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On the hypothesis of the obtuse angle

Published online by Cambridge University Press:  24 October 2008

M. J. M. Hill
M. J. M. Hill
Affiliation:
Peterhouse
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Extract

Engel and Stäckel in their Theorie der Parallellinien von Euklid bis auf Gauss (1895) call attention to the fact that Saccheri and Lambert have used in their argument the property of the triangle that the external angle is greater than either of the interior and opposite angles (Euc. I. 16), which is not true for triangles of all magnitudes when the hypothesis of the obtuse angle holds good. They do not show how starting from the point of view of Saccheri and Lambert the proofs should be amended so that they would be valid in the Elliptic Plane. Bonola (Non-Euclidean Geometry, translated by Carslaw, 1912, p. 31) in his account of the subject also uses Euc. I. 16. It is the object of this paper to obtain the main results proved by Saccheri and Lambert when the hypothesis of the obtuse angle holds without using Euc. I. 16. It is not suggested that this is a convenient method of procedure but it may perhaps have some historical interest. The key to the procedure here adopted is found in the reversal of the Euclidean order in a certain group of propositions, viz. I. 16, 18, 19, 20.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 23 , Issue 1 , February 1926 , pp. 2 - 18
Copyright
Copyright © Cambridge Philosophical Society 1926

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References

* Pp. 52, 144 and elsewhere.Google Scholar

† This is the only kind of right-angled triangle which Saccheri considered.Google Scholar

* The proofs of the theorems of this group are valid whether Euclid's Postulate of Parallels holds good or not.Google Scholar

* A similar demonstration would hold good in Elliptic Geometry if the angles BAC, B′A′C′ were equal and obtuse.Google Scholar

* The only other proof of this theorem, known to me, which is valid for triangles of any magnitude in the Elliptic Plane, will be found in Coolidge's Non-Euclidean Geometry, p. 35, Theorem 28. It involves an axiom of continuity, an infinite limiting process, and a reductio ad absurdum.Google Scholar

* [In Elliptic Geometry this involves the sides of this triangle, each being less than L.]Google Scholar