If as many numbers as we please beginning from a unit be in continued proportion, and the number after the unit be square, all the rest will also be square; and if the number after the unit be cube, all the rest will also be cube.
Proof
In 1,r,r2,r3,…, if r is square then every rk is
square; if r is cube then every rk is cube — by VIII.22 /
VIII.23 propagated through the sequence.