Girolamo Saccheri (1667−1733) is credited with being the first to successfully study the logical consequences of a denial of the Parallel Postulate. Though he gets the credit, his is a sad tale. Saccheri tried to prove the Parallel Postulate by contradiction: He assumed that the Parallel Postulate was false, developed the resulting consequences, and looked for a contradiction. Much to his disdain, he was unable to establish a contradiction of any definition, theorem, or postulate. He did manage to violate several intuitive ideas, but a proof by contradiction must involve a violation of a postulate, theorem, or definition, not intuition. If he had admitted that his reasoning did not lead to a contradiction (and had been willing to explore the resulting paradoxical ideas), he would have anticipated the discovery of one form of non-Euclidean geometry by at least 100 years. This non-Euclidean geometry might have even been named in his honor. As it happened, he downplayed his results and concluded that, based on his work, the Parallel Postulate must be true. As a result of his incorrect conclusion, his results were not widely read and did not have much influence on the mathematical community until almost 150 years after his death.

Tangent Line

The denial of the Parallel Postulate leads to two alternatives:

• Given a line and a point not on the line, there are no lines through the point parallel to the original line. In other words, parallel lines do not exist.
• Given a line and a point not on the line, there are at least two lines through the point parallel to the original line.

Spherical geometry originates from the first alternative, hyperbolic geometry from the second.

It is ironic that Saccheri thought he was helping Euclidean geometry by trying to prove the Parallel Postulate. Had he delved into these paradoxical results, he would have been able to put to rest any doubts about the Parallel Postulate. He would have shown that the Parallel Postulate was in fact independent of Euclid's other postulates. That result would have helped Euclidean geometry more than anything.

This is all wishful thinking, though, because Saccheri did not have confidence in his results and he drew the wrong conclusion from his work. As it stands, Janos Bolyai (1802−1860) and Nikolai Lobachevsky (1793−1856) are the two mathematicians credited with the development of a logically consistent geometry based on a denial of the Parallel Postulate. As is typical, they never received full recognition of the value of their work during their lifetimes. This particular branch of non-Euclidean geometry is called hyperbolic geometry, or saddle-point geometry. The postulate that generates hyperbolic geometry is the Lobachevskian Postulate.

Tangent Line

Janos Bolyai published his results in an appendix in his father's book that was appropriately titled (in translation) An Attempt to Introduce Studious Youth to the Elements of Pure Mathematics. A copy of his work was also sent to the highly celebrated mathematician Carl Gauss. Gauss had already worked out the ideas written by Janos Bolyai, but had never published them. In a letter that he wrote to Janos's father, he mentioned that he was happy that these results would be published and that he was spared the trouble of writing them up himself. Rather than interpreting Gauss' comment as a compliment, Janos thought Gauss was trying to take credit for his work. Janos Bolyai lost interest in mathematics and never published again.

• Lobachevskian Postulate: Through a point not on a line, there are infinitely many lines parallel to the given line.

In this geometry, several anti-intuitive results follow.

• The sum of the measures of the interior angles of a triangle is less than 180º.
• The sum of the measures of the interior angles of a triangle increase as the area decreases.
• Rectangles do not exist.
• Two triangles with congruent corresponding angles are congruent.
• There is an upper limit on the area of a triangle.

This is only the tip of the iceberg. Believe me when I say that strange things happen in hyperbolic geometry. Even stranger is the fact that a large group of scientists believe that the shape of the universe is best modeled using hyperbolic geometry!

Excerpted from The Complete Idiot's Guide to Geometry © 2004 by Denise Szecsei, Ph.D.. All rights reserved including the right of reproduction in whole or in part in any form. Used by arrangement with Alpha Books, a member of Penguin Group (USA) Inc.

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