Proposition IX.9 — If as many numbers as we please beginning from a unit be in …

Proof

1, r, r^{2}, r^{3}, \dots

, if

r

is square then every

r^{k}

is square; if

r

is cube then every

r^{k}

is cube — by VIII.22 / VIII.23 propagated through the sequence.

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Depends on (3)

VIII.22Proposition VIII.22If three numbers be in continued proportion, and the first be square, the third will also be square.
VIII.23Proposition VIII.23If four numbers be in continued proportion, and the first be cube, the fourth will also be cube.
IX.8Proposition IX.8If as many numbers as we please beginning from a unit be in continued proportion, the third from the unit will be…

Required by (dependents) (1)

IX.10Proposition IX.10If as many numbers as we please beginning from a unit be in continued proportion, and the number after the unit be not…

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