Euclid Book 5 Proposition 16
Statement
If a first magnitude is to a second as a third is to a fourth (), then the first magnitude is to the third as the second to the fourth ().
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Computational Explanation
GeometricScene{{A.,B.,C.,D.,E.,F.,G.,H.},{a.,b.,c.,d.}},Style[Line[{A.,B.}],Red],Style[Line[{C.,D.}],Red],Style[Line[{E.,F.}],Blue],Style[Line[{G.,H.}],Blue],EuclideanDistance[A.,B.]a.,EuclideanDistance[C.,D.]b.,EuclideanDistance[E.,F.]c.,EuclideanDistance[G.,H.]d.,,,
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EuclideanDistance[A.,B.]
EuclideanDistance[E.,F.]
EuclideanDistance[C.,D.]
EuclideanDistance[G.,H.]
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Explanations
Let , , , be four proportional magnitudes, so that, as is to , so is to ; I say that they will also be so alternately, that is, as is to , so is to .
For of, let equimultiples , be taken, and of , other, chance, equimultiples , .
Then, since is the same multiple of that is of , and parts have the same ratio as the same multiples of them, therefore, as is to , so is to .
But as is to , so is to ; therefore also, as is to , so is to .
Again, since, are equimultiples of , , therefore, as is to , so is to .
But, as is to , so is to ; therefore also, as is to, so is to .
But, if four magnitudes be proportional, and the first be greater than the third, the second will also be greater than the fourth; if equal, equal; and if less, less.
Therefore, if is in excess of , is also in excess of , if equal, equal, and if less, less.
Now, are equimultiples of , , and , other, chance, equimultiples of , ; therefore, as is to , so is to .
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For of
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Then, since
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But as
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Again, since
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But, as
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But, if four magnitudes be proportional, and the first be greater than the third, the second will also be greater than the fourth; if equal, equal; and if less, less.
Therefore, if
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Now
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