Talk:Saccheri quadrilateral
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(redirect)
edit- Added title. —Nils von Barth (nbarth) (talk) 21:33, 19 April 2009 (UTC)
This new page is to direct article text links of "Saccheri quadrilateral" to the "Saccheri Quadrilateral" article.
Capitalization matters in Wiki article links... perhaps to be considered a Wikiflaw.
in case of obtuse angle
editThis article claims that in the case of obtuse angle, the quadrilateral leads to elliptical geometry, but both Morris Kline (Mathematical Thought From Ancient to Modern Times, chapter XXXVI) and H.S.M. Coxeter (Non-Euclidean Geometry, 1999, page 5, ISBN 978-0883855225) state (without proof) that this case leads to accept the fifth postulate. Which is correct?--Nickanc (talk) 22:10, 9 April 2012 (UTC)
- I can see why you are confused. The article is not being precise enough to clarify this. The assumption of the obtuse angle, in the presence of the first four Euclidean postulates, leads to a contradiction. Both Saccheri and Omar Khayyam were able to show this, so they talk about rejecting the assumption of the obtuse angle. When properly phrased this contradiction can be turned into a proof of the parallel postulate. This is what Kline and Coxeter are talking about. On the other hand, in elliptic geometry (in which you need to toss out both Euclid's second and fifth postulates) it can be shown that the summit angles of a Saccheri quadrilateral are obtuse. This is what the article is saying without mentioning the additional modifications needed to make the statement true. So, the answer to your question is, they both are ... but under different sets of assumptions. Bill Cherowitzo (talk) 06:43, 10 April 2012 (UTC)
Ibn Qurra
editAccording to Cooke's history of math text, Jeremy Gray calls this a Thabit quadrilateral and that it thus predates Khayyam by a few centuries. If so, perhaps some Wikipedia guru could fix that; I am not familiar enough with the editing protocol and syntax to do so myself without spending more time than I can devote to it. But it would be good to have the right people attributed to this concept. — Preceding unsigned comment added by 50.227.112.182 (talk) 13:58, 29 October 2019 (UTC)