Proposition IX.15 — If three numbers in continued proportion be the least of tho…

Proof

For

a, b, c

with

a : b = b : c

in lowest terms: by VIII.3 the extremes are coprime; combining with VII.28 the sums

a + b

b + c

a + c

are each coprime to the third term.

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Depends on (2)

VII.28Proposition VII.28If two numbers be prime to one another, the sum will also be prime to each of them; and if the sum of two numbers be…
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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