Proposition VIII.22 — If three numbers be in continued proportion, and the first b…

Proof

a^{2} : m = m : c

, then

m^{2} = a^{2} c

c = (m / a)^{2}

, hence

c

is square. VII.19 ensures the division produces an integer.

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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…
VIII.11Proposition VIII.11Between two square numbers there is one mean proportional number, and the square has to the square the ratio duplicate…

Required by (dependents) (3)

VIII.23Proposition VIII.23If four numbers be in continued proportion, and the first be cube, the fourth will also be cube.
VIII.24Proposition VIII.24If two numbers have to one another the ratio which a square number has to a square number, and the first be square, the…
IX.9Proposition IX.9If as many numbers as we please beginning from a unit be in continued proportion, and the number after the unit be…

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