In explaining why those curves were as they were, the Fellows of Cambridge would instinctively use Euclid's geometry: the earth is a sphere. Its orbit around the sun is an ellipse – you get an ellipse by constructing a vast imaginary cone in space and then cutting through it with an imaginary plane; the intersection of the cone and the plane is the ellipse. Beginning with these primitive objects (viz. the tiny sphere revolving around the place where the gigantic cone was cut by the imaginary plane), these geometers would add on more spheres, cones, planes, lines, and other elements – so many that if you could look up and see 'em, the heavens would turn nearly black with them – until at last they had found a way to account for the curves that Newton had drawn on the wall. Along the way, every step would be verified by applying one or the other of the rules that Euclid had proved to be true, two thousand years earlier, in Alexandria, where everyone had been a genius.