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Euclid: Elementa

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Euclid: Elementa
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PROPOSITION 4. 
 
 
If a cube number by multiplying a cube number make some number, the product will be cube. 
 
 
For let the cube number A by multiplying the cube number B make C;  I say that C is cube. 
   
   
For let A by multiplying itself make D;  therefore D is cube. [IX. 3]  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]  And, since A, B are cube numbers, A, B are similar solid numbers.  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]  And D is cube;  therefore C is also cube [VIII. 23]  Q. E. D. 
                   
                   
 
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