| 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. | VII.33
VII.20 | |
| 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. | VII.20 | |
| Let there be as many units in B 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. | VII.17 | |
| 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. | VII.17 | |
| 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. | ||
Book VIII Introduction - Proposition VIII.19 - Proposition VIII.21.