If a whole is to a whole as a part subtracted is to a part subtracted, then the remainder is also to the remainder as the whole is to the whole.

Let the whole *AB* be to the whole *CD* as the part *AE* subtracted is to the part *CF* subtracted.

I say that the remainder *EB* is also to the remainder *FD* as the whole *AB* is to the whole *CD.*

Since *AB* is to *CD* as *AE* is to *CF,* therefore alternately, *BA* is to *AE* as *DC* is to *CF.*

And, since the magnitudes are proportional taken jointly, they are also proportional taken separately, that is, *BE* is to *EA* as *DF* is to *CF,* and, alternately, *BE* is to *DF* as *EA* is to *FC.*

But, by hypothesis, *AE* is to *CF* as is the whole *AB* to the whole *CD.*

Therefore the remainder *EB* is also to the remainder *FD* as the whole *AB* is to the whole *CD.*

Therefore *if a whole is to a whole as a part subtracted is to a part subtracted, then the remainder is also to the remainder as the whole is to the whole.*

Q.E.D.

From this it is manifest that, *if magnitudes are proportional taken jointly, then they are also proportional in conversion.*

The transformations of proportions taken jointly, taken separately, and in conversion are summarized in the Guide for V.Def.14.

The magnitudes in this proposition must all be of the same kind, but those in the corollary can be of two different kinds. Thus, the corollary is out of place. It should probably be after the last proposition since it follows from the previous two propositions by inversion. As Heiberg and Heath agree, the corollary was probably interpolated before Theon’s time.

This proposition relies on using V.Def.4 as an axiom of comparability. (Infinitesimal counterexample: when *y* is infinitesimal with respect to *x,* then 2*x* : (2*x* + 2*y*) equals *x* : (*x* + 2*y*) but does not equal *x* : *x.*) The corollary, however, does not rely on an axiom of comparability.