Euclid’s Mysteries

Following a proof in Euclid feels a lot like watching Holmes or Poirot solve a murder. If the sum of the squares on AB and CD are equal to the value of the gold cufflinks, then the line from the library door at Gaveston House to the dock is greater than EF. And therefore Miss Emily Ross committed the crime with a regular pentagon inscribed in circle GHI. Like many a good mystery, each proof of Euclid’s puts all the facts before the reader, who, nevertheless, hasn’t the ability to see all the connections until the final reveal.

OK, I exaggerate. I’ve usually jumped on Euclid’s Orient Express of thought a bit before the very end. I love the moment, usually about three-fourths of the way through the proof, when I finally see the predestined conclusion coming. I still follow the clues through to the last deduction, though, and the Q.E.D. announcement never fails to overwhelm me.

And it is a deduction in the proper sense. Euclid unfolds a picture of spatial relationships every detail of which is absolutely certain given his very few postulates and the rules of logic. This makes Euclid more like Poirot than Holmes. For all of his famous talk about eliminating the impossible, Sherlock makes an abductive leap in just about every story that would make Agent Mulder proud.

The experience of enjoying Euclid’s trails of inevitable bread crumbs was especially interesting this year after I had just finished reading Kant’s Critique of Teleological Reason. I learned for instance that length of the side of a hexagon and that of a decagon inscribed in the same circle exhibit the Golden Ratio (what Euclid calls “extreme-and-mean ratio”), the greater segment being the side of the hexagon. The properties of the irrational phi are mind-bogglingly beautiful to begin with. But to think that inserting something as clear and comprehensible as a figure with six equal sides and a figure with ten equal sides inside a circle, itself formed by another wonderful and mysterious irrational, brings about the Golden Ratio makes one ponder if we’re seeing Proverbs 25:2 being demonstrated before our eyes. Did God conceal this beautiful network of mental constructs in order for Euclid, surely one of the glorious kings of human history, to discover it?