2GA2 2000 (539.231)
Euclidean and Non-Euclidean Geometry
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Historically, plane and solid bodies were studied in the Babylonian, Egyptian,
Chinese and Indian cultures. Some of their properties depending on similarity
were deduced: for example, area being proportional to the square of the linear
dimension; volume being proportional to the cube of the linear dimension, and
other properties relevant to surveying and astronomy.
Formal methods of proof began with the Greeks around 600 BC with the idea of
starting from a limited number of `self-evident postulates' and deducing
everything else from these by explicit rules of logic. This culminated with
Euclid's `Elements' around 300 BC. Euclid's 13 books contain definitions and
postulates (or axioms) from which all of plane and solid geometry can be
deduced. They were intended to give a description of physical space (the `real
world').
Euclid's Five Postulates
1. Any two points determine a unique line containing them.
2. Any line segment may be extended.
3. Given a point P and a distance r, there is a circle
with centre P and radius r.
4. All right angles are equal.
5. If a line meets two other lines, and the two interior angles on
one side, a and b, satisfy a+ b < two right
angles, then the lines meet on that side.
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Euclid's
Elements online.
A very interesting website containing the full text of Elements
with illustrations written in Java. Worth having a look.
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Because the 5th postulate was less self-evident than the others, many
mathematicians from 300 BC to 1820 tried to show that it followed logically from
the other four postulates.
One way to attempt this is to assume that postulates 1-4, and the negation of 5
are true, and then try to deduce a contradiction. No one was able to do this,
but this method of attack lead to the discovery of properties of `Non-Euclidean
Geometry', that is, the geometry that is a consequence of postulates 1-4, and
the negation of postulate 5.
Finally, all around 1820, the Russian mathematician Lobachevsky, the Hungarian
mathematician Bolyai, and the German mathematician Gauss independently proved
that there could be no contradiction to `Non-Euclidean Geometry', because it is
actually true in some models of space.
So the question changed to:
Since Euclidean and Non-Euclidean Geometry are equally logically
correct, which is a model of the real world?
General Relativity provides the answer: locally, that is, in a limited
domain, Euclidean Geometry is correct, but the whole universe is Non-Euclidean.
C. E. Praeger & Csaba Schneider
Pictures are from The MacTutor History of Mathematics Archive at
St Andrews, Scotland.
File translated from
TEX
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TTH,
version 2.71.
On 12 Jun 2000, 10:43.
