I'm reading something about "degrees of freedom" trying to learn what exactly it means, and there's this one sentence I'm running into that I can't really understand...

What is this "the two independent ratios {a : b : c}" ???

They talk a lot about how a line on a plane is represented by the equation [tex]ax+by+c=0[/tex]. But I know from learning about [tex]y=mx+b[/tex] in grade school that you only need two numbers to specify a line.. If anybody could explain that sentence to me I'd really appreciate it.

The point that the author was trying to make is that to include ALL lines, you need to allow vertical lines (x=k). The form being used in the text allows for this (b=0). The two degrees of freedom is a way of saying that multiplying a,b,c by a constant doesn't change the line.

{a:b:c} is shorthand for the proportion a/b= b/c. There are "two degrees of freedom" because you are "free" to choose two of the numbers to be almost anything you like and then could solve for the third.

The notation is not new to me, but the concept of two degrees of freedom for a straight line is new (to me). I always thought a caterpillar walking along a wire had only one degree of freedom, same as all straight lines regardless of where they are. How does introducing more constants into the equation change that? Halls, can you expand a bit on your explanation?

Well, you are not a caterpillar, are you? If you were constrained to a specific straight line, but could pick any point on that line, yes, that would be "one degree of freedom". Here, however, If we write a line as "ax+ by+ c= 0", we could multiply or divide each of the coefficients by any number (except 0 of course) and still have the same line: "rax+ rby+ rc= 0" is satisfied by exactly the same (x,y) and so is the same line. Notice that ra/rb= a/b and rb/rc= b/c no matter what r is. In the formula "ax+ by+ c= 0" two of the numbers can be chosen any way we want but the other is then fixed.