Non-Euclidean Geometry

History of Hyperbolic Geometry

Euclid

It is arguable that Euclid's The Elements is the most famous account of geometry from any point in history. The best known part of The Elements is Euclids list of the five postulates of geometry.

1. To draw a straight line from any point to any other.
2. To produce a finite straight line continuously in a straight line.
3. To describe a circle with any centre and distance.
4. That all right angles are equal to each other.
5. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

It was widely believed for many years that this fifth postulate could actually be proven by the other four. Many great mathemeticians tried to do just that. Yet many "proofs" were devised and shot down soon afterward. Yet the attempts to prove the fifth postulate were not totally a waste of time. Many developments came from these exploits, the most important of which is the development of Non-Euclidean Geometry. Among the mathematicians who failed to find this proof are Giovanni Saccheri, Johann Lambert, Proclus Diadochus, and Claudius Ptolemy.

So what is Non-Euclidean Geometry? It's any form of geometry which uses the first four of Euclid's postulates and the negation of the fifth. To fully understand the concepts which are being addressed, it is important that we put Euclid's fifth postulate into modern English. John Playfair reworded Euclid's fifth postulate to the form it is currently known by.

Playfair's Axiom:- Given a line and a point not on the line, it is possible to draw exactly one line through the given point parallel to the line.

Thus, the negation of this would be that there exists some point and line not containing that point such that there are at least two lines through the given point which do not intersect the given line. Assuming this to be true, it is fairly easy to prove that given a line L and a point p not on the line, there are an infinite number of lines through p and parallel to L. It may be hard to imagine that this could be so, but carefully crafted rules can allow for this Non-Euclidean geometry to exist.

So where did this "Non-Euclidean" geometry come from? There is evidence that Johann Gauss had speculated this type of geometric system as early as 1813, but he never published anything so he doesn't get the credit. The credit for devising Non-Euclidean geometry goes to one of two mathematicians, both of whom published their works at about the same time. Nikolay Lobachevsky was the first to publish accounts of Non-Euclidean geometry in 1829. In 1823, János Bolyai had written a letter to his father, also a mathematician, claiming "I have discovered things so wonderful that I was astounded ... out of nothing I have created a strange new world." Bolyai's reference was to Non-Euclidean geometry. Unfortunately, he did not publish anything about it until 1832, but many still credit him with at least partially discovering it.

Want to know more? You can either come with me to find out More About Non-Euclidean Geometry: Hyperbolic Geometry (my page 2) or you can dig into some more history with the links at the bottom of this page.

Other sites which explore hyperbolic geometry:

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