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Difference between revisions of "Saccheri quadrangle"

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A quadrangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083010/s0830101.png" />, right-angled at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083010/s0830102.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083010/s0830103.png" /> and with equal sides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083010/s0830104.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083010/s0830105.png" />. It was discussed by G. Saccheri (1733) in attempts to prove Euclid's fifth postulate about parallel lines. Of the three possibilities regarding the angles at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083010/s0830106.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083010/s0830107.png" />: they are right angles, they are obtuse angles or they are acute angles, the first is equivalent to Euclid's [[Fifth postulate|fifth postulate]], and the second leads to spherical or elliptic geometry. As regards the third possibility, Saccheri made the erroneous deduction that it also contradicts the other axioms and postulates of Euclid.
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{{TEX|done}}{{MSC|51M10|01A50}}
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====References====
 
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<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,  "Foundations of geometry" , Noordhoff  (1966)  (Translated from Russian)</TD></TR></table>
 
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====Comments====
 
  
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A quadrangle $ABCD$, right-angled at $A$ and $B$ and with equal sides $AD$ and $BC$. It was discussed by G. Saccheri (1733) in attempts to prove Euclid's [[fifth postulate]] about parallel lines. Of the three possibilities regarding the angles at $C$ and $D$: they are right angles, they are obtuse angles or they are acute angles, the first is equivalent to the fifth postulate, and the second leads to [[Spherical geometry|spherical]] or [[elliptic geometry]]. As regards the third possibility, Saccheri made the erroneous deduction that it also contradicts the other axioms and postulates of Euclid.
  
 
====References====
 
====References====
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Bonola,   "Non-Euclidean geometry" , Dover, reprint  (1955)  pp. 23  (Translated from Italian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.S.M. Coxeter,   "Non-Euclidean geometry" , Univ. Toronto Press  (1965)  pp. 5, 190</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  N.V. Efimov,   "Higher geometry" , MIR  (1980)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  K. Borsuk,   W. Szmielew,   "Foundations of geometry" , North-Holland  (1960)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  V.F. Kagan,  "Foundations of geometry" , '''1''' , Moscow-Leningrad  (1949)  (In Russian)</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  A.V. Pogorelov, "Foundations of geometry" , Noordhoff  (1966)  (Translated from Russian)</TD></TR>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Bonola, "Non-Euclidean geometry" , Dover, reprint  (1955)  pp. 23  (Translated from Italian)</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  H.S.M. Coxeter, "Non-Euclidean geometry" , Univ. Toronto Press  (1965)  pp. 5, 190</TD></TR>
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<TR><TD valign="top">[a3]</TD> <TD valign="top">  N.V. Efimov, "Higher geometry" , MIR  (1980)  (Translated from Russian)</TD></TR>
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<TR><TD valign="top">[a4]</TD> <TD valign="top">  K. Borsuk, W. Szmielew, "Foundations of geometry" , North-Holland  (1960)</TD></TR>
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</table>

Latest revision as of 14:10, 15 April 2023

2020 Mathematics Subject Classification: Primary: 51M10 Secondary: 01A50 [MSN][ZBL]

A quadrangle $ABCD$, right-angled at $A$ and $B$ and with equal sides $AD$ and $BC$. It was discussed by G. Saccheri (1733) in attempts to prove Euclid's fifth postulate about parallel lines. Of the three possibilities regarding the angles at $C$ and $D$: they are right angles, they are obtuse angles or they are acute angles, the first is equivalent to the fifth postulate, and the second leads to spherical or elliptic geometry. As regards the third possibility, Saccheri made the erroneous deduction that it also contradicts the other axioms and postulates of Euclid.

References

[1] V.F. Kagan, "Foundations of geometry" , 1 , Moscow-Leningrad (1949) (In Russian)
[2] A.V. Pogorelov, "Foundations of geometry" , Noordhoff (1966) (Translated from Russian)
[a1] R. Bonola, "Non-Euclidean geometry" , Dover, reprint (1955) pp. 23 (Translated from Italian)
[a2] H.S.M. Coxeter, "Non-Euclidean geometry" , Univ. Toronto Press (1965) pp. 5, 190
[a3] N.V. Efimov, "Higher geometry" , MIR (1980) (Translated from Russian)
[a4] K. Borsuk, W. Szmielew, "Foundations of geometry" , North-Holland (1960)
How to Cite This Entry:
Saccheri quadrangle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Saccheri_quadrangle&oldid=15635
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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