Difference between revisions of "Lambert quadrangle"
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A quadrangle with right angles at three of the vertices. It was considered by J.H. Lambert (1766) in attempts to prove Euclid's parallelism postulate (cf. [[Fifth postulate|Fifth postulate]]). Of the three possible assumptions about the size of the fourth angle, that it is a right, an obtuse or an acute angle, the first is equivalent to Euclid's postulate and the second leads to a contradiction with the other axioms and postulates of Euclid. As for the third, Lambert conjectured that it is satisfied on an imaginary sphere. | A quadrangle with right angles at three of the vertices. It was considered by J.H. Lambert (1766) in attempts to prove Euclid's parallelism postulate (cf. [[Fifth postulate|Fifth postulate]]). Of the three possible assumptions about the size of the fourth angle, that it is a right, an obtuse or an acute angle, the first is equivalent to Euclid's postulate and the second leads to a contradiction with the other axioms and postulates of Euclid. As for the third, Lambert conjectured that it is satisfied on an imaginary sphere. | ||
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> V.F. Kagan, "Foundations of geometry" , '''1''' , Moscow-Leningrad (1949) (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.V. Pogorelov, "Lectures on the foundations of geometry" , Noordhoff (1966) (Translated from Russian)</TD></TR> | + | <table> |
| − | + | <TR><TD valign="top">[1]</TD> <TD valign="top"> V.F. Kagan, "Foundations of geometry" , '''1''' , Moscow-Leningrad (1949) (In Russian)</TD></TR> | |
| − | + | <TR><TD valign="top">[2]</TD> <TD valign="top"> A.V. Pogorelov, "Lectures on the foundations of geometry" , Noordhoff (1966) (Translated from Russian)</TD></TR> | |
| − | + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Busemann, P.J. Kelly, "Projective geometry and projective metrics" , Acad. Press (1953)</TD></TR> | |
| − | + | <TR><TD valign="top">[a2]</TD> <TD valign="top"> H.S.M. Coxeter, "Introduction to geometry" , Wiley (1963) {{ZBL|0095.34502}}</TD></TR> | |
| − | + | <TR><TD valign="top">[a3]</TD> <TD valign="top"> M. Greenberg, "Euclidean and non-Euclidean geometries" , Freeman (1974)</TD></TR> | |
| − | + | <TR><TD valign="top">[a4]</TD> <TD valign="top"> N.W. [N.V. Efimov] Efimow, "Höhere Geometrie" , Deutsch. Verlag Wissenschaft. (1960) (Translated from Russian)</TD></TR> | |
| − | + | <TR><TD valign="top">[a5]</TD> <TD valign="top"> A.P. Norden, "Elementare Einführung in die Lobatschewskische Geometrie" , Deutsch. Verlag Wissenschaft. (1958) (Translated from Russian)</TD></TR> | |
| − | + | <TR><TD valign="top">[a6]</TD> <TD valign="top"> R. Bonola, "Non-Euclidean geometry" , Dover, reprint (1955) (Translated from Italian)</TD></TR> | |
| + | </table> | ||
Latest revision as of 07:34, 19 March 2023
2020 Mathematics Subject Classification: Primary: 51-03 Secondary: 01A50 [MSN][ZBL]
A quadrangle with right angles at three of the vertices. It was considered by J.H. Lambert (1766) in attempts to prove Euclid's parallelism postulate (cf. Fifth postulate). Of the three possible assumptions about the size of the fourth angle, that it is a right, an obtuse or an acute angle, the first is equivalent to Euclid's postulate and the second leads to a contradiction with the other axioms and postulates of Euclid. As for the third, Lambert conjectured that it is satisfied on an imaginary sphere.
References
| [1] | V.F. Kagan, "Foundations of geometry" , 1 , Moscow-Leningrad (1949) (In Russian) |
| [2] | A.V. Pogorelov, "Lectures on the foundations of geometry" , Noordhoff (1966) (Translated from Russian) |
| [a1] | H. Busemann, P.J. Kelly, "Projective geometry and projective metrics" , Acad. Press (1953) |
| [a2] | H.S.M. Coxeter, "Introduction to geometry" , Wiley (1963) Zbl 0095.34502 |
| [a3] | M. Greenberg, "Euclidean and non-Euclidean geometries" , Freeman (1974) |
| [a4] | N.W. [N.V. Efimov] Efimow, "Höhere Geometrie" , Deutsch. Verlag Wissenschaft. (1960) (Translated from Russian) |
| [a5] | A.P. Norden, "Elementare Einführung in die Lobatschewskische Geometrie" , Deutsch. Verlag Wissenschaft. (1958) (Translated from Russian) |
| [a6] | R. Bonola, "Non-Euclidean geometry" , Dover, reprint (1955) (Translated from Italian) |
How to Cite This Entry:
Lambert quadrangle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lambert_quadrangle&oldid=15714
Lambert quadrangle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lambert_quadrangle&oldid=15714
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article