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.

VII.33
VII.20

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.

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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.

VII.20

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.

VII.17

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
VII.13

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.

Guide

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

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.

Outline of the proof

Suppose two numbers a and b have a mean proportional c. Reduce the ratio a : c to lowest terms d : e. Then d divides a the same number of times e divides c; call that number f. Then a is a plane number with sides d and f.

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.

Use of this proposition

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