Sunday, May 11, 2014

Daily 2: Propositions 5-11

The wording on Euclid's book VII, propoitions 5 through 11 seemed very murky. There were no translations for those propositions on our class google.doc. So for my post, I wanted to "decode" propositions 5 through 11, without looking it up on google or something. That proved to take more than a few hours. So for this post I will be translating proposition 5. Though the rest of the 6 through 11 have very similar wording; so translating 5 can help build on those. Here it is,
(Caution: this is just an attempt, I'm not 100% sure on the translation.)

Proposition 5: If a number is part of a number, and another is the same part of another, then the sum is also the same part of the sum that the one is of the one.

Translation: Let x,y,z,x1,y1,z1 be non-zero real numbers. If xy = z, x1y1 = z1, and x/z = x1/z1, then
(x + x1)/(z + z1) = x/z.

Now let's break down some of Euclid's terms. It seems like if we have a number (x) being a part of another number (z), then the term "part" means z/x. For instance, 2 is a part of 8 because 2 is 8/2 = 4 parts of 8. Also, the phrase "the one is of the one" is referring to (x) of (z), where (x) is z/x parts of (z)
Now using the notation above, I will restate the proposition, adding in notes as necessary.

New Proposition 5: If a number (x) is part of a number (z, where xy = z), and another (x1) is the same part of another (y1, where x1y1 = z1 and z/x = z1/x1), then the sum (x + x1) is also the same part of the sum (z + z1) that the one is of the one ((z + z1)/(x + x1) = z/x).

I had to read the proposition like 10 times to get it in my mind. When Euclid says "the one is of the one," it's very confusing and it's not so clear what he is referring to. I actually found the translation by ignoring the one to one part. The propositions 6-11 follow similar wording make similar algebraic claims. So after translating 5, they don't seem so bad. As part of the main daily 2 work, I wrote a proof for this proposition.

1 comment:

  1. 5C's +
    Nice work! Including the proof is a good thing here for coherence and completeness. You might include a little of what you found good, satisfying or helpful out of what you did. (Consolidation)

    In general, we think about Euclid meaning natural numbers when he says number. Sometimes people connect this proposition to distributivity because he uses that in the middle of his proof, too.