Gerard Desargues
(1593  1662) was a French mathematician who gave much of the
beginnings of projective geometry. Projective geometry's first
formal text, Traité des propriétés des
figures by J. V. Poncelet (1788  1867), was published in
1822. Interesting fact is that Poncelet did most of his work on
this text while in prison in Russia after the Napoleonic
wars.
Definitions
for the Finite Geometry of Desargues

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Axioms for
the Finite Geometry of Desargues
1. There exists at least one point
2. Each point has at least one polar.
3. Every line has at most one pole.
4. Each two distinct points are on at most
one line.
5. Every line has exactly three distinct
points on it.
6. If a line does not contain a certain
point, then there exists a point on both the
line and any polar of the point. In symbols,
if
is a line, P is a point, and P ,
then
and the polar of P intersect.


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Model for
the Finite Geometry of Desargues

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Theorems for
the Finite Geometry of Desargues
1. Every line has exactly one pole.
2. Every point has exactly one polar.


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Questions
for the Finite Geometry of Desargues
HINT: Parallel lines exist
in this geometry, but their properties are
different. sometimes there are more than
one line parallel to a given line and
sometimes there is only one line parallel.
to a given line.
For example, in the model above, three
different lines can be drawn parallel to
line R, C, B through point A', but only
one line can be drawn through A' parallel
to line A, B, T.


from Modern
Geometries
by James Smart
pp. 28  31.
