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Proposition 20

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 VII.13
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.

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This proposition is used in the next two propositions and also IX.2.

Book VIII Introduction - Proposition VIII.19 - Proposition VIII.21.

© 1996
D.E.Joyce
Clark University