If one mean proportional number falls between two numbers, then the numbers are similar plane numbers.

Let one mean proportional number *C* fall between the two numbers *A* and *B.*

I say that *A* and *B* are similar plane numbers.

Take *D* and *E,* the least numbers of those which have the same ratio with *A* and *C.* Then *D* measures *A* the same number of times that *E* measures *C.*

Let there be as many units in *F* as times that *D* measures *A.* Then *F* multiplied by *D* makes *A,* so that *A* is plane, and *D* and *F* are its sides.

Again, since *D* and *E* are the least of the numbers which have the same ratio with *C* and *B,* therefore *D* measures *C* the same number of times that *E* measures *B.*

Let there be as many units in *G* as times that *E* measures *B.* Then *E* measures *B* according to the units in *G.* Therefore *G* multiplied by *E* makes *B.*

Therefore *B* is plane, and *E* and *G* are its sides. Therefore *A* and *B* are plane numbers.

I say next that they are also similar.

Since *F* multiplied by *D* makes *A,* and multiplied by *E* makes *C,* therefore *D* is to *E* as *A* is to *C,* that is, *C* to *B.*

Again, since *E* multiplied by *F* and *G* makes *C* and *B* respectively, therefore *F* is to *G* as *C* is to *B.* But *C* is to *B* as *D* is to *E,* therefore *D* is to *E* as *F* is to *G.* And alternately *D* is to *F* as *E* is to *G.*

Therefore *A* and *B* are similar plane numbers, for their sides are proportional.

Therefore, *if one mean proportional number falls between two numbers, then the numbers are similar plane numbers.*

Q.E.D.

This is a partial converse of VIII.18. It says that if two numbers have a mean proportional, then they can be viewed as two similar plane numbers.

An example might clarify the details. The variables refer to the outline of the proof below. The numbers *a* = 18 and *b* = 50 have a mean proportional *c* = 30. We’ll see *a* and *b* as the sides of the plane numbers, 3 by 6 and 5 by 10, as follows.

When *a* : *c* is converted to lowest terms, the result
is *d* : *e* = 3 : 5.
Then *f,* which is *a*/*d,* equals 6, and the number *a* = 18 is seen as the plane number *d* = 3 by *f* = 6. Also *g,*
which is *d*/*c,* equals 10, and the number *b* = 50 is seen as the plane number *e* = 5 by *g* = 10. The sides of these plane numbers, 3 by 6 and 5 by 10, are proportional.

Now since, *c* : *b* is the same ratio as *a* : *c,* it also reduces to the ratio *d* : *e* in lowest terms. Therefore, *d* divides *c* the same number of times that *e* divides *b*; call that number *g.* Then *b* is a plane number with sides *e* and *g.*

Furthermore, the two plane numbers *a* and *b* are similar since we can show their sides are proportional as follows. From the three proportions
*d* : *e* = *a* : *c*
(which follows from *a* = *fd* and *c* = *fe*),
*a* : *c* = *c* : *b*
(since *c* is a mean proportional),
and *c* : *b* = *f* : *g*
(which follows from *g* = *ef* and *b* = *ec*), therefore,
*d* : *e* = *f* : *g,*
and alternately,
*d* : *f* = *e* : *g.*
Thus, the two plane numbers have proportional sides.