Euclid Book 5 Proposition 24
Statement
If a first magnitude has to a second the same ratio as a third has to a fourth (), and also a fifth has to the second the same ratio as a sixth to the fourth (), then the sum of the first and fifth has to the second the same ratio as the sum of the third and sixth has to the fourth ().
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GH
Computational Explanation
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EuclideanDistance[A.,I.]
EuclideanDistance[C.,D.]
EuclideanDistance[E.,J.]
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a.+b.
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Explanations
Let a first magnitude have to a second the same ratio as a third has to a fourth ; and let also a fifth have to the second the same ratio as a sixth has to the fourth ; I say that the first and fifth added together, , will have to the second the same ratio as the third and sixth, , has to the fourth .
For since, as is to , so is to , inversely, as is to , so is to .
Since, then, as is to , so is to , and, as is to , so is to , therefore, ex aequali, as is to , so is to .
And, since the magnitudes are proportional separando, they will also be proportional componendo; therefore, as is to , so is to .
But also, as is to , so is to ; therefore, ex aequali, as is to , so is to .
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For since, as
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Since, then, as
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C
BG
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EH
AB
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And, since the magnitudes are proportional separando, they will also be proportional componendo; therefore, as
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But also, as
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AG
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F