Book 8
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Let there be as many numbers as we please, A, B, C, in continued proportion, so that, as A is to B, so is B to C; let A, B, C by multiplying themselves make D, E, F, and by multiplying D, E, F let them make G, H, K; I say that D, E, F and G, H, K are in continued proportion.
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For let A by multiplying B make L, and let the numbers A, B by multiplying L make M. N respectively.
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And again let B by multiplying C make O, and let the numbers B, C by multiplying O make P, Q respectively.
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Then, in manner similar to the foregoing, we can prove that D, L, E and G, M, N, H are continuously proportional in the ratio of A to B, and further E, O, F and H, P, Q, K are continuously proportional in the ratio of B to C.
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Now, as A is to B, so is B to C; therefore D, L, E are also in the same ratio with E, O, F, and further G, M, N, H in the same ratio with H, P, Q, K.
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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.
