If a magnitude is the same multiple of a magnitude that a (part) taken away (is) of a (part) taken away (respectively) then the remainder will also be the same multiple of the remainder as that which the whole (is) of the whole (respectively).
If we are given two positive real numbers \(\alpha\), \(\beta\),^{1} and the following multiples of aliquot parts \(m\ge 1\), \(n\ge 1\): \[\alpha m=\beta m,\quad \alpha n=\beta n\quad\quad( * )\] then adding both equations gives us \[\alpha(m-n)=\beta m-\beta n.\]
See distributivity law for real numbers.
Proofs: 1
From a geometrical point of view, \(\alpha,\beta\) could e.g. mean the lengths of some segments, the areas of some plane figures or the volumes of some solids. ↩