PROPOSITION 20.If one mean proportional number fall between two numbers, the numbers will be similar plane numbers. 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. 1