Proposition IX.16 — If two numbers be prime to one another, the second will not …

Proof

a : b = b : c

with

g cd (a, b) = 1

, then

b^{2} = a c

, forcing

a ∣ b^{2}

which (Euclid's lemma) forces

a ∣ b

– contradicting coprimality unless

a = 1

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Depends on (2)

VII.19Proposition VII.19If four numbers be proportional, the number produced from the first and fourth will be equal to the number produced…
VII.30Proposition VII.30If two numbers by multiplying one another make some number, and any prime number measure the product, it will also…

Required by (dependents) (2)

IX.17Proposition IX.17If as many numbers as we please be in continued proportion, and the extremes of them be prime to one another, the last…
IX.18Proposition IX.18Given two numbers, to investigate whether it is possible to find a third proportional to them.

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