Knowledge of The Ancients

I've been checking a calculation of late. It's a bear, 150 pages and every detail has to be correct. Anyway, the engineer and I were spinning in circles on a force that went through three separate coordinate transformations (same force, different perspective), but had to end up perfectly vertical. We just couldn't figure out why we were about 20 degrees off on the vector direction. Equation upon equation, Microsoft Excel, MathCad... it wasn't working but we didn't know why. To muddy the waters, I had made a recommendation a month ago to try a different approach, which changed the answers by only a few pounds. Then, I remembered something that Professor Shapton taught. I drew the vectors... and proved that we were off and by how much. Then, I drew them with the equations I had recommended... and the lines stopped perfectly on the vertical line I was shooting for. Problem solved with a pencil, protractor, and a ruler. We're not always so much smarter than our predecessors as we would like to imagine.