My understanding is that magnitude to AG mathematicians meant the actual line segments and plane regions (not the size of the line segment or the area of the plane region) the concept of ratio was then used to talk about the 'size' of different magnitudes. Euclid defines ratio as 'sort of a relation in respect of size between two magnitudes of the same kind'. It seems this definition attaches no numerical meaning to the concept of ratio.

My questions are:

  1. Was there a particular reason magnitudes were defined this way?
  2. With no numerical meaning, how was this concept of ratio used to talk about size?


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