Book 8
14
Therefore etc. Q. E. D.
PROPOSITION 15.
15
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.
15
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.
15
For let C by multiplying itself make E, and let D by multiplying itself make G; further, let C by multiplying D make F, and let C, D by multiplying F make H, K respectively.
15
Now it is manifest that E, F, G and A, H, K, B are continuously proportional in the ratio of C to D. [VIII. 11, 12]
15
And, since A, H, K, B are continuously proportional, and A measures B, therefore it also measures H. [VIII. 7]
15
And, as A is to H, so is C to D; therefore C also measures D. [VII. Def. 20]
15
Next, let C measure D; I say that A will also measure B.
15
For, with the same construction, we can prove in a similar manner that A, H, K, B are continuously proportional in the ratio of C to D.
15
And, since C measures D, and, as C is to D, so is A to H, therefore A also measures H, [VII. Def. 20] so that A measures B also. Q. E. D.
PROPOSITION 16.
16
If a square number do not measure a square number, neither will the side measure the side; and, if the side do not measure the side, neither will the square measure the square.
16
Let A, B be square numbers, and let C, D be their sides; and let A not measure B; I say that neither does C measure D.
16
For, if C measures D, A will also measure B. [VIII. 14]
16
But A does not measure B; therefore neither will C measure D.
16
Again, let C not measure D; I say that neither will A measure B.
16
For, if A measures B, C will also measure D. [VIII. 14]
16
But C does not measure D; therefore neither will A measure B. Q. E. D.
PROPOSITION 17.
17
If a cube number do not measure a cube number, neither will the side measure the side; and, if the side do not measure the side, neither will the cube measure the cube.
