I remember reading somewhere that in ancient times they were not treating a ratio like a division as we do.
I was wondering is there a subtle distinction between the concept of the ratio and the idea of division that made them cautious about this?
[Math] Ratio vs division
divisibilitymath-historyratio
Best Answer
They did not work so much with either ratio or division as with proportion. This was denoted $a:b::c:d$. For example, altitudes $a$ and $b$ will be in the same proportion as the corresponding triangles with the same base. Today we would write this as $\frac{a}{b}=\frac{c}{d}$ but for the Greeks the only numbers were natural and proportion was a relation among actual entities rather than abstract numbers.