Ratios which are the same with the same ratio are also the same with one another.

Let *A* be to *B* as *C* is to *D,* and let *C* be to *D* as *E* is to *F.*

I say that *A* is to *B* as *E* is to *F.*

Take equimultiples *G, H,* and *K* of *A, C,* and *E,* and take other, arbitrary, equimultiples *L, M,* and *N* of *B, D,* and *F.*

Then since *A* is to *B* as *C* is to *D,* and of *A* and *C* equimultiples *G* and *H* have been taken, and of *B* and *D* other, arbitrary, equimultiples *L* and *M,* therefore, if *G* is in excess of *L, H* is also in excess of *M*; if equal, equal; and if less, less.

Again, since *C* is to *D* as *E* is to *F,* and of *C* and *E* equimultiples *H* and *K* have been taken, and of *D* and *F* other, arbitrary, equimultiples *M* and *N,* therefore, if *H* is in excess of *M, K* is also in excess of *N*; if equal, equal; and if less, less.

But we saw that, if *H* was in excess of *M, G* was also in excess of *L*; if equal, equal; and if less, less, so that, in addition, if *G* is in excess of *L, K* is also in excess of *N*; if equal, equal; and if less, less.

And *G* and *K* are equimultiples of *A* and *E,* while *L* and *N* are other, arbitrary, equimultiples of *B* and *F,* therefore *A* is to *B* as *E* is to *F.*

Therefore, *ratios which are the same with the same ratio are also the same with one another.*

Q.E.D.

The magnitudes may be of three different kinds with *A* and *B* of one kind, *C* and *D* of a second kind, and *E* and *F* of a third kind.

This proposition is used very frequently whenever ratios are used.