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PROP. XI. THEO.

Ratios that are the same to the same ratio are the same to each

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PROP. XII. THEO.

If any number of magnitudes be proportionals, as one of the antecedents is to its consequent, so shall all the antecedents taken together be to all the consequents.

Let:=:=:=0: A= ▲ : 0;

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In the same way it may be shown, if M times one of the antecedents be equal to or less than m times one of the consequents, M times all the antecedents taken together, will be equal to or less than m times all the consequents taken together. Therefore,

by the fifth definition, as one of the antecedents is to its consequent, so is all the antecedents taken together to all the consequents taken together.

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and therefore ab + ad + aƒ + a h = ba + b c + be + b g,

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PROP. XIII. THEO.

If the first has to the second the same ratio which the third has to the fourth, but the third to the fourth a greater ratio than the fifth has to the sixth; the first shall also have to the second a greater ratio than the fifth to the sixth.

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Let these multiples be taken, and take the same multiples of

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