Euclid, Draw Back Your Bow

This post originally appeared July 6, 2018, on the Chile Today Hot Tamale! website. (www.chiletodayhottamale.net)

The summer is going by so quickly. That much hasn’t changed since the dark and dismal days of my youth. In about a month I will be back at school, declaiming on the wonders of mathematics and physics.

Despite the heat, the wildfires, the humidity, the floods, the divisiveness in our political discourse, and the fact that Tractor Supply still can’t seem to produce the latch that I ordered (and paid for) back on May 21, I am having a good summer. I’m spending it with an old friend.

A qualifications is in order. Although Kathy is my dearest friend, I am not referring to her. She is not “old” and thus does not qualify. (Whew!)

The friend is Euclid (fl. 300 BC). I was introduced to Euclid back in the 1960s, but ours was a shallow friendship until the early part of this century, when I began teaching geometry. At that point I began to appreciate his genius.

Euclid gave his name to a branch of geometry, and for his insight into geometry he is revered to this day. But if you open his text, “The Elements of Geometry,” you will find that only seven of the thirteen books deal with geometry, either plane geometry, or solid geometry (including a treatment of the Platonic solids). The other six books represent, to me, the most fascinating aspect of Euclid’s work. In them, he deals with a variety of mathematical topics (quadratics, proportion, number theory, irrational numbers) in an era before the discovery of algebra. He does so the only way he knows how: geometrically, using a straight edge and a compass.

One example should give you an appreciation of just how difficult it was to do math in the days before the discovery of algebra. The picture at the top of this blog is from Euclid’s Book II Proposition 4, which reads: “If a straight line be cut at random, the square on the whole is equal to the squares on the segments and twice the rectangle contained by the segments.” His proof is logical, each step along the way is justified, and the whole thing takes about a page in the textbook to prove. Given our knowledge of algebra, we would instead write

(x + y)^2 = x^2 + 2xy + y^2

and the proof would hardly take a full page in a textbook.

I used Euclid’s textbook for the past several years in my geometry classes. The students found it a heavy slog to read, so I began translating Euclid into more modern English, and along the way used liberal doses of algebra. The order of the propositions, or theorems, in Euclid did not depend upon any knowledge of algebra, so when we apply algebra to his theorems, we find that his propositions are not in the order we would present them today. That was a problem.

I’ve looked at other textbooks, and while they are fine in their own ways, they seem to me to stray a tad bit far afield from the classic text, Euclid’s “Elements”. So, finding no suitable alternative, I began writing my own textbook, a variation on Euclid. It adheres to the basic text, but juggles the order a bit to make the information presented seem a bit more logical, especially as the book uses algebra where possible.

I figure I’m halfway through. I should have the whole thing finished in another year or so. (It usually takes me 18 months to write a book.) There will be errors to correct, problems to solve, and lots of feedback from my students to help me polish the book off. But I must confess that spending all my free time for the last couple of months in the company of Euclid and his remarkable mind has been a most enjoyable aspect of the summer.

And it appears that the fun will not end. I have been asked to teach a course on the mathematics of Descartes, and I agreed enthusiastically. I was introduced to Descartes decades ago, but I haven’t spent the quality time with him that I have with Euclid. I’m excited about getting to know him better.

Which reminds me of a joke: Descartes goes into a bar, orders a drink, and guzzles it down. The bartender says “Would you like another drink?” Descartes says, “I think not,” and, in a puff of smoke, disappears!

I’ll tell you, that one always cracks me up!