Tuesday, November 09, 2010

Beauty bare -- Part 1

If there is a sacred scripture of science it is surely Euclid's Elements. The book (actually 13 "books") has been continuously "in print" since it was written in about 300 B.C., in Alexandria, Egypt. For over two thousand years it was the standard text from which students of mathematics and the exact sciences learned the ropes. Even today it is the basis for every high-school geometry text.

I can't say that I have read all 13 books. At one time I did work my way step-by-step through Book One, using Thomas Heath's profusely annotated edition. It was an exhilarating experience, and a masterful illustration of what rigorous thinking is all about.

Euclid begins with definitions of a point, a line, and so on.

Then he offers 5 Common Notions and 5 Postulates, all of which he assumes the reader will accept as self-evidently true.

For example:

Common Notion 1: Things equal to the same thing are equal to each other.

Postulate 1: One may draw a straight line from any point to any other point.

Postulate 3: One may describe a circle with any center and any radius.

Basic stuff, really. Who's to quibble?

Now the fun begins, as Euclid deduces Propositions from his first principles.

For example, his first Proposition is to construct an equilateral triangle on a given straight line, for which he evokes Common Notion 1 and Postulates 1 and 3.

And on he goes, building a magnificent compilation of Propositions on what has gone before. Proposition 9: To bisect a given triangle. Proposition 37: Triangles on the same base and in the same parallels have equal areas. And so on.

As I worked my way through the book, I kept track of the logic. Here is a diagram I drew at the time showing the pathway to Proposition 47, the Pythagorean Theorem (the square on the hypotenuse of a right triangle is equal to the sum of the squares on the other two sides). Click to enlarge.

The Pythagorean Theorem is by no means self-evidently true, but its truth is hidden in the self-evidently true Common Notions and Postulates, to be revealed by logical thinking.

If the Elements is the sacred scriptures of science, it is not because of its particular contents of propositions, but as a model for a way of thinking -- a way of thinking that has guided us into the universe of the DNA and the galaxies, and out of the gabble and hiss of parochial prejudice. In effect, Euclid is saying: "Here is a way, a truth, and a light."

Tomorrow: Another avenue to the Pythagorean Theorem and some thoughts on truth and beauty.

My daughter Mo and daughter-in-law Patty were recently in Alexandria, walking the streets walked by Euclid, and were honored with a personal tour through the new Alexandrian Library, once the greatest library in the world. You can see some of her photos here.