Proposition IX.18 — Given two numbers, to investigate whether it is possible to …

Proof

A third proportional to

a

b

exists iff

b^{2}

is divisible by

a

(i.e.\

b^{2} / a

is an integer); apply VII.19.

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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…
IX.16Proposition IX.16If two numbers be prime to one another, the second will not be to any other number as the first is to the second.

Required by (dependents) (1)

IX.19Proposition IX.19Given three numbers, to investigate when it is possible to find a fourth proportional to them.

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