19 October 2013

Two theorems by Thales

Most ancient peoples on earth could count, the Egyptians had geometrical rules of thumb, the Chinese had notions of geometrical proof, but geometry as a part of abstract mathematics did not exist anywhere before the Greeks. They, and only they, created the impressive abstract system of definitions, axioms and proofs which, with all its imperfections, has remained one of the major achievements of the human mind. The difference with basic everyday math is huge; it's the difference between counting moons (as all peoples do) and proving that there are infinitely many prime numbers (as the Greeks did). 

For those who think that geometry, following the dogms of political correctness, must have originated in any continent and culture, we give two theorems attributed to the Greek philosopher Thales. Consider them for a while, then try to find out what your ancestors were dealing with in 550 BC, when Thales died.

Theorem 1. Three parallel lines (yellow) cut a transversal (white, 3 examples) in a proportion which is independent of the position of the transversal. For the three examples, we have AB:BC=DE:EF=GH:HI.

Theorem 2. Consider two points on a circle (yellow). Any point on the circle forms with the given points an angle (white, 3 examples) whose magnitude is independent of its position, as long as it remains on the same side of the straight line through the given points. 
It's both instructive and sobering, to try and prove this old stuff from scratch. Enjoy!

(add grin)