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Proposition 14

If a square measures a square, then the side also measures the side; and, if the side measures the side, then the square also measures the square.
Let A and B be square numbers, let C and D be their sides, and let A measure B. I say that C also measures D.
| Multiply C by D to make E. Then A, E, and B are continuously proportional in the ratio of C to D. | VIII.11
And, since A, E, and B are continuously proportional, and A measures B, therefore A also measures E. And A is to E as C is to D, therefore C measures D. | VIII.7 VII.Def.20
Next, let C measure D. I say that A also measures B.
With the same construction, we can in a similar manner prove that A, E, and B are continuously proportional in the ratio of C to D. And since C is to D as A is to E, and C measures D, therefore A also measures E. | VII.Def.20
And A, E, and B are continuously proportional, therefore A also measures B.
Therefore, if a square measures a square, then the side also measures the side; and, if the side measures the side, then the square also measures the square.
Q.E.D.

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This proposition is used in VIII.16.

Book VIII Introduction - Proposition VIII.13 - Proposition VIII.15.

© 1996
D.E.Joyce
Clark University