From: Allan Adler <ara@nestle.ai.mit.edu>
Subject: certain axioms for geometry
Date: 16 Sep 2000 09:30:12 -0500
Newsgroups: sci.math.research
Summary: [missing]

At the end of Krause's book Taxicab Geometry, he gives some axioms for
Euclidean geometry. The axioms involve a metric and a function which measures
angles and he explicitly considers them to be part of a modern formulation
of geometry. The axioms seem to be somewhat in the spirit of Hilbert's axioms
but are also somewhat different. One correspondent suggested that they might
be due to Birkhoff around 1940. I looked up some work of Birkhoff from that
time and he does have a metric approach to geometry in which he characterizes
elliptic, euclidean and hyperbolic geometries by simple metric properties,
but I didn't see a place for an angle measurement function in his system.

In some treatments of geometry, for example that of Szmielew and Borsuk, there
is some discussion of angle measurement but it isn't really axiomatized and
they seem to determine a canonical one uniquely from metric properties.
I think that in "length spaces" there is also a definition of angle in terms
of metric properties. But I think there is really a more general set of
axioms that people work with that gives a more independent status to angle
measurement. The reason I think so is that in Yaglom's book "A simple
non-euclidean geometry and its physical basis", he discusses 9 plane
geometries, where 9=3 x 3, by choosing a kind of metric in 3 ways and by
choosing a kind of angle measurement in 3 ways. Similarly, he remarks that
one can get 27 = 3 x 3 x 3 solid geometries by also having 3 ways of defining
the measure of dihedral angles.

So my question is this: where is this more general axiom system, based on
a "metric space" together with an independent notion of angle measurement,
discussed in a general setting?

Ignorantly,
Allan Adler
ara@zurich.ai.mit.edu

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From: lrudolph@panix.com (Lee Rudolph)
Subject: Re: certain axioms for geometry
Date: 16 Sep 2000 15:30:04 -0500
Newsgroups: sci.math.research

Allan Adler <ara@nestle.ai.mit.edu> writes:

>At the end of Krause's book Taxicab Geometry, he gives some axioms for
>Euclidean geometry. The axioms involve a metric and a function which measures
>angles and he explicitly considers them to be part of a modern formulation
>of geometry. The axioms seem to be somewhat in the spirit of Hilbert's axioms
>but are also somewhat different. One correspondent suggested that they might
>be due to Birkhoff around 1940. I looked up some work of Birkhoff from that
>time and he does have a metric approach to geometry in which he characterizes
>elliptic, euclidean and hyperbolic geometries by simple metric properties,
>but I didn't see a place for an angle measurement function in his system.

You may have been looking too late.  The stuff you want is (it seems 
to me) in "A set of postulates for plane geometry, based on scale and
protractor", Annals of Mathemeatics, April, 1932, vol. 33, pp. 329-345.

Lee Rudolph