Re: for the classics profs

From: Stephen C. Carlson (scarlson@mindspring.com)
Date: Fri Jan 23 1998 - 08:15:06 EST


At 10:16 1/22/98 +0000, Brian E. Wilson wrote:
>Amazingly, as I discovered decades later, some of Euclid's theorems were
>not provable from Euclid's axioms. For instance the theorem that the sum
>of the angles of a triangle is two right angles cannot be demonstrated
>from Euclid's axioms. In traditional Euclidian Geometry, QED would not
>be true, in its Latin meaning, at the end of this theorem. Euclid was
>wrong at this point.

I believe that all of Eucicd's theorems are provable from Euclid's five
axioms (also called postulates). While most of the theorems can be
proved using only the first four postulates, some of the theorems, which
include the one cited about the sum of the interior angles of a triangle,
require the application of the fifth postulate.

For those who memory of geometry is a bit fuzzy, this is the text of
the fifth postulate: KAI EAN EIS DUO EUQEIAS EMPIPTOUSA TAS ENTOS
KAI EPI TA AUTA MERH GWNIAS DUO ORQWN ELASSONAS POIHi, EKBALLOMENAS
TAS DUO EUQEIAS EP' APEIRON SUMPIPTEIN, EF' hA MERH EISIN hAI TWN
DUO ORQWN ELASSONES. A modern equivalent to this postulate is that,
through a point not on a given line, there passes exactly one parallel
to the line. Concern about the fifth postulate has lead to mathematical
anxiety over the centuries and, ultimately, to the establishment of
non-Euclidean geometries, which postulate either no parallels (Riemann)
or a plurality of parallels (Lobaschevsky).

For one web page on the topic, see
        http://sunset.backbone.olemiss.edu/~rpagejr/euclid.html

Stephen Carlson
(Hoping that the marginal relevence of this message is attenuated by
the quotation of actual Greek text)

--
Stephen C. Carlson                   : Poetry speaks of aspirations,
scarlson@mindspring.com              : and songs chant the words.
http://www.mindspring.com/~scarlson/ :               -- Shujing 2.35


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