Proposition VIII.2 — To find numbers in continued proportion, as many as may be p…

Proof

For

a : b

with

g cd (a, b) = 1

, the sequence

a^{n - 1}, a^{n - 2} b, \dots, b^{n - 1}

is in continued proportion with ratio

a : b

; by VII.27 the extremes are coprime, so by VIII.1 the sequence is least in its ratio.

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Depends on (2)

VII.27Proposition VII.27If two numbers be prime to one another, and each by multiplying itself make a certain number, the products will be…
VIII.1Proposition VIII.1If there be as many numbers as we please in continued proportion, and the extremes of them be prime to one another, the…

Required by (dependents) (4)

VIII.4Proposition VIII.4Given as many ratios as we please in least numbers, to find numbers in continued proportion which are the least in the…
VIII.8Proposition VIII.8If between two numbers there fall numbers in continued proportion with them, then, however many numbers fall between…
VIII.9Proposition VIII.9If two numbers be prime to one another, and numbers fall between them in continued proportion, then, however many…
VIII.13Proposition VIII.13If there be as many numbers as we please in continued proportion, and each by multiplying itself make some number, the…

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