Proposition IX.8 — If as many numbers as we please beginning from a unit be in …

Proof

The sequence

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

has third term a square, fourth a cube, sixth both square and cube, by VIII.11 / VIII.12.

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Depends on (2)

VIII.11Proposition VIII.11Between two square numbers there is one mean proportional number, and the square has to the square the ratio duplicate…
VIII.12Proposition VIII.12Between two cube numbers there are two mean proportional numbers, and the cube has to the cube the ratio triplicate of…

Required by (dependents) (3)

IX.9Proposition IX.9If as many numbers as we please beginning from a unit be in continued proportion, and the number after the unit be…
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…
IX.11Proposition IX.11If as many numbers as we please beginning from a unit be in continued proportion, the less measures the greater…

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