to Western Europe until c. 1120, when the English monk [[Adelard of Bath]] translated it into Latin from an Arabic translation.<ref>One older work claims Adelard disguised himself as a Muslim student in order to obtain a copy in Muslim Córdoba (Rouse Ball, p. 165). However, more recent biographical work has turned up no clear documentation that Adelard ever went to Muslim-ruled Spain, although he spent time in Norman-ruled Sicily and Crusader-ruled Antioch, both of which had Arabic-speaking populations. Charles Burnett, ''Adelard of Bath: Conversations with his Nephew'' (Cambridge, 1999); Charles Burnett, ''Adelard of Bath'' (University of London, 1987).</ref>
| |
| | |
| [[File:Euclid's Elements 1573 Edition.JPG|thumb|Euclidis – Elementorum libri XV Paris, Hieronymum de Marnef & Guillaume Cavelat, 1573 (second edition after the 1557 ed.); in-8, 350, (2)pp. THOMAS-STANFORD, Early Editions of Euclid's Elements, n°32. Mentioned in T.L. Heath's translation. Private collection Hector Zenil.]] The first printed edition appeared in 1482 (based on [[Campanus of Novara|Campanus of Novara's]] 1260 edition),<ref name=Busard>{{cite book |last1=Busard |first1=H.L.L. |authorlink1= |last2= |first2= |authorlink2= |editor1-first= |editor1-last= |editor1-link= |others= |title=Campanus of Novara and Euclid's Elements |trans_title= |url= |archiveurl= |archivedate= |format= |accessdate= |type= |edition= |series= |volume=I |date= |year=2005 |month= |origyear= |publisher=Franz Steiner Verlag |location=Stuttgart |language= |isbn=978-3-515-08645-5 |oclc= |doi= |id= |page= |pages= |at= |trans_chapter= |chapter=Introduction to the Text |chapterurl= |quote= |ref= |bibcode= |laysummary= |laydate= |separator= |postscript= |lastauthoramp=}}</ref> and since then it has been translated into many languages and published in about a thousand different editions. Theon's Greek edition was recovered in 1533. In 1570, [[John Dee (mathematician)|John Dee]] provided a widely respected "Mathematical Preface", along with copious notes and supplementary material, to the first English edition by [[Henry Billingsley]].
| |
| | |
| Copies of the Greek text still exist, some of which can be found in the [[Vatican Library]] and the [[Bodleian Library]] in Oxford. The manuscripts available are of variable quality, and invariably incomplete. By careful analysis of the translations and originals, hypotheses have been made about the contents of the original text (copies of which are no longer available).
| |
| | |
| Ancient texts which refer to the ''Elements'' itself, and to other mathematical theories that were current at the time it was written, are also important in this process. Such analyses are conducted by [[Johan Ludvig Heiberg (historian)|J. L. Heiberg]] and Sir [[Thomas Little Heath]] in their editions of the text.
| |
| | |
| Also of importance are the [[scholia]], or annotations to the text. These additions, which often distinguished themselves from the main text (depending on the manuscript), gradually accumulated over time as opinions varied upon what was worthy of explanation or further study. | |
| | |
| ==Influence==
| |
| The ''Elements'' is still considered a masterpiece in the application of [[logic]] to [[mathematics]]. In historical context, it has proven enormously influential in many areas of [[science]]. Scientists [[Nicolaus Copernicus]], [[Johannes Kepler]], [[Galileo Galilei]], and Sir [[Isaac Newton]] were all influenced by the ''Elements'', and applied their knowledge of it to their work. Mathematicians and philosophers, such as [[Bertrand Russell]], [[Alfred North Whitehead]], and [[Baruch Spinoza]], have attempted to create their own foundational "Elements" for their respective disciplines, by adopting the axiomatized deductive structures that Euclid's work introduced.
| |
| | |
| The austere beauty of Euclidean geometry has been seen by many in western culture as a glimpse of an otherworldly system of perfection and certainty. Abraham Lincoln kept a copy of Euclid in his saddlebag, and studied it late at night by lamplight; he related that he said to himself, "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".<ref>Henry Ketcham, The Life of Abraham Lincoln, at Project Gutenberg, http://www.gutenberg.org/ebooks/6811</ref> [[Edna St. Vincent Millay]] wrote in her sonnet ''Euclid Alone Has Looked on Beauty Bare'', "O blinding hour, O holy, terrible day, When first the shaft into his vision shone Of light anatomized!". [[Einstein]] recalled a copy of the ''Elements'' and a magnetic compass as two gifts that had a great influence on him as a boy, referring to the Euclid as the "holy little geometry book".<ref>Dudley Herschbach, "Einstein as a Student," Department of Chemistry and Chemical Biology, Harvard University, Cambridge, MA, USA, page 3, web: [http://www.chem.harvard.edu/herschbach/Einstein_Student.pdf HarvardChem-Einstein-PDF]: about Max Talmud visited on Thursdays for six years.</ref>
| |
| | |
| The success of the ''Elements'' is due primarily to its logical presentation of most of the mathematical knowledge available to Euclid. Much of the material is not original to him, although many of the proofs are his. However, Euclid's systematic development of his subject, from a small set of axioms to deep results, and the consistency of his approach throughout the ''Elements'', encouraged its use as a textbook for about 2,000 years. The ''Elements'' still influences modern geometry books. Further, its logical axiomatic approach and rigorous proofs remain the cornerstone of mathematics.
| |
| | |
| == Outline of ''Elements'' ==
| |
| [[File:Euclid-proof.svg|thumb|300px|A proof from Euclid's ''Elements'' that, given a line segment, an equilateral triangle exists that includes the segment as one of its sides. The proof is by construction: an equilateral triangle ΑΒΓ is made by drawing circles Δ and Ε centered on the points Α and Β, and taking one intersection of the circles as the third vertex of the triangle.]]
| |
| [[Image:Oxyrhynchus papyrus with Euclid's Elements.jpg|thumb|300px|A [[Papyrus Oxyrhynchus 29|fragment]] of Euclid's elements found at [[Oxyrhynchus]], which is dated to circa 100 AD. The diagram accompanies Proposition 5 of Book II of the ''Elements''.]]
| |
| | |
| === Contents of the books ===
| |
| Books 1 through 4 deal with [[plane geometry]]:
| |
| * Book 1 contains Euclid's 10 axioms (5 named postulates—including the [[parallel postulate]]—and 5 named axioms) and the basic propositions of geometry: the ''[[pons asinorum]]'' (proposition 5), the [[Pythagorean theorem]] (Proposition 47), equality of angles and [[area]]s, parallelism, the sum of the angles in a triangle, and the three cases in which triangles are "equal" (have the same area).
| |
| * Book 2 is commonly called the "book of [[Greek geometric algebra|geometric algebra]]" because most of the propositions can be seen as geometric interpretations of algebraic identities, such as {{nowrap|''a''(''b'' + ''c'' + ...)}} = ''ab'' + ''ac'' + ... or (2''a'' + ''b'')<sup>2</sup> + ''b''<sup>2</sup> = 2(''a''<sup>2</sup> + (''a'' + ''b'')<sup>2</sup>). It also contains a method of finding the [[square root]] of a given number.
| |
| * Book 3 deals with circles and their properties: [[inscribe]]d angles, [[tangent]]s, the power of a point, [[Thales' theorem]].
| |
| * Book 4 constructs the [[incircle]] and [[circumcircle]] of a triangle, and constructs [[regular polygon]]s with 4, 5, 6, and 15 sides.
| |
| | |
| Books 5 through 10 introduce [[ratio]]s and [[proportionality (mathematics)|proportions]]:
| |
| | |
| * Book 5 is a treatise on proportions of [[magnitude (mathematics)|magnitudes]]. Proposition 25 has as a special case the [[inequality of arithmetic and geometric means]].
| |
| * Book 6 applies proportions to geometry: Similar figures.
| |
| * Book 7 deals strictly with elementary number theory: [[divisibility]], [[prime number]]s, [[Euclid's algorithm]] for finding the [[greatest common divisor]], [[least common multiple]]. Propositions 30 and 32 together are essentially equivalent to the [[fundamental theorem of arithmetic]] stating that every positive integer can be written as a product of primes in an essentially unique way, though Euclid would have had trouble stating it in this modern form as he did not use the product of more than 3 numbers.
| |
| * Book 8 deals with proportions in number theory and [[geometric progression|geometric sequences]].
| |
| * Book 9 applies the results of the preceding two books and gives the infinitude of prime numbers (proposition 20), the sum of a [[geometric series]] (proposition 35), and the construction of even [[perfect number]]s (proposition 36).
| |
| * Book 10 attempts to classify [[commensurability (mathematics)|incommensurable]] (in modern language, [[irrational number|irrational]]) magnitudes by using the [[method of exhaustion]], a precursor to [[integral|integration]].
| |
| | |
| Books 11 through to 13 deal with spatial geometry:
| |
| | |
| * Book 11 generalizes the results of Books 1–6 to space: perpendicularity, parallelism, volumes of [[parallelepiped]]s.
| |
| * Book 12 studies volumes of [[cone (geometry)|cones]], [[Pyramid (geometry)|pyramids]], and [[cylinder (geometry)|cylinders]] in detail, and shows for example that the volume of a cone is a third of the volume of the corresponding cylinder. It concludes by showing the volume of a [[sphere]] is proportional to the cube of its radius by approximating it by a union of many pyramids.
| |
| * Book 13 constructs the five regular [[Platonic solid]]s inscribed in a sphere, calculates the ratio of their edges to the radius of the sphere, and proves that there are no further regular solids.
| |
| | |
| ==Euclid's method and style of presentation==
| |
| [[File:Euclid Vat ms no 190 I prop 47.jpg|thumb|Codex Vaticanus 190]]
| |
| Euclid's [[Axiomatic system#Axiomatic method|axiomatic approach]] and [[Euclidean geometry#Methods of proof|constructive methods]] were widely influential.
| |
| | |
| As was common in ancient mathematical texts, when a proposition needed [[Mathematical proof|proof]] in several different cases, Euclid often proved only one of them (often the most difficult), leaving the others to the reader. Later editors such as [[Theon of Alexandria|Theon]] often interpolated their own proofs of these cases.
| |
| | |
| Euclid's presentation was limited by the mathematical ideas and notations in common currency in his era, and this causes the treatment to seem awkward to the modern reader in some places. For example, there was no notion of an angle greater than two right angles,<ref>Ball, p. 55</ref> the number 1 was sometimes treated separately from other positive integers, and as multiplication was treated geometrically he did not use the product of more than 3 different numbers. The geometrical treatment of number theory may have been because the alternative would have been the extremely awkward [[Greek numerals|Alexandrian system of numerals]].<ref>Ball, pp. 58, 127</ref>
| |
| | |
| The presentation of each result is given in a stylized form, which, although not invented by Euclid, is recognized as typically classical. It has six different parts: First is the ''enunciation'' which states the result in general terms (i.e. the statement of the proposition). Then the ''setting-out'', which gives the figure and denotes particular geometrical objects by letters. Next comes the ''definition'' or ''specification'' which restates the enunciation in terms of the particular figure. Then the ''construction'' or ''machinery'' follows. It is here that the original figure is extended to forward the proof. Then, the ''proof'' itself follows. Finally, the ''conclusion'' connects the proof to the enunciation by stating the specific conclusions drawn in the proof, in the general terms of the enunciation.<ref>Heath (1963), p. 216</ref>
| |
| | |
| No indication is given of the method of reasoning that led to the result, although the ''[[Data (Euclid)|Data]]'' does provide instruction about how to approach the types of problems encountered in the first four books of the ''Elements''.<ref>Ball, p. 54</ref> Some scholars have tried to find fault in Euclid's use of figures in his proofs, accusing him of writing proofs that depended on the specific figures drawn rather than the general underlying logic, especially concerning Proposition II of Book I. However, Euclid's original proof of this proposition is general, valid, and does not depend on the figure used as an example to illustrate one given configuration.<ref>[[Godfried Toussaint]], "A new look at Euclid's second proposition," ''The Mathematical Intelligencer'', Vol. 15, No. 3, 1993, pp. 12–23.</ref>
| |
| | |
| == Criticism ==
| |
| Euclid's list of axioms in the ''Elements'' was not exhaustive, but represented the principles that were the most important. His proofs often invoke axiomatic notions which were not originally presented in his list of axioms. Later editors have interpolated Euclid's implicit axiomatic assumptions in the list of formal axioms.<ref>Heath (1956) (vol. 1), p. 62</ref>
| |
| | |
| For example, in the first construction of Book 1, Euclid used a premise that was neither postulated nor proved: that two circles with centers at the distance of their radius will intersect in two points.<ref>Heath (1956) (vol. 1), p. 242</ref> Later, in the fourth construction, he used superposition (moving the triangles on top of each other) to prove that if two sides and their angles are equal then they are congruent; during these considerations he uses some properties of superposition, but these properties are not described explicitly in the treatise. If superposition is to be considered a valid method of geometric proof, all of geometry would be full of such proofs. For example, propositions I.1 – I.3 can be proved trivially by using superposition.<ref>Heath (1956) (vol. 1), p. 249</ref>
| |
| | |
| Mathematician and historian [[W. W. Rouse Ball]] put the criticisms in perspective, remarking that "the fact that for two thousand years [the ''Elements''] was the usual text-book on the subject raises a strong presumption that it is not unsuitable for that purpose."<ref>Ball (1960) p. 55.</ref>
| |
| | |
| == Apocrypha ==
| |
| It was not uncommon in ancient time to attribute to celebrated authors works that were not written by them. It is by these means that the [[apocryphal]] books XIV and XV of the ''Elements'' were sometimes included in the collection.<ref name="Boyer Apocrypha"/> The spurious Book XIV was probably written by [[Hypsicles]] on the basis of a treatise by [[Apollonius of Perga|Apollonius]]. The book continues Euclid's comparison of regular solids inscribed in spheres, with the chief result being that the ratio of the surfaces of the [[dodecahedron]] and [[icosahedron]] inscribed in the same sphere is the same as the ratio of their volumes, the ratio being
| |
| :<math>\sqrt{\tfrac{10}{3(5-\sqrt{5})}} = \sqrt{\tfrac{5+\sqrt{5}}{6}}.\ </math>
| |
| | |
| The spurious Book XV was probably written, at least in part, by [[Isidore of Miletus]]. This book covers topics such as counting the number of edges and solid angles in the regular solids, and finding the measure of dihedral angles of faces that meet at an edge.<ref name="Boyer Apocrypha">{{cite book|last=Boyer|authorlink=Carl Benjamin Boyer|title=|year=1991|chapter=Euclid of Alexandria|pages=118–119|quote=In ancient times it was not uncommon to attribute to a celebrated author works that were not by him; thus, some versions of Euclid's ''Elements'' include a fourteenth and even a fifteenth book, both shown by later scholars to be apocryphal. The so-called Book XIV continues Euclid's comparison of the regular solids inscribed in a sphere, the chief results being that the ratio of the surfaces of the dodecahedron and icosahedron inscribed in the same sphere is the same as the ratio of their volumes, the ratio being that of the edge of the cube to the edge of the icosahedron, that is, <math>\sqrt{10/[3(5-\sqrt{5})]}</math>. It is thought that this book may have been composed by Hypsicles on the basis of a treatise (now lost) by Apollonius comparing the dodecahedron and icosahedron. [...] The spurious Book XV, which is inferior, is thought to have been (at least in part) the work of Isidore of Miletus (fl. ca. A.D. 532), architect of the cathedral of Holy Wisdom (Hagia Sophia) at Constantinople. This book also deals with the regular solids, counting the number of edges and solid angles in the solids, and finding the measures of the dihedral angles of faces meeting at an edge.}}</ref>
| |
| | |
| ==Editions==
| |
| [[Image:Ricci Guangqi 2.jpg|thumb|The [[Italy|Italian]] [[Jesuit]] [[Matteo Ricci]] (left) and the [[Chinese mathematics|Chinese mathematician]] [[Xu Guangqi]] (right) published the [[Chinese language|Chinese]] edition of ''Euclid's Elements'' (幾何原本) in 1607.]]
| |
| *1460s, [[Regiomontanus]] (incomplete)
| |
| *1482, Erhard Ratdolt (Venice), first printed edition<ref>{{harvnb|Alexanderson|Greenwalt|2012|loc= pg. 163}}</ref>
| |
| *1533, ''[[editio princeps]]'' by Simon Grynäus
| |
| *1557, by Jean Magnien and Pierre de Montdoré, reviewed by Stephanus Gracilis (only propositions, no full proofs, includes original Greek and the Latin translation)
| |
| *1572, Commandinus Latin edition
| |
| *1574, [[Christoph Clavius]]
| |
| | |
| ===Translations===
| |
| *1505, Bartolomeo Zamberti (Latin)
| |
| *1543, [[Niccolò Tartaglia]] (Italian)
| |
| *1557, Jean Magnien and Pierre de Montdoré, reviewed by Stephanus Gracilis (Greek to Latin)
| |
| *1558, Johann Scheubel (German)
| |
| *1562, Jacob Kündig (German)
| |
| *1562, Wilhelm Holtzmann (German)
| |
| *1564–1566, Pierre Forcadel de Béziers (French)
| |
| *1570, [[Henry Billingsley]] (English)
| |
| *1575, Commandinus (Italian)
| |
| *1576, [[Rodrigo de Zamorano]] (Spanish)
| |
| *1594, Typografia Medicea (edition of the Arabic translation of [[Nasir al-Din al-Tusi]])
| |
| *1604, Jean Errard de Bar-le-Duc (French)
| |
| *1606, Jan Pieterszoon Dou (Dutch)
| |
| *1607, [[Matteo Ricci]], [[Xu Guangqi]] (Chinese)
| |
| *1613, [[Pietro Cataldi]] (Italian)
| |
| *1615, [[Denis Henrion]] (French)
| |
| *1617, Frans van Schooten (Dutch)
| |
| *1637, L. Carduchi (Spanish)
| |
| *1639, [[Pierre Hérigone]] (French)
| |
| *1651, Heinrich Hoffmann (German)
| |
| *1651, [[Thomas Rudd]] (English)
| |
| *1660, [[Isaac Barrow]] (English)
| |
| *1661, John Leeke and Geo. Serle (English)
| |
| *1663, Domenico Magni (Italian from Latin)
| |
| *1672, Claude François Milliet Dechales (French)
| |
| *1680, Vitale Giordano (Italian)
| |
| *1685, William Halifax (English)
| |
| *1689, Jacob Knesa (Spanish)
| |
| *1690, Vincenzo Viviani (Italian)
| |
| *1694, Ant. Ernst Burkh v. Pirckenstein (German)
| |
| *1695, C. J. Vooght (Dutch)
| |
| *1697, [[Samuel Reyher]] (German)
| |
| *1702, Hendrik Coets (Dutch)
| |
| *1705, Edmund Scarburgh (English)
| |
| *1708, John Keill (English)
| |
| *1714, Chr. Schessler (German)
| |
| *1714, W. Whiston (English)
| |
| *1720s [[Jagannatha Samrat]] (Sanskrit, based on the Arabic translation of Nasir al-Din al-Tusi)<ref>{{citation | year=1997 | title = Encyclopaedia of the history of science, technology, and medicine in non-western cultures | author1=[[K. V. Sarma]] | editor1=Helaine Selin | editor1-link=Helaine Selin | publisher=Springer | isbn=978-0-7923-4066-9 | pages=460–461 | url=http://books.google.com/books?id=yoiXDTXSHi4C&pg=PA460&dq=rekhaganita}}</ref>
| |
| *1731, Guido Grandi (abbreviation to Italian)
| |
| *1738, Ivan Satarov (Russian from French)
| |
| *1744, [[Mårten Strömer]] (Swedish)
| |
| *1749, Dechales (Italian)
| |
| *1745, Ernest Gottlieb Ziegenbalg (Danish)
| |
| *1752, Leonardo Ximenes (Italian)
| |
| *1756, [[Robert Simson]] (English)
| |
| *1763, Pubo Steenstra (Dutch)
| |
| *1768, Angelo Brunelli (Portuguese)
| |
| *1773, 1781, J. F. Lorenz (German)
| |
| *1780, [[Baruch Schick of Shklov]] (Hebrew)<ref>[http://web.archive.org/web/20090622080437/http://aleph500.huji.ac.il/nnl/dig/books/bk001139706.html JNUL Digitized Book Repository]</ref>
| |
| *1781, 1788 James Williamson (English)
| |
| *1781, William Austin (English)
| |
| *1789, Pr. Suvoroff nad Yos. Nikitin (Russian from Greek)
| |
| *1795, John Playfair (English)
| |
| *1803, H.C. Linderup (Danish)
| |
| *1804, F. Peyrard (French)
| |
| *1807, Józef Czech (Polish based on Greek, Latin and English editions)
| |
| *1807, J. K. F. Hauff (German)
| |
| *1818, Vincenzo Flauti (Italian)
| |
| *1820, Benjamin of Lesbos (Modern Greek)
| |
| *1826, George Phillips (English)
| |
| *1828, Joh. Josh and Ign. Hoffmann (German)
| |
| *1828, [[Dionysius Lardner]] (English)
| |
| *1833, E. S. Unger (German)
| |
| *1833, [[Thomas Perronet Thompson]] (English)
| |
| *1836, H. Falk (Swedish)
| |
| *1844, 1845, 1859 P. R. Bråkenhjelm (Swedish)
| |
| *1850, F. A. A. Lundgren (Swedish)
| |
| *1850, H. A. Witt and M. E. Areskong (Swedish)
| |
| *1862, [[Isaac Todhunter]] (English)
| |
| *1865, [[Sámuel Brassai]] (Hungarian)
| |
| *1873, Masakuni Yamada (Japanese)
| |
| *1880, [[Mykhailo Vaschenko-Zakharchenko|Vachtchenko-Zakhartchenko]] (Russian)
| |
| *1901, Max Simon (German)
| |
| *1908, [[Thomas Little Heath]] (English)
| |
| *1939, [[R. Catesby Taliaferro]] (English)
| |
| | |
| ===Currently in print===
| |
| *''Euclid's Elements – All thirteen books in one volume'', Based on Heath's translation, Green Lion Press ISBN 1-888009-18-7.
| |
| *''The Elements: Books I-XIII-Complete and Unabridged,'' (2006) Translated by Sir Thomas Heath, Barnes & Noble ISBN 0-7607-6312-7.
| |
| *''The Thirteen Books of Euclid's Elements'', translation and commentaries by Heath, Thomas L. (1956) in three volumes. Dover Publications. ISBN 0-486-60088-2 (vol. 1), ISBN 0-486-60089-0 (vol. 2), ISBN 0-486-60090-4 (vol. 3)
| |
| | |
| == Notes ==
| |
| {{reflist|30em}}
| |
| | |
| ==References==
| |
| *{{citation|last1=Alexanderson|first1=Gerald L.|last2=Greenwalt|first2=William S.| title=About the cover: Billingsley's Euclid in English|journal=Bulletin (New Series) of the American Mathematical Society|volume=49|issue=1|year=2012|pages=163–167}}
| |
| *{{cite book
| |
| | last = Ball
| |
| | first = W.W. Rouse
| |
| | authorlink = W. W. Rouse Ball
| |
| | title = A Short Account of the History of Mathematics
| |
| | origyear =
| |
| | url =
| |
| | edition = 4th ed. [Reprint. Original publication: London: Macmillan & Co., 1908]
| |
| | year = 1960
| |
| | publisher = Dover Publications
| |
| | location = New York
| |
| | isbn = 0-486-20630-0
| |
| | pages = 50–62
| |
| }}
| |
| *{{cite book
| |
| | last = Heath
| |
| | first = Thomas L.
| |
| | authorlink = T. L. Heath
| |
| | title = The Thirteen Books of Euclid's Elements
| |
| | edition = 2nd ed. [Facsimile. Original publication: Cambridge University Press, 1925]
| |
| | year = 1956
| |
| | publisher = Dover Publications
| |
| | location = New York
| |
| }}
| |
| : (3 vols.): ISBN 0-486-60088-2 (vol. 1), ISBN 0-486-60089-0 (vol. 2), ISBN 0-486-60090-4 (vol. 3). Heath's authoritative translation plus extensive historical research and detailed commentary throughout the text.
| |
| * {{cite book| last = Heath
| |
| | first = Thomas L.
| |
| | authorlink = T. L. Heath
| |
| | title=A Manual of Greek Mathematics|url=http://books.google.com/books?id=_HZNr_mGFzQC|year=1963|publisher=Dover Publications|isbn=978-0-486-43231-1}}
| |
| *{{cite book
| |
| | first=Carl B.
| |
| | last=Boyer
| |
| | authorlink=Carl Benjamin Boyer
| |
| | title=A History of Mathematics
| |
| | edition=Second Edition
| |
| | publisher=John Wiley & Sons, Inc.
| |
| | year=1991
| |
| | isbn=0-471-54397-7
| |
| }}
| |
| | |
| == External links ==
| |
| {{Wikisource|The Elements of Euclid}}
| |
| {{commons category|Elements of Euclid}}
| |
| *{{cite book
| |
| | last = Euclid | title = Elements | origyear = c. 300 BC
| |
| | url = http://aleph0.clarku.edu/~djoyce/java/elements/toc.html
| |
| | accessdate = 2006-08-30 | year = David E. Joyce, ed. 1997
| |
| }} In HTML with Java-based interactive figures.
| |
| * [http://farside.ph.utexas.edu/euclid/Elements.pdf Euclid's Elements in English and Greek] (PDF), utexas.edu
| |
| *Richard Fitzpatrick [http://farside.ph.utexas.edu/euclid.html a bilingual edition] (typset in PDF format, with the original Greek and an English translation on facing pages; free in PDF form, available in print) ISBN 978-0-615-17984-1
| |
| * [<!-- http://old.perseus.tufts.edu/cgi-bin/ptext?lookup=Euc.+1 -->http://www.perseus.tufts.edu/hopper/text?doc=Euc.+1 Heath's English translation] (HTML, without the figures, public domain) (accessed February 4, 2010)
| |
| ** Heath's English translation and commentary, with the figures (Google Books): [http://books.google.com/books?id=UhgPAAAAIAAJ vol. 1], [http://books.google.com/books?id=lxkPAAAAIAAJ vol. 2], [http://books.google.com/books?id=xhkPAAAAIAAJ vol. 3], [http://books.google.com/books?id=KHMDAAAAYAAJ vol. 3 c. 2]
| |
| * [http://www.math.ubc.ca/~cass/Euclid/byrne.html Oliver Byrne's 1847 edition] (also hosted at [http://archive.org/details/firstsixbooksofe00eucl archive.org])– an unusual version by [[Oliver Byrne (mathematician)]] who used color rather than labels such as ABC (scanned page images, public domain)
| |
| * [http://gutenberg.org/ebooks/21076 The First Six Books of the Elements] by John Casey and Euclid scanned by [[Project Gutenberg]].
| |
| * [<!-- http://www.du.edu/~etuttle/classics/nugreek/contents.htm -->http://mysite.du.edu/~etuttle/classics/nugreek/contents.htm Reading Euclid] – a course in how to read Euclid in the original Greek, with English translations and commentaries (HTML with figures)
| |
| *[[Thomas More|Sir Thomas More]]'s [http://www.columbia.edu/acis/textarchive/rare/24.html manuscript]
| |
| *[http://www.columbia.edu/acis/textarchive/rare/6.html Latin translation] by [[Aethelhard of Bath]]
| |
| *[<!-- http://www.physics.ntua.gr/Faculty/mourmouras/euclid/index.html -->http://www.physics.ntua.gr/~mourmouras/euclid/index.html Euclid Elements – The original Greek text] Greek HTML
| |
| *[[Clay Mathematics Institute]] Historical Archive – [http://www.claymath.org/library/historical/euclid/ The thirteen books of Euclid's Elements] copied by Stephen the Clerk for Arethas of Patras, in Constantinople in 888 AD
| |
| *[http://pds.lib.harvard.edu/pds/view/13079270 Kitāb Taḥrīr uṣūl li-Ūqlīdis] Arabic translation of the thirteen books of Euclid's Elements by Nasīr al-Dīn al-Ṭūsī. Published by Medici Oriental Press(also, Typographia Medicea). Facsimile hosted by [http://ocp.hul.harvard.edu/ihp/ Islamic Heritage Project].
| |
| *[http://starrhorse.com/euclid/ Euclid's "Elements" Redux], an open textbook based on the "Elements"
| |
| *[http://archive.org/search.php?query=%E5%B9%BE%E4%BD%95%E5%8E%9F%E6%9C%AC 1607 Chinese translations] reprinted as part of [[Siku Quanshu]], or "Complete Library of the Four Treasuries."
| |
| | |
| {{Greek mathematics}}
| |
| | |
| [[Category:3rd-century BC books]]
| |
| [[Category:Euclidean geometry|*]]
| |
| [[Category:Mathematics books]]
| |
| [[Category:Ancient Greek mathematical works]]
| |
| [[Category:Works by Euclid]]
| |
| [[Category:History of geometry]]
| |
| | |
| {{Link FA|pl}}
| |
| {{Link FA|he}}
| |