Smart Greeks
by rrusczyk, Jan 28, 2007, 10:29 PM
Imagine you don't have algebraic symbols or equations.
Then try to prove this:
In algebra speak: if (a+b)/b = (c+d)/d, then a/b = c/d. Pretty simple, right? Just subtract one from both sides. But the Greeks didn't have variables or symbolic manipulation of equations. They had geometry. And here's what Euclid did.
Just plain terrifying. And this is the power of mathematical notation - it gives us the ability to state and use complex ideas more simply. Imagine the brainpower it took to come up with Euclid's argument. Perhaps growing up in the intellectual framework the Greeks had, it wasn't so difficult. But still, it's a far cry harder than 'subtract one from both sides'.
Then try to prove this:
Euclid wrote:
If magnitudes are proportional taken jointly, then they are also proportional taken separately.
In algebra speak: if (a+b)/b = (c+d)/d, then a/b = c/d. Pretty simple, right? Just subtract one from both sides. But the Greeks didn't have variables or symbolic manipulation of equations. They had geometry. And here's what Euclid did.
Just plain terrifying. And this is the power of mathematical notation - it gives us the ability to state and use complex ideas more simply. Imagine the brainpower it took to come up with Euclid's argument. Perhaps growing up in the intellectual framework the Greeks had, it wasn't so difficult. But still, it's a far cry harder than 'subtract one from both sides'.
February 2012
January 2012