VIII.20

If one mean proportional number fall between two numbers, the numbers will be similar plane numbers.

Proof

For let one mean proportional number C fall between the two numbers A, B; I say that A, B are similar plane numbers.

Let D, E, the least numbers of those which have the same ratio with A, C, be taken; VII. 33 therefore D measures A the same number of times that E measures C. VII. 20

Now, as many times as D measures A, so many units let there be in F; therefore F by multiplying D has made A, so that A is plane, and D, F are its sides.

Again, since D, E are the least of the numbers which have the same ratio with C, B, therefore D measures C the same number of times that E measures B. VII. 20

As many times, then, as E measures B, so many units let there be in G; therefore E measures B according to the units in G; therefore G by multiplying E has made B.

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

Therefore A, B are plane numbers.

I say next that they are also similar.

For, † since F by multiplying D has made A, and by multiplying E has made C, therefore, as D is to E, so is A to C, that is, C to B. VII. 17

Again, † since E by multiplying F, G has made C, B respectively, therefore, as F is to G, so is C to B. VII. 17

But, as C is to B, so is D to E; therefore also, as D is to E, so is F to G.

And alternately, as D is to F, so is E to G. VII. 13

Therefore A, B are similar plane numbers; for their sides are proportional. Q. E. D.