Plane numbers have to one another the ratio compounded of the ratios of their sides.

Let *A* and *B* be plane numbers, and let the numbers *C* and *D* be the sides of *A,* and *E* and *F* the sides of *B.*

I say that *A* has to *B* the ratio compounded of the ratios of the sides.

The ratios being given which *C* has to *E* and *D* to *F,* take the least numbers *G, H,* and *K* that are continuously in the ratios *C, E, D,* and *F,* so that *C* is to *E* as *G* is to *H,* and *D* is to *F* as *H* is to *K.*

Multiply *D* by *E* to make *L.*

Now, since *D* multiplied by *C* makes *A,* and multiplied by *E* makes *L,* therefore *C* is to *E* as *A* is to *L.* But *C* is to *E* as *G* is to *H,* therefore *G* is to *H* as *A* is to *L.*

Again, since *E* multiplied by *D* makes *L,* and further multiplied by *F* makes *B,* therefore *D* is to *F* as *L* is to *B.* But *D* is to *F* as *H* is to *K,* therefore *H* is to *K* as *L* is to *B.*

But it was also proved that, *H* as *G* is to *H* as *A* is to *L,* therefore, *ex aequali, L* as *G* is to *K* as *A* is to *B.*

But *G* has to *K* the ratio compounded of the ratios of the sides, therefore *A* also has to *B* the ratio compounded of the ratios of the sides.

Therefore, *plane numbers have to one another the ratio compounded of the ratios of their sides.*

Q.E.D.

The ratio *compounded* from two given ratios *a* : *b* and *b* : *c* is just the ratio *a* : *c.* But if the middle term *b* is not shared by the two given ratios, then equal ratios must be found that do have a shared middle term.

To find the ratio compounded from two given ratios *a* : *b* and *c* : *d,* first find *e, f,* and *g* so that *e* : *f* = *a* : *b*
and *f* : *g* = *c* : *d.* Then, the ratio compounded from the ratios *a* : *b* and *c* : *d* will be the same as the ratio compounded from the ratios
*e* : *f* and *f* : *g,* namely *e* : *g.* For numbers, this construction was done in the previous proposition VIII.4.

therefore, the ratio of the plane numbers *cd* : *ef* will do.

Euclid’s proof is complicated because (1) his symbolic notation is limited, and (2) the application of VIII.4 to find the least numbers continuously in the ratios *c* : *d* and *e* : *f* makes the proof more difficult.

Since *a* = *cd,* therefore *c* : *e* = *a* : *de,* and so *g* : *h* = *a* : *de.* Since *b* = *ef,* therefore *d* : *f* = *de* : *b,* and so *h* : *k* = *de* : *b.* From the two proportions
*g* : *h* = *a* : *de* and
*h* : *k* = *de* : *b* therefore, *ex aequali,*
*g* : *k* = *a* : *b.* Thus, ratio the plane numbers is the ratio compounded of the ratios of their sides.