Therefore, as the unit is to A, so is B to C. [VII. Def. 20] But, as the unit is to A, so is A to B; therefore also, as A is to B, so is B to C. And, since B, C are cube, they are similar solid numbers. Therefore there are two mean proportional numbers between B, C. [VIII. 19] And, as B is to C, so is A to B. Therefore there are two mean proportional numbers between A, B also. [VIII. 8] And B is cube; therefore A is also cube. [cf. VIII. 23] Q. E. D.
Therefore, as the unit is to A, so is B to C. [VII. Def. 20] But, as the unit is to A, so is A to B; therefore also, as A is to B, so is B to C. And, since B, C are cube, they are similar solid numbers. Therefore there are two mean proportional numbers between B, C. [VIII. 19] And, as B is to C, so is A to B. Therefore there are two mean proportional numbers between A, B also. [VIII. 8] And B is cube; therefore A is also cube. [cf. VIII. 23] Q. E. D.
Euclid. Euclid's Elements. Sir Thomas Little Heath. New York. Dover. 1956.
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