(We should be inclined to say that the proof of the impossibility of trisecting the angle with ruler and compasses analyses our idea of the trisection of an angle. But the proof gives us a new idea of trisection, one which we didn't have before the proof constructed it. The proof led us a road which we were inclined to go; but it led us away from where we were, and didn't just show us clearly the place where we had been all the time.)