I've been thinking about this for a few days now, and it has come to my attention, how do you prove your answer is correct? Formulas and geometric proof put to one side; I'm talking about the answer to your problem in general. If one ever watches the MIT math "You-Tube" videos, then you will notice how a correction for an error that was made pop up every now and again. Back in the day, there is no doubt i'm sure, a small error can "somehow" get magnified through cumbersome black boards full of math. My point is, what does your answer depend upon? Is it competence, a back-up mathemetician, double-checking your work, mathematical reasoning? There is always room for error and somehow it happens, but back then computer software did not exist, it was purely down to competence in one respect.