Why aren't mixed ratios allowed in Euclid's "Elements"? Why did Euclid forbid the comparing of different kinds of magnitudes? And was it the same for numbers? Or were ratios specifically meant for magnitudes (lines, planes, solids) What are the consequences of comparing 2 different kinds of magnitudes? I mean, we eventually figured out that the concept of velocity and rate change could be useful but why did it take so long to realize this?