Algebra, when invented, existed not to deliver proofs, but to help translate the description of proofs. The actual proofs were geometric. We didn't get an algebraic reformulation of geometry for many, many centuries. Why not? What separated DaVinci from Kepler?
I observe, as a computer programmer, that algebra works for solving equations, but - absent a TRS-80 Model 2 computer - is impractical for difficult equations.
First, multiplication and more so the Long Division is a O(N2) algorithm so a pain in every problem-solver's rear. And then, in the end, you got the Squaring The Circle problem, with pi. You could literally waste an eternity calculating decimal places and not finish your proof. But with geometry: state your case, draw a few circles and lines and Q.E.D.
For day to day work, that putative Middle Egyptian drew out his equation just as far as he needed, to get the immediate job done, and then he prayed to Bas that it wouldn't blow up in his patron's face until he was safely retired in some other province. And so it went until Kepler came to Prague. Geometry and rough calculations were good enough for the pyramids. For Notre Dame Cathedral, too.
Not so much for the exact positions of the planets against the sidereal field over time. By 1600, they had a new tool, though: the logarithmic table. Base ten or base two, as long as we all agreed on the same base, O(N2) became O(N). Calculations to arbitrary precision became faster and less prone to error along the way. Algebra became practical.
Which meant that algebra grew popular. It could be expanded to proofs hitherto not considered by geometers.
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