Bibliotheca Polyglotta

Let A, B be equal magnitudes and C any other, chance, magnitude; I say that each of the magnitudes A, B has the same ratio to C, and C has the same ratio to each of the magnitudes A, B.

For let equimultiples D, E of A, B be taken, and of C another, chance, multiple F.

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. But F is another, chance, magnitude. If therefore D is in excess of F, E is also in excess of F, if equal to it, equal; and, if less, less. 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]

I say next that C also has the same ratio to each of the magnitudes A, B.

For, with the same construction, we can prove similarly that D is equal to E; and F is some other magnitude. If therefore F is in excess of D, it is also in excess of E, if equal, equal; and, if less, less. 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]

Therefore etc.

PORISM.
From this it is manifest that, if any magnitudes are proportional, they will also be proportional inversely. Q. E. D.

後論與本篇第四題之系。同用反理。如甲與丙。若乙與丙。反推之。丙與甲。亦若丙與乙也。

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