Euclid Book 5 Proposition 13
Statement
If a first magnitude has to a second the same ratio as a third to a fourth (), and the third has to the fourth a greater ratio than a fifth has to a sixth (>), then the first also has to the second a greater ratio than the fifth to the sixth (>).
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Computational Explanation
GeometricScene{{A.,B.,C.,D.,E.,F.,G.,H.,I.,J.,K.,L.},{a.,b.,c.,d.,e.,f.}},Style[Line[{A.,B.}],Red],Style[Line[{C.,D.}],Red],Style[Line[{E.,F.}],Blue],Style[Line[{G.,H.}],Blue],Style[Line[{I.,J.}],Green],Style[Line[{K.,L.}],Green],EuclideanDistance[A.,B.]a.,EuclideanDistance[C.,D.]b.,EuclideanDistance[E.,F.]c.,EuclideanDistance[G.,H.]d.,EuclideanDistance[I.,J.]e.,EuclideanDistance[K.,L.]f.,,>,>,>
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Explanations
For let a first magnitude have to a second the same ratio as a third has to a fourth , and let the third have to the fourth a greater ratio than a fifth has to a sixth ; I say that the first will also have to the second a greater ratio than the fifth to the sixth .
For, since there are some equimultiples of, , and of , other, chance, equimultiples, such that the multiple of is in excess of the multiple of , while the multiple of is not in excess of the multiple of ,let them be taken, and let , be equimultiples of , , and , other, chance, equimultiples of , , so that is in excess of , but is not in excess of ; and, whatever multiple is of , let be also that multiple of , and, whatever multiple is of , let be also that multiple of .
Now, since, as is to , so is to , and of , equimultiples , have been taken, and of , other, chance, equimultiples , , therefore, if is in excess of , is also in excess of , if equal, equal, and if less, less.
But is in excess of ; therefore is also in excess of .
But is not in excess of ; and , are equimultiples of , , and , other, chance, equimultiples of , ; therefore has to a greater ratio than has to .
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For, since there are some equimultiples of
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Now, since, as
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