Kitap 9
3
But, if two mean proportional numbers fall between two numbers, and the first be cube, the second will also be cube. [VIII. 23]
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And A is cube; therefore B is also cube. Q. E. D.
PROPOSITION 4.
4
If a cube number by multiplying a cube number make some number, the product will be cube.
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For let the cube number A by multiplying the cube number B make C; I say that C is cube.
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For let A by multiplying itself make D; therefore D is cube. [IX. 3]
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And, since A by multiplying itself has made D, and by multiplying B has made C therefore, as A is to B, so is D to C. [VII. 17]
4
And, since A, B are cube numbers, A, B are similar solid numbers.
4
Therefore two mean proportional numbers fall between A, B; [VIII. 19] so that two mean proportional numbers will fall between D, C also. [VIII. 8]
4
And D is cube; therefore C is also cube [VIII. 23] Q. E. D.
PROPOSITION 5.
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If a cube number by multiplying any number make a cube number, the multiplied number will also be cube.
5
For let the cube number A by multiplying any number B make the cube number C; I say that B is cube.
5
For let A by multiplying itself make D; therefore D is cube. [IX. 3]
5
Now, since A by multiplying itself has made D, and by multiplying B has made C, therefore, as A is to B, so is D to C. [VII. 17]
5
And since D, C are cube, they are similar solid numbers.
5
Therefore two mean proportional numbers fall between D, C. [VIII. 19]
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And, as D is to C, so is A to B; therefore two mean proportional numbers fall between A, B also. [VIII. 8]
