Proposition VIII.7 — If there be as many numbers as we please in continued propor…

Proof

a_{1} ∣ a_{n}

, reduce

a_{1}, \dots, a_{n}

to lowest terms (VIII.3); since the lowest extremes are coprime but

a_{1}

divides

a_{n}

, the ratio

a_{1} : a_{n}

must be 1:1 in lowest terms, forcing

a_{1} ∣ a_{2}

via VII.20.

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Depends on (2)

VII.20Proposition VII.20The least numbers of those which have the same ratio with them measure those which have the same ratio the same number…
VIII.3Proposition VIII.3If as many numbers as we please in continued proportion be the least of those which have the same ratio with them, the…

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