Proposition VIII.11 — Between two square numbers there is one mean proportional nu…

Proof

For squares

a^{2}

b^{2}

: the mean proportional is

ab

(since

a^{2} : ab = ab : b^{2} = a : b

), and

a^{2} : b^{2}

is the duplicate of

a : b

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Depends on (3)

VII.17Proposition VII.17If a number by multiplying two numbers make certain numbers, the numbers so produced will have the same ratio as the…
VII.18Proposition VII.18If two numbers by multiplying any number make certain numbers, the numbers so produced will have the same ratio as the…
VII.18Definition VII.18A square number is equal multiplied by equal, or a number which is contained by two equal numbers.

Required by (dependents) (4)

VIII.14Proposition VIII.14If a square measure a square, the side will also measure the side; and if the side measure the side, the square will…
VIII.22Proposition VIII.22If three numbers be in continued proportion, and the first be square, the third will also be square.
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.8Proposition IX.8If as many numbers as we please beginning from a unit be in continued proportion, the third from the unit will be…

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