Tuesday, August 01, 2006

A Supreme Beauty

We all have our favorite math books. Well, some of us do. One of mine is Coxeter's Introduction to Geometry, a book I did not discover until after I got out of college. I did not discover Euclid until after college either. In fact, I wasn't much interested in geometry while taking math in college although I enjoyed my topology courses a lot. Coxeter and Euclid changed my mind about the subject. Coxeter's book is not really an introduction to geometry in the sense that Euclid's book is an introduction to geometry, or the course we took in high school is an introduction. You might call Coxeter's book an introduction to the geometry that follows Euclid--the geometry developed after Euclid.

Here is some of what Coxeter has to say about Euclid in Introduction to Geometry.
About 300 B.C., Euclid of Alexandria wrote a treatise in thirteen books called the Elements. Of the author (sometimes regrettably confused with the earlier philosopher, Euclid of Megara) we know very little. Proclus (410-485 A.D.) said that he "put together the Elements, collecting many of Eudoxus's theorems, perfecting many of Theaetetus's and also bringing to irrefragable demonstration the things which were only somewhat loosely proved by his predecessors. This man lived in the time of the first Ptolemy, [who] once asked him if there was in geometry any shorter way than that of the Elements, and he answered that there was no royal road to geometry." Heath quotes a story by Stobaeus, to the effect that someone who had begun to read geometry with Euclid asked him "What shall I get by learning these things?" Euclid called his slave and said, "Give him a dime, since he must make gain out of what he learns."

Coxeter prefaces his book with an oft quoted statement by Bertrand Russell.
Mathematics possesses not only truth, but supreme beauty--a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature ... sublimely pure, and capable of stern perfection such as only the greatest art can show.

Nobody has summarized the nature of mathematics better than that. One of the better parts of my life is that I got to study Euclid and Coxeter and some other good mathematics.


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