Book 10
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Porism. From this it is manifest that, if a magnitude measure three magnitudes, it will also measure their greatest common measure.
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Similarly too, with more magnitudes, the greatest common measure can be found, and the porism can be extended. Q. E. D.
PROPOSITION 5.
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Commensurable magnitudes have to one another the ratio which a number has to a number.
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Let A, B be commensurable magnitudes; I say that A has to B the ratio which a number has to a number.
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For, since A, B are commensurable, some magnitude will measure them.
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Let it measure them, and let it be C.
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And, as many times as C measures A, so many units let there be in D; and, as many times as C measures B, so many units let there be in E.
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Since then C measures A according to the units in D, while the unit also measures D according to the units in it, therefore the unit measures the number D the same number of times as the magnitude C measures A; therefore, as C is to A, so is the unit to D; [VII. Def. 20] therefore, inversely, as A is to C, so is D to the unit. [cf. V. 7, Por.]
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Again, since C measures B according to the units in E, while the unit also measures E according to the units in it, therefore the unit measures E the same number of times as C measures B; therefore, as C is to B, so is the unit to E.
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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 B, so is the number D to E. [V. 22]
