Kitap 5
15
Separation of a ratio means taking the excess by which the antecedent exceeds the consequent in relation to the consequent by itself.
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16
Conversion of a ratio means taking the antecedent in relation to the excess by which the antecedent exceeds the consequent.
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17
A ratio ex aequali arises when, there being several magnitudes and another set equal to them in multitude which taken two and two are in the same proportion, as the first is to the last among the first magnitudes, so is the first to the last among the second magnitudes;
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Or, in other words, it means taking the extreme terms by virtue of the removal of the intermediate terms.
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18
A perturbed proportion arises when, there being three magnitudes and another set equal to them in multitude, as antecedent is to consequent among the first magnitudes, so is antecedent to consequent among the second magnitudes, while, as the consequent is to a third among the first magnitudes, so is a third to the antecedent among the second magnitudes.
BOOK V. PROPOSITIONS.
PROPOSITION I.
1
If there be any number of magnitudes whatever which are, respectively, equimultiples of any magnitudes equal in multitude, then, whatever multiple one of the magnitudes is of one, that multiple also will all be of all.
1
Let any number of magnitudes whatever AB, CD be respectively equimultiples of any magnitudes E, F equal in multitude; I say that, whatever multiple AB is of E, that multiple will AB, CD also be of E, F.
1
For, since AB is the same multiple of E that CD is of F, as many magnitudes as there are in AB equal to E, so many also are there in CD equal to F.
1
Let AB be divided into the magnitudes AG, GB equal to E, and CD into CH, HD equal to F; then the multitude of the magnitudes AG, GB will be equal to the multitude of the magnitudes CH, HD.
1
Now, since AG is equal to E, and CH to F, therefore AG is equal to E, and AG, CH to E, F.
1
For the same reason
