Book 9
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And, as C is to D, so is B to C; therefore B also has to C the ratio which a cube has to a cube.
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And C is cube; therefore B is also cube. [VIII. 25]
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And since, as the unit is to A, so is A to B, and the unit measures A according to the units in it, therefore A also measures B according to the units in itself; therefore A by multiplying itself has made the cube number B.
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But, if a number by multiplying itself make a cube number, it is also itself cube. [IX. 6]
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Therefore A is also cube: which is contrary to the hypothesis.
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Therefore D is not cube.
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Similarly we can prove that neither is any other of the numbers cube except the fourth from the unit and those which leave out two. Q. E. D.
PROPOSITION II.
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If as many numbers as we please beginning from an unit be in continued proportion, the less measures the greater according to some one of the numbers which have place among the proportional numbers.
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Let there be as many numbers as we please, B, C, D, E, beginning from the unit A and in continued proportion; I say that B, the least of the numbers B, C, D, E, measures E according to some one of the numbers C, D.
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For since, as the unit A is to B, so is D to E, therefore the unit A measures the number B the same number of times as D measures E; therefore, alternately, the unit A measures D the same number of times as B measures E. [VII. 15]
