Bibliotheca Polyglotta

Complete text

PROPOSITION 9.

第九題二支

Magnitudes which have the same ratio to the same are equal to one another; and magnitudes to which the same has the same ratio are equal.

兩幾何、與一幾何。各為比例、而等。則兩幾何必等。一幾何、與兩幾何。各為比例、而等。則兩幾何亦等。
先解曰。甲、乙、兩幾何。各與丙為比例、等。題言甲與乙等。

For let each of the magnitudes A, B have the same ratio to C; I say that A is equal to B.

For, otherwise, each of the magnitudes A, B would not have had the same ratio to C; [V. 8] but it has; therefore A is equal to B.

Again, let C have the same ratio to each of the magnitudes A, B; I say that A is equal to B.

For, otherwise, C would not have had the same ratio to each of the magnitudes A, B; [V. 8] but it has; therefore A is equal to B.

Therefore etc. Q. E. D.

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