Methodical relativity

I think it's important to keep in mind that all of these astronomical viewpoints represented entirely different approaches and dogmas. In response to some of the previously posted comments, it would have been doctrinal madness to try to combine them, blasphemous in some cases. What I'm confused about is how the Tychonic system could ultimately have the same mathematical results as the Ptolemaic one. It's kind of funny to see how much importance astronomers gave to particular method and calculations, and yet see their products end up the same despite having conjured entirely different hypotheses. It's also interesting that theoretical knowledge could have held so much more value than empirical knowledge did at some point in time, in comparison with the modern scientific approach.