Book 10
5
Therefore the commensurable magnitudes A, B have to one another the ratio which the number D has to the number E. Q. E. D.
PROPOSITION 6.
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If two magnitudes have to one another the ratio which a number has to a number, the magnitudes will be commensurable.
6
For let the two magnitudes A, B have to one another the ratio which the number D has to the number E; I say that the magnitudes A, B are commensurable.
6
For let A be divided into as many equal parts as there are units in D, and let C be equal to one of them; and let F be made up of as many magnitudes equal to C as there are units in E.
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Since then there are in A as many magnitudes equal to C as there are units in D, whatever part the unit is of D, the same part is C of A also; therefore, as C is to A, so is the unit to D. [VII. Def. 20]
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But the unit measures the number D; therefore C also measures A.
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And since, as C is to A, so is the unit to D, therefore, inversely, as A is to C, so is the number D to the unit. [cf. V. 7, Por.]
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Again, since there are in F as many magnitudes equal to C as there are units in E, therefore, as C is to F, so is the unit to E. [VII. Def. 20]
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But it was also proved that, as A is to C, so is D to the unit; therefore, ex aequali, as A is to F, so is D to E. [v. 22]
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But, as D is to E, so is A to B; therefore also, as A is to B, so is it to F also. [V. 11]
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Therefore A has the same ratio to each of the magnitudes B, F; therefore B is equal to F. [V. 9]
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But C measures F; therefore it measures B also.
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Further it measures A also; therefore C measures A, B.
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Therefore A is commensurable with B.
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Therefore etc.
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Porism. From this it is manifest that, if there be two numbers, as D, E, and a straight line, as A, it is possible to make a straight line [F] such that the given straight line is to it as the number D is to the number E.
6
And, if a mean proportional be also taken between A, F, as B,
