A couple of days ago, I said that Euclid talks about ratios of line segments without using numbers to measure them. So how does he define ratios of line segments if he doesn't use numbers? Definition 5 of book 5 gives his method, an abstract method of imagination: The ratio of segments A and B is equal to that of C and D if, when A and C are each multiplied by the same arbitrary number and B and D are each multiplied by another arbitrary number, the "equimultiples" of A and C are both greater than or both equal to or both less than the equimultiples of B and D, respectively.
Yeah. I didn't get it at first, either. Trying to understand the significance, I thought about two pairs of line segments, each pair in a ratio of 1:2. Maybe the first pair measured 6 inches and 12 inches, while the second pair measured 5 and 10. Multiply each of the first pair by 5 and each of the second pair by 6, and you come up with 30 and 60 in each case: the two pairs are equal. Why bother with less-than and greater-than?
Only partway through book 5 as Euclid used this definition did I understand that it meant that if you performed this operation using every imaginable pair of multipliers (an infinity of possibilities) the results would always come out equivalent: both greater than or both equal to or both less than. Anachronistically using decimal fractions to measure segments, let's imagine a unit square and its diameter, whose length is the square root of 2: 1.4142135623. . . . Now imagine a square built on a side twice as long. Its diameter is also twice as long: 2.8284271247 . . . . Knowing as we do that the lengths of the diameters are irrational, we can never find a pair of integers to use as multipliers that will make the results equal. But imagine multiplying the sides (1 and 2) by 14 (results: 14 and 28), and imagine multiplying the diameters each by 10 (results: 14.1421. . . and 28.28427 . . . ). Each of the first numbers is less than its corresponding number in the second pair. Now try multiplying the sides by 141 (141 and 282) and the diameters by 100 (141.421 . . . and 282.842 . . . ). The first pair is still less than the second pair, but the difference is proportionally less. We could go on and on, getting the two pairs as close as we like to each other, but they will never be equal. That wording of our situation reminds me of what I've learned at the beginning of several calculus books. (I've never made it very far in any of them.) So Euclid doesn't exactly anticipate Descartes's geometry on a measurable grid, but definition 5 does give hints of the theory of limits.
Imagining an infinite number of monkeys multiplying two numbers by an infinite combination of integers over an infinite amount of time makes for a fun thought experiment (and probably reproduces Hamlet), but it hardly provides a practical way to reproduce a given ratio. Fortunately, Euclid offers a quick, practical method. If you have one line segment (A) divided at some point (P) and you want to divide another line segment (B) by the same ratio, make any triangle out of those two lines and a third (C), and draw a new line (D) parallel to C going through point B. The point at which line D intersects line B divides B into the desired ratio.
Forgive this ludicrous post. No one wants to read about geometry with no pictures. I could learn to insert graphics into my blog, or I could just apologize, and I'm much better at the latter. I'm just recording the things I've been thinking about prompted by my reading, and recently, this is it, I'm afraid. And on top of all these problems with today's post, I could be totally wrong. Music theorists don't usually have to count higher than 12, so my math skills are rusty -- although not as rusty as they would be if I didn't read Euclid every year and try a calculus book every once in a while.
Since I'm close to the end of book 6, I peeked ahead yesterday at the beginning of book 7 (next year's assignment) and saw immediately that I was wrong about one thing in my last post. Book 7 starts with definitions about numbers and primes and such, and the proofs all use line segments to represent the numbers, something I thought Euclid never did. My new guess is that while he thought any sets of numbers (by which he meant counting numbers: 1, 2, 3, and so on) could be represented by corresponding line segments, he knew that not every set of line segments could be assigned to numbers, since some line segments (like the diameter of the square) could not be measured by integers.
I have now exceeded all proper proportions and will stop for the day.