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Chapter 6
Alternative Concepts for Parallelism:
Non
Euclidean Geometry
6.1
Historical Background of NonEuclidean Geometry
6.2
An Improbable Logical Case
6.3
Hyperbolic Geometry:
Angle Sum Theorem
6.3 ½
Hyperbolic Geometry Enrichment and Workbook
Homework 7
1
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View Full Document6.1
Historical Background of NonEuclidean Geometry
6.2
An Improbable Logical Case
About 575 B.C. Pythagoras wrote his book on Geometry.
Some of the material was
known in other cultures centuries before he wrote it down, of course.
His was the first
axiomatic approach to organizing the material.
Interestingly, he did as much work as he
could before introducing the Parallel Postulate.
Many people have interpreted this
progression in his work as indicating a level of discomfort with the Parallel Postulate.
It’s not really possible to know what he was really thinking.
About 400 years after the
birth of Christ, Proclus, a Greek philosopher and head of Plato’s Academy, wrote a
“proof” that derived the Parallel Postulate from the first 4 Postulates. thereby setting the
tone for research in Geometry for the next 1400 years.
Johann Gauss, the great German
mathematician, actually realized that there was another choice of axiom but didn’t choose
to publish his work for fear of getting into the same trouble as other scholars had with the
Catholic Church.
Around 1830, two young mathematicians published works on Hyperbolic Geometry –
independently of one another.
The world took no note of them.
In 1868, the Italian
Beltrami found the first model of Hyperbolic Geometry and in 1882, Henri Poincare
developed the model we’ll study.
120 years later, Hyperbolic Geometry is finally making it into high school textbooks.
My
favorite is a text that St. Pius X used in the 90’s.
If you ever get a chance to look at it –
it’s just terrific.
And it includes a section on Spherical Geometry as well:
Geometry, second edition by Harold Jacobs.
ISBN:
071671745X (copyright 1987).
Note that St. Pius isn’t a flagship diocese school – they actually have a very full section
of “Algebra half”, the course for the kids not ready for Algebra I.
In the 1400 years of work on the axioms of Absolute Geometry there was always a
special group of people who endeavored to prove that the Parallel Postulate was actually
a theorem.
In fact, Saccheri and Lambert who both came so close to realizing that there
was an alternate geometry out there waiting to be discovered never got all the way past
believing that the axiom was a theorem.
On page 427 is a list of statements that are
equivalent to P1, the Euclidean Parallel Postulate.
Here they are:
•
The area of a triangle can be made arbitrarily large.
•
The angle sum of all triangles is a constant.
•
The angle sum of any triangle is 180.
•
Rectangles exist.
•
A circle can be passed through any 3 noncollinear points.
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 Spring '08
 Staff
 Logic, NonEuclidean Geometry

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