Gauß Und Die Anfänge Der Nicht-Euklidischen Geometrie

Name: Gauß Und Die Anfänge Der Nicht-Euklidischen Geometrie
Brand: Springer
SKU: 9783211958223
Price: 3336.0 INR
Availability: InStock

by H. Reichardt , J. Bolyai , N. I. Lobatschewski

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ISBN13: 9783211958223
Binding: Paperback
Subject: N/A
Publisher: Springer
Publisher Imprint: Springer
Publication Date: February 24, 1986
Pages: 250
Original Price: EUR 45.41
Language: German
Edition: N/A
Item Weight: 304 grams
BISAC Subject(s): Geometry / General

About The Book

von RIEMANNS Habilitationsvortrag (aus den 1876 bei Teubner erschienenen "Gesam- melten Mathematischen Werken" [11]), so daB wir hier darauf verzichten konnen. Uber die philosophischen Betrachtungen, die im Zusammenhang mit der Entwick- lung der Theorie der Parallelen und der nicht-euklidischen Geometrie angestellt wor- den sind, ist so viel geschrieben worden (schon GAUSS hat Bemerkungen dazu ge- macht; siehe etwa [35, S. 27/28; vgl. S. 33/34 dieses Bandes]), daB es unmoglich ist, im Rahmen dieses Buches darauf einzugehen. Zwei der Hauptfragen, nlimlich wie weit die euklidische oder die nicht-euklidische Geometrie unsere rliumliche Situation er- fassen konnen und wie es mit der inneren Widerspruchsfreiheit der nicht-euklidischen Geometrie steht, sind im Laufe meines Textes [35] immer wieder behandelt worden, so daB diese Betrachtungen hier nicht erweitert werden. Die Vorgehensweisen von GAUSS, BOLYAI und LoBATSCHEWSKI einerseits und KLEIN andererseits waren einander entgegengesetzt. Die ersteren gingen rein hypothetisch vor: Sie untersuchten die Frage, wie eine Geometrie aussehen miisse, in der das Paral- lelenaxiom nicht gelte, setzten also voraus, daB es eine solche Geometrie gibt, und muBten damit rechnen, daB bei nOlh weitergehenden Untersuchungen Widerspruche auftauchen wiirden. Sie zeigten also: Es gibt im wesentlichen hiichstens eine solche Geometrie. KLEIN dagegen gab ein konkretes Beispiel fUr eine solche Geometrie an, indem er die auf der projektiven MaBbestimmung beruhende Cayleysche Geometrie als Modell fUr eine nicht-euklidische Geometrie erkannte. Da dieses Modell auf der projektiven Geometrie beruhte, die man als widerspruchsfrei ansieht, hatte er damit ein Modell fUr die nicht-euklidische Geometrie angegeben.

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