Greeks, e.g., Euclid, had done all of these things long ago and discovered various more or less interesting properties of conic sections (as this family of curves was called) and of other geometric constructions such as circles and triangles. But they'd done so as an exploration of pure thought, as a mathematician might compute the sum of two numbers. Every assertion that Euclid, et al., made concerning geometry was backed up by a chain of logical proofs that could be followed all the way back to a few axioms that were obviously true, e.g., 'the shortest distance between two points is a straight line.' The truths of geometry were necessary truths; the human mind could imagine a universe in which Daniel's name was David, or in which Ipswich had been built on the other side of the Orwell, but geometry and math had to be true, there was no conceivable universe in which 2 + 3 was equal to 2 + 2.