Sunday, July 3, 2011

Out of Proportion

I have an intelligent friend who says she's "bad at math." She has a sad case of arithmophobia, but she's so smart, I sometimes wonder if she wouldn't have had strong skills with numbers if she just hadn't developed the fear. But in any case, there it is: she can't do math. This problem became known to me early in our acquaintance when, for one reason or another, our conversation hit upon geometry and I sad something about similar triangles or parallel lines or something of that sort. My friend said she didn't know what I was talking about. "You took geometry in high school, didn't you?" "Sure, everyone took geometry, but I never learned that." "You must have just forgotten, because this subject comes up early in any geometry course." "No," she insisted, "we never studied this."

I assumed she didn't even remember what she had forgotten until one day a few years later when I was helping another friend's son with his homework. He asked the group of adults a question about an equation to express a series, and they all looked at me, so I looked at his book to see what I could contribute. But the book gave no instructions on how to find the equation; we were left to the mercy of trial and error. Frustrated, I looked at the cover so I could at least know the names of the authors I was angry with, only to receive a shock upon seeing the title: Modern Geometry. I flipped through the book looking for axioms, for proofs, for inscribed polygons, but I found none of these things. One chapter gave the formulas for areas of various plane figures, but otherwise, the book had nothing to do with the noble ancient art of Earth Measuring or its classic logical method. Such is public education in our times. At least my innumerate friend was right about never having studied it; she went to the same high school and probably studied the same book.

Somewhere in some defense of the Great Books program, Mortimer Adler uses Euclid as his prime example: why study geometry from anyone but the classic master? Not foreseeing our local school district's advanced policies, Adler didn't know he had to defend the study of geometry itself. But assuming that I want to study actual geometry rather than guessing at equations, Adler's advice isn't all bad. For a high-school class or a first introduction to the subject, Euclid's Elements would need a little page-layout makeover, some extra notation on the diagrams, and a workbook of exercises, but otherwise it would work perfectly. Its topics are immaculately organized, and the presentation is almost always crystal clear, although Thomas Heath's century-old translation could use a little polishing and updating.

Reading Euclid gives me an audience in the court of brilliance. He nimbly juggles a variety of ways of approaching a problem, for instance. Some proofs he conducts by straight deduction from the premises, some by first making a construction and then making deductions. At other times, he poses a counterhypothesis and then argues the reductio ad absurdum. If you're in the mood for it, the slowly building procession of theorems can have the exciting effect of a mystery novel's plot: some parts may at first seem random (although entertaining), but by the end of any given book, each element plays its part in the drama of the whole. The first time I read book 1, as I reached the last demonstration, proving the Pythagorean Theorem, I had the feeling the classic philosophers speak of when they say that contemplating such truths gives us a glimpse of God's mind -- the most powerful instance of this sensation I've ever experienced. For a moment I saw the whole edifice, from its foundation in a handful of axioms to its cupola of right triangles and squares, and I felt my mind lifted from some earthly burden I hadn't even noticed before.

The portion on my schedule for this year, bk. 6, deals with similar triangles and other similar polygons. Similar figures have equal angles and proportional sides, so before dealing with these figures, Euclid had to cover proportionality of line segments, the subject of book 5. Last year, I went a long way through that book writing my notes in something like algebraic notation (If a > b, a:c > b:c and c:b > c:a, for instance) and wondering why it took Euclid so many words and pictures to say the same thing. I'll have to check with my friend Kerry, a science historian, but I think I only got it right when I finally realized that Euclid "measured" (at least mentally and abtractly) these line segments as magnitudes, not as quantities, in other words, not as numbers in the way we'd think of them. I think it was Descartes who first worked geometry by numbers and arithmetic.

If anybody can help me out on this, please share. Right now, I understand just a few differences between Euclid's view and ours. First, I think of ratios as convertible with fractions: a:c > b:c is true if and only if a/c > b/c. But I don't believe Euclid had knowledge of fractions or worked with them. From my study of the history of music theory, I know writers dealing with string lengths use only ratios of whole numbers until the seventeenth century. In ancient thinking, 1 cannot be divided. It is unity, an atom. How can you have half of a person or half of a hole? Numbers start with 1 and go up. Magnitudes, on the other hand, can be divided, perhaps indefinitely. Another difference is that without thinking of the magnitudes as represented by some particular number, Euclid can have "ratios" involving sides of "irrational" length, as for instance between one side of a square and its diagonal.

My high school may have done English and history all wrong, but they did mathematics right. I loved learning geometry, and I've often made use of both its content and its logical method ever since. Now I enjoy studying it all over again with Euclid. This year, I'm again inspired by Euclid's creative intelligence and by the beauty of the material, yet I'm also thankful again for Descartes and fractions.