In last year’s posts about Euclid, I reported on my slow journey of discovery through book V of the Elements. I began by assuming that what appeared to be geometry was really algebra – that the ratios and proportions at issue were ratios of numbers Euclid merely represented as line segments. Then I slowly figured out (so I thought) that what appeared to be algebra in the guise of geometry was really geometry – that Euclid couldn’t think of some hypotenuses, for instance, as representing numbers since his universe of numbers included only whole, positive integers, not irrationals like the square root of 2. But then I peeked ahead at book VII, where Euclid talks about ratios of numbers and does indeed represent them by line segments (although never in triangles with potentially irrational sides).
This year I read book VII and must admit that I learned a few things. For instance, I don’t know that it had ever occurred to me that one could prove the commutative and distributive properties of multiplication. Some good teacher, possibly my dad, showed me to think of multiplication in terms of rows and columns of squares; these properties become absolutely obvious when you look at this kind of representation of a multiplication problem. Also, Euclid had methods I didn’t know for finding the greatest common factor and the least common multiple of two and even three numbers. For all I know, there could have been ten “New Maths” between Euclid’s time and the 1970s, when I took algebra in high school. Having read Euclid’s explanations, I’m grateful for the changes in math pedagogy and newly appreciative of at least one aspect of the high-school education I still strive to overcome. But then I’ve never complained about the math curriculum at my high school; math and science were the weapons with which we were winning the Cold War, so these subjects were taught fairly well. (I have, though, complained about some math teachers I had. I once “proved” to one that division by zero was possible.)
But Euclid confused me just as often as he enlightened me this year. He gave definitions for terms he never used. He repeated himself and proved special cases after already proving more general cases. And he worded some proofs so strangely, and in apparent contradiction to his definitions, that I couldn’t make even any consistent guesses as to what they were about. I can’t help but wonder if the genius who proved the Pythagorean Theorem hasn’t suffered in transmission and translation here.
Speaking of the Pythagorean Theorem, I found out more about that famous proposition this past spring as I tutored a student in geometry. She had to write a short history paper on the Theorem and include a critique of three proofs. Three? The one I knew about, which I assumed was as clear and straightforward as Euclid could make it, was so complex, I felt religious exaltation when I first understood it, as if I were looking directly into the mind of God. What must the others be like, and how was I ever going to help this student compare three of them?
As it turns out, this most unintuitive of geometric truths has at least ninety-seven proofs, as shown on this website, and many of them are much simpler than Euclid’s I.47. One was concocted by James Garfield four years before his tragically short Presidency. One was submitted by a fourteen-year-old girl from Iran. After going through a few of the ninety-seven with the student, I came up with one in my head on my walk home. As it turns out, I “found” number 4. So I’ve learned some about geometry this year, but surprisingly enough, mostly not from Euclid.