Kitap 8
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And the multitude of D, L, E is equal to the multitude of E, O, F, and that of G, M, N, H to that of H, P, Q, K; therefore, ex acquali, as D is to E, so is E to F, and, as G is to H, so is H to K. [VII. 14] Q. E. D.
PROPOSITION 14.
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If a square measure a square, the side will also measure the side; and, if the side measure the side, the square will also measure the square.
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Let A, B be square numbers, let C, D be their sides, and let A measure B; I say that C also measures D.
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For let C by multiplying D make E; therefore A, E, B are continuously proportional in the ratio of C to D. [VIII. 11]
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And, since A, E, B are continuously proportional, and A measures B, therefore A also measures E. [VIII. 7]
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And, as A is to E, so is C to D; therefore also C measures D. [VII. Def. 20]
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Again, let C measure D; I say that A also measures B.
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For, with the same construction, we can in a similar manner prove that A, E, B are continuously proportional in the ratio of C to D.
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And since, as C is to D, so is A to E, and C measures D, therefore A also measures E. [VII. Def. 20]
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And A, E, B are continuously proportional; therefore A also measures B.
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Therefore etc. Q. E. D.
PROPOSITION 15.
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If a cube number measure a cube number, the side will also measure the side; and, if the side measure the side, the cube will also measure the cube.
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For let the cube number A measure the cube B, and let C be the side of A and D of B; I say that C measures D.
