Wikipedia - Benutzerbeiträge [de]

Stephen Smale

2016-04-13T17:52:30Z

MathKeduor7:

[[File:Stephen Smale2.jpg|thumb|Stephen Smale (2008)]]
'''Stephen Smale''' (* [[15. Juli]] [[1930]] in [[Flint (Michigan)|Flint]], [[Michigan]], [[USA]]) ist ein US-amerikanischer [[Mathematiker]], der hauptsächlich durch seine Arbeiten über dynamische Systeme und für seinen Beweis der [[Poincaré-Vermutung]] für den Fall <math>n > 4</math> bekannt wurde. Er ist Träger der [[Fields-Medaille]] und war Professor an der [[University of California, Berkeley]].

== Leben ==
Smale begann sein Studium an der Universität von [[Michigan]] 1948, mit anfangs eher mäßigen Noten – er interessierte sich eher für Reisen und politische Aktivitäten auf dem Campus. Er trat der kommunistischen Partei bei. Wegen seiner nachlassenden Noten erhielt er sogar eine Ermahnung seines Fakultätsleiters Hildebrandt. 1952 graduierte er und 1957 machte er seine Doktorarbeit (''Regular Curves on Riemannian [[Mannigfaltigkeit|Manifolds]]'') unter [[Raoul Bott]], dessen erster Doktorand er war. Mit dieser Arbeit verallgemeinerte er ältere Resultate von [[Hassler Whitney]], der 1937 reguläre geschlossene Kurven in der Ebene durch ihre [[Windungszahl]] klassifizierte. 1956 besuchte er die [[Topologie (Mathematik)|Topologie]]-Konferenz in [[Mexiko-Stadt]], an der die weltweit führenden Topologen teilnahmen.

1959 sorgte er an der Universität von [[Chicago]] mit dem Beweis der Möglichkeit, eine Sphäre im dreidimensionalen Raum von innen nach außen zu stülpen, ohne „Risse“ zu erzeugen (''Sphere Eversion''), für Aufsehen. Eine anschauliche Vorgehensweise zeigte später z. B. der blinde französische Mathematiker [[Bernard Morin]]. Genauer zeigte Smale, dass alle stetigen Einbettungen (Immersionen) von <math>S^{2}</math> in den <math>R^{3}</math> regulär [[homotop]] waren, also auch die Standard-Einbettung zur Einbettung der invertierten Sphäre („von außen nach innen gestülpt“).

Mit diesen Arbeiten gewann er ein Stipendium der National Science Foundation und erhielt eine Einladung an das [[Institute for Advanced Study]], er ging aber 1960 nach [[Rio de Janeiro]] an das [[IMPA]] zu [[Mauricio Peixoto]], der auf dynamische Systeme spezialisiert war und den er schon 1958 getroffen hatte. Hier „am Strand von Rio“ kamen ihm die Ideen für seine [[Hufeisen-Abbildung]] und für den Beweis der verallgemeinerten Poincaré-Vermutung für Dimensionen größer als 4. Dabei benutzte er Ideen aus der [[Morsetheorie]]. Ideen aus seinem Beweis verallgemeinerte er später und leitete aus ihnen das [[Kobordismus|h-Kobordismus-Theorem]] her. Heute wird meist umgekehrt die Poincaré-Vermutung in d>4 als Folge dieses h-Kobordismus Theorems bewiesen. Ein etwa gleichzeitiger Beweis einer Version der Poincaré-Vermutung in d>4 durch [[John Stallings]] führte zu einem Prioritätsstreit.

Schon Anfang der 1960er Jahre begann er, sich mit [[Dynamisches System|dynamischen Systemen]] wie seiner berühmten Hufeisen-Abbildung zu beschäftigen, die chaotisch, aber „strukturell stabil“ ist. Damit verallgemeinerte er Untersuchungen über Störungen stabiler Bewegungen der russischen Mathematiker [[Alexander Alexandrowitsch Andronow|Andronov]] und [[Lew Semjonowitsch Pontrjagin|Pontrjagin]] und begann seine eigenen qualitativen, topologischen Untersuchungen dynamischer Systeme. Smale fasste chaotische Systeme wie das Hufeisen oder auch geodätische Flüsse auf Mannigfaltigkeiten negativer Krümmung zusammen als „hyperbolische“ Systeme, gekennzeichnet durch lokales Stauchen und Strecken. Anfangs glaubte er, dass diese Systeme „typisch“ sind (ihre Bahnen „dicht“ liegen), was sich aber als falsch herausstellte. Smale knüpfte auch – damals unüblich – Kontakte zu den traditionell in der Theorie dynamischer Systeme starken sowjetischen Mathematikern wie [[Wladimir Arnold]], z. B. 1961 in Moskau und auf dem Internationalen Mathematikerkongress 1966 in Moskau, wo er die [[Fields-Medaille]] bekam.

In den 1970er Jahren begann er, Anwendungen dynamischer Systeme zu untersuchen, z. B. das n-Körperproblem, elektrische Schwingkreise oder die Gleichgewichte von Systemen aus den Wirtschaftswissenschaften. Daraus ergab sich die Frage nach der Konvergenz der Annäherung an Gleichgewichtspunkte, was Smale zu [[Algorithmus|algorithmischen]] Untersuchungen führte, die er ebenfalls global anging.

Ab den 1990er Jahren versucht er, die Numerische Analysis und das auf Turing-Automaten beruhende Berechnungsmodell der theoretischen Informatik zu vereinigen (Arbeiten mit [[Lenore Blum]], [[Mike Shub]]).

Als er sich einmal dahingehend äußerte, dass seine besten Arbeiten „am Strand von Rio“ entstanden, nahm die [[National Science Foundation]] dies in den 1960er Jahren zum Anlass, ihm Gelder kürzen zu wollen, sie nahmen aber später wieder davon Abstand. Der wissenschaftliche Berater von Präsident Johnson [[Donald Hornig]] nahm Smales Äußerungen 1968 in Science als Beispiel für eine leichtfertige Einstellung von Mathematikern anzunehmen, dass sie das Geld der Steuerzahler für mathematische Forschungen an den Stränden von Rio verwenden könnten<ref>Smale ''Finding a Horseshoe on the beaches of Rio''</ref>. Auch mit seinen linksgerichteten politischen Aktivitäten besonders in den 1960er Jahren erregte er Aufsehen. 1960 und dann wieder 1964–1995 war er Professor in [[University of California, Berkeley|Berkeley]], also im Zentrum der amerikanischen Studentenbewegung, und im Mai 1965 war er maßgeblich an der Organisation der Anti[[vietnamkrieg]]s-Tage beteiligt. 1966 erregte er bei der US-amerikanischen Administration Unwillen, als er sich in Moskau, wo er die Fields-Medaille in Empfang nahm, öffentlich gegen den Vietnamkrieg äußerte. Zur selben Zeit versuchte das Haus-Komitee gegen Unamerikanische Umtriebe (HUAC) ihn vorzuladen. Smale war in seiner Studentenzeit Mitglied der Jugendorganisation (Labor Youth League) der kommunistischen Partei (und später auch insgeheim Mitglied der Kommunistischen Partei)<ref>Batterson, ''Stephen Smale'', AMS 2000</ref>.

Nach seiner Emeritierung war er an der [[Universität Hongkong]] und ist zurzeit am Toyota Institut für Technologie in Chicago.

1998 stellte er eine Liste von 18 noch ungelösten Problemen für das 21. Jahrhundert auf ([[Mathematical Intelligencer]] 1998 Nr. 2). Diese ist von [[David Hilbert|Hilberts]] [[Hilberts Liste von 23 mathematischen Problemen|23 Problemen]] inspiriert, die dieser im Jahr 1900 aufstellte. Zwei von ihnen kommen auch wieder bei Smale vor, zum einen die [[Riemannsche Vermutung]], zum anderen eine moderne Version eines Teils von Hilberts 16. Problem. Manche von Smales Problemen sind auch unter den [[Millennium-Probleme]]n (Riemannvermutung, [[Navier-Stokes-Gleichung]], [[P-NP-Problem]], [[Poincaré-Vermutung]]). Viele seiner Probleme sind aus der Theorie dynamischer Systeme oder haben einen Algorithmen-Hintergrund, sein letztes Problem fragt allgemein nach den Grenzen künstlicher und menschlicher Intelligenz.

Smale wurde für seine Arbeit mehrfach ausgezeichnet, insbesondere mit der [[Fields-Medaille]] und dem [[Oswald-Veblen-Preis]] (beide 1966). 2007 erhielt er den [[Wolf-Preis]]. Er war Invited Speaker (Plenarvortrag) auf dem [[Internationaler Mathematikerkongress|ICM]] 1986 in Berkeley (''Complexity aspects of numerical analysis''), in Stockholm 1962 (''Dynamical systems and the topological conjugacy problem for diffeomorphisms'') und in Moskau 1966 (''Differentiable dynamical systems''). 1968 wurde er in die [[American Academy of Arts and Sciences]] und 1970 in die [[National Academy of Sciences]] aufgenommen.

Smale besitzt eine große Mineralien- und Edelsteinsammlung und ist auch als Fotograf von Mineralien hervorgetreten.

Zu seinen Doktoranden gehören [[Michael Shub]], [[Robert Devaney]], [[Morris Hirsch]], [[John Guckenheimer]], [[Nancy Kopell]], [[Zbigniew Nitecki]], [[Jacob Palis]], [[Sheldon Newhouse]].<ref>[http://www.genealogy.math.ndsu.nodak.edu/id.php?id=5086 Mathematics Genealogy Project]</ref>

== Siehe auch ==
* [[Hufeisen-Abbildung]]

== Literatur ==
von Stephen Smale:
* ''The Story of the Higher Dimensional Poincaré Conjecture (what actually happened on the beaches of Rio).'' Mathematical Intelligencer 1990
* ''A classification of immersions of the 2-sphere.'' Bulletin AMS 1958 (wie man eine Sphäre von innen nach außern stülpt ohne Risse), und Transactions AMS 1958
* ''Generalized Poincaré's conjecture in dimensions greater than four.'' Annals of Math., Bd.74, 1961, S.391
* ''A survey of some recent results in differential topology.'' Bulletin AMS 1963
* ''Finding a horseshoe on the beaches of Rio.'' Mathematical Intelligencer 1998
* ''Differentiable dynamical systems.'' Bulletin AMS Bd.73, 1967, S.747–817
* ''On the problem of reviving the ergodic hypothesis of Boltzmann and Birkhoff.'' In: Helleman (Hrsg.): ''Nonlinear dynamics.'' Annales NY Academy of Sciences 1979
* mit Morris Hirsch: ''Differential equations, dynamical systems and linear algebra.'' Academic Press 1974

zu ihm und seinen Arbeiten:
* Steve Batterson: ''The mathematician who broke the dimension barrier.'' American Mathematical Society 2000
* Phillips: ''Turning a sphere inside out.'' Scientific American, Mai 1966
* Hirsch: ''The work of Stephen Smale in differential topology.'' In: Hirsch (Hrsg.): ''From topology to computation: Proceedings of the Smalefest, Berkeley 1990''. Springer 1993
* Shub: ''What is a Horseshoe?'' Notices AMS, Mai 2005, online hier: [http://www.ams.org/notices/200505/200505-toc.html]
*Donald J. Albers, G. L. Alexanderson, [[Constance Reid]] ''More Mathematical People - Contemporary Conversations'', Academic Press 1994

== Weblinks ==
* {{DNB-Portal|119122472}}
* [http://mathworld.wolfram.com/SmalesProblems.html Smales 18 Probleme]
* {{MacTutor Biography|id=Smale}}
* [http://www.ams.org/notices/200708/ Interview Notices AMS 2007, pdf]
* [http://gdz.sub.uni-goettingen.de/de/dms/load/img/?PPN=GDZPPN002088525 Smale: ''Topology and mechanics 1.'' Inv.Math. 1970], Teil 2 ist online hier: [http://gdz.sub.uni-goettingen.de/de/dms/load/img/?PPN=GDZPPN002088592]
* {{Webarchiv | url=http://www.maa.org/reviews/smale.html | wayback=20100302211942 | text=Shirley Gray, Review von Battersons Biographie, engl.}}
* [http://www.ams.org/notices/200011/200011-toc.html Kirby, Review von Battersons Biographie, engl., pdf]
* [http://www.ams.org/notices/200305/200305-toc.html Smale, Poggio: ''The mathematics of learning- dealing with data.'' Notices AMS 2003]
* [http://www.numdam.org/numdam-bin/fitem?id=SB_1969-1970__12__177_0 Smale: ''Stability and genericity of dynamical systems.'' Sem.Bourbaki 1969/70]

== Einzelnachweise ==
<references />

{{Navigationsleiste Träger der Fields-Medaille}}
{{Navigationsleiste Träger des Wolf-Preises in Mathematik}}

{{Normdaten|TYP=p|GND=119122472|LCCN=n/80/102173|NDL=00456840|VIAF=71452582}}

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[[Kategorie:Mathematiker (20. Jahrhundert)]]
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[[Kategorie:Person (Universität Hongkong)]]
[[Kategorie:Träger der Fields-Medaille]]
[[Kategorie:Mitglied der American Academy of Arts and Sciences]]
[[Kategorie:Mitglied der National Academy of Sciences der Vereinigten Staaten]]
[[Kategorie:US-Amerikaner]]
[[Kategorie:Geboren 1930]]
[[Kategorie:Mann]]

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Great Books of the Western World

2015-04-27T05:01:04Z

MathKeduor7: wikification

{{italic title}}
[[Image:Great Books.jpg|thumb|300px|The Great Books (second edition)]]

'''''Great Books of the Western World''''' is a series of books originally published in the [[United States]] in 1952, by [[Encyclopædia Britannica Inc.]], to present the [[Great Books]] in a 54-volume set; the second edition of the series comprises 60 volumes.

The original editors had three criteria for including a book in the series: the book must be relevant to contemporary matters, and not only important in its historical context; it must be rewarding to re-read; and it must be a part of "the great conversation about the great ideas", relevant to at least 25 of the 102 great ideas identified by the editors. The books were not chosen on the basis of ethnic and cultural inclusiveness, historical influence, or the editors' agreement with the views expressed by the authors.<ref name=Adler />{{Citation broken|date=April 2015}}

==History==
The project for the ''Great Books of the Western World'' began at the [[University of Chicago]], where the president, [[Robert Hutchins]], collaborated with [[Mortimer Adler]] to develop a course — generally aimed at businesspeople — for the purpose of filling the gaps in their [[Liberal arts|liberal education]]; to render the reader as an [[Intellectualism|intellectually]] rounded man or woman familiar with the [[Great Books]] of the Western canon, and knowledgeable of the great ideas developed in the course of three millennia. An original student of the project was [[William Benton (senator)|William Benton]] (later a U.S. senator, and then chief executive officer of the ''[[Encyclopædia Britannica]]'' publishing company) who proposed selecting the greatest books of the Western canon, and that Hutchins and Adler produce unabridged editions for publication, by Encyclopædia Britannica. Yet, Hutchins was wary of such a business endeavour, fearing that the books would be sold as a product, thereby devaluing them as cultural artefacts; nevertheless, he agreed to the business deal, and was paid $60,000 for the project.

After deciding what subjects and authors to include, and how to present the materials, the project was begun, with a budget of $2,000,000. On April 15, 1952, the ''Great Books of the Western World'' were presented at a publication party in the [[Waldorf-Astoria Hotel]], in New York City. In his speech, Hutchins said, "This is more than a set of books, and more than a liberal education. ''Great Books of the Western World'' is an act of piety. Here are the sources of our being. Here is our heritage. This is [[the West]]. This is its meaning for mankind." The first two sets of books were given to [[Elizabeth II]], Queen of the U.K., and to [[Harry S. Truman]], the incumbent U.S. President.

The initial sales of the book sets were poor, with only 1,863 sets sold in 1952, and less than one-tenth of that number of book sets were sold in 1953. A financial debacle loomed until Encyclopædia Britannica altered the sales strategy, and sold the book set through experienced door-to-door encyclopædia-salesmen, as Hutchins had feared; but, through that method, 50,000 sets were sold in 1961. In 1963 the editors published ''[[Gateway to the Great Books]]'', a ten-volume set of readings meant to introduce the authors and the subjects of the ''Great Books''. Each year, from 1961 to 1998, the editors published ''[[The Great Ideas Today]]'', an annual updating about the applicability of the ''Great Books'' to contemporary life.<ref>{{cite web|url=http://ark.cdlib.org/ark:/13030/ft4w10061d/|title=''Robert Maynard Hutchins: A Memoir''|author=Milton Meyer|publisher=University of California Press|year=1993|accessdate=2007-05-30}} This biography of Robert M. Hutchins contains an extensive discussion of the Great Books project.</ref><ref>{{cite web|url=http://chronicle.uchicago.edu/020711/greatbooks.shtml|title=Special Collections tells the story of a cornerstone of American education|author=Carrie Golus|date=2002-07-11|accessdate=2007-05-30|publisher=''The University of Chicago Chronicle''}}</ref> The Internet and the [[E-book reader]] have made available some of the ''Great Books of the Western World'' in an on-line format.<ref>{{cite web|title=Great Books of the Western World (eBooks @ University of Adelaide)|url=http://ebooks.adelaide.edu.au/l/literature/gbww/index.html|publisher=University of Adelaide|accessdate=7 June 2012}}</ref>

==Volumes==
Originally published in 54 volumes, ''The Great Books of the Western World'' covers categories including [[fiction]], [[history]], [[poetry]], [[natural science]], [[mathematics]], [[philosophy]], [[drama]], [[politics]], [[religion]], [[economics]], and [[ethics]]. Hutchins wrote the first volume, titled ''[[Great Conversation|The Great Conversation]]'', as an introduction and discourse on [[liberal arts|liberal education]]. Adler sponsored the next two volumes, "The Great Ideas: [[A Syntopicon: An Index to The Great Ideas|A Syntopicon]]", as a way of emphasizing the unity of the set and, by extension, of Western thought in general. A team of indexers spent months compiling references to such topics as "Man's freedom in relation to the will of God" and "The denial of void or vacuum in favor of a [[Plenism|plenum]]". They grouped the topics into 102 chapters, for which Adler wrote 102 introductions. Four colors identify each volume by subject area -- Imaginative Literature, Mathematics and the Natural Sciences, History and Social Science, and Philosophy and Theology. The volumes contained the following works:

'''Volume 1'''
* [[Great Conversation|The Great Conversation]]

'''Volume 2'''
* [[Syntopicon]] I: [[Angel]], [[Animal]], [[Aristocracy]], [[Art]], [[Astronomy]], [[Beauty]], [[Being]], [[Cause]], [[Chance (philosophy)|Chance]], [[wikt:change|Change]], [[Citizen]], [[Constitution]], [[Courage]], [[Custom (law)|Custom]] and [[Convention (norm)|Convention]], [[Definition]], [[Democracy]], [[Interpersonal attraction|Desire]], [[Dialectic]], [[Duty]], [[Education]], [[Classical element|Element]], [[Emotion]], [[Eternity]], [[Evolution]], [[Experience]], [[Family]], [[destiny|Fate]], [[wikt:Form|Form]], [[God]], [[goodness and value theory|Good]] and [[Evil]], [[Government]], [[Habituation|Habit]], [[Happiness]], [[History]], [[Honor]], [[Hypothesis]], [[Idea]], [[Immortality]], [[Inductive reasoning|Induction]], [[Infinity]], [[Judgment]], [[Justice]], [[Knowledge]], [[Labour (economics)|Labor]], [[Language]], [[Law]], [[Liberty]], [[Life]] and [[Death]], [[Logic]], and [[Love]]

'''Volume 3'''
* [[Syntopicon]] II: [[Man]], [[Mathematics]], [[Matter]], [[Mechanics]], [[Medicine]], [[Memory]] and [[Imagination]], [[Metaphysics]], [[Mind]], [[Monarchy]], [[Nature]], [[Necessity]] and [[Contingency]], [[Oligarchy]], [[1 (number)|One]] and [[wikt:many|Many]], [[Opinion]], [[wikt:Opposition|Opposition]], [[Philosophy]], [[Physics]], [[Pleasure]] and [[Pain]], [[Poetry]], [[Principle]], [[Progress (history)|Progress]], [[Prophecy]], [[Prudence]], [[Punishment]], [[Quality (philosophy)|Quality]], [[Quantity]], [[Reasoning]], [[Relation (disambiguation)|Relation]], [[Religion]], [[Revolution]], [[Rhetoric]], [[Sameness|Same]] and [[Other]], [[Science]], [[Sense]], [[Sign (linguistics)|Sign]] and [[Symbol]], [[Sin]], [[Slavery]], [[Soul]], [[Space]], [[sovereign state|State]], [[Temperance (virtue)|Temperance]], [[Theology]], [[Time]], [[Truth]], [[Tyranny]], [[Universal (metaphysics)|Universal]] and [[Particular]], [[Virtue]] and [[Vice]], [[War]] and [[Peace]], [[Wealth]], [[Will (philosophy)|Will]], [[Wisdom]], and [[World]]

'''Volume 4'''
* [[Homer]] (rendered into English prose by [[Samuel Butler (novelist)|Samuel Butler]])
** ''[[The Iliad]]''
** ''[[The Odyssey]]''

'''Volume 5'''
* [[Aeschylus]] (translated into English verse by [[G.M. Cookson]])
** ''[[The Suppliants (Aeschylus)|The Suppliant Maidens]]''
** ''[[The Persians]]''
** ''[[Seven Against Thebes]]''
** ''[[Prometheus Bound]]''
** ''[[The Oresteia]]''
*** ''[[Agamemnon (play)|Agamemnon]]''
*** ''[[The Libation Bearers|Choephoroe]]''
*** ''[[The Eumenides]]''
* [[Sophocles]] (translated into English prose by [[Sir Richard C. Jebb]])
** ''[[Oedipus Cycle|The Oedipus Cycle]]''
*** ''[[Oedipus the King]]''
*** ''[[Oedipus at Colonus]]''
*** ''[[Antigone (Sophocles)|Antigone]]''
** ''[[Ajax (Sophocles)|Ajax]]''
** ''[[Electra (Sophocles)|Electra]]''
** ''[[The Trachiniae]]''
** ''[[Philoctetes (Sophocles)|Philoctetes]]''
* [[Euripides]] (translated into English prose by [[Edward P. Coleridge]])
** ''[[Rhesus (play)|Rhesus]]''
** ''[[Medea (play)|Medea]]''
** ''[[Hippolytus (play)|Hippolytus]]''
** ''[[Alcestis (play)|Alcestis]]''
** ''[[Heracleidae (play)|Heracleidae]]''
** ''[[The Suppliants (Euripides)|The Suppliants]]''
** ''[[Trojan Women]]''
** ''[[Ion (play)|Ion]]''
** ''[[Helen (play)|Helen]]''
** ''[[Andromache (play)|Andromache]]''
** ''[[Electra (Euripides)|Electra]]''
** ''[[The Bacchae|Bacchantes]]''
** ''[[Hecuba (play)|Hecuba]]''
** ''[[Heracles (Euripides)|Heracles Mad]]''
** ''[[Phoenician Women]]''
** ''[[Orestes (play)|Orestes]]''
** ''[[Iphigeneia in Tauris]]''
** ''[[Iphigeneia at Aulis]]''
** ''[[Cyclops (play)|Cyclops]]''
* [[Aristophanes]] (translated into English verse by [[Benjamin Bickley Rogers]])
** ''[[The Acharnians]]''
** ''[[The Knights]]''
** ''[[The Clouds]]''
** ''[[The Wasps]]''
** ''[[Peace (play)|Peace]]''
** ''[[The Birds (play)|The Birds]]''
** ''[[The Frogs]]''
** ''[[Lysistrata]]''
** ''[[Thesmophoriazusae]]''
** ''[[Assemblywomen|Ecclesiazousae]]''
** ''[[Plutus (play)|Plutus]]''

'''Volume 6'''
* [[Herodotus]]
** ''[[Histories (Herodotus)|The History]] ''(translated by [[George Rawlinson]])
* [[Thucydides]]
** ''[[History of the Peloponnesian War]] ''(translated by [[Richard Crawley]] and revised by [[R. Feetham]])

'''Volume 7'''
* [[Plato]]
** The Dialogues (translated by [[Benjamin Jowett]])
*** ''[[Charmides (dialogue)|Charmides]]''
*** ''[[Lysis (dialogue)|Lysis]]''
*** ''[[Laches (dialogue)|Laches]]''
*** ''[[Protagoras (dialogue)|Protagoras]]''
*** ''[[Euthydemus (dialogue)|Euthydemus]]''
*** ''[[Cratylus (dialogue)|Cratylus]]''
*** ''[[Phaedrus (dialogue)|Phaedrus]]''
*** ''[[Ion (dialogue)|Ion]]''
*** ''[[Symposium (Plato dialogue)|Symposium]]''
*** ''[[Meno]]''
*** ''[[Euthyphro]]''
*** ''[[Apology (Plato)|Apology]]''
*** ''[[Crito]]''
*** ''[[Phaedo]]''
*** ''[[Gorgias (dialogue)|Gorgias]]''
*** ''[[Republic (Plato)|The Republic]]''
*** ''[[Timaeus (dialogue)|Timaeus]]''
*** ''[[Critias (dialogue)|Critias]]''
*** ''[[Parmenides (dialogue)|Parmenides]]''
*** ''[[Theaetetus (dialogue)|Theaetetus]]''
*** ''[[Sophist (dialogue)|Sophist]]''
*** ''[[Statesman (dialogue)|Statesman]]''
*** ''[[Philebus]]''
*** ''[[Laws (dialogue)|Laws]]''
** ''[[Seventh Letter (Plato)|The Seventh Letter]] ''(translated by [[J. Harward]])

'''Volume 8'''
* [[Aristotle]]
** ''[[Categories (Aristotle)|Categories]]''
** ''[[De Interpretatione|On Interpretation]]''
** ''[[Prior Analytics]]''
** ''[[Posterior Analytics]]''
** ''[[Topics (Aristotle)|Topics]]''
** ''[[Sophistical Refutations]]''
** ''[[Physics (Aristotle)|Physics]]''
** ''[[On the Heavens]]''
** ''[[On Generation and Corruption]]''
** ''[[Meteorology (Aristotle)|Meteorology]]''
** ''[[Metaphysics (Aristotle)|Metaphysics]]''
** ''[[On the Soul]]''
** Minor biological works

'''Volume 9'''
* [[Aristotle]]
** ''[[History of Animals]]''
** ''[[Parts of Animals]]''
** ''[[Movement of Animals|On the Motion of Animals]]''
** ''[[Progression of Animals|''On the Gait of Animals'']]
** ''[[On the Generation of Animals]]''
** ''[[Nicomachean Ethics]]''
** ''[[Politics (Aristotle)|Politics]]''
** ''[[Constitution of the Athenians|The Athenian Constitution]]''
** ''[[Rhetoric (Aristotle)|Rhetoric]]''
** ''[[Poetics (Aristotle)|Poetics]]''

'''Volume 10'''
* [[Hippocrates]]
** Works
* [[Galen]]
** ''On the Natural Faculties''

'''Volume 11'''
* [[Euclid]]
** The Thirteen Books of ''[[Euclid's Elements]]''
* [[Archimedes]]
** ''[[On the Sphere and Cylinder]]''
** ''[[Measurement of a Circle]]''
** ''On Conoids and Spheroids''
** ''[[On Spirals]]''
** ''On the Equilibrium of Planes''
** ''[[The Sand Reckoner]]''
** ''[[The Quadrature of the Parabola]]''
** ''On Floating Bodies''
** ''[[Book of Lemmas]]''
** ''[[The Method of Mechanical Theorems|The Method Treating of Mechanical Problems]]''
* [[Apollonius of Perga]]
** ''[[On Conic Sections]]''
* [[Nicomachus of Gerasa]]
** ''[[Introduction to Arithmetic]]''

'''Volume 12'''
* [[Lucretius]]
** ''[[De rerum natura|On the Nature of Things]] ''(translated by [[H.A.J. Munro]])
* [[Epictetus]]
** ''[[Discourses of Epictetus|The Discourses]] ''(translated by [[George Long (scholar)|George Long]])
* [[Marcus Aurelius]]
** ''[[Meditations|The Meditations]] ''(translated by [[George Long (scholar)|George Long]])

'''Volume 13'''
* [[Virgil]]
** ''[[Eclogues]]''
** ''[[Georgics]]''
** ''[[Aeneid]]''

'''Volume 14'''
* [[Plutarch]]
** ''[[Parallel Lives|The Lives of the Noble Grecians and Romans]]''

'''Volume 15'''
* [[Tacitus|P. Cornelius Tacitus]] (translated by [[Alfred John Church]] and [[William Jackson Brodribb]])
** ''[[Annals (Tacitus)|The Annals]] ''
** ''[[Histories (Tacitus)|The Histories]]''

'''Volume 16'''
* [[Ptolemy]]
** ''[[Almagest]], part 1 ''(translated by [[R. Catesby Taliaferro]])
* [[Nicolaus Copernicus]]
** ''[[De revolutionibus orbium coelestium|On the Revolutions of Heavenly Spheres]] ''(translated by [[Charles Glenn Wallis]])
* [[Johannes Kepler]] (translated by [[Charles Glenn Wallis]])
** ''Epitome of Copernican Astronomy'' (Books IV–V)
** ''[[Harmonices Mundi|The Harmonies of the World]]'' (Book V)

'''Volume 17'''
* [[Plotinus]]
** ''[[Enneads|The Six Enneads]]''

'''Volume 18'''
* [[Augustine of Hippo]]
** ''[[Confessions (St. Augustine)|The Confessions]]''
** ''[[City of God (book)|The City of God]]''
** ''[[On Christian Doctrine]]''

'''Volume 19'''
* [[Thomas Aquinas]]
** ''[[Summa Theologica]]'' (First part complete, selections from second part, translated by the [[Fathers of the English Dominican Province]] and revised by [[Daniel J. Sullivan]])

'''Volume 20'''
* [[Thomas Aquinas]]
** ''[[Summa Theologica]]'' (Selections from second and third parts and supplement, translated by the [[Fathers of the English Dominican Province]] and revised by [[Daniel J. Sullivan]])

'''Volume 21'''
* [[Dante Alighieri]]
** ''[[The Divine Comedy]] ''(Translated by [[Charles Eliot Norton]])

'''Volume 22'''
* [[Geoffrey Chaucer]]
** ''[[Troilus and Criseyde]]''
** ''[[The Canterbury Tales]]''

'''Volume 23'''
* [[Niccolò Machiavelli]]
** ''[[The Prince]]''
* [[Thomas Hobbes]]
** ''[[Leviathan (book)|Leviathan]]''

'''Volume 24'''
* [[François Rabelais]]
** ''[[Gargantua and Pantagruel]]''

'''Volume 25'''
* [[Michel Eyquem de Montaigne]]
** [[Essays (Montaigne)|Essays]]

'''Volume 26'''
* [[William Shakespeare]]
** ''[[Henry VI, Part 1|The First Part of King Henry the Sixth]]''
** ''[[Henry VI, Part 2|The Second Part of King Henry the Sixth]]''
** ''[[Henry VI, Part 3|The Third Part of King Henry the Sixth]]''
** ''[[Richard III (play)|The Tragedy of Richard the Third]]''
** ''[[The Comedy of Errors]]''
** ''[[Titus Andronicus]]''
** ''[[The Taming of the Shrew]]''
** ''[[The Two Gentlemen of Verona]]''
** ''[[Love's Labour's Lost]]''
** ''[[Romeo and Juliet]]''
** ''[[Richard II (play)|The Tragedy of King Richard the Second]]''
** ''[[A Midsummer Night's Dream]]''
** ''[[The Life and Death of King John]]''
** ''[[The Merchant of Venice]]''
** ''[[Henry IV, Part 1|The First Part of King Henry the Fourth]]''
** ''[[Henry IV, Part 2|The Second Part of King Henry the Fourth]]''
** ''[[Much Ado About Nothing]]''
** ''[[Henry V (play)|The Life of King Henry the Fifth]]''
** ''[[Julius Caesar (play)|Julius Caesar]]''
** ''[[As You Like It]]''

'''Volume 27'''
* [[William Shakespeare]]
** ''[[Twelfth Night|''Twelfth Night; or, What You Will'']]
** ''[[Hamlet|The Tragedy of Hamlet, Prince of Denmark]]''
** ''[[The Merry Wives of Windsor]]''
** ''[[Troilus and Cressida]] ''
** ''[[All's Well That Ends Well]]''
** ''[[Measure for Measure]]''
** ''[[Othello|Othello, the Moor of Venice]]''
** ''[[King Lear]]''
** ''[[Macbeth]]''
** ''[[Antony and Cleopatra]]''
** ''[[Coriolanus]]''
** ''[[Timon of Athens]]''
** ''[[Pericles, Prince of Tyre]]''
** ''[[Cymbeline]]''
** ''[[The Winter's Tale]]''
** ''[[The Tempest]]''
** ''[[The Famous History of the Life of King Henry the Eighth]]''
** [[Shakespeare's sonnets|Sonnets]]

'''Volume 28'''
* [[William Gilbert (astronomer)|William Gilbert]]
** ''[[De Magnete|On the Loadstone and Magnetic Bodies]]''
* [[Galileo Galilei]]
** ''[[Two New Sciences|Dialogues Concerning the Two New Sciences]]''
* [[William Harvey]]
** ''[[On the Motion of the Heart and Blood in Animals]]''
** ''On the Circulation of Blood''
** ''[[On the Generation of Animals]]''

'''Volume 29'''
* [[Miguel de Cervantes]]
** ''[[Don Quixote|The History of Don Quixote de la Mancha]]''

'''Volume 30'''
* [[Sir Francis Bacon]]
** ''[[The Advancement of Learning]]''
** ''[[Novum Organum]]''
** ''[[New Atlantis]]''

'''Volume 31'''
* [[René Descartes]]
** ''[[Rules for the Direction of the Mind]]''
** ''[[Discourse on the Method]]''
** ''[[Meditations on First Philosophy]]''
** ''[[Meditations on First Philosophy|Objections Against the Meditations and Replies]]''
** ''[[La Géométrie|The Geometry]]''
* [[Benedict de Spinoza]]
** ''[[Ethics (book)|Ethics]]''

'''Volume 32'''
* [[John Milton]]
** English Minor Poems
** ''[[Paradise Lost]]''
** ''[[Samson Agonistes]]''
** ''[[Areopagitica]]''

'''Volume 33'''
* [[Blaise Pascal]]
** ''[[Lettres provinciales|The Provincial Letters]]''
** ''[[Pensées]]''
** Scientific and mathematical essays

'''Volume 34'''
* [[Sir Isaac Newton]]
** ''[[Mathematical Principles of Natural Philosophy]]''
** ''[[Opticks|Optics]]''
* [[Christian Huygens]]
** ''[[Treatise on Light]]''

'''Volume 35'''
* [[John Locke]]
** ''[[A Letter Concerning Toleration]]''
** ''[[Two Treatises of Government|Concerning Civil Government, Second Essay]]''
** ''[[An Essay Concerning Human Understanding]]''
* [[George Berkeley]]
** ''[[Treatise Concerning the Principles of Human Knowledge|The Principles of Human Knowledge]]''
* [[David Hume]]
** ''[[An Enquiry Concerning Human Understanding]]''

'''Volume 36'''
* [[Jonathan Swift]]
** ''[[Gulliver's Travels]]''
* [[Laurence Sterne]]
** ''[[The Life and Opinions of Tristram Shandy, Gentleman]]''

'''Volume 37'''
* [[Henry Fielding]]
** ''[[The History of Tom Jones, a Foundling]]''

'''Volume 38'''
* [[Charles de Secondat, Baron de Montesquieu]]
** ''[[The Spirit of the Laws]]''
* [[Jean Jacques Rousseau]]
** ''[[Discourse on Inequality|A Discourse on the Origin of Inequality]]''
** ''A Discourse on Political Economy''
** ''[[The Social Contract]]''

'''Volume 39'''
* [[Adam Smith]]
** ''[[The Wealth of Nations|An Inquiry into the Nature and Causes of the Wealth of Nations]]''

'''Volume 40'''
* [[Edward Gibbon]]
** ''[[The History of the Decline and Fall of the Roman Empire|The Decline and Fall of the Roman Empire]]'' (Part 1)

'''Volume 41'''
* [[Edward Gibbon]]
** ''[[The History of the Decline and Fall of the Roman Empire|The Decline and Fall of the Roman Empire]]'' (Part 2)

'''Volume 42'''
* [[Immanuel Kant]]
** ''[[Critique of Pure Reason]]''
** ''[[Groundwork of the Metaphysic of Morals|Fundamental Principles of the Metaphysic of Morals]]''
** ''[[Critique of Practical Reason]]''
** Excerpts from ''[[Metaphysics of Morals|The Metaphysics of Morals]]''
*** ''Preface and Introduction to the Metaphysical Elements of Ethics with a note on Conscience''
*** ''General Introduction to the Metaphysic of Morals''
*** ''The Science of Right''
** ''[[Critique of Judgment|The Critique of Judgement]]''

'''Volume 43'''
* American State Papers
** [[United States Declaration of Independence|Declaration of Independence]]
** [[Articles of Confederation]]
** [[United States Constitution|The Constitution of the United States of America]]
* [[Alexander Hamilton]], [[James Madison]], [[John Jay]]
** ''[[The Federalist Papers|The Federalist]]''
* [[John Stuart Mill]]
** ''[[On Liberty]]''
** ''[[Considerations on Representative Government]]''
** ''[[Utilitarianism (book)|Utilitarianism]]''

'''Volume 44'''
* [[James Boswell]]
** ''[[The Life of Samuel Johnson|The Life of Samuel Johnson, LL.D.]]''

'''Volume 45'''
* [[Antoine Laurent Lavoisier]]
** ''[[Traité Élémentaire de Chimie|Elements of Chemistry]]''
* [[Jean Baptiste Joseph Fourier]]
** ''Analytical Theory of Heat''
* [[Michael Faraday]]
** ''Experimental Researches in Electricity''

'''Volume 46'''
* [[Georg Wilhelm Friedrich Hegel]]
** ''[[The Philosophy of Right]]''
** ''[[Lectures on the Philosophy of History|The Philosophy of History]]''

'''Volume 47'''
* [[Johann Wolfgang von Goethe]]
** ''[[Goethe's Faust|Faust]]''

'''Volume 48'''
* [[Herman Melville]]
** ''[[Moby Dick|Moby Dick; or, The Whale]]''

'''Volume 49'''
* [[Charles Darwin]]
** ''[[On the Origin of Species|The Origin of Species by Means of Natural Selection]]''
** ''[[The Descent of Man, and Selection in Relation to Sex]]''

'''Volume 50'''
* [[Karl Marx]]
** ''[[Das Kapital|Capital]]''
* [[Karl Marx]] and [[Friedrich Engels]]
** ''[[The Communist Manifesto|Manifesto of the Communist Party]]''

'''Volume 51'''
* [[Count Leo Tolstoy]]
** ''[[War and Peace]]''

'''Volume 52'''
* [[Fyodor Mikhailovich Dostoevsky]]
** ''[[The Brothers Karamazov]]''

'''Volume 53'''
* [[William James]]
** ''[[The Principles of Psychology]]''

'''Volume 54'''
* [[Sigmund Freud]]
** ''The Origin and Development of Psycho-Analysis''
** ''Selected Papers on Hysteria''
** ''The Sexual Enlightenment of Children''
** ''The Future Prospects of Psycho-Analytic Therapy''
** ''Observations on "Wild" Psycho-Analysis''
** ''[[The Interpretation of Dreams]]''
** ''[[On Narcissism]]''
** ''Instincts and Their Vicissitudes''
** ''Repression''
** ''The Unconscious''
** ''[[Introduction to Psychoanalysis|A General Introduction to Psycho-Analysis]]''
** ''[[Beyond the Pleasure Principle]]''
** ''[[Group Psychology and the Analysis of the Ego]]''
** ''[[The Ego and the Id]]''
** ''Inhibitions, Symptoms, and Anxiety''
** ''[[Thoughts for the Times on War and Death]]''
** ''[[Civilization and Its Discontents]]''
** ''New Introductory Lectures on Psycho-Analysis''

==Second edition==
In 1990 a second edition of ''Great Books of the Western World'' was published, with updated translations and six more volumes of material covering the 20th century, an era of which the first edition was nearly devoid. A number of pre-20th century books were also added, and four were dropped: [[Apollonius of Perga|Apollonius']] ''[[On Conic Sections]]'', [[Laurence Sterne|Laurence Sterne's]] ''[[The Life and Opinions of Tristram Shandy, Gentleman|Tristram Shandy]]'', [[Henry Fielding|Henry Fielding's]] ''[[The History of Tom Jones, a Foundling|Tom Jones]]'', and [[Joseph Fourier|Joseph Fourier's]] ''[[Analytical Theory of Heat]]''. Adler later expressed regret about dropping ''[[On Conic Sections]]'' and ''[[The History of Tom Jones, a Foundling|Tom Jones]]''. Adler also voiced disagreement with the addition of [[Voltaire|Voltaire's]] ''[[Candide]]'', and said that the Syntopicon should have included references to the [[Koran]]. He addressed criticisms that the set was too heavily Western European and did not adequately represent women and minority authors.<ref name=Adler>{{cite web|url=http://books.mirror.org/gb.sel1990.html|title=Selecting works for the 1990 edition of Great Books of the Western World|author=Mortimer Adler|date=September 1997|accessdate=2007-05-29|publisher=Great Books Index|quote=We did not base our selections on an author's nationality, religion, politics, or field of study; nor on an author's race or gender. Great books were not chosen to make up quotas of any kind; there was no "affirmative action" in the process.}}</ref>

The pre-20th century books added (volume numbering is not strictly compatible with the first edition due to rearrangement of some books):

'''Volume 20'''
* [[John Calvin]]
** ''[[Institutes of the Christian Religion]]'' (Selections)

'''Volume 23'''
* [[Erasmus]]
** ''[[The Praise of Folly]]''

'''Volume 31'''
* [[Molière]]
** ''[[The School for Wives]]''
** ''The Critique of the School for Wives''
** ''[[Tartuffe]]''
** ''[[Dom Juan|Don Juan]]''
** ''[[The Miser]]''
** [[Le Bourgeois gentilhomme|''The Would-Be Gentleman'']]
** ''[[The Imaginary Invalid]]''
* [[Jean Racine]]
** ''[[Bérénice]]''
** ''[[Phèdre]]''

'''Volume 34'''
* [[Voltaire]]
** ''[[Candide]]''
* [[Denis Diderot]]
** ''[[Rameau's Nephew]]''

'''Volume 43'''
* [[Søren Kierkegaard]]
** ''[[Fear and Trembling]]''
* [[Friedrich Nietzsche]]
** ''[[Beyond Good and Evil]]''

'''Volume 44'''
* [[Alexis de Tocqueville]]
** ''[[Democracy in America]]''

'''Volume 45'''
* [[Honoré de Balzac]]
** ''[[Cousin Bette]]''

'''Volume 46'''
* [[Jane Austen]]
** ''[[Emma (novel)|Emma]]''
* [[George Eliot]]
** ''[[Middlemarch]]''

'''Volume 47'''
* [[Charles Dickens]]
** ''[[Little Dorrit]]''

'''Volume 48'''
* [[Mark Twain]]
** ''[[Adventures of Huckleberry Finn|Huckleberry Finn]]''

'''Volume 52'''
* [[Henrik Ibsen]]
** ''[[A Doll's House]]''
** ''[[The Wild Duck]]''
** ''[[Hedda Gabler]]''
** ''[[The Master Builder]]''

The six volumes of 20th century material consisted of the following:

'''Volume 55'''
* [[William James]]
** ''[[Pragmatism]]''
* [[Henri Bergson]]
** "[[Introduction to Metaphysics (Bergson)|An Introduction to Metaphysics]]"
* [[John Dewey]]
** ''[[Experience and Education (book)|Experience and Education]]''
* [[Alfred North Whitehead]]
** ''Science and the Modern World''
* [[Bertrand Russell]]
** ''[[The Problems of Philosophy]]''
* [[Martin Heidegger]]
** ''What Is Metaphysics?''
* [[Ludwig Wittgenstein]]
** ''[[Philosophical Investigations]]''
* [[Karl Barth]]
** ''The Word of God and the Word of Man''

'''Volume 56'''
* [[Henri Poincaré]]
** ''Science and Hypothesis''
* [[Max Planck]]
** ''Scientific Autobiography and Other Papers''
* [[Alfred North Whitehead]]
** ''An Introduction to Mathematics''
* [[Albert Einstein]]
** ''Relativity: The Special and the General Theory''
* [[Arthur Eddington]]
** ''The Expanding Universe''
* [[Niels Bohr]]
** ''Atomic Theory and the Description of Nature'' (selections)
** ''Discussion with Einstein on Epistemology''
* [[G. H. Hardy]]
** ''[[A Mathematician's Apology]]''
* [[Werner Heisenberg]]
** ''Physics and Philosophy''
* [[Erwin Schrödinger]]
** ''[[What is Life? (Schrödinger)|What Is Life?]]''
* [[Theodosius Dobzhansky]]
** ''[[Genetics and the Origin of Species]]''
* [[C. H. Waddington]]
** ''The Nature of Life''

'''Volume 57'''
* [[Thorstein Veblen]]
** ''[[The Theory of the Leisure Class]]''
* [[R. H. Tawney]]
** ''The Acquisitive Society''
* [[John Maynard Keynes]]
** ''[[The General Theory of Employment, Interest and Money]]''

'''Volume 58'''
* [[Sir James George Frazer]]
** ''[[The Golden Bough]]'' (selections)
* [[Max Weber]]
** ''Essays in Sociology'' (selections)
* [[Johan Huizinga]]
** ''[[The Autumn of the Middle Ages]]''
* [[Claude Lévi-Strauss]]
** ''Structural Anthropology'' (selections)

'''Volume 59'''
* [[Henry James]]
** ''[[The Beast in the Jungle]]''
* [[George Bernard Shaw]]
** ''[[Saint Joan (play)|Saint Joan]]''
* [[Joseph Conrad]]
** ''[[Heart of Darkness]]''
* [[Anton Chekhov]]
** ''[[Uncle Vanya]]''
* [[Luigi Pirandello]]
** ''[[Six Characters in Search of an Author]]''
* [[Marcel Proust]]
** ''[[In Search of Lost Time|Remembrance of Things Past]]'': "[[Swann in Love]]"
* [[Willa Cather]]
** ''[[A Lost Lady]]''
* [[Thomas Mann]]
** ''[[Death in Venice]]''
* [[James Joyce]]
** ''[[A Portrait of the Artist as a Young Man]]''

'''Volume 60'''
* [[Virginia Woolf]]
** ''[[To the Lighthouse]]''
* [[Franz Kafka]]
** ''[[The Metamorphosis]]''
* [[D. H. Lawrence]]
** ''[[The Prussian Officer]]''
* [[T. S. Eliot]]
** ''[[The Waste Land]]''
* [[Eugene O'Neill]]
** ''[[Mourning Becomes Electra]]''
* [[F. Scott Fitzgerald]]
** ''[[The Great Gatsby]]''
* [[William Faulkner]]
** ''[[A Rose for Emily]]''
* [[Bertolt Brecht]]
** ''[[Mother Courage and Her Children]]''
* [[Ernest Hemingway]]
** ''[[The Short Happy Life of Francis Macomber]]''
* [[George Orwell]]
** ''[[Animal Farm]]''
* [[Samuel Beckett]]
** ''[[Waiting for Godot]]''

==Criticisms and responses==

===Criticisms of the authors selected===
Criticism has attended ''Great Books of the Western World'' since publication. The stress Hutchins placed on the monumental importance of these works was an easy target for those who dismissed the project as a celebration of dead European males, ignoring contributions of women and non-European authors.<ref>{{cite web|url=http://www.encyclopedia.com/doc/1P2-4603568.html|author=Sabrina Walters|title=Great Books won Adler fame, scorn|publisher=''[[Chicago Sun-Times]]''|date=2001-07-01|accessdate=2007-07-01}}</ref><ref>{{cite web|url=http://findarticles.com/p/articles/mi_qn4155/is_20010703/ai_n13917760|author=Peter Temes|title=Death of a Great Reader and Philosopher|publisher=''[[Chicago Sun-Times]]''|date=2001-07-03|accessdate=2007-07-11 |archiveurl = http://web.archive.org/web/20071104012348/http://findarticles.com/p/articles/mi_qn4155/is_20010703/ai_n13917760  |archivedate = 2007-11-04}}</ref> The criticism swelled in tandem with the [[feminist]] and [[civil rights movement]]s.<ref>{{cite journal|url=http://www.greatbooksacademy.org/newsroom/what-happened-to-the-great-ideas-by-john-berlau/|title=What Happened to the Great Ideas? – Mortimer J. Adler's Great Books programs|author=[[John Berlau]]|date=August 2001|accessdate=March 2014|work=[[Insight Magazine]] Insight on the News|volume=17 |issue =32|page=16|quote=Harvard University's [[Henry Louis Gates]] blasted the Great Books for showing 'profound disrespect for the intellectual capacities of people of color – red, brown or yellow.'}}</ref>

In his ''[[Europe: A History]]'', [[Norman Davies]] criticizes the compilation for overrepresenting selected parts of the western world, especially [[UK|Britain]] and the U.S., while ignoring the other, particularly [[Central Europe|Central]] and [[Eastern Europe]]. According to his calculation, in 151 authors included in both editions, there are 49 English or American authors, 27 Frenchmen, 20 Germans, 15 ancient Greeks, 9 ancient Romans, 6 Russians, 4 Scandinavians, 3 Spaniards, 3 Italians, 3 Irishmen, 3 Scots, and 3 Eastern Europeans. Prejudices and preferences, he concludes, are self-evident.

In response, such criticisms have been derided as ''[[ad hominem]]'' and biased in themselves. The counter-argument maintains that such criticisms discount the importance of books solely because of generic, imprecise and possibly irrelevant characteristics of the books' authors, rather than because of the content of the books themselves.<ref name=Adler /> In France there appeared several criticisms arguing that writers included in the list such as [[John Milton|Milton]], [[William Harvey|Harvey]], [[William Gilbert (astronomer)|Gilbert]] or [[Herman Melville|Melville]] weren't universally as relevant as some other writers such as [[John Calvin]] and [[Voltaire]], who were initially excluded; also, that it excluded many non-British or US authors from the early 20th century who were better known to French readers, such as [[Robert Musil|Musil]], [[Joseph Roth|Roth]] or [[Stefan Zweig|Zweig]].{{Citation needed|date=September 2011}}

===Criticisms of the works selected===
Others thought that while the selected authors were worthy, too much emphasis was placed on the complete works of a single author rather than a wider selection of authors and representative works (for instance, all of [[Shakespeare]]'s plays are included). The second edition of the set already contained 130 authors and 517 individual works. The editors point out that the guides to additional reading for each topic in the ''Syntopicon'' refer the interested reader to many more authors.<ref>{{cite book|author=Mortimer J. Adler|title=The Syntopicon: II|edition=2nd edition|series=Great Books of the Western World, vol. 1-2|publisher=Encyclopædia Britannica, Inc.|year=1990|isbn=0-85229-531-6|pages=909–996|chapter=Bibliography of Additional Readings}}</ref>

===Criticisms of difficulty===
The scientific and mathematical selections also came under criticism for being incomprehensible to the average reader, especially with the absence of any sort of critical apparatus. The second edition did drop two scientific works, by [[Apollonius of Perga|Apollonius]] and [[Joseph Fourier|Fourier]], in part because of their perceived difficulty for the average reader. Nevertheless, the editors steadfastly maintain that average readers are capable of understanding far more than the critics deem possible. Robert Hutchins stated this view in the introduction to the first edition:

:Because the great bulk of mankind have never had the chance to get a liberal education, it cannot be "proved" that they can get it. Neither can it be "proved" that they cannot. The statement of the ideal, however, is of value in indicating the direction that education should take.<ref>{{cite book|author=Robert M. Hutchins|title=The Great Conversation|year=1952|publisher=Encyclopædia Britannica, Inc.|chapter=Chapter VI: Education for All|page=44}}</ref>

===Criticisms of the set's rationale===
Since the great majority of the works were still in print, one critic noted that the company could have saved two million dollars and simply written a list. Encyclopædia Britannica's aggressive promotion produced solid sales. Dense formatting also did not help readability.<ref>{{cite web|url=http://www.writing.upenn.edu/~afilreis/50s/macdonald-great-books.html|title=The Book-of-the-Millennium Club|author=Dwight Macdonald|date=1952-11-29 with later appendix|accessdate=2007-05-29|publisher=''[[The New Yorker]]''|quote=I also wonder how many of the over 100,000 customers who have by now caved in under the pressure of Mr. Harden and his banner-bearing colleagues are doing much browsing in these upland pastures?}}</ref>

The second edition selected translations that were generally considered an improvement, though the cramped typography remained. Through reading plans and the ''Syntopicon'', the editors have attempted to guide readers through the set.<ref>{{cite book|author=Mortimer J. Adler|title=The Great Conversation|edition=2nd edition|publisher=Encyclopædia Britannica, Inc.|year=1990|isbn=0-85229-531-6|pages=33–34 for discussion of new translations, pp.74–98 for reading plans and guides}}</ref>

===Response to criticisms===

The editors respond that the set contains wide-ranging debates representing many viewpoints on significant issues, not a monolithic school of thought. Mortimer Adler argued in the introduction to the second edition:

:Presenting a wide variety and divergence of views or opinions, among which there is likely to be some truth but also much more error, the ''Syntopicon'' [and by extension the larger set itself] invites readers to think for themselves and make up their own minds on every topic under consideration.<ref>{{cite book|author=Mortimer J. Adler|year=1990|title=The Great Conversation|edition=2nd edition|publisher=Encyclopædia Britannica, Inc.|isbn=0-85229-531-6|chapter=Section 1: The Great Books and the Great Ideas|page=27}}</ref>

==See also==
* [[John Erskine (educator)|John Erskine]]
* [[Charles William Eliot|Charles W. Eliot]]
* [[Robert Maynard Hutchins]]
* [[Mortimer J. Adler]]
* [[Educational perennialism]]
* [[Western canon]]
* [[Great Books]]
* [[Harvard Classics]]
* [[Liberal arts]]

==References==
{{Reflist}}

==External links==
* [http://britannicashop.britannica.co.uk/epages/Store.sf/Shops/Britannicashop/Products/ENC_BOOK_0123.html Official Britannica web page for the Great Books]
* [http://www.thegreatideas.org/index.html Center for the Study of the Great Ideas] Mortimer Adler web pages with extensive discussion of the Great Books
* [http://readingthegreat.com/ The Great Conversation: Confessions of an Eavesdropper] – a blog detailing the experiences of reading through the great books of the Western World.
* [http://www.greaterbooks.com Greater Books] - a site documenting lists of "great books," classics, canons, including the Great Books of the Western World

[[Category:Series of books]]
[[Category:1952]]
[[Category:Encyclopædia Britannica]]
[[Category:Great Books]]

Cornelis Zwikker

2015-01-12T18:58:21Z

MathKeduor7: seiner berühmtesten Bücher

'''Cornelis Zwikker''' (* [[19. August]] [[1900]] in [[Zaandam]]; † [[20. April]] [[1985]] in [[Zwijndrecht (Niederlande)|Zwijndrecht]]) war ein niederländischer Physiker.

Der Sohn von Klaartje und Klaas Zwikker studierte ab 1918 an der [[Universität von Amsterdam]] Chemie und Naturkunde. Von 1923 bis 1929 war er wissenschaftlicher Assistent an der NV Philips in Eindhoven. 1925 wurde er mit einer Arbeit zu [[Wolfram]]-Eigenschaften bei hohen Temperaturen promoviert. 1929 wurde er an die [[Technische Universität Delft|Technische Hochschule Delft]] zum Professor für theoretische Physik berufen. Er forschte zur akustischen Gestaltung von Gebäuden. 1945 nahm er bei [[Philips]] die Position eines Technischen Direktors ein. 1956–1970 war er Professor in [[Technische Universität Eindhoven|Eindhoven]].

== Schriften ==
* mit Cornelis Willem Kosten, ''Sound Absorbing Materials'' (1949)
* ''The Advanced Geometry of Plane Curves and Their Applications'' (1963)

== Weblinks ==
* [http://www.webcitation.org/6P3TJsfT7 Kurzbiografie bei hogenda.nl]
* {{Worldcat id|LCCN=n/86/863764}}

{{Normdaten|TYP=p|GND=|LCCN=n/86/863764|VIAF=47130449}}

{{SORTIERUNG:Zwikker, Cornelis}}
[[Kategorie:Physiker (20. Jahrhundert)]]
[[Kategorie:Hochschullehrer (Delft)]]
[[Kategorie:Hochschullehrer (Eindhoven)]]
[[Kategorie:Niederländer]]
[[Kategorie:Geboren 1900]]
[[Kategorie:Gestorben 1985]]
[[Kategorie:Mann]]

{{Personendaten
|NAME=Zwikker, Cornelis
|ALTERNATIVNAMEN=
|KURZBESCHREIBUNG=niederländischer Physiker
|GEBURTSDATUM=19. August 1900
|GEBURTSORT=[[Zaandam]]
|STERBEDATUM=20. April 1985
|STERBEORT=[[Zwijndrecht (Niederlande)]]
}}

Fernando Codá Marques

2014-08-05T13:30:33Z

MathKeduor7:

'''Fernando Codá Marques''' (* [[8. Oktober]] [[1979]]) ist ein brasilianischer Mathematiker, der sich mit Differentialgeometrie mit Anwendungen in geometrischer Theorie partieller Differentialgleichungen und Allgemeiner Relativitätstheorie befasst. Er ist am [[Instituto de Matemática Pura e Aplicada]] (IMPA) in [[Rio de Janeiro]].

Marques studierte Mathematik an der [[Universidade Federal de Alagoas Maceio]] in [[Alagoas]] und am IMPA mit dem Diplom-Abschluss 1999 bei Bruno César Scardua und wurde 2003 an der [[Cornell University]] bei [[José F. Escobar]] promoviert (''Existence and compactness theorems on conformal deformations of metrics''). Danach war er Assistant Professor am IMPA mit einer vollen Professur seit 2010. Als [[Post-Doktorand]] war er 2005/6 an der [[Stanford University]].

2012 kündigte er den Beweis der [[Willmore-Energie|Willmore-Vermutung]] mit [[André Neves]] an<ref>[http://arxiv.org/abs/1202.6036 Min-Max Theory and the Willmore Conjecture]. Erscheint in Annals of Mathematics.</ref><ref>Neves (Imperial College) erhielt dafür 2013 den [[Whitehead-Preis]]</ref>. [[Thomas Willmore]] vermutete, dass die ''Willmore-Energie'' für im dreidimensionalen euklidischen Raum [[Immersion (Mathematik)|immersierte]] Tori größer oder gleich <math>2 {\pi}^2</math> ist. Neves und Marques bewiesen das mit der Min-Max-Theorie von [[Minimalfläche]]n. Teilweise in Zusammenarbeit mit [[Simon Brendle]] befasst er sich mit Problemen im Umkreis des (inzwischen gelösten) Yamabe-Problems (benannt nach [[Hidehiko Yamabe]]) und wandte [[Grigori Perelman|Perelman]]s und [[Richard Hamilton|Hamilton]]s Theorie des Ricci-Flusses an, zum Beispiel im Beweis, dass der Modulraum der Metriken mit positiver Skalarkrümmung auf kompakten orientierbaren 3-Mannigfaltigkeiten wegzusammenhängend ist.

2008 war er am [[Institute for Advanced Study]] und er war Gastwissenschaftler in Stanford und an der [[Princeton University]], am Institut Fourier in Grenoble, am [[Institut Henri Poincaré]], an der [[Ecole Polytechnique]] und der Universität Paris-Ost.

2012 erhielt er den Preis der ''Union Matematica de America Latina y el Caribe'' (UMALCA), den Mathematikpreis der ''[[Third World Academy of Sciences]]'' (TWAS) und den Ramanujan-Preis des [[International Centre for Theoretical Physics|ICTP]]. 2009 wurde er Mitglied der Brasilianischen Akademie der Wissenschaften.

Er war Invited Speaker auf dem [[Internationaler Mathematikerkongress|Internationalen Mathematikerkongress]] 2010 in Hyderabad (''Scalar curvature, conformal geometry and the Ricci flow with surgery'') und wurde als Plenarsprecher für den ICM 2014 in [[Seoul]] ausgewählt.
==Schriften==

*mit Simon Brendle, André Neves ''Deformations of the hemisphere that increase scalar curvature'', Inventiones Mathematicae 185, 2011, 175-197(Widerlegung der Vermutung von Min-Oo in 3 und mehr Dimensionen), [http://arxiv.org/abs/1004.3088/ Arxiv]
*mit Simon Brendle ''Recent progress on the Yamabe Problem'', in ''Surveys in Geometric Analysis and Relativity'', 2011 (Richard Schoen zum 60. Geburtstag), [http://arxiv.org/abs/1010.4960 Arxiv]
* ''Deforming three-manifolds with positive scalar curvature'', Annals of Mathematics 176, 2012, 815-863, [http://arxiv.org/abs/0907.2444 Arxiv]
* mit Neves: ''Min-max theory and the Willmore conjecture.'' Ann. of Math. (2) 179 (2014), no. 2, 683–782

==Weblinks==
*[http://w3.impa.br/~coda/ Homepage am IMPA]
==Einzelnachweise==
<references />
{{SORTIERUNG:Marques, Fernando Coda}}
[[Kategorie:Mathematiker (21. Jahrhundert)]]
[[Kategorie:Brasilianer]]
[[Kategorie:Geboren 1979]]
[[Kategorie:Mann]]

{{Personendaten
|NAME=Marques, Fernando Codà
|ALTERNATIVNAMEN=
|KURZBESCHREIBUNG=brasilianischer Mathematiker
|GEBURTSDATUM=8. Oktober 1979
|GEBURTSORT=
|STERBEDATUM=
|STERBEORT=
}}

Diskussion:Fernando Codá Marques

2014-08-05T13:29:37Z

MathKeduor7: Vinícius Machado Vogt verschob die Seite Diskussion:Fernando Codà Marques nach Diskussion:Fernando Codá Marques: Korrektur.

Ist Codà der zweite Vorname oder der erste Nachname?--[[Benutzer:Café Bene|Café Bene]] ([[Benutzer Diskussion:Café Bene|Diskussion]]) 18:45, 14. Mai 2014 (CEST)

Da er F. C. Marques zitiert wird (siehe [http://w3.impa.br/~coda/publicacoes.html Publikationsverz.]) nehme ich an ist Marques der Nachname.--[[Benutzer:Claude J|Claude J]] ([[Benutzer Diskussion:Claude J|Diskussion]]) 18:49, 14. Mai 2014 (CEST)

Fernando Codà Marques

2014-08-05T13:29:36Z

MathKeduor7: Vinícius Machado Vogt verschob die Seite Fernando Codà Marques nach Fernando Codá Marques: Korrektur.

#WEITERLEITUNG [[Fernando Codá Marques]]

Fernando Codá Marques

2014-08-05T13:29:36Z

MathKeduor7: Vinícius Machado Vogt verschob die Seite Fernando Codà Marques nach Fernando Codá Marques: Korrektur.

'''Fernando Codà Marques''' (* [[8. Oktober]] [[1979]]) ist ein brasilianischer Mathematiker, der sich mit Differentialgeometrie mit Anwendungen in geometrischer Theorie partieller Differentialgleichungen und Allgemeiner Relativitätstheorie befasst. Er ist am [[Instituto de Matemática Pura e Aplicada]] (IMPA) in [[Rio de Janeiro]].

Marques studierte Mathematik an der [[Universidade Federal de Alagoas Maceio]] in [[Alagoas]] und am IMPA mit dem Diplom-Abschluss 1999 bei Bruno César Scardua und wurde 2003 an der [[Cornell University]] bei [[José F. Escobar]] promoviert (''Existence and compactness theorems on conformal deformations of metrics''). Danach war er Assistant Professor am IMPA mit einer vollen Professur seit 2010. Als [[Post-Doktorand]] war er 2005/6 an der [[Stanford University]].

2012 kündigte er den Beweis der [[Willmore-Energie|Willmore-Vermutung]] mit [[André Neves]] an<ref>[http://arxiv.org/abs/1202.6036 Min-Max Theory and the Willmore Conjecture]. Erscheint in Annals of Mathematics.</ref><ref>Neves (Imperial College) erhielt dafür 2013 den [[Whitehead-Preis]]</ref>. [[Thomas Willmore]] vermutete, dass die ''Willmore-Energie'' für im dreidimensionalen euklidischen Raum [[Immersion (Mathematik)|immersierte]] Tori größer oder gleich <math>2 {\pi}^2</math> ist. Neves und Marques bewiesen das mit der Min-Max-Theorie von [[Minimalfläche]]n. Teilweise in Zusammenarbeit mit [[Simon Brendle]] befasst er sich mit Problemen im Umkreis des (inzwischen gelösten) Yamabe-Problems (benannt nach [[Hidehiko Yamabe]]) und wandte [[Grigori Perelman|Perelman]]s und [[Richard Hamilton|Hamilton]]s Theorie des Ricci-Flusses an, zum Beispiel im Beweis, dass der Modulraum der Metriken mit positiver Skalarkrümmung auf kompakten orientierbaren 3-Mannigfaltigkeiten wegzusammenhängend ist.

2008 war er am [[Institute for Advanced Study]] und er war Gastwissenschaftler in Stanford und an der [[Princeton University]], am Institut Fourier in Grenoble, am [[Institut Henri Poincaré]], an der [[Ecole Polytechnique]] und der Universität Paris-Ost.

2012 erhielt er den Preis der ''Union Matematica de America Latina y el Caribe'' (UMALCA), den Mathematikpreis der ''[[Third World Academy of Sciences]]'' (TWAS) und den Ramanujan-Preis des [[International Centre for Theoretical Physics|ICTP]]. 2009 wurde er Mitglied der Brasilianischen Akademie der Wissenschaften.

Er war Invited Speaker auf dem [[Internationaler Mathematikerkongress|Internationalen Mathematikerkongress]] 2010 in Hyderabad (''Scalar curvature, conformal geometry and the Ricci flow with surgery'') und wurde als Plenarsprecher für den ICM 2014 in [[Seoul]] ausgewählt.
==Schriften==

*mit Simon Brendle, André Neves ''Deformations of the hemisphere that increase scalar curvature'', Inventiones Mathematicae 185, 2011, 175-197(Widerlegung der Vermutung von Min-Oo in 3 und mehr Dimensionen), [http://arxiv.org/abs/1004.3088/ Arxiv]
*mit Simon Brendle ''Recent progress on the Yamabe Problem'', in ''Surveys in Geometric Analysis and Relativity'', 2011 (Richard Schoen zum 60. Geburtstag), [http://arxiv.org/abs/1010.4960 Arxiv]
* ''Deforming three-manifolds with positive scalar curvature'', Annals of Mathematics 176, 2012, 815-863, [http://arxiv.org/abs/0907.2444 Arxiv]
* mit Neves: ''Min-max theory and the Willmore conjecture.'' Ann. of Math. (2) 179 (2014), no. 2, 683–782

==Weblinks==
*[http://w3.impa.br/~coda/ Homepage am IMPA]
==Einzelnachweise==
<references />
{{SORTIERUNG:Marques, Fernando Coda}}
[[Kategorie:Mathematiker (21. Jahrhundert)]]
[[Kategorie:Brasilianer]]
[[Kategorie:Geboren 1979]]
[[Kategorie:Mann]]

{{Personendaten
|NAME=Marques, Fernando Codà
|ALTERNATIVNAMEN=
|KURZBESCHREIBUNG=brasilianischer Mathematiker
|GEBURTSDATUM=8. Oktober 1979
|GEBURTSORT=
|STERBEDATUM=
|STERBEORT=
}}

Toeplitz-Vermutung

2012-07-01T22:59:37Z

MathKeduor7:

[[Image:Inscribed square.svg|thumb|right|Example black dashed curve goes through corners of several blue squares.]]
The '''inscribed square problem''' is an unsolved question in [[geometry]]: ''Does every [[Jordan curve|plane simple closed curve]] contain all four vertices of some [[Square (geometry)|square]]?'' This is known to be true if the curve is [[convex set|convex]] or piecewise smooth and in other special cases. The problem was proposed by [[Otto Toeplitz]] in 1911. Some early positive results were obtained by Arnold Emch<ref>{{citation
| last = Emch | first = Arnold
| doi = 10.2307/2370541
| issue = 1
| journal = American Journal of Mathematics
| mr = 1506274
| pages = 6–18
| title = On some properties of the medians of closed continuous curves formed by analytic arcs
| volume = 38
| year = 1916}}.</ref> and [[Lev Schnirelmann]].<ref>{{citation
| last = Šnirel'man | first = L. G. | author-link = Lev Schnirelmann
| journal = Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk
| mr = 0012531
| pages = 34–44
| title = On certain geometrical properties of closed curves
| volume = 10
| year = 1944}}.</ref> As of 2007, the general case remains open.

== Overview ==

Let ''C'' be a [[Jordan curve]]. A [[polygon]] ''P'' is '''inscribed in ''C'' ''' if all vertices of ''P'' belong to ''C''. The '''inscribed square problem''' asks:

: ''Does every Jordan curve admit an inscribed square?''

It is ''not'' required that the vertices of the square appear along the curve in any particular order.

Some figures, such [[circle]]s and [[Square (geometry)|square]]s, admit infinitely many inscribed squares. If ''C'' is an [[obtuse triangle]] then it admits exactly one inscribed square.

The most encompassing result to date is due to Stromquist, who proved that every ''local monotone'' plane simple curve admits an inscribed square.<ref name="stromquist">{{citation
| last = Stromquist | first = Walter
| doi = 10.1112/S0025579300013061
| issue = 2
| journal = Mathematika
| mr = 1045781
| pages = 187–197
| title = Inscribed squares and square-like quadrilaterals in closed curves
| volume = 36
| year = 1989}}.</ref> The condition is that for any point ''p'', the curve ''C'' can be locally represented as a graph of a function ''y'' = ''f''(''x''). More precisely, for any point ''p'' on ''C'' there is a neighborhood ''U''(''p'') such that no chord of ''C'' in this neighborhood is parallel to a fixed direction ''n''(''p'') (the direction of the "''y''-axis"). Locally monotone curves includes all closed convex curves and all piecewise-''C''1 curves without cusps.

The affirmative answer is also known for centrally symmetric curves.<ref name="nielsen-wright">{{citation
| last1 = Nielsen | first1 = Mark J.
| last2 = Wright | first2 = S. E.
| doi = 10.1007/BF01263570
| issue = 3
| journal = Geometriae Dedicata
| mr = 1340790
| pages = 285–297
| title = Rectangles inscribed in symmetric continua
| volume = 56
| year = 1995}}.</ref>

== Variants and generalizations ==

One may ask whether other shapes can be inscribed into an arbitrary Jordan curve. It is known that for any triangle ''T'' and Jordan curve ''C'', there is a triangle similar to ''T'' and inscribed in ''C''.<ref>{{citation
| last = Meyerson | first = Mark D.
| issue = 1
| journal = Fundamenta Mathematicae
| mr = 600575
| pages = 1–9
| title = Equilateral triangles and continuous curves
| volume = 110
| year = 1980}}.</ref><ref>{{citation
| last1 = Kronheimer | first1 = E. H.
| last2 = Kronheimer | first2 = P. B. | author2-link = Peter B. Kronheimer
| doi = 10.1112/jlms/s2-24.1.182
| issue = 1
| journal = The Journal of the London Mathematical Society | series = Second Series
| mr = 623685
| pages = 182–192
| title = The tripos problem
| volume = 24
| year = 1981}}.</ref> Moreover, the set of the vertices of such triangles is [[dense set|dense]] in ''C''.<ref>{{citation
| last = Nielsen | first = Mark J.
| doi = 10.1007/BF00151519
| issue = 3
| journal = Geometriae Dedicata
| mr = 1181760
| pages = 291–297
| title = Triangles inscribed in simple closed curves
| volume = 43
| year = 1992}}.</ref> In particular, there is always an inscribed [[equilateral triangle]]. It is also known that any Jordan curve admits an inscribed [[rectangle]].

Some generalizations of the inscribed square problem consider inscribed polygons for curves and even more general [[continuum (topology)|continua]] in higher dimensional [[Euclidean space]]s. For example, Stromquist proved that every continuous closed curve ''C'' in '''R'''''n'' satisfying "Condition A" that no two chords of ''C'' in a suitable neighborhood of any point are perpendicular admits an inscribed quadrilateral with equal sides and equal diagonals.<ref name="stromquist"/> This class of curves includes all ''C''2 curves. Nielsen and Wright proved that any symmetric continuum ''K'' in '''R'''''n'' contains many inscribed rectangles.<ref name="nielsen-wright"/> H.W. Guggenheimer proved that every hypersurface ''C''3-[[diffeomorphic]] to the [[n-sphere|sphere]] ''S''''n''−1 contains 2''n'' vertices of a regular Euclidean [[n-cube|''n''-cube]].<ref>{{citation
| last = Guggenheimer | first = H.
| doi = 10.1007/BF02760036
| journal = Israel Journal of Mathematics
| mr = 0188898
| pages = 104–112
| title = Finite sets on curves and surfaces
| volume = 3
| year = 1965}}.</ref>

== References ==
{{reflist|colwidth=30em}}

== Additional reading ==
* [[Victor Klee]] and [[Stan Wagon]], ''Old and New Unsolved Problems in Plane Geometry and Number Theory'', The Dolciani Mathematical Expositions, Number 11, [[Mathematical Association of America]], 1991

== External links ==
* Mark J. Nielsen, [http://www.webpages.uidaho.edu/~markn/squares/ Figures Inscribed in Curves. A short tour of an old problem]
* [http://quomodocumque.wordpress.com/2007/08/31/inscribed-squares-denne-speaks/ Inscribed squares: Denne speaks] at Jordan Ellenberg's blog

[[Category:Curves]]
[[Category:Unsolved problems in mathematics]]

[[pt:Problema do quadrado inscrito]]