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Euclid: Elementa

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Euclid: Elementa
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PROPOSITION 5. 
 
 
If a cube number by multiplying any number make a cube number, the multiplied number will also be cube. 
 
 
For let the cube number A by multiplying any number B make the cube number C;  I say that B is cube. 
   
   
For let A by multiplying itself make D;  therefore D is cube. [IX. 3]  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]  And since D, C are cube, they are similar solid numbers.  Therefore two mean proportional numbers fall between D, C. [VIII. 19]  And, as D is to C, so is A to B;  therefore two mean proportional numbers fall between A, B also. [VIII. 8]  And A is cube;  therefore B is also cube. [VIII. 23]   
                     
                     
 
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