Proposition VII.17 — If a number by multiplying two numbers make certain numbers,…

Proof

c \cdot a : c \cdot b = a : b

by gathering the

c

copies of each side; the equimultiples test of definition V.5 (in its discrete specialisation, VII.20) reduces to the test on

(a, b)

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Depends on (2)

VII.16Proposition VII.16If two numbers by multiplying one another make certain numbers, the numbers so produced will be equal to one another.
VII.20Definition VII.20Numbers are proportional when the first is the same multiple, or the same part, or the same parts, of the second that…

Required by (dependents) (7)

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.19Proposition VII.19If four numbers be proportional, the number produced from the first and fourth will be equal to the number produced…
VIII.5Proposition VIII.5Plane numbers have to one another the ratio compounded of the ratios of their sides.
VIII.11Proposition VIII.11Between two square numbers there is one mean proportional number, and the square has to the square the ratio duplicate…
VIII.12Proposition VIII.12Between two cube numbers there are two mean proportional numbers, and the cube has to the cube the ratio triplicate of…
IX.3Proposition IX.3If a cube number by multiplying itself make some number, the product will be cube.
IX.7Proposition IX.7If a composite number by multiplying any number make some number, the product will be solid.

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