By geometric, I mean it was expressed in geometric terms, as was Ptolemy’s, or any scientist for still another hundred years after Galileo until modern annaliyical tools were developed. But don’t despise induction: this is really the approach that St. Thomas did when he was trying to “prove” the existence of God.
You have misinterpreted. I didn't say anything about despising inductive reasoning, only that inductive reasoning is not a proof except under very limited conditions, namely complete enumeration. There are never proofs in science.
You used the word "proof". It's not a proof.