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Source — Euclid's Elements, encoded as an rrxiv paper — rrxiv

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Source — Euclid's Elements, encoded as an rrxiv paper

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VIII.7 Proposition VIII.7

If there be as many numbers as we please in continued proportion, and the first measure the last, it will measure the second also.

Proof

If

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.

lines 74–74 in main.tex