Talk:Euclidean geometry: Difference between revisions
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Probably also the concept of distance? (Somewhat implicitly, I guess, in the form of |AB|=|CD|?) [[User:Boris Tsirelson|Boris Tsirelson]] 19:12, 28 March 2010 (UTC) | Probably also the concept of distance? (Somewhat implicitly, I guess, in the form of |AB|=|CD|?) [[User:Boris Tsirelson|Boris Tsirelson]] 19:12, 28 March 2010 (UTC) | ||
== History == | |||
The following paragraph (by Paul Wormer), moved from "Plane", could be used in the history of Euclidean geometry. [[User:Boris Tsirelson|Boris Tsirelson]] 07:24, 30 July 2010 (UTC) | |||
In 1899 [[David Hilbert]] published his seminal book ''Grundlagen der Geometrie'' [Foundations of Geometry]<noinclude><ref name="Hilbert"> | |||
D. Hilbert, ''Grundlagen der Geometrie'', B. G. Teubner, Leipzig (1899) [http://www.archive.org/stream/grunddergeovon00hilbrich#page/n9/mode/2up 2nd German edition]</ref></noinclude> in which he re-investigated and rephrased Euclid's two-millennia-old axioms and propositions. Hilbert begins with listing undefined concepts, among which are "point", "line", and "plane". In terms of these undefined concepts Hilbert formulates sets of axioms. The first axiom regarding the plane is axiom '''I4''': Three points ''A'', ''B'', ''C'' that are not on one and the same line determine always a plane α. He adds that this is expressed as "''A'', ''B'', and ''C'' lie in α", or "''A'', ''B'', and ''C'' are points of α". His axiom '''I5''' is a subtle extension of I4: Any three points in plane α that are not on one line determine plane α. | |||
Revision as of 02:24, 30 July 2010
Distance
Probably also the concept of distance? (Somewhat implicitly, I guess, in the form of |AB|=|CD|?) Boris Tsirelson 19:12, 28 March 2010 (UTC)
History
The following paragraph (by Paul Wormer), moved from "Plane", could be used in the history of Euclidean geometry. Boris Tsirelson 07:24, 30 July 2010 (UTC)
In 1899 David Hilbert published his seminal book Grundlagen der Geometrie [Foundations of Geometry][1] in which he re-investigated and rephrased Euclid's two-millennia-old axioms and propositions. Hilbert begins with listing undefined concepts, among which are "point", "line", and "plane". In terms of these undefined concepts Hilbert formulates sets of axioms. The first axiom regarding the plane is axiom I4: Three points A, B, C that are not on one and the same line determine always a plane α. He adds that this is expressed as "A, B, and C lie in α", or "A, B, and C are points of α". His axiom I5 is a subtle extension of I4: Any three points in plane α that are not on one line determine plane α.
- ↑ D. Hilbert, Grundlagen der Geometrie, B. G. Teubner, Leipzig (1899) 2nd German edition