"Euclid alone has looked on beauty bare," wrote Edna St. Vincent Millay. Well, maybe so, maybe not. Surely beauty is not confined to mathematics. But, yes, upon completing Book One of Elements I had the sensation of having been released "from dusty bondage into luminous air." And that was long ago when I was a young man.
Not everyone looking at Elements will see the beauty. It is a kind of deductive beauty that only reveals itself to the initiated. But there is another proof of the Pythagorean Theorem, sometimes (perhaps erroneously) attributed to Pythagoras himself, several centuries before Euclid, that has about it such a spare simplicity that even the mathematically unpracticed can feel she is in the presence of something fine and beautiful.
Here it is, in two big panels I made many long years ago -- nearly 40 years ago, I suppose -- and that for a while were hanging in my living room, then in my office at the college. When I was about to toss them out they were rescued by a friend, and so I am able to photograph them now.
I will leave it to the reader to grasp the proof (four identical right triangles and the square on the hypotenuse equals the same four triangles and the squares on the other two sides).
It is unlikely that Pythagoras and his students were the first to see the truth of the Pythagorean Theorem, but they certainly knew they were looking at something that connected mysteriously and profoundly to the order of the cosmos. They made a religion of what Keats would later proclaim: "Beauty is truth, truth beauty, -- that is all/ Ye know on earth, and all ye need to know."