Book 5
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For let equimultiples D, E of A, B be taken, and of C another, chance, multiple F.
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Then, since D is the same multiple of A that E is of B, while A is equal to B, therefore D is equal to E.
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But F is another, chance, magnitude.
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If therefore D is in excess of F, E is also in excess of F, if equal to it, equal; and, if less, less.
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And D, E are equimultiples of A, B, while F is another, chance, multiple of C; therefore, as A is to C, so is B to C. [V. Def. 5]
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I say next that C also has the same ratio to each of the magnitudes A, B.
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For, with the same construction, we can prove similarly that D is equal to E; and F is some other magnitude.
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If therefore F is in excess of D, it is also in excess of E, if equal, equal; and, if less, less.
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And F is a multiple of C, while D, E are other, chance, equimultiples of A, B; therefore, as C is to A, so is C to B. [V. Def. 5]
7
Therefore etc.
7
Porism. From this it is manifest that, if any magnitudes are proportional, they will also be proportional inversely. Q. E. D.
PROPOSITION 8.
8
Of unequal magnitudes, the greater has to the same a greater ratio than the less has; and the same has to the less a greater ratio than it has to the greater.
8
Let AB, C be unequal magnitudes, and let AB be greater; let D be another, chance, magnitude; I say that AB has to D a greater ratio than C has to D, and D has to C a greater ratio than it has to AB.
