Two ratios count as the same when, for every choice of whole-number multiples, the comparisons come out alike: both larger, both equal, or both smaller. This defines sameness of proportion for any sizes, even ones no fraction can express. You compare by testing every possible scaling instead of measuring.
Instead of computing areas with numbers, you compare shapes by constructing a square equal in area to a given figure. Facts we now write as algebra appear as statements about rectangles and squares on lines. The best-known example says the square on a right triangle's longest side equals the squares on the other two sides combined.
If you keep removing more than half of a quantity, then more than half of what remains, you will eventually be left with less than any fixed amount you name. Repeated finite steps can therefore pin down a quantity as closely as you like. This lets you prove things about curved or hard-to-measure shapes by squeezing them.
One starting assumption, that two lines tilted toward each other must eventually meet, was less obvious than the rest. People spent centuries trying to show it followed from the others. Their failure revealed that dropping it gives consistent alternative geometries, so testing whether a rule is truly needed can create whole new systems.